From 579f03b95a9b0caa7e68adc597bb631cbc228229 Mon Sep 17 00:00:00 2001
From: grants-deployer <grants-deployer@users.noreply.github.com>
Date: Thu, 2 Nov 2023 09:06:49 +0000
Subject: [PATCH] Deploy website - based on
 8fbeeead095925d97efcbecd0637856e58d63840

---
 404.html                                      |   2 +-
 Implementation Guide.html                     |   2 +-
 Implementation Guide/faq.html                 |   2 +-
 Polkadot_Protocol_Spec.pdf                    | Bin 2998320 -> 3095150 bytes
 Support Docs/privacy_policy.html              |   2 +-
 __docusaurus/debug.html                       |   2 +-
 __docusaurus/debug/config.html                |   2 +-
 __docusaurus/debug/content.html               |   2 +-
 __docusaurus/debug/globalData.html            |   2 +-
 __docusaurus/debug/metadata.html              |   2 +-
 __docusaurus/debug/registry.html              |   2 +-
 __docusaurus/debug/routes.html                |   2 +-
 ..._Spec-32cc34d410af00038105107bcd191836.pdf | Bin 0 -> 3095150 bytes
 ..._Spec-cf1a953ff60b738f59f19c4d2552f805.pdf | Bin 2998320 -> 0 bytes
 ...93b7f.041469af.js => 1df93b7f.6a9bf7b1.js} |   4 ++--
 ...E.txt => 1df93b7f.6a9bf7b1.js.LICENSE.txt} |   0
 assets/js/1e5ed2e1.1e403103.js                |   1 -
 assets/js/1e5ed2e1.c81a7a1f.js                |   1 +
 ...n.a2453290.js => runtime~main.23d931e1.js} |   2 +-
 chap-host-api.html                            |   2 +-
 chap-networking.html                          |   2 +-
 chap-overview.html                            |   2 +-
 chap-runtime-api.html                         |   2 +-
 chap-state.html                               |   2 +-
 chap-sync.html                                |   2 +-
 chapter-anv.html                              |   2 +-
 id-consensus.html                             |   2 +-
 id-cryptography-encoding.html                 |   2 +-
 id-extrinsics.html                            |   2 +-
 id-glossary.html                              |   2 +-
 id-polkadot-protocol.html                     |   2 +-
 id-weights.html                               |   2 +-
 index.html                                    |   4 ++--
 part-polkadot-host.html                       |   2 +-
 part-polkadot-runtime.html                    |   2 +-
 sect-block-production.html                    |   2 +-
 sect-finality.html                            |   6 +++---
 sect-lightclient.html                         |   2 +-
 sect-metadata.html                            |   2 +-
 39 files changed, 38 insertions(+), 38 deletions(-)
 create mode 100644 assets/files/Polkadot_Protocol_Spec-32cc34d410af00038105107bcd191836.pdf
 delete mode 100644 assets/files/Polkadot_Protocol_Spec-cf1a953ff60b738f59f19c4d2552f805.pdf
 rename assets/js/{1df93b7f.041469af.js => 1df93b7f.6a9bf7b1.js} (99%)
 rename assets/js/{1df93b7f.041469af.js.LICENSE.txt => 1df93b7f.6a9bf7b1.js.LICENSE.txt} (100%)
 delete mode 100644 assets/js/1e5ed2e1.1e403103.js
 create mode 100644 assets/js/1e5ed2e1.c81a7a1f.js
 rename assets/js/{runtime~main.a2453290.js => runtime~main.23d931e1.js} (97%)

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index 8ef212f05..734a027f3 100644
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diff --git a/Implementation Guide.html b/Implementation Guide.html
index e9e7bb961..23d036a45 100644
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diff --git a/Implementation Guide/faq.html b/Implementation Guide/faq.html
index 21b21b831..21e0bfe81 100644
--- a/Implementation Guide/faq.html	
+++ b/Implementation Guide/faq.html	
@@ -6,7 +6,7 @@
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diff --git a/Support Docs/privacy_policy.html b/Support Docs/privacy_policy.html
index f2079e06b..2df39cf16 100644
--- a/Support Docs/privacy_policy.html	
+++ b/Support Docs/privacy_policy.html	
@@ -6,7 +6,7 @@
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index 6139700b1..f3fde0236 100644
--- a/__docusaurus/debug.html
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index 795067ea0..418a5f236 100644
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index 3f89b345e..886747ac7 100644
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diff --git a/__docusaurus/debug/globalData.html b/__docusaurus/debug/globalData.html
index 8ed214107..5a9fe13ce 100644
--- a/__docusaurus/debug/globalData.html
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index cb3f47618..d86ebbcea 100644
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index c4274fd4c..d2761aacf 100644
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diff --git a/__docusaurus/debug/routes.html b/__docusaurus/debug/routes.html
index 0360ca634..dc6f34d1e 100644
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index 37ad62b37..435fe8186 100644
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diff --git a/assets/js/1df93b7f.041469af.js.LICENSE.txt b/assets/js/1df93b7f.6a9bf7b1.js.LICENSE.txt
similarity index 100%
rename from assets/js/1df93b7f.041469af.js.LICENSE.txt
rename to assets/js/1df93b7f.6a9bf7b1.js.LICENSE.txt
diff --git a/assets/js/1e5ed2e1.1e403103.js b/assets/js/1e5ed2e1.1e403103.js
deleted file mode 100644
index 65b36ac62..000000000
--- a/assets/js/1e5ed2e1.1e403103.js
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1, \\textrm{Fin}}($\\call{Best-Final-Candidate}{$r - 1$}$)$}\n      \\IF{\\call{Best-Final-Candidate}{$r - 1$} $\\geqslant$ \\textsc{Last-Finalized-Block}}\n        \\STATE \\call{Broadcast}{$M_{v}^{r - 1, \\textrm{Prim}}($\\call{Best-Final-Candidate}{$r - 1$}$)$}\n      \\ENDIF\n    \\ENDIF\n    \\STATE \\call{Receive-Messages}{\\textbf{until} Time $\\geqslant t_{r_,v} + 2 \\times T$ \\or $r$ \\textbf{is} completable}\n    \\STATE $L \\leftarrow$ \\call{Best-Final-Candidate}{$r - 1$}\n    \\STATE $N \\leftarrow$ \\call{Best-PreVote-Candidate}{$r$}\n    \\STATE \\call{Broadcast}{$M_v^{r, \\textrm{pv}} (N)$}\n    \\STATE \\call{Receive-Messages}{\\textbf{until} $B^{r,\\textrm{pv}}_v \\geqslant L$ \\and $($ Time $\\geqslant t_{r_,v} + 4 \\times T$ \\or $r$ \\textbf{is} completable $)$}\n    \\STATE \\call{Broadcast}{$M_v^{r, \\textrm{pc}}(B_v^{r, \\textrm{pv}})$}\n    \n    \\REPEAT\n      \\STATE \\call{Receive-Messages}{}\n      \\STATE \\call{Attempt-To-Finalize-At-Round}{$r$}\n    \\UNTIL{$r$ \\textbf{is} completable \\and \\call{Finalizable}{$r$} \\and \\textsc{Last-Finalized-Block} $\\geqslant$ \\call{Best-Final-Candidate}{$r - 1$}}\n    \n    \\STATE \\call{Play-Grandpa-round}{$r + 1$}\n    \n    \\REPEAT\n      \\STATE \\call{Receive-Messages}{}\n      \\STATE \\call{Attempt-To-Finalize-At-Round}{$r$}\n    \\UNTIL{\\textsc{Last-Finalized-Block} $\\geqslant$ \\call{Best-Final-Candidate}{$r$}}\n    \n    \\IF{$r > $ \\textsc{Last-Completed-Round}}\n      \\STATE \\textsc{Last-Completed-Round} $\\leftarrow r$\n    \\ENDIF\n\\end{algorithmic}\n\\end{algorithm}",N="\\begin{algorithm}\n\\caption{Derive-Primary}\n\\begin{algorithmic}\n    \\input $r$\n    \\return $r \\bmod |\\mathbb{V}|$\n\\end{algorithmic}\n\\end{algorithm}",i="\\begin{algorithm}\n\\caption{Best-Final-Candidate}\n\\begin{algorithmic}\n    \\input $r$\n    \\state $B_v^{r, pv} \\leftarrow$ \\call{GRANDPA-GHOST}{$r$}\n    \\if{$r = 0$}\n        \\return $B_v^{r, pv}$\n    \\else\n        \\state $\\mathcal{C} \\leftarrow \\{ B' | B' \\leqslant B_v^{r,pv} | \\#V^{r, pc}_{\\operatorname{obv}(v), pot}(B') > \\frac{2}{3} |\\mathbb{V}| \\}$\n        \\if{$\\mathcal{C} = \\phi$}\n            \\return $B_v^{r, pv}$\n        \\else\n            \\return $E \\in \\mathcal{C} : H_n (E) = \\operatorname{max}\\left(H_n (B') | B' \\in \\mathcal{C}\\right)$\n        \\endif\n    \\endif\n\\end{algorithmic}\n\\end{algorithm}",l="\\begin{algorithm}\n\\caption{GRANDPA-GHOST}\n\\begin{algorithmic}\n    \\input $r$\n    \\if{$r = 0$}\n        \\state $G \\leftarrow B_{last}$\n    \\else\n        \\state $L \\leftarrow$ \\call{Best-Final-Candidate}{$r - 1$}\n        \\state $\\mathcal{G} = \\{ \\forall B > L | \\#V_{\\operatorname{obs}(v)}^{r, pv}(B) \\geqslant \\frac{2}{3} |\\mathbb{V}| \\}$\n        \\if{$\\mathcal{G} = \\phi$}\n            \\state $G \\leftarrow L$\n        \\else\n            \\state $G \\in \\mathcal{G} | H_n(G) = \\operatorname{max}\\left( H_n (B) | \\forall B \\in \\mathcal{G} \\right)$\n        \\endif\n    \\endif\n    \\return $G$\n\\end{algorithmic}\n\\end{algorithm}",k="\\begin{algorithm}\n\\caption{Best-PreVote-Candidate}\n\\begin{algorithmic}\n    \\input $r$\n    \\state $B^{r, pv}_v \\leftarrow$ \\call{GRANDPA-GHOST}{$r$}\n    \\if{\\call{Received}{$M_{v_{primary}}^{r, prim}(B))$ \\and $B^{r, pv}_v \\geqslant B > L$}}\n        \\state $N \\leftarrow B$\n    \\else\n        \\state $N \\leftarrow B^{r, pv}_v$\n    \\endif\n\\end{algorithmic}\n\\end{algorithm}",o="\\begin{algorithm}\n\\caption{Attempt-To-Finalize-At-Round}\n\\begin{algorithmic}\n    \\REQUIRE($r$)\n    \\STATE $L \\leftarrow$ \\textsc{Last-Finalized-Block}\n    \\STATE $E \\leftarrow$ \\call{Best-Final-Candidate}{$r$}\n    \\IF{$E \\geqslant L$ \\and ${V^{r, \\textrm{pc}}_{\\textrm{obs}(v)}}(E) > 2 / 3 |\\mathbb{V}|$}\n    \\STATE{\\textsc{Last-Finalized-Block}$\\leftarrow E$}\n    \\IF{$M_v^{r, \\textrm{Fin}} (E) \\notin $\\textsc{Received-Messages}}\n        \\STATE \\call{Broadcast}{$M_v^{r, \\textrm{Fin}}(E)$}\n        \\RETURN\n    \\ENDIF\n    \\ENDIF\n\\end{algorithmic}\n\\end{algorithm}",c="\\begin{algorithm}\n\\caption{Finalizable}\n\\begin{algorithmic}\n    \\REQUIRE($r$)\n    \\IF{$r$ \\textbf{is not} Completable}\n        \\RETURN \\textbf{False}\n    \\ENDIF\n    \\STATE $G \\leftarrow$ \\call{GRANDPA-GHOST}{$J^{r, pv}(B)$}\n    \\IF{$G = \\phi$}\n        \\RETURN \\textbf{False}\n    \\ENDIF\n    \\STATE $E_r \\leftarrow$ \\call{Best-Final-Candidate}{$r$}\n    \\IF{$E_r \\neq \\phi$ \\and \\call{Best-Final-Candidate}{$r - 1$} $\\leqslant E_r \\leqslant G$}\n        \\RETURN \\textbf{True}\n    \\ELSE\n        \\RETURN \\textbf{False}\n    \\ENDIF\n\\end{algorithmic}\n\\end{algorithm}",h="\\begin{algorithm}\n\\caption{Process-Catchup-Request}\n\\begin{algorithmic}\n    \\input $M_{i, v}^\\text{Cat-q}(\\text{id}_\\mathbb{V}, r)$\n    \\if{$M_{i, v}^\\text{Cat-q}(\\text{id}_\\mathbb{V}, r).\\text{id}_\\mathbb{V} \\neq \\text{id}_\\mathbb{V}$}\n        \\state \\textbf{error} ``Catching up on different set''\n    \\endif\n    \\if{$i \\notin \\mathbb{P}$}\n        \\state \\textbf{error} ``Requesting catching up from a non-peer''\n    \\endif\n    \\if{$r >$ \\textsc{Last-Completed-Round}}\n        \\state \\textbf{error} ``Catching up on a round in the future''\n    \\endif\n    \\state \\call{Send}{$i, M_{v, i}^\\text{Cat-s}(\\text{id}_\\mathbb{V}, r)$}\n\\end{algorithmic}\n\\end{algorithm}",g="\\begin{algorithm}\n\\caption{Process-Catchup-Response}\n\\begin{algorithmic}\n    \\input $M_{v,i}^\\text{Cat-s}(\\text{id}_{\\mathbb{V}}, r)$\n    \\state $M_{v,i}^\\text{Cat-s}(\\text{id}_{\\mathbb{V}}, r).\\text{id}_{\\mathbb{V}}, r, J^{r, pv}(B), J^{r, pc}(B), H_h(B'), H_i(B') \\leftarrow \\text{Dec}_{SC}(M_{v, i}^{Cat-s}(\\text{id}_{\\mathbb{V}}, r)$\n    \\if{$M_{v, i}^\\text{Cat-s}(\\text{id}_{\\mathbb{V}}, r).\\text{id}_{\\mathbb{V}} \\neq \\text{id}_{\\mathbb{V}}$}\n        \\state \\textbf{error} ``Catching up on different set''\n    \\endif\n    \\if{$r \\leqslant$ \\textsc{Leading-Round}}\n        \\state \\textbf{error} ``Catching up in to the past''\n    \\endif\n    \\if{$J^{r, pv}(B)$ \\textbf{is not} valid}\n        \\state \\textbf{error} ``Invalid pre-vote justification''\n    \\endif\n    \\if{$J^{r, pc}(B)$ \\textbf{is not} valid}\n        \\state \\textbf{error} ``Invalid pre-commit justification''\n    \\endif\n    \\state $G \\leftarrow$ \\call{GRANDPA-GHOST}{$J^{r, pv}(B)$}\n    \\if{$G = \\phi$}\n        \\state \\textbf{error} ``GHOST-less Catch-up''\n    \\endif\n    \\if{$r$ \\textbf{is not} completable}\n        \\state \\textbf{error} ``Catch-up round is not completable''\n    \\endif\n    \\if{$J^{r, pc}(B)$ justifies $B'$ finalization}\n        \\state \\textbf{error} ``Unjustified Catch-up target finalization''\n    \\endif\n    \\state \\textsc{Last-Completed-Round} $\\leftarrow r$\n    \\if{$i \\in \\mathbb{V}$}\n        \\state \\call{Play-Grandpa-round}{$r + 1$}\n    \\endif\n\\end{algorithmic}\n\\end{algorithm}",d={title:"6. Finality"},u=void 0,w={unversionedId:"sect-finality",id:"sect-finality",title:"6. Finality",description:"6.1. Introduction",source:"@site/docs/sect-finality.md",sourceDirName:".",slug:"/sect-finality",permalink:"/sect-finality",draft:!1,unlisted:!1,editUrl:"https://github.com/w3f/polkadot-spec/blob/main/docs/sect-finality.md",tags:[],version:"current",frontMatter:{title:"6. Finality"},sidebar:"tutorialSidebar",previous:{title:"5. Block Production",permalink:"/sect-block-production"},next:{title:"7. Light Clients",permalink:"/sect-lightclient"}},y={},v=[{value:"6.1. Introduction",id:"id-introduction-4",level:2},{value:"Definition 77. GRANDPA Voter",id:"defn-grandpa-voter",level:6},{value:"Definition 78. Authority Set Id",id:"defn-authority-set-id",level:6},{value:"Definition 79. GRANDPA State",id:"defn-grandpa-state",level:6},{value:"Definition 80. GRANDPA Vote",id:"defn-vote",level:6},{value:"Definition 81. Voting Rounds",id:"defn-voting-rounds",level:6},{value:"Definition 82. Vote Signature",id:"defn-sign-round-vote",level:6},{value:"Definition 83. Justification",id:"defn-grandpa-justification",level:6},{value:"Definition 84. Finalizing Justification",id:"defn-finalizing-justification",level:6},{value:"Definition 85. Equivocation",id:"defn-voter-equivocation",level:6},{value:"Definition 86. Set of Observed Direct Votes",id:"defn-observed-direct-votes",level:6},{value:"Definition 87. Set of Total Observed Votes",id:"defn-observed-votes",level:6},{value:"Definition 88. Set of Total Potential Votes",id:"defn-total-potential-votes",level:6},{value:"Definition 89. Current Pre-Voted Block",id:"defn-grandpa-ghost",level:6},{value:"Definition 90. Completable Round",id:"defn-grandpa-completable",level:6},{value:"Definition 91. GRANDPA Consensus Message",id:"defn-consensus-message-grandpa",level:6},{value:"6.2. Initiating the GRANDPA State",id:"id-initiating-the-grandpa-state",level:2},{value:"6.2.1. Voter Set Changes",id:"id-voter-set-changes",level:3},{value:"Algorithm 13. Initiate Grandpa",id:"algo-initiate-grandpa",level:6},{value:"6.3. Rejoining the Same Voter Set",id:"id-rejoining-the-same-voter-set",level:2},{value:"6.4. Voting Process in Round {r}",id:"id-voting-process-in-round-r",level:2},{value:"Algorithm 14. Play Grandpa Round",id:"algo-grandpa-round",level:6},{value:"Algorithm 15. Derive Primary",id:"algo-derive-primary",level:6},{value:"Algorithm 16. Best Final Candidate",id:"algo-grandpa-best-candidate",level:6},{value:"Algorithm 17. GRANDPA GHOST",id:"algo-grandpa-ghost",level:6},{value:"Algorithm 18. Best PreVote Candidate",id:"algo-best-prevote-candidate",level:6},{value:"Algorithm 19. Attempt To Finalize At Round",id:"algo-attempt-to\u2013finalize",level:6},{value:"Algorithm 20. Finalizable",id:"algo-finalizable",level:6},{value:"Definition 92. Unfinalized Candidate",id:"defn-unfinalized-candidate",level:6},{value:"6.5. Forced Authority Set Changes",id:"sect-finality-forced-changes",level:2},{value:"Image 6. Applying a scheduled change",id:"img-scheduled-change",level:6},{value:"Image 7. Applying a forced change",id:"img-forced-change",level:6},{value:"6.6. Block Finalization",id:"sect-block-finalization",level:2},{value:"Definition 93. Justified Block Header",id:"defn-justified-block-header",level:6},{value:"Definition 94. Finalized",id:"defn-finalized-block",level:6},{value:"6.6.1. Catching up",id:"sect-grandpa-catchup",level:3},{value:"6.6.1.1. Sending the catch-up requests",id:"sect-sending-catchup-request",level:4},{value:"6.6.1.2. Processing the catch-up requests",id:"id-processing-the-catch-up-requests",level:4},{value:"Algorithm 21. Process Catchup Request",id:"algo-process-catchup-request",level:6},{value:"6.6.1.3. Processing catch-up responses",id:"id-processing-catch-up-responses",level:4},{value:"Algorithm 22. Process Catchup Response",id:"algo-process-catchup-response",level:6},{value:"6.7. Bridge design (BEEFY)",id:"sect-grandpa-beefy",level:2},{value:"6.7.1. Motivation",id:"id-motivation-beefy-1",level:3},{value:"6.7.2. Protocol Overview",id:"id-overview-beefy-1",level:3},{value:"6.7.3. Preliminaries",id:"id-preliminaries-2",level:3},{value:"Definition 95. BEEFY Session Keys",id:"defn-beefy-key",level:6},{value:"6.7.4. Merkle Mountain Ranges",id:"id-beefy-merkle-mountain-ranges",level:3},{value:"Definition 96. Merkle Mountain Ranges",id:"defn-mmr",level:6},{value:"Definition 98. MMR operations",id:"defn-mmr-operations",level:6}],x={toc:v},b="wrapper";function f(a){let{components:e,...d}=a;return(0,n.kt)(b,(0,s.Z)({},x,d,{components:e,mdxType:"MDXLayout"}),(0,n.kt)("h2",{id:"id-introduction-4"},"6.1. Introduction"),(0,n.kt)("p",null,"The Polkadot Host uses GRANDPA Finality protocol to finalize blocks. Finality is obtained by consecutive rounds of voting by the validator nodes. 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In that regard, we have ","[","To do: change function name, only call at genesis, adjust V_B over the sections","]"),(0,n.kt)("div",{parentName:"admonition",className:"math 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We refer to ",(0,n.kt)("span",{parentName:"p",className:"math 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mathbb"},"V"))),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3283em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.05017em"}},"B")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," as ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow",mathvariant:"double-struck"},"V")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\mathbb{{V}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6889em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathbb"},"V")))))))," when there is no chance of ambiguity."),(0,n.kt)("p",{parentName:"admonition"},"Analogously, we say that a Polkadot node is a ",(0,n.kt)("strong",{parentName:"p"},"non-voter node")," for block ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"B")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"))))))," if it does not own any of the key pairs in ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub",mathvariant:"double-struck"},"V"),(0,n.kt)("mi",{parentName:"msub"},"B"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\mathbb{{V}}}_{{B}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8389em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathbb"},"V"))),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3283em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.05017em"}},"B")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),".")),(0,n.kt)("h6",{id:"defn-authority-set-id"},"Definition 78. Authority Set Id"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"The ",(0,n.kt)("strong",{parentName:"p"},"authority set Id")," (",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mtext",{parentName:"msub"},"id"),(0,n.kt)("mi",{parentName:"msub",mathvariant:"double-struck"},"V"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\text{id}_{{{\\mathbb{{V}}}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8444em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord text"},(0,n.kt)("span",{parentName:"span",className:"mord"},"id")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3322em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathbb mtight"},"V")))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),") is an incremental counter which tracks the amount of authority list changes that occurred (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-consensus-message-grandpa"},"Definition 91"),"). Starting with the value of zero at genesis, the Polkadot Host increments this value by one every time a ",(0,n.kt)("strong",{parentName:"p"},"Scheduled Change")," or a ",(0,n.kt)("strong",{parentName:"p"},"Forced Change")," occurs. The authority set ID is an unsigned 64-bit integer.")),(0,n.kt)("h6",{id:"defn-grandpa-state"},"Definition 79. GRANDPA State"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"The ",(0,n.kt)("strong",{parentName:"p"},"GRANDPA state"),", ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mtext",{parentName:"mrow"},"GS")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\text{GS}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord text"},(0,n.kt)("span",{parentName:"span",className:"mord"},"GS")))))),", is defined as:"),(0,n.kt)("div",{parentName:"admonition",className:"math 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mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathbb mtight"},"V")))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))),(0,n.kt)("span",{parentName:"span",className:"mpunct"},","),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"mclose delimcenter",style:{top:"0em"}},"}")))))))),(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"where"),"  "),(0,n.kt)("ul",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow",mathvariant:"double-struck"},"V")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\mathbb{{V}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6889em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathbb"},"V"))))))),": is the set of voters.")),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mtext",{parentName:"msub"},"id"),(0,n.kt)("mi",{parentName:"msub",mathvariant:"double-struck"},"V"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\text{id}_{{{\\mathbb{{V}}}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8444em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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mtight"},"V")))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),": is the authority set ID (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-authority-set-id"},"Definition 78"),").")),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("span",{parentName:"p",className:"math 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In contrast, a voter node can only be assured of the finality (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-finalized-block"},"Definition 94"),") of block ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"B")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"))))))," by actively participating in the voting process. That is by invoking ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-grandpa-round"},"Play-Grandpa-Round"),"."),(0,n.kt)("p",{parentName:"admonition"},"The GRANDPA protocol dictates how an honest voter should vote in each sub-round, which is described by ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-grandpa-round"},"Play-Grandpa-Round"),". After defining what constitutes a vote in GRANDPA, we define how GRANDPA counts votes.")),(0,n.kt)("h6",{id:"defn-voter-equivocation"},"Definition 85. Equivocation"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"Voter ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"v")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{v}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"v"))))))," ",(0,n.kt)("strong",{parentName:"p"},"equivocates")," if they broadcast two or more valid votes to blocks during one voting sub-round. In such a situation, we say that ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"v")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{v}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"v"))))))," is an ",(0,n.kt)("strong",{parentName:"p"},"equivocator")," and any vote ",(0,n.kt)("span",{parentName:"p",className:"math 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text"},(0,n.kt)("span",{parentName:"span",className:"mord"},"GRANDPA-GHOST")),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"minner"},(0,n.kt)("span",{parentName:"span",className:"mopen delimcenter",style:{top:"0em"}},"("),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"mclose delimcenter",style:{top:"0em"}},")")))))))),(0,n.kt)("p",{parentName:"admonition"},"Finally, we define when a voter ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"v")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{v}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"v"))))))," sees a round as completable, that is, when they are confident that ",(0,n.kt)("span",{parentName:"p",className:"math 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mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"mpunct"},","),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"m"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"+"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.7519em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"\u2032"))))))))),(0,n.kt)("span",{parentName:"span",className:"mclose"},")"),(0,n.kt)("span",{parentName:"span",className:"mord"},"\u2225"))))))," where ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup"},"B"),(0,n.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B'}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.7519em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.7519em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"\u2032"))))))))))))))," is the block where the change is applied. The earliest digest of this type in a block will be respected. ",(0,n.kt)("br",null),(0,n.kt)("br",null)," Forced changes are explained further in ",(0,n.kt)("a",{parentName:"td",href:"sect-finality#sect-finality-forced-changes"},"Section 6.5."),".")),(0,n.kt)("tr",{parentName:"tbody"},(0,n.kt)("td",{parentName:"tr",align:null},(0,n.kt)("strong",{parentName:"td"},"3")),(0,n.kt)("td",{parentName:"tr",align:null},"Implies ",(0,n.kt)("strong",{parentName:"td"},"on disabled"),": An index to the individual authority in the current authority list (",(0,n.kt)("a",{parentName:"td",href:"chap-sync#defn-authority-list"},"Definition 33"),") that should be immediately disabled until the next authority set changes. When an authority gets disabled, the node should stop performing any authority functionality from that authority, including authoring blocks and casting GRANDPA votes for finalization. Similarly, other nodes should ignore all messages from the indicated authority which pertain to their authority role.")),(0,n.kt)("tr",{parentName:"tbody"},(0,n.kt)("td",{parentName:"tr",align:null},(0,n.kt)("strong",{parentName:"td"},"4")),(0,n.kt)("td",{parentName:"tr",align:null},"Implies ",(0,n.kt)("strong",{parentName:"td"},"pause"),": A signal to pause the current authority set after the given delay of ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"N"),(0,n.kt)("mtext",{parentName:"msub"},"delay")),(0,n.kt)("mo",{parentName:"mrow"},":"),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2225"),(0,n.kt)("mtext",{parentName:"mrow"},"SubChain"),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},"("),(0,n.kt)("mi",{parentName:"mrow"},"B"),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup"},"B"),(0,n.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032")),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},")"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2225")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{N_{\\text{delay}} := \\|\\text{SubChain}(B,B')\\|}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1.038em",verticalAlign:"-0.2861em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10903em"}},"N"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 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mtight"},"\u2032"))))))))),(0,n.kt)("span",{parentName:"span",className:"mclose"},")"),(0,n.kt)("span",{parentName:"span",className:"mord"},"\u2225"))))))," where ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup"},"B"),(0,n.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B'}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.7519em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.7519em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"\u2032"))))))))))))))," is a block where the change is applied. Once applied, the authorities should stop voting.")),(0,n.kt)("tr",{parentName:"tbody"},(0,n.kt)("td",{parentName:"tr",align:null},(0,n.kt)("strong",{parentName:"td"},"5")),(0,n.kt)("td",{parentName:"tr",align:null},"Implies ",(0,n.kt)("strong",{parentName:"td"},"resume"),": A signal to resume the current authority set after the given delay of ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"N"),(0,n.kt)("mtext",{parentName:"msub"},"delay")),(0,n.kt)("mo",{parentName:"mrow"},":"),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2225"),(0,n.kt)("mtext",{parentName:"mrow"},"SubChain"),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},"("),(0,n.kt)("mi",{parentName:"mrow"},"B"),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup"},"B"),(0,n.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032")),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},")"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2225")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{N_{\\text{delay}} := \\|\\text{SubChain}(B,B')\\|}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1.038em",verticalAlign:"-0.2861em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10903em"}},"N"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 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mtight"},"\u2032"))))))))),(0,n.kt)("span",{parentName:"span",className:"mclose"},")"),(0,n.kt)("span",{parentName:"span",className:"mord"},"\u2225"))))))," where ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup"},"B"),(0,n.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B'}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.7519em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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Once applied, the authorities should resume voting."))))),(0,n.kt)("h2",{id:"id-initiating-the-grandpa-state"},"6.2. Initiating the GRANDPA State"),(0,n.kt)("p",null,"In order to participate coherently in the voting process, a validator must initiate its state and sync it with other active validators. In particular, considering that voting is happening in different distinct rounds where each round of voting is assigned a unique sequential round number ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"r"),(0,n.kt)("mi",{parentName:"msub"},"v"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}_{{v}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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mtight",style:{marginRight:"0.03588em"}},"v")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),", it needs to determine and set its round counter ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r"))))))," equal to the voting round ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"r"),(0,n.kt)("mi",{parentName:"msub"},"n"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}_{{n}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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currently undergoing in the network. The mandated initialization procedure for the GRANDPA protocol for a joining validator is described in detail in ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-initiate-grandpa"},"Initiate-Grandpa"),"."),(0,n.kt)("p",null,"The process of joining a new voter set is different from the one of rejoining the current voter set after a network disconnect. The details of this distinction are described further in this section."),(0,n.kt)("h3",{id:"id-voter-set-changes"},"6.2.1. Voter Set Changes"),(0,n.kt)("p",null,"A GRANDPA voter node which is initiating GRANDPA protocol as part of joining a new authority set is required to execute ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-initiate-grandpa"},"Initiate-Grandpa"),". The algorithm mandates the initialization procedure for GRANDPA protocol."),(0,n.kt)("admonition",{type:"info"},(0,n.kt)("p",{parentName:"admonition"},"The GRANDPA round number reset to 0 for every authority set change.")),(0,n.kt)("p",null,"Voter set changes are signaled by Runtime via a consensus engine message (",(0,n.kt)("a",{parentName:"p",href:"chap-sync#sect-consensus-message-digest"},"Section 3.3.2."),"). When Authorities process such messages they must not vote on any block with a higher number than the block at which the change is supposed to happen. The new authority set should reinitiate GRANDPA protocol by executing ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-initiate-grandpa"},"Initiate-Grandpa"),"."),(0,n.kt)("h6",{id:"algo-initiate-grandpa"},"Algorithm 13. Initiate Grandpa"),(0,n.kt)("admonition",{type:"algorithm"},(0,n.kt)(m.Z,{content:p,algID:"initiateGrandpa",options:{lineNumber:!0},mdxType:"Pseudocode"}),(0,n.kt)("p",{parentName:"admonition"},"where ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mtext",{parentName:"msub"},"last"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{\\text{last}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"last"))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," is the last block which has been finalized on the chain (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-finalized-block"},"Definition 94"),"). ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"r"),(0,n.kt)("mtext",{parentName:"msub"},"last"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}_{{\\text{last}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"last"))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," is equal to the latest round the voter has observed that other voters are voting on. The voter obtains this information through various gossiped messages including those mentioned in ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-finalized-block"},"Definition 94"),". ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"r"),(0,n.kt)("mtext",{parentName:"msub"},"last"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}_{{\\text{last}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"last"))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," is set to ",(0,n.kt)("em",{parentName:"p"},"0")," if the GRANDPA node is initiating the GRANDPA voting process as a part of a new authority set. This is because the GRANDPA round number resets to ",(0,n.kt)("em",{parentName:"p"},"0")," for every authority set change.")),(0,n.kt)("h2",{id:"id-rejoining-the-same-voter-set"},"6.3. Rejoining the Same Voter Set"),(0,n.kt)("p",null,"When a voter node rejoins the network after a disconnect from the voter set and with the condition that there has been no change to the voter set at the time of the disconnect, the node must continue performing the GRANDPA protocol at the same state as before getting disconnected from the network, ignoring any possible progress in GRANDPA finalization. Following reconnection, the node eventually gets updated to the current GRANDPA round and synchronizes its state with the rest of the voting set through the process called Catchup (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#sect-grandpa-catchup"},"Section 6.6.1."),")."),(0,n.kt)("h2",{id:"id-voting-process-in-round-r"},"6.4. Voting Process in Round ",(0,n.kt)("span",{parentName:"h2",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r"))))))),(0,n.kt)("p",null,"For each round ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")))))),", an honest voter ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"v")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{v}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"v"))))))," must participate in the voting process by following ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-grandpa-round"},"Play-Grandpa-Round"),"."),(0,n.kt)("h6",{id:"algo-grandpa-round"},"Algorithm 14. Play Grandpa Round"),(0,n.kt)("admonition",{type:"algorithm"},(0,n.kt)(m.Z,{content:r,algID:"playGrandpaRound",options:{lineNumber:!0},mdxType:"Pseudocode"}),(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"where"),"  "),(0,n.kt)("ul",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"T")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{T}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.13889em"}},"T"))))))," is sampled from a log-normal distribution whose mean and standard deviation are equal to the average network delay for a message to be sent and received from one validator to another.")),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mtext",{parentName:"mrow"},"Derive-Primary")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\text{Derive-Primary}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,n.kt)("span",{parentName:"span",className:"mord text"},(0,n.kt)("span",{parentName:"span",className:"mord"},"Derive-Primary"))))))," is described in ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-derive-primary"},"Derive-Primary"),".")),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"The condition of ",(0,n.kt)("em",{parentName:"p"},"completablitiy")," is defined in ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-grandpa-completable"},"Definition 90"),".")),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("span",{parentName:"p",className:"math 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Best PreVote Candidate"),(0,n.kt)("admonition",{type:"algorithm"},(0,n.kt)(m.Z,{content:k,algID:"bestPrevoteCandidate",options:{lineNumber:!0},mdxType:"Pseudocode"})),(0,n.kt)("h6",{id:"algo-attempt-to\u2013finalize"},"Algorithm 19. Attempt To Finalize At Round"),(0,n.kt)("admonition",{type:"algorithm"},(0,n.kt)(m.Z,{content:o,algID:"attemptToFinalizeAtRound",options:{lineNumber:!0},mdxType:"Pseudocode"})),(0,n.kt)("h6",{id:"algo-finalizable"},"Algorithm 20. Finalizable"),(0,n.kt)("admonition",{type:"algorithm"},(0,n.kt)(m.Z,{content:c,algID:"finalizable",options:{lineNumber:!0},mdxType:"Pseudocode"}),(0,n.kt)("p",{parentName:"admonition"},"where the condition for ",(0,n.kt)("em",{parentName:"p"},"completability")," is defined in ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-grandpa-completable"},"Definition 90"),".")),(0,n.kt)("p",null,"Note that we might not always succeed in finalizing our best candidate due to the possibility of equivocation. We might even not finalize anything in a round (although ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-grandpa-round"},"Play-Grandpa-Round")," prevents us from moving to the round ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mo",{parentName:"mrow"},"+"),(0,n.kt)("mn",{parentName:"mrow"},"1")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}+{1}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"+"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"1"))))))," before finalizing the best final candidate of round ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mo",{parentName:"mrow"},"\u2212"),(0,n.kt)("mn",{parentName:"mrow"},"1")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}-{1}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"\u2212"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"1")))))),") The example in ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-unfinalized-candidate"},"Definition 92")," serves to demonstrate a situation where the best final candidate of a round cannot be finalized during its own round:"),(0,n.kt)("h6",{id:"defn-unfinalized-candidate"},"Definition 92. Unfinalized Candidate"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"Let us assume that we have 100 voters and there are two blocks in the chain (",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1")),(0,n.kt)("mo",{parentName:"mrow"},"<"),(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}<{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"<"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),"). At round 1, we get 67 pre-votes for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," and at least one pre-vote for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," which means that ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mtext",{parentName:"mrow"},"GRANDPA-GHOST"),(0,n.kt)("mrow",{parentName:"mrow"},(0,n.kt)("mo",{parentName:"mrow",fence:"true"},"("),(0,n.kt)("mn",{parentName:"mrow"},"1"),(0,n.kt)("mo",{parentName:"mrow",fence:"true"},")")),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\text{GRANDPA-GHOST}{\\left({1}\\right)}={B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord text"},(0,n.kt)("span",{parentName:"span",className:"mord"},"GRANDPA-GHOST")),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"minner"},(0,n.kt)("span",{parentName:"span",className:"mopen delimcenter",style:{top:"0em"}},"("),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"1")),(0,n.kt)("span",{parentName:"span",className:"mclose delimcenter",style:{top:"0em"}},")"))),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),"."),(0,n.kt)("p",{parentName:"admonition"},"Subsequently, potentially honest voters who could claim not seeing all the pre-votes for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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the pre-votes for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," would pre-commit to ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),". In this way, we receive 66 pre-commits for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," and 1 pre-commit for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),". Henceforth, we finalize ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," since we have a threshold commit (67 votes) for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),"."),(0,n.kt)("p",{parentName:"admonition"},"At this point, though, we have ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mtext",{parentName:"mrow"},"Best-Final-Candidate"),(0,n.kt)("mrow",{parentName:"mrow"},(0,n.kt)("mo",{parentName:"mrow",fence:"true"},"("),(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mo",{parentName:"mrow",fence:"true"},")")),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\mathtt{\\text{Best-Final-Candidate}}}{\\left({r}\\right)}={B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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delimcenter",style:{top:"0em"}},")"))),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," as ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"#"),(0,n.kt)("msubsup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msubsup"},"V"),(0,n.kt)("mrow",{parentName:"msubsup"},(0,n.kt)("mtext",{parentName:"mrow"},"obs"),(0,n.kt)("mrow",{parentName:"mrow"},(0,n.kt)("mo",{parentName:"mrow",fence:"true"},"("),(0,n.kt)("mi",{parentName:"mrow"},"v"),(0,n.kt)("mo",{parentName:"mrow",fence:"true"},")")),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("mtext",{parentName:"mrow"},"pot")),(0,n.kt)("mrow",{parentName:"msubsup"},(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("mtext",{parentName:"mrow"},"stage"))),(0,n.kt)("mrow",{parentName:"mrow"},(0,n.kt)("mo",{parentName:"mrow",fence:"true"},"("),(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2")),(0,n.kt)("mo",{parentName:"mrow",fence:"true"},")")),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("mn",{parentName:"mrow"},"67")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\#{{V}_{{\\text{obs}{\\left({v}\\right)},\\text{pot}}}^{{{r},\\text{stage}}}}{\\left({B}_{{2}}\\right)}={67}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1.4267em",verticalAlign:"-0.5152em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"#"),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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delimcenter",style:{top:"0em"}},(0,n.kt)("span",{parentName:"span",className:"mtight"},"(")),(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.03588em"}},"v")),(0,n.kt)("span",{parentName:"span",className:"mclose mtight delimcenter",style:{top:"0em"}},(0,n.kt)("span",{parentName:"span",className:"mtight"},")")))),(0,n.kt)("span",{parentName:"span",className:"mpunct mtight"},","),(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"pot")))))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.1809em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord 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mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))),(0,n.kt)("span",{parentName:"span",className:"mclose delimcenter",style:{top:"0em"}},")"))),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"67"))))))," and ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mn",{parentName:"mrow"},"2"),(0,n.kt)("mo",{parentName:"mrow"},">"),(0,n.kt)("mn",{parentName:"mrow"},"1")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{2}>{1}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6835em",verticalAlign:"-0.0391em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"2")),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},">"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"1")))))),"."),(0,n.kt)("p",{parentName:"admonition"},"However, at this point, the round is already completable as we know that we have ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mtext",{parentName:"mrow"},"GRANDPA-GHOST"),(0,n.kt)("mrow",{parentName:"mrow"},(0,n.kt)("mo",{parentName:"mrow",fence:"true"},"("),(0,n.kt)("mn",{parentName:"mrow"},"1"),(0,n.kt)("mo",{parentName:"mrow",fence:"true"},")")),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\mathtt{\\text{GRANDPA-GHOST}}}{\\left({1}\\right)}={B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord text"},(0,n.kt)("span",{parentName:"span",className:"mord"},"GRANDPA-GHOST"))),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"minner"},(0,n.kt)("span",{parentName:"span",className:"mopen delimcenter",style:{top:"0em"}},"("),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"1")),(0,n.kt)("span",{parentName:"span",className:"mclose delimcenter",style:{top:"0em"}},")"))),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," as an upper limit on what we can finalize and nothing greater than ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," can be finalized at ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("mn",{parentName:"mrow"},"1")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}={1}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"1")))))),". Therefore, the condition of ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-grandpa-round"},"Play-Grandpa-Round")," is satisfied and we must proceed to round 2."),(0,n.kt)("p",{parentName:"admonition"},"Nonetheless, we must continue to attempt to finalize round ",(0,n.kt)("em",{parentName:"p"},"1")," in the background as the condition of ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-attempt-to%E2%80%93finalize"},"Attempt-To-Finalize-At-Round")," has not been fulfilled."),(0,n.kt)("p",{parentName:"admonition"},"This prevents us from proceeding to round 3 until either:"),(0,n.kt)("ul",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"We finalize ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," in round 2, or")),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"We receive an extra pre-commit vote for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," in round 1. This will make it impossible to finalize ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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matter to whom the remaining pre-commits are going to be cast for (even with considering the possibility of 1/3 of voter equivocating) and therefore we have ",(0,n.kt)("span",{parentName:"p",className:"math 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Bridge design (BEEFY)"),(0,n.kt)("admonition",{type:"caution"},(0,n.kt)("p",{parentName:"admonition"},"The BEEFY protocol specifications are Work-In-Progress. It is currently being tested on test-nets like Rococco and has not yet been deployed on Polkadot. ")),(0,n.kt)("p",null,"The BEEFY (Bridge Efficiency Enabling Finality Yielder) is a secondary protocol to GRANDPA to support efficient bridging between the Polkadot network (relay chain) and remote, segregated blockchains, such as Ethereum, which were not built with the Polkadot interchain operability in mind. BEEFY\u2019s aim is to enable clients to efficiently follow a chain that has GRANDPA finality, a finality gadget created for Substrate/Polkadot ecosystem. This is useful for bridges (e.g., Polkadot->Ethereum), where a chain can follow another chain and light clients suitable for low storage devices such as mobile phones."),(0,n.kt)("p",null,"The protocol allows participants of the remote network to verify finality proofs created by the Polkadot relay chain validators. In other words: clients in the target network (e.g., Ethereum) can verify that the Polkadot network is at a specific state."),(0,n.kt)("p",null,"Storing all the information necessary to verify the state of the remote chain, such as the block headers, is too expensive. BEEFY stores the information in a space-efficient way, and clients can request additional information over the protocol."),(0,n.kt)("h3",{id:"id-motivation-beefy-1"},"6.7.1. Motivation"),(0,n.kt)("p",null,"A client could just follow GRANDPA using GRANDPA justifications, sets of signatures from validators. This is used for some substrate-substrate bridges and in light clients such as the Substrate Connect browser extension. GRANDPA was designed for fast and secure finality. Certain design decisions, like validators voting for different blocks in a justification and using ED25519 signatures, allow fast finality. However, GRANDPA justifications are large and are expensive to verify on other chains like Ethereum that do not support some cryptographic signature schemes. Thus, BEEFY adds an extra layer of finality that allows lighter bridges and clients for Polkadot."),(0,n.kt)("p",null,"To summarise, the goals of BEEFY are:"),(0,n.kt)("ul",null,(0,n.kt)("li",{parentName:"ul"},"Allow customization of signature schemes to adapt to different target chains. "),(0,n.kt)("li",{parentName:"ul"},"Minimize the size of the finality proof and the effort required by a light client to follow finality. "),(0,n.kt)("li",{parentName:"ul"},"Unify data types and use backward-compatible versioning so that the protocol can be extended (additional payload, different signature schemes) without breaking existing light clients.")),(0,n.kt)("h3",{id:"id-overview-beefy-1"},"6.7.2. Protocol Overview"),(0,n.kt)("p",null,"Since BEEFY runs on top of GRANDPA, similarly to how GRANDPA is lagging behind the best produced (non-finalized) block, BEEFY finalized block lags behind the best GRANDPA (finalized) block. "),(0,n.kt)("ul",null,(0,n.kt)("li",{parentName:"ul"},"The BEEFY validator set is the same as GRANDPA's. However, they might be using different types of session keys to sign BEEFY messages. "),(0,n.kt)("li",{parentName:"ul"},"From a single validator perspective, BEEFY has at most one active voting round. "),(0,n.kt)("li",{parentName:"ul"},"Since GRANDPA validators are reaching finality, we assume they are online and well-connected and have a similar view of the state of the blockchain.")),(0,n.kt)("p",null,"BEEFY consists of two components:"),(0,n.kt)("p",null,"a. ",(0,n.kt)("strong",{parentName:"p"},"Consensus Extension")," on GRANDPA finalization that is a voting round. "),(0,n.kt)("p",null,"The consensus extension serves to have a smaller consensus justification than GRANDPA and alternative cryptography, which helps the light client side of the BEEFY protocol described below. "),(0,n.kt)("p",null,"b. ",(0,n.kt)("strong",{parentName:"p"},"Light client")," protocol for convincing the other chain/device efficiently about the finality vote."),(0,n.kt)("p",null,"In the BEEFY light client, a prover, a full node, or a bridge relayer, wants to convince a verifier, a light client that may be a bridge implementation on the target chain, of the outcome of a BEEFY vote. The prover has access to all voting data from the BEEFY voting round. In the light client part, the prover may generate a proof and send it to the verifier or they may engage in an interactive protocol with several rounds of communication."),(0,n.kt)("h3",{id:"id-preliminaries-2"},"6.7.3. Preliminaries"),(0,n.kt)("h6",{id:"defn-beefy-key"},"Definition 95. BEEFY Session Keys"),(0,n.kt)("p",null,"Validators use an ",(0,n.kt)("inlineCode",{parentName:"p"},"ECDSA")," key scheme for signing Beefy messages. This is different from schemes like ",(0,n.kt)("inlineCode",{parentName:"p"},"sr25519")," and ",(0,n.kt)("inlineCode",{parentName:"p"},"ed25519"),", which are commonly used in Substrate for other components like BABE, and GRANDPA. The most noticeable difference is that an ",(0,n.kt)("inlineCode",{parentName:"p"},"ecdsa"),"\npublic key is ",(0,n.kt)("inlineCode",{parentName:"p"},"33")," bytes long, instead of ",(0,n.kt)("inlineCode",{parentName:"p"},"32")," bytes for a ",(0,n.kt)("inlineCode",{parentName:"p"},"sr25519")," based public key. As a consequence, the ",(0,n.kt)("inlineCode",{parentName:"p"},"AccountId")," (32-bytes) matches the ",(0,n.kt)("inlineCode",{parentName:"p"},"PublicKey")," for other session keys, but note that it's not the case for BEEFY."),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"BEEFY session key pair")," ",(0,n.kt)("span",{parentName:"p",className:"math 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mtight",style:{marginRight:"0.05017em"}},"B"))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3948em"}},(0,n.kt)("span",{parentName:"span"}))))))),(0,n.kt)("span",{parentName:"span",className:"mclose delimcenter",style:{top:"0em"}},(0,n.kt)("span",{parentName:"span",className:"delimsizing size1"},")"))))))))," is a ",(0,n.kt)("inlineCode",{parentName:"p"},"secp256k1")," key pair which the BEEFY authority node ",(0,n.kt)("span",{parentName:"p",className:"math 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Merkle Mountain Ranges"),(0,n.kt)("h6",{id:"defn-mmr"},"Definition 96. Merkle Mountain Ranges"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"Merkle Mountain Ranges, ",(0,n.kt)("strong",{parentName:"p"},"MMR"),", are used as an efficient way to send block headers and signatures to light clients. Merkle Mountain Ranges (",(0,n.kt)("inlineCode",{parentName:"p"},"MMR"),") is an improvement of the traditional Merkle tree data structure. Just like a Merkle tree, an ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR")," is a binary tree where each leaf node represents a data element and each non-leaf node is the hash of its child nodes. The key difference between a traditional Merkle tree and a MMR lies in the way nodes are organized. In traditional Merkle trees, whenever a leaf node is appended or removed, the tree must be rebuilt and the hashes of the non-leaf nodes must be recalculated. The overhead of recomputing the hashes up to the root makes traditional Merkle unsuitable for handling dynamic data. The ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR")," is designed to optimize the appending and removal of elements without requiring a complete rebuild of the tree, which makes it more efficient to handle growing lists of leaf nodes."),(0,n.kt)("p",{parentName:"admonition"},"MMR structure "),(0,n.kt)("p",{parentName:"admonition"},"A ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR")," structure can be seen as a list of perfectly balanced binary sub-trees in descending order of height. It is a strictly append-only structure where nodes are added from left to right, such that a parent node is added as soon as two children exist. The following representation shows a ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR")," with 11 elements, 7 leaf nodes and 4 non-leaf nodes, where the value of each node corresponds to the order in which it was inserted into the tree."),(0,n.kt)("pre",{parentName:"admonition"},(0,n.kt)("code",{parentName:"pre"},"       7\n      / \\\n     /   \\\n    /     \\\n   3       6        10\n  / \\     / \\       / \\   \n /   \\   /   \\     /   \\   \n1     2 4     5   8     9    11\n"))),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("h6",{parentName:"admonition",id:"defn-mmr-root"},"Definition 97. Merkle Mountain Ranges root (MMR-root)"),(0,n.kt)("p",{parentName:"admonition"},"An ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR")," does not have a single ",(0,n.kt)("inlineCode",{parentName:"p"},"root")," by design, as a conventional Merkle tree. Every sub-tree has a separate sub-root, which we refer to as the ",(0,n.kt)("inlineCode",{parentName:"p"},"peak")," of the sub-tree.\nBagging the peaks is the process used for hashing the peaks in order to compute the ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR-root"),". It is important to define the order in which the peaks are hashed to ensure that a given sub-set of peaks will always derive a unique ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR-root"),". Here we state that peaks are merged from right to left and bagged via ",(0,n.kt)("span",{parentName:"p",className:"math 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p_2,...,p_n")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"p"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))),(0,n.kt)("span",{parentName:"span",className:"mpunct"},","),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"p"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))),(0,n.kt)("span",{parentName:"span",className:"mpunct"},","),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"..."),(0,n.kt)("span",{parentName:"span",className:"mpunct"},","),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"p"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.1514em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight"},"n")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),": represents the list of current peaks in decreasing order of heights."),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("span",{parentName:"li",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"h"),(0,n.kt)("mi",{parentName:"mrow"},"a"),(0,n.kt)("mi",{parentName:"mrow"},"s"),(0,n.kt)("mi",{parentName:"mrow"},"h")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"hash")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"ha"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"s"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"h"))))),": corresponds to the 256-bit ",(0,n.kt)("a",{parentName:"li",href:"chap-host-api#sect-finality#id-ext_hashing_keccak_256"},"Keccak")," hash function used to merge the peaks."))),(0,n.kt)("p",null,"A distinguished feature of this process is that whenever new leaf nodes are added to the tree, the earlier hash computations of peaks are reused, making new leaf nodes less expensive to insert and to prove (i.e., to verify the integrity of leaf data). "),(0,n.kt)("h6",{id:"defn-mmr-operations"},"Definition 98. MMR operations"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"Here are the basic operations we should be able to perform on the MMR:"),(0,n.kt)("ul",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Append Leaf Node (appendData):"),(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"Signature: append(data: T) -> None"),(0,n.kt)("li",{parentName:"ul"},"Description: appends a new leaf element with the provided data to the MMR.\n"))),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Create MMR root (bagPeaks):"),(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"Signature: baggingPeaks(peaksIndexes: List","[int]",") -> str"),(0,n.kt)("li",{parentName:"ul"},"Description: creates the single MMR root based on the list of peaks, and returns the hash string of the  MMR root corresponding to the current state of the tree. "))),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Verify Node (verifyProof):"),(0,n.kt)("pre",{parentName:"li"},(0,n.kt)("code",{parentName:"pre"},"- Signature: verifyNode(nodeHash: str, requiredProofNodes: List[str], MMRroot: str) -> bool\n- Description: verifies if the given node hash can be proved based on the list of required proof nodes and the MMR root hash.\n")),(0,n.kt)("p",{parentName:"li"},":::"))),(0,n.kt)("h6",{parentName:"admonition",id:"defn-beefy-payload"},"Definition 99. Payload"),(0,n.kt)("admonition",{parentName:"admonition",type:"definition"},(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"Payload:")," is the Merkle root of the MMR generated where the leaf data contains the following fields for each block:"),(0,n.kt)("ul",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("em",{parentName:"li"},"LeafVersion"),": a byte indicating the current version number of the Leaf Format. The first 3 bits are for major versions and the last 5 bits are for minor versions. "),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("em",{parentName:"li"},"BeefyNextAuthoritySetInfo"),": It is a tuple consisting of:",(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"ValidatorSetID"),(0,n.kt)("li",{parentName:"ul"},"Len (u32): length of the validator set"),(0,n.kt)("li",{parentName:"ul"},"Merkle Root of the list of Next Beefy Authority Set (ECDSA public keys). The exact format depends on the implementation.  "))),(0,n.kt)("li",{parentName:"ul"},"Parent Block number and Parent Block Hash."),(0,n.kt)("li",{parentName:"ul"},"Extra Leaf Data: Currently the Merkle root of the list of (ParaID, ParaHeads)"))),(0,n.kt)("h6",{parentName:"admonition",id:"defn-beefy-Commitment"},"Definition 100. Signed Commitment"),(0,n.kt)("admonition",{parentName:"admonition",type:"definition"},(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"Signed Commitment:"),"\n",(0,n.kt)("inlineCode",{parentName:"p"},"commitment")," is a tuple of ",(0,n.kt)("inlineCode",{parentName:"p"},"(payload, Block Number, ValidatorSetID)"),". A ",(0,n.kt)("inlineCode",{parentName:"p"},"signed commitment")," is a tuple ",(0,n.kt)("inlineCode",{parentName:"p"},"(commitment, signatures)"),", where ",(0,n.kt)("inlineCode",{parentName:"p"},"signatures")," is a list of optional signatures of the validator set on the SCALE encoded ",(0,n.kt)("inlineCode",{parentName:"p"},"commitment"),". Note that the number of signatures in ",(0,n.kt)("inlineCode",{parentName:"p"},"signatures")," may be less than the length of the Validator Set. ")),(0,n.kt)("h6",{parentName:"admonition",id:"defn-beefy-witness-data"},"Definition 101. Witness Data"),(0,n.kt)("admonition",{parentName:"admonition",type:"definition"},(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"Signed Commitment Witnesses")," contains the commitment and an array indicating which validator of the Polkadot network voted for the payload (but not the signatures themselves). The indicators of which validator voted for the payload are just claims and provide no proof. It also contains the signature of one validator on the commitment, which is used only by the subsampling-based Light Clients. The network message is defined in ",(0,n.kt)("a",{parentName:"p",href:"chap-networking#defn-grandpa-beefy-signed-commitment-witness"},"Definition 58")," and the relayer saves it on the chain of the remote network.")),(0,n.kt)("h6",{parentName:"admonition",id:"defn-beefy-light-client"},"Definition 102. Light Client"),(0,n.kt)("admonition",{parentName:"admonition",type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"A ",(0,n.kt)("strong",{parentName:"p"},"light client")," is an abstract entity in a remote network such as Ethereum. It can be a node or a smart contract with the intent of requesting finality proofs from the Polkadot network. A light client reads the witness data (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-witness-data"},"Definition 101")," from the chain, then requests the signatures directly from the relayer in order to verify those.")),(0,n.kt)("h6",{parentName:"admonition",id:"defn-beefy-relayer"},"Definition 103. Relayer"),(0,n.kt)("admonition",{parentName:"admonition",type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"A ",(0,n.kt)("strong",{parentName:"p"},"relayer"),' (or "prover") is an abstract entity that takes finality proofs from the Polkadot network and makes those available to the light clients. The relayer attempts to convince the light clients that the finality proofs have been voted for by the Polkadot relay chain validators. The relayer operates off-chain and can for example be a node or a collection of nodes.')),(0,n.kt)("h3",{parentName:"admonition",id:"id-voting-on-payloads"},"6.7.5. Voting on Payloads"),(0,n.kt)("p",{parentName:"admonition"},"The Polkadot Host signs the MMR payload (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-payload"},"Definition 99"),") and gossips it as part of a vote (",(0,n.kt)("a",{parentName:"p",href:"chap-networking#defn-msg-beefy-gossip"},"Definition 56"),") to its peers on every new finalized block. The Polkadot Host uses ECDSA for signing the payload since Ethereum has better compatibility for it compared to SR25519 or ED25519."),(0,n.kt)("h3",{parentName:"admonition",id:"sect-beefy-committing-witnesses"},"6.7.6. Committing Witnesses"),(0,n.kt)("p",{parentName:"admonition"},"The relayer (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-relayer"},"Definition 103"),") participates in the Polkadot network by collecting the gossiped votes (",(0,n.kt)("a",{parentName:"p",href:"chap-networking#defn-msg-beefy-gossip"},"Definition 56"),"). Those votes are converted into the witness data structure (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-witness-data"},"Definition 101"),"). The relayer saves the data on the chain of the remote network. The occurrence of saving witnesses on remote networks is undefined."),(0,n.kt)("h3",{parentName:"admonition",id:"id-requesting-signed-commitments"},"6.7.7. Requesting Signed Commitments"),(0,n.kt)("p",{parentName:"admonition"},"A light client (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-light-client"},"Definition 102"),") fetches the Signed Commitment Witness (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-witness-data"},"Definition 101"),") from the chain. Once the light client knows which validators apparently voted for the specified payload, it needs to request the signatures from the relayer to verify whether the claims are actually true. This is achieved by requesting signed commitments (",(0,n.kt)("a",{parentName:"p",href:"chap-networking#defn-grandpa-beefy-signed-commitment"},"Definition 57"),")."),(0,n.kt)("p",{parentName:"admonition"},"How those signed commitments are requested by the light client and delivered by the relayer varies among networks or implementations."),(0,n.kt)("h6",{parentName:"admonition",id:"defn-consensus-message-beefy"},"Definition 104. BEEFY Consensus Message"),(0,n.kt)("admonition",{parentName:"admonition",type:"definition"},(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("span",{parentName:"p",className:"math 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mathnormal",style:{marginRight:"0.22222em"}},"V")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3283em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.05017em"}},"B")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," that should be immediately disabled until the next authority change.")),(0,n.kt)("tr",{parentName:"tbody"},(0,n.kt)("td",{parentName:"tr",align:null},"3"),(0,n.kt)("td",{parentName:"tr",align:null},"implies ",(0,n.kt)("strong",{parentName:"td"},"MMR root"),": a 32-byte array containing the MMR root payload."))))),(0,n.kt)("h3",{parentName:"admonition",id:"id-consensus-mechanism-beefy-1"},"6.7.8. Consensus Mechanism"),(0,n.kt)("p",{parentName:"admonition"},"Role of various Actors in BEEFY:"),(0,n.kt)("ul",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Validators are expected to additionally:"),(0,n.kt)("ol",{parentName:"li"},(0,n.kt)("li",{parentName:"ol"},"Produce & broadcast vote for the current round."))),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Regular nodes perform the following tasks:"),(0,n.kt)("ol",{parentName:"li"},(0,n.kt)("li",{parentName:"ol"},"Receive & validate votes for the current round and broadcast them to their peers."),(0,n.kt)("li",{parentName:"ol"},"Receive & validate BEEFY Justifications and broadcast them to their peers."),(0,n.kt)("li",{parentName:"ol"},"Return BEEFY Justifications for ",(0,n.kt)("strong",{parentName:"li"},"Mandatory Blocks")," on demand."),(0,n.kt)("li",{parentName:"ol"},"Optionally return BEEFY Justifications for non-mandatory blocks on demand.")))),(0,n.kt)("p",{parentName:"admonition"},"A ",(0,n.kt)("strong",{parentName:"p"},"round")," is an attempt by BEEFY validators to produce a BEEFY Justification. ",(0,n.kt)("strong",{parentName:"p"},"Round number")," is simply defined as a block number the validators are voting for, or to be more precise, the Commitment for that block number. Round ends when the next round is started, which may happen when one of the events occur:"),(0,n.kt)("ol",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ol"},"Either the node collects ",(0,n.kt)("inlineCode",{parentName:"li"},"2/3rd + 1")," valid votes for that round."),(0,n.kt)("li",{parentName:"ol"},"Or the node receives a BEEFY Justification for a block greater than the current best BEEFY block.")),(0,n.kt)("p",{parentName:"admonition"},'In both cases the node proceeds to determine the new round number using "Round Selection"\nprocedure.'),(0,n.kt)("p",{parentName:"admonition"},"Both kinds of actors are expected to fully participate in the protocol ONLY IF they believe they are up-to-date with the rest of the network, i.e. they are fully synced. Before this happens, the node should continue processing imported BEEFY Justifications and votes without actively voting themselves."),(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"Round Selection")),(0,n.kt)("p",{parentName:"admonition"},"Every node (both regular nodes and validators) determines locally what it believes\nthe current round number is. The choice is based on their knowledge of:"),(0,n.kt)("ol",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ol"},"Best GRANDPA finalized block number (",(0,n.kt)("inlineCode",{parentName:"li"},"best_grandpa"),")."),(0,n.kt)("li",{parentName:"ol"},"Best BEEFY finalized block number (",(0,n.kt)("inlineCode",{parentName:"li"},"best_beefy"),")."),(0,n.kt)("li",{parentName:"ol"},"Starting block of the current session (",(0,n.kt)("inlineCode",{parentName:"li"},"session_start"),").")),(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"Session")," means a period of time (or rather a number of blocks) where the validator set (keys) does not change. Session are synonymous to epochs (",(0,n.kt)("a",{parentName:"p",href:"sect-block-production#defn-epoch-slot"},"Definition 59"),"). Since the BEEFY authority set is the same as the GRANDPA authority set for any GRANDPA finalized block, the session boundaries for BEEFY are exactly the same as the ones for GRANDPA."),(0,n.kt)("p",{parentName:"admonition"},"We define two kinds of blocks from the perspective of the BEEFY protocol:"),(0,n.kt)("ol",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("strong",{parentName:"li"},"Mandatory Blocks")),(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("strong",{parentName:"li"},"Non-mandatory Blocks"))),(0,n.kt)("p",{parentName:"admonition"},"Mandatory blocks are the ones that MUST have BEEFY justification. That means that the validators\nwill always start and conclude a round at mandatory blocks. For non-mandatory blocks, there may\nor may not be a justification and validators may never choose these blocks to start a round."),(0,n.kt)("p",{parentName:"admonition"},"Every ",(0,n.kt)("strong",{parentName:"p"},"first block in")," each ",(0,n.kt)("strong",{parentName:"p"},"session")," is considered a ",(0,n.kt)("strong",{parentName:"p"},"mandatory block"),". All other blocks\nin the session are non-mandatory, however validators are encouraged to finalize as many blocks as\npossible to enable lower latency for light clients and hence end users. Since GRANDPA is\nconsidering session boundary blocks as mandatory as well, ",(0,n.kt)("inlineCode",{parentName:"p"},"session_start")," block will always have\nboth GRANDPA and BEEFY Justification."),(0,n.kt)("h6",{parentName:"admonition",id:"defn-beefy-round-number"},"Definition 105. BEEFY Round NUmber"),(0,n.kt)("admonition",{parentName:"admonition",type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"The formula for determining the current round number is defined as:"),(0,n.kt)("pre",{parentName:"admonition"},(0,n.kt)("code",{parentName:"pre"},"round_number =\n      (1 - M) * session_start\n   +        M * Minimum(next_session_start, (best_beefy + NEXT_POWER_OF_TWO((best_grandpa - best_beefy + 1) / 2)))\n")),(0,n.kt)("p",{parentName:"admonition"},"where:"),(0,n.kt)("ul",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("inlineCode",{parentName:"li"},"M")," is ",(0,n.kt)("inlineCode",{parentName:"li"},"1")," if the mandatory block in the current session is already finalized and ",(0,n.kt)("inlineCode",{parentName:"li"},"0")," otherwise."),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("inlineCode",{parentName:"li"},"NEXT_POWER_OF_TWO(x)")," returns the smallest number greater or equal to ",(0,n.kt)("inlineCode",{parentName:"li"},"x")," that is a power of two."))),(0,n.kt)("p",{parentName:"admonition"},"Intuitively, the next round number should be the oldest mandatory block without a justification,\nor the highest GRANDPA-finalized block, whose block number difference with ",(0,n.kt)("inlineCode",{parentName:"p"},"best_beefy")," block is\na power of two. The mental model for round selection is to first finalize the mandatory block and\nthen to attempt to pick a block taking into account how fast BEEFY catches up with GRANDPA.\nIn case GRANDPA makes progress, but BEEFY seems to be lagging behind, validators are changing\nrounds less often to increase the chance of concluding them."),(0,n.kt)("p",{parentName:"admonition"},"As mentioned earlier, every time the node picks a new ",(0,n.kt)("inlineCode",{parentName:"p"},"round_number")," (and the validator casts a vote)\nit ends the previous one, no matter if finality was reached (i.e. the round concluded) or not.\nVotes for an inactive round should not be propagated."),(0,n.kt)("p",{parentName:"admonition"},"Note that since BEEFY only votes for GRANDPA-finalized blocks, ",(0,n.kt)("inlineCode",{parentName:"p"},"session_start"),' here actually means:\n"the latest session for which the start of is GRANDPA-finalized", i.e. block production might\nhave already progressed, but BEEFY needs to first finalize the mandatory block of the older\nsession.'),(0,n.kt)("p",{parentName:"admonition"},"While it is useful to finalize non-mandatory blocks frequently, in good networking conditions BEEFY may end up finalizing each and every block GRANDPA finalized block. Practically, with short block times, it's going to be rare and might be excessive, so\nit's suggested for implementations to introduce a ",(0,n.kt)("inlineCode",{parentName:"p"},"min_delta")," parameter which will limit the frequency with which new rounds are started. The affected component of the formula would be:\n",(0,n.kt)("inlineCode",{parentName:"p"},"best_beefy + MAX(min_delta, NEXT_POWER_OF_TWO(...))"),", so we start a new round only if the\nthe power-of-two component is greater than the min delta. Note that if ",(0,n.kt)("inlineCode",{parentName:"p"},"round_number > best_grandpa")," the validators are not expected to start any round."),(0,n.kt)("h3",{parentName:"admonition",id:"id-light-client-beefy-1"},"6.7.9. BEEFY Light Client"),(0,n.kt)("p",{parentName:"admonition"},"A light client following BEEFY could request ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"N"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"/"),(0,n.kt)("mn",{parentName:"mrow"},"3"),(0,n.kt)("mo",{parentName:"mrow"},"+"),(0,n.kt)("mn",{parentName:"mrow"},"1")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"N/3 +1")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10903em"}},"N"),(0,n.kt)("span",{parentName:"span",className:"mord"},"/3"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"+"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"1")))))," signatures to be checked, where ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"N")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"N")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10903em"}},"N")))))," is the number of validators on Polkadot. Assuming a maximum of ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"N"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"/"),(0,n.kt)("mn",{parentName:"mrow"},"3")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"N/3")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10903em"}},"N"),(0,n.kt)("span",{parentName:"span",className:"mord"},"/3")))))," malicious validators, the light client can be certain of the payloads finality if all the signatures it requested are valid. "),(0,n.kt)("h3",{parentName:"admonition",id:"id-subsampling-light-client-beefy-1"},"6.7.10. Subsampling Light Client"),(0,n.kt)("p",{parentName:"admonition"},"It is an interactive protocol between the light-client (verifier) and the relayer (prover) to convince the Light Client with a high probability that the payload sent by the prover is signed by honest Polkadot validators. The protocol prioritizes efficiency and tries to minimize the number (",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mo",{parentName:"mrow"},"<"),(0,n.kt)("mo",{parentName:"mrow"},"<"),(0,n.kt)("mi",{parentName:"mrow"},"N"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"/"),(0,n.kt)("mn",{parentName:"mrow"},"3")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"<< N/3")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.5782em",verticalAlign:"-0.0391em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"<<"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10903em"}},"N"),(0,n.kt)("span",{parentName:"span",className:"mord"},"/3"))))),") of signature checks (computationally expensive operations) on the light client side. "),(0,n.kt)("h3",{parentName:"admonition",id:"id-APK-light-client-beefy-1"},"6.7.11. APK Proof based Light Clients"),(0,n.kt)("p",{parentName:"admonition"},"TODO: Section on using Aggregatable Signatures for efficient verification on light clients.")))}f.isMDXComponent=!0},2043:(a,e,t)=>{t.d(e,{Z:()=>s});const s=t.p+"assets/images/forced-change-2944560a62e2b9898252b1c1fa7b727f.png"},482:(a,e,t)=>{t.d(e,{Z:()=>s});const s=t.p+"assets/images/scheduled-change-6f4cf547417d699e492e9b563ccd5867.png"}}]);
\ No newline at end of file
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1, \\textrm{Fin}}($\\call{Best-Final-Candidate}{$r - 1$}$)$}\n      \\IF{\\call{Best-Final-Candidate}{$r - 1$} $\\geqslant$ \\textsc{Last-Finalized-Block}}\n        \\STATE \\call{Broadcast}{$M_{v}^{r - 1, \\textrm{Prim}}($\\call{Best-Final-Candidate}{$r - 1$}$)$}\n      \\ENDIF\n    \\ENDIF\n    \\STATE \\call{Receive-Messages}{\\textbf{until} Time $\\geqslant t_{r_,v} + 2 \\times T$ \\or $r$ \\textbf{is} completable}\n    \\STATE $L \\leftarrow$ \\call{Best-Final-Candidate}{$r - 1$}\n    \\STATE $N \\leftarrow$ \\call{Best-PreVote-Candidate}{$r$}\n    \\STATE \\call{Broadcast}{$M_v^{r, \\textrm{pv}} (N)$}\n    \\STATE \\call{Receive-Messages}{\\textbf{until} $B^{r,\\textrm{pv}}_v \\geqslant L$ \\and $($ Time $\\geqslant t_{r_,v} + 4 \\times T$ \\or $r$ \\textbf{is} completable $)$}\n    \\STATE \\call{Broadcast}{$M_v^{r, \\textrm{pc}}(B_v^{r, \\textrm{pv}})$}\n    \n    \\REPEAT\n      \\STATE \\call{Receive-Messages}{}\n      \\STATE \\call{Attempt-To-Finalize-At-Round}{$r$}\n    \\UNTIL{$r$ \\textbf{is} completable \\and \\call{Finalizable}{$r$} \\and \\textsc{Last-Finalized-Block} $\\geqslant$ \\call{Best-Final-Candidate}{$r - 1$}}\n    \n    \\STATE \\call{Play-Grandpa-round}{$r + 1$}\n    \n    \\REPEAT\n      \\STATE \\call{Receive-Messages}{}\n      \\STATE \\call{Attempt-To-Finalize-At-Round}{$r$}\n    \\UNTIL{\\textsc{Last-Finalized-Block} $\\geqslant$ \\call{Best-Final-Candidate}{$r$}}\n    \n    \\IF{$r > $ \\textsc{Last-Completed-Round}}\n      \\STATE \\textsc{Last-Completed-Round} $\\leftarrow r$\n    \\ENDIF\n\\end{algorithmic}\n\\end{algorithm}",N="\\begin{algorithm}\n\\caption{Derive-Primary}\n\\begin{algorithmic}\n    \\input $r$\n    \\return $r \\bmod |\\mathbb{V}|$\n\\end{algorithmic}\n\\end{algorithm}",i="\\begin{algorithm}\n\\caption{Best-Final-Candidate}\n\\begin{algorithmic}\n    \\input $r$\n    \\state $B_v^{r, pv} \\leftarrow$ \\call{GRANDPA-GHOST}{$r$}\n    \\if{$r = 0$}\n        \\return $B_v^{r, pv}$\n    \\else\n        \\state $\\mathcal{C} \\leftarrow \\{ B' | B' \\leqslant B_v^{r,pv} | \\#V^{r, pc}_{\\operatorname{obv}(v), pot}(B') > \\frac{2}{3} |\\mathbb{V}| \\}$\n        \\if{$\\mathcal{C} = \\phi$}\n            \\return $B_v^{r, pv}$\n        \\else\n            \\return $E \\in \\mathcal{C} : H_n (E) = \\operatorname{max}\\left(H_n (B') | B' \\in \\mathcal{C}\\right)$\n        \\endif\n    \\endif\n\\end{algorithmic}\n\\end{algorithm}",l="\\begin{algorithm}\n\\caption{GRANDPA-GHOST}\n\\begin{algorithmic}\n    \\input $r$\n    \\if{$r = 0$}\n        \\state $G \\leftarrow B_{last}$\n    \\else\n        \\state $L \\leftarrow$ \\call{Best-Final-Candidate}{$r - 1$}\n        \\state $\\mathcal{G} = \\{ \\forall B > L | \\#V_{\\operatorname{obs}(v)}^{r, pv}(B) \\geqslant \\frac{2}{3} |\\mathbb{V}| \\}$\n        \\if{$\\mathcal{G} = \\phi$}\n            \\state $G \\leftarrow L$\n        \\else\n            \\state $G \\in \\mathcal{G} | H_n(G) = \\operatorname{max}\\left( H_n (B) | \\forall B \\in \\mathcal{G} \\right)$\n        \\endif\n    \\endif\n    \\return $G$\n\\end{algorithmic}\n\\end{algorithm}",k="\\begin{algorithm}\n\\caption{Best-PreVote-Candidate}\n\\begin{algorithmic}\n    \\input $r$\n    \\state $B^{r, pv}_v \\leftarrow$ \\call{GRANDPA-GHOST}{$r$}\n    \\if{\\call{Received}{$M_{v_{primary}}^{r, prim}(B))$ \\and $B^{r, pv}_v \\geqslant B > L$}}\n        \\state $N \\leftarrow B$\n    \\else\n        \\state $N \\leftarrow B^{r, pv}_v$\n    \\endif\n\\end{algorithmic}\n\\end{algorithm}",o="\\begin{algorithm}\n\\caption{Attempt-To-Finalize-At-Round}\n\\begin{algorithmic}\n    \\REQUIRE($r$)\n    \\STATE $L \\leftarrow$ \\textsc{Last-Finalized-Block}\n    \\STATE $E \\leftarrow$ \\call{Best-Final-Candidate}{$r$}\n    \\IF{$E \\geqslant L$ \\and ${V^{r, \\textrm{pc}}_{\\textrm{obs}(v)}}(E) > 2 / 3 |\\mathbb{V}|$}\n    \\STATE{\\textsc{Last-Finalized-Block}$\\leftarrow E$}\n    \\IF{$M_v^{r, \\textrm{Fin}} (E) \\notin $\\textsc{Received-Messages}}\n        \\STATE \\call{Broadcast}{$M_v^{r, \\textrm{Fin}}(E)$}\n        \\RETURN\n    \\ENDIF\n    \\ENDIF\n\\end{algorithmic}\n\\end{algorithm}",c="\\begin{algorithm}\n\\caption{Finalizable}\n\\begin{algorithmic}\n    \\REQUIRE($r$)\n    \\IF{$r$ \\textbf{is not} Completable}\n        \\RETURN \\textbf{False}\n    \\ENDIF\n    \\STATE $G \\leftarrow$ \\call{GRANDPA-GHOST}{$J^{r, pv}(B)$}\n    \\IF{$G = \\phi$}\n        \\RETURN \\textbf{False}\n    \\ENDIF\n    \\STATE $E_r \\leftarrow$ \\call{Best-Final-Candidate}{$r$}\n    \\IF{$E_r \\neq \\phi$ \\and \\call{Best-Final-Candidate}{$r - 1$} $\\leqslant E_r \\leqslant G$}\n        \\RETURN \\textbf{True}\n    \\ELSE\n        \\RETURN \\textbf{False}\n    \\ENDIF\n\\end{algorithmic}\n\\end{algorithm}",h="\\begin{algorithm}\n\\caption{Process-Catchup-Request}\n\\begin{algorithmic}\n    \\input $M_{i, v}^\\text{Cat-q}(\\text{id}_\\mathbb{V}, r)$\n    \\if{$M_{i, v}^\\text{Cat-q}(\\text{id}_\\mathbb{V}, r).\\text{id}_\\mathbb{V} \\neq \\text{id}_\\mathbb{V}$}\n        \\state \\textbf{error} ``Catching up on different set''\n    \\endif\n    \\if{$i \\notin \\mathbb{P}$}\n        \\state \\textbf{error} ``Requesting catching up from a non-peer''\n    \\endif\n    \\if{$r >$ \\textsc{Last-Completed-Round}}\n        \\state \\textbf{error} ``Catching up on a round in the future''\n    \\endif\n    \\state \\call{Send}{$i, M_{v, i}^\\text{Cat-s}(\\text{id}_\\mathbb{V}, r)$}\n\\end{algorithmic}\n\\end{algorithm}",g="\\begin{algorithm}\n\\caption{Process-Catchup-Response}\n\\begin{algorithmic}\n    \\input $M_{v,i}^\\text{Cat-s}(\\text{id}_{\\mathbb{V}}, r)$\n    \\state $M_{v,i}^\\text{Cat-s}(\\text{id}_{\\mathbb{V}}, r).\\text{id}_{\\mathbb{V}}, r, J^{r, pv}(B), J^{r, pc}(B), H_h(B'), H_i(B') \\leftarrow \\text{Dec}_{SC}(M_{v, i}^{Cat-s}(\\text{id}_{\\mathbb{V}}, r)$\n    \\if{$M_{v, i}^\\text{Cat-s}(\\text{id}_{\\mathbb{V}}, r).\\text{id}_{\\mathbb{V}} \\neq \\text{id}_{\\mathbb{V}}$}\n        \\state \\textbf{error} ``Catching up on different set''\n    \\endif\n    \\if{$r \\leqslant$ \\textsc{Leading-Round}}\n        \\state \\textbf{error} ``Catching up in to the past''\n    \\endif\n    \\if{$J^{r, pv}(B)$ \\textbf{is not} valid}\n        \\state \\textbf{error} ``Invalid pre-vote justification''\n    \\endif\n    \\if{$J^{r, pc}(B)$ \\textbf{is not} valid}\n        \\state \\textbf{error} ``Invalid pre-commit justification''\n    \\endif\n    \\state $G \\leftarrow$ \\call{GRANDPA-GHOST}{$J^{r, pv}(B)$}\n    \\if{$G = \\phi$}\n        \\state \\textbf{error} ``GHOST-less Catch-up''\n    \\endif\n    \\if{$r$ \\textbf{is not} completable}\n        \\state \\textbf{error} ``Catch-up round is not completable''\n    \\endif\n    \\if{$J^{r, pc}(B)$ justifies $B'$ finalization}\n        \\state \\textbf{error} ``Unjustified Catch-up target finalization''\n    \\endif\n    \\state \\textsc{Last-Completed-Round} $\\leftarrow r$\n    \\if{$i \\in \\mathbb{V}$}\n        \\state \\call{Play-Grandpa-round}{$r + 1$}\n    \\endif\n\\end{algorithmic}\n\\end{algorithm}",d={title:"6. Finality"},u=void 0,w={unversionedId:"sect-finality",id:"sect-finality",title:"6. Finality",description:"6.1. Introduction",source:"@site/docs/sect-finality.md",sourceDirName:".",slug:"/sect-finality",permalink:"/sect-finality",draft:!1,unlisted:!1,editUrl:"https://github.com/w3f/polkadot-spec/blob/main/docs/sect-finality.md",tags:[],version:"current",frontMatter:{title:"6. Finality"},sidebar:"tutorialSidebar",previous:{title:"5. Block Production",permalink:"/sect-block-production"},next:{title:"7. Light Clients",permalink:"/sect-lightclient"}},y={},v=[{value:"6.1. Introduction",id:"id-introduction-4",level:2},{value:"Definition 77. GRANDPA Voter",id:"defn-grandpa-voter",level:6},{value:"Definition 78. Authority Set Id",id:"defn-authority-set-id",level:6},{value:"Definition 79. GRANDPA State",id:"defn-grandpa-state",level:6},{value:"Definition 80. GRANDPA Vote",id:"defn-vote",level:6},{value:"Definition 81. Voting Rounds",id:"defn-voting-rounds",level:6},{value:"Definition 82. Vote Signature",id:"defn-sign-round-vote",level:6},{value:"Definition 83. Justification",id:"defn-grandpa-justification",level:6},{value:"Definition 84. Finalizing Justification",id:"defn-finalizing-justification",level:6},{value:"Definition 85. Equivocation",id:"defn-voter-equivocation",level:6},{value:"Definition 86. Set of Observed Direct Votes",id:"defn-observed-direct-votes",level:6},{value:"Definition 87. Set of Total Observed Votes",id:"defn-observed-votes",level:6},{value:"Definition 88. Set of Total Potential Votes",id:"defn-total-potential-votes",level:6},{value:"Definition 89. Current Pre-Voted Block",id:"defn-grandpa-ghost",level:6},{value:"Definition 90. Completable Round",id:"defn-grandpa-completable",level:6},{value:"Definition 91. GRANDPA Consensus Message",id:"defn-consensus-message-grandpa",level:6},{value:"6.2. Initiating the GRANDPA State",id:"id-initiating-the-grandpa-state",level:2},{value:"6.2.1. Voter Set Changes",id:"id-voter-set-changes",level:3},{value:"Algorithm 13. Initiate Grandpa",id:"algo-initiate-grandpa",level:6},{value:"6.3. Rejoining the Same Voter Set",id:"id-rejoining-the-same-voter-set",level:2},{value:"6.4. Voting Process in Round {r}",id:"id-voting-process-in-round-r",level:2},{value:"Algorithm 14. Play Grandpa Round",id:"algo-grandpa-round",level:6},{value:"Algorithm 15. Derive Primary",id:"algo-derive-primary",level:6},{value:"Algorithm 16. Best Final Candidate",id:"algo-grandpa-best-candidate",level:6},{value:"Algorithm 17. GRANDPA GHOST",id:"algo-grandpa-ghost",level:6},{value:"Algorithm 18. Best PreVote Candidate",id:"algo-best-prevote-candidate",level:6},{value:"Algorithm 19. Attempt To Finalize At Round",id:"algo-attempt-to\u2013finalize",level:6},{value:"Algorithm 20. Finalizable",id:"algo-finalizable",level:6},{value:"Definition 92. Unfinalized Candidate",id:"defn-unfinalized-candidate",level:6},{value:"6.5. Forced Authority Set Changes",id:"sect-finality-forced-changes",level:2},{value:"Image 6. Applying a scheduled change",id:"img-scheduled-change",level:6},{value:"Image 7. Applying a forced change",id:"img-forced-change",level:6},{value:"6.6. Block Finalization",id:"sect-block-finalization",level:2},{value:"Definition 93. Justified Block Header",id:"defn-justified-block-header",level:6},{value:"Definition 94. Finalized",id:"defn-finalized-block",level:6},{value:"6.6.1. Catching up",id:"sect-grandpa-catchup",level:3},{value:"6.6.1.1. Sending the catch-up requests",id:"sect-sending-catchup-request",level:4},{value:"6.6.1.2. Processing the catch-up requests",id:"id-processing-the-catch-up-requests",level:4},{value:"Algorithm 21. Process Catchup Request",id:"algo-process-catchup-request",level:6},{value:"6.6.1.3. Processing catch-up responses",id:"id-processing-catch-up-responses",level:4},{value:"Algorithm 22. Process Catchup Response",id:"algo-process-catchup-response",level:6},{value:"6.7. Bridge design (BEEFY)",id:"sect-grandpa-beefy",level:2},{value:"6.7.1. Motivation",id:"id-motivation-beefy-1",level:3},{value:"6.7.2. Protocol Overview",id:"id-overview-beefy-1",level:3},{value:"6.7.3. Preliminaries",id:"id-preliminaries-2",level:3},{value:"Definition 95. BEEFY Session Keys",id:"defn-beefy-key",level:6},{value:"6.7.4. Merkle Mountain Ranges",id:"id-beefy-merkle-mountain-ranges",level:3},{value:"Definition 96. Merkle Mountain Ranges",id:"defn-mmr",level:6},{value:"Definition 98. MMR operations",id:"defn-mmr-operations",level:6},{value:"Definition 99. Payload",id:"defn-beefy-payload",level:6},{value:"Definition 100. Signed Commitment",id:"defn-beefy-Commitment",level:6},{value:"Definition 101. Witness Data",id:"defn-beefy-witness-data",level:6},{value:"Definition 102. Light Client",id:"defn-beefy-light-client",level:6},{value:"Definition 103. Relayer",id:"defn-beefy-relayer",level:6},{value:"6.7.5. Voting on Payloads",id:"id-voting-on-payloads",level:3},{value:"6.7.6. Committing Witnesses",id:"sect-beefy-committing-witnesses",level:3},{value:"6.7.7. Requesting Signed Commitments",id:"id-requesting-signed-commitments",level:3},{value:"Definition 104. BEEFY Consensus Message",id:"defn-consensus-message-beefy",level:6},{value:"6.7.8. Consensus Mechanism",id:"id-consensus-mechanism-beefy-1",level:3},{value:"Definition 105. BEEFY Round NUmber",id:"defn-beefy-round-number",level:6},{value:"6.7.9. BEEFY Light Client",id:"id-light-client-beefy-1",level:3},{value:"6.7.10. Subsampling Light Client",id:"id-subsampling-light-client-beefy-1",level:3},{value:"6.7.11. APK Proof based Light Clients",id:"id-APK-light-client-beefy-1",level:3}],x={toc:v},b="wrapper";function f(a){let{components:e,...d}=a;return(0,n.kt)(b,(0,s.Z)({},x,d,{components:e,mdxType:"MDXLayout"}),(0,n.kt)("h2",{id:"id-introduction-4"},"6.1. 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reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"pr"))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.247em"}},(0,n.kt)("span",{parentName:"span"})))))))))))," represents an ",(0,n.kt)("em",{parentName:"p"},"ed25519")," private key, is a node running a GRANDPA protocol and broadcasting votes to finalize blocks in a Polkadot Host-based chain. 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In that regard, we have ","[","To do: change function name, only call at genesis, adjust V_B over the sections","]"),(0,n.kt)("div",{parentName:"admonition",className:"math 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math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"g"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"r"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"a"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"n"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"d"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"p"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"a"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"_"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"a"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"u"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"t"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"h"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"o"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"r"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"i"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"t"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"i"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"e"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"monospace"},"s")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\tt{grandpa\\_authorities}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathtt"},"grandpa_authorities"))))))))," is a function entry point of the Runtime described in ",(0,n.kt)("a",{parentName:"p",href:"chap-runtime-api#sect-rte-grandpa-auth"},"Section C.10.1."),". We refer to ",(0,n.kt)("span",{parentName:"p",className:"math 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mathbb"},"V"))),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3283em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.05017em"}},"B")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," as ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow",mathvariant:"double-struck"},"V")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\mathbb{{V}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6889em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathbb"},"V")))))))," when there is no chance of ambiguity."),(0,n.kt)("p",{parentName:"admonition"},"Analogously, we say that a Polkadot node is a ",(0,n.kt)("strong",{parentName:"p"},"non-voter node")," for block ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"B")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"))))))," if it does not own any of the key pairs in ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub",mathvariant:"double-struck"},"V"),(0,n.kt)("mi",{parentName:"msub"},"B"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\mathbb{{V}}}_{{B}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8389em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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Voting Rounds"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"Voters engage in a maximum of two sub-rounds of voting for each round ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")))))),". 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reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"mpunct mtight"},","),(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"pc"))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.247em"}},(0,n.kt)("span",{parentName:"span"})))))))))))," we refer to the vote cast by voter ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"v")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{v}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"v"))))))," in round ",(0,n.kt)("span",{parentName:"p",className:"math 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(",(0,n.kt)("a",{parentName:"p",href:"chap-networking#sect-msg-grandpa"},"Section 4.8.7."),") to the network. Validators inform their peers about the block finalized in round ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r"))))))," by broadcasting a commit message (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-grandpa-round"},"Play-Grandpa-Round"),").")),(0,n.kt)("h6",{id:"defn-sign-round-vote"},"Definition 82. 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mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"pc"))))))))))),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"minner"},(0,n.kt)("span",{parentName:"span",className:"mopen delimcenter",style:{top:"0em"}},"("),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"mclose delimcenter",style:{top:"0em"}},")")))))))," can only be used by a non-voter node to finalize a block. In contrast, a voter node can only be assured of the finality (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-finalized-block"},"Definition 94"),") of block ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"B")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"))))))," by actively participating in the voting process. That is by invoking ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-grandpa-round"},"Play-Grandpa-Round"),"."),(0,n.kt)("p",{parentName:"admonition"},"The GRANDPA protocol dictates how an honest voter should vote in each sub-round, which is described by ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-grandpa-round"},"Play-Grandpa-Round"),". After defining what constitutes a vote in GRANDPA, we define how GRANDPA counts votes.")),(0,n.kt)("h6",{id:"defn-voter-equivocation"},"Definition 85. Equivocation"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"Voter ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"v")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{v}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"v"))))))," ",(0,n.kt)("strong",{parentName:"p"},"equivocates")," if they broadcast two or more valid votes to blocks during one voting sub-round. In such a situation, we say that ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"v")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{v}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"v"))))))," is an ",(0,n.kt)("strong",{parentName:"p"},"equivocator")," and any vote ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msubsup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msubsup"},"V"),(0,n.kt)("mi",{parentName:"msubsup"},"v"),(0,n.kt)("mrow",{parentName:"msubsup"},(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("mtext",{parentName:"mrow"},"stage"))),(0,n.kt)("mrow",{parentName:"mrow"},(0,n.kt)("mo",{parentName:"mrow",fence:"true"},"("),(0,n.kt)("mi",{parentName:"mrow"},"B"),(0,n.kt)("mo",{parentName:"mrow",fence:"true"},")"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{{V}_{{v}}^{{{r},\\text{stage}}}}{\\left({B}\\right)}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1.0436em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"mpunct mtight"},","),(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"stage"))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.247em"}},(0,n.kt)("span",{parentName:"span"}))))))),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"minner"},(0,n.kt)("span",{parentName:"span",className:"mopen delimcenter",style:{top:"0em"}},"("),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"mclose delimcenter",style:{top:"0em"}},")")))))))," cast by ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"v")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{v}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"v"))))))," in that sub-round is an ",(0,n.kt)("strong",{parentName:"p"},"equivocatory vote"),", and"),(0,n.kt)("div",{parentName:"admonition",className:"math math-display"},(0,n.kt)("span",{parentName:"div",className:"katex-display"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup",mathvariant:"script"},"E"),(0,n.kt)("mrow",{parentName:"msup"},(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("mtext",{parentName:"mrow"},"stage")))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\mathcal{{E}}}^{{{r},\\text{stage}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8436em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"stage")))))))))))))))),(0,n.kt)("p",{parentName:"admonition"},"represents the set of all equivocators voters in sub-round ",(0,n.kt)("em",{parentName:"p"},"stage")," of round ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")))))),". When we want to refer to the number of equivocators whose equivocation has been observed by voter ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"v")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{v}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"v")))))),", we refer to it by:"),(0,n.kt)("div",{parentName:"admonition",className:"math math-display"},(0,n.kt)("span",{parentName:"div",className:"katex-display"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msubsup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msubsup",mathvariant:"script"},"E"),(0,n.kt)("mrow",{parentName:"msubsup"},(0,n.kt)("mtext",{parentName:"mrow"},"obs"),(0,n.kt)("mrow",{parentName:"mrow"},(0,n.kt)("mo",{parentName:"mrow",fence:"true"},"("),(0,n.kt)("mi",{parentName:"mrow"},"v"),(0,n.kt)("mo",{parentName:"mrow",fence:"true"},")"))),(0,n.kt)("mrow",{parentName:"msubsup"},(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("mtext",{parentName:"mrow"},"stage")))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{{\\mathcal{{E}}}_{{\\text{obs}{\\left({v}\\right)}}}^{{{r},\\text{stage}}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1.4267em",verticalAlign:"-0.5152em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"mpunct"},","),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.7519em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"\u2032"))))))))),(0,n.kt)("span",{parentName:"span",className:"mclose"},")"),(0,n.kt)("span",{parentName:"span",className:"mord"},"\u2225"))))))," where ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup"},"B"),(0,n.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B'}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.7519em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.7519em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"\u2032"))))))))))))))," is the block where the change is applied. The earliest digest of this type in a single block will be respected, unless a force change is present, in which case the force change takes precedence.")),(0,n.kt)("tr",{parentName:"tbody"},(0,n.kt)("td",{parentName:"tr",align:null},(0,n.kt)("strong",{parentName:"td"},"2")),(0,n.kt)("td",{parentName:"tr",align:null},"Implies ",(0,n.kt)("strong",{parentName:"td"},"forced change"),": Schedule a forced authority set change after the given delay of ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"N"),(0,n.kt)("mtext",{parentName:"msub"},"delay")),(0,n.kt)("mo",{parentName:"mrow"},":"),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2225"),(0,n.kt)("mtext",{parentName:"mrow"},"SubChain"),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},"("),(0,n.kt)("mi",{parentName:"mrow"},"B"),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("mi",{parentName:"mrow"},"m"),(0,n.kt)("mo",{parentName:"mrow"},"+"),(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup"},"B"),(0,n.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032")),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},")"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2225")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{N_{\\text{delay}} := \\|\\text{SubChain}(B,m + B')\\|}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1.038em",verticalAlign:"-0.2861em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10903em"}},"N"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"delay")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.2861em"}},(0,n.kt)("span",{parentName:"span"})))))),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},":="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"\u2225"),(0,n.kt)("span",{parentName:"span",className:"mord text"},(0,n.kt)("span",{parentName:"span",className:"mord"},"SubChain")),(0,n.kt)("span",{parentName:"span",className:"mopen"},"("),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"mpunct"},","),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"m"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"+"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.7519em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"\u2032"))))))))),(0,n.kt)("span",{parentName:"span",className:"mclose"},")"),(0,n.kt)("span",{parentName:"span",className:"mord"},"\u2225"))))))," where ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup"},"B"),(0,n.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B'}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.7519em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.7519em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"\u2032"))))))))))))))," is the block where the change is applied. The earliest digest of this type in a block will be respected. ",(0,n.kt)("br",null),(0,n.kt)("br",null)," Forced changes are explained further in ",(0,n.kt)("a",{parentName:"td",href:"sect-finality#sect-finality-forced-changes"},"Section 6.5."),".")),(0,n.kt)("tr",{parentName:"tbody"},(0,n.kt)("td",{parentName:"tr",align:null},(0,n.kt)("strong",{parentName:"td"},"3")),(0,n.kt)("td",{parentName:"tr",align:null},"Implies ",(0,n.kt)("strong",{parentName:"td"},"on disabled"),": An index to the individual authority in the current authority list (",(0,n.kt)("a",{parentName:"td",href:"chap-sync#defn-authority-list"},"Definition 33"),") that should be immediately disabled until the next authority set changes. When an authority gets disabled, the node should stop performing any authority functionality from that authority, including authoring blocks and casting GRANDPA votes for finalization. Similarly, other nodes should ignore all messages from the indicated authority which pertain to their authority role.")),(0,n.kt)("tr",{parentName:"tbody"},(0,n.kt)("td",{parentName:"tr",align:null},(0,n.kt)("strong",{parentName:"td"},"4")),(0,n.kt)("td",{parentName:"tr",align:null},"Implies ",(0,n.kt)("strong",{parentName:"td"},"pause"),": A signal to pause the current authority set after the given delay of ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"N"),(0,n.kt)("mtext",{parentName:"msub"},"delay")),(0,n.kt)("mo",{parentName:"mrow"},":"),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2225"),(0,n.kt)("mtext",{parentName:"mrow"},"SubChain"),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},"("),(0,n.kt)("mi",{parentName:"mrow"},"B"),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup"},"B"),(0,n.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032")),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},")"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2225")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{N_{\\text{delay}} := \\|\\text{SubChain}(B,B')\\|}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1.038em",verticalAlign:"-0.2861em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10903em"}},"N"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"delay")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.2861em"}},(0,n.kt)("span",{parentName:"span"})))))),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},":="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"\u2225"),(0,n.kt)("span",{parentName:"span",className:"mord text"},(0,n.kt)("span",{parentName:"span",className:"mord"},"SubChain")),(0,n.kt)("span",{parentName:"span",className:"mopen"},"("),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"mpunct"},","),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.7519em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"\u2032"))))))))),(0,n.kt)("span",{parentName:"span",className:"mclose"},")"),(0,n.kt)("span",{parentName:"span",className:"mord"},"\u2225"))))))," where ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup"},"B"),(0,n.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B'}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.7519em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B"),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.7519em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"\u2032"))))))))))))))," is a block where the change is applied. Once applied, the authorities should stop voting.")),(0,n.kt)("tr",{parentName:"tbody"},(0,n.kt)("td",{parentName:"tr",align:null},(0,n.kt)("strong",{parentName:"td"},"5")),(0,n.kt)("td",{parentName:"tr",align:null},"Implies ",(0,n.kt)("strong",{parentName:"td"},"resume"),": A signal to resume the current authority set after the given delay of ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"N"),(0,n.kt)("mtext",{parentName:"msub"},"delay")),(0,n.kt)("mo",{parentName:"mrow"},":"),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2225"),(0,n.kt)("mtext",{parentName:"mrow"},"SubChain"),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},"("),(0,n.kt)("mi",{parentName:"mrow"},"B"),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("msup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msup"},"B"),(0,n.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032")),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},")"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2225")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{N_{\\text{delay}} 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Once applied, the authorities should resume voting."))))),(0,n.kt)("h2",{id:"id-initiating-the-grandpa-state"},"6.2. Initiating the GRANDPA State"),(0,n.kt)("p",null,"In order to participate coherently in the voting process, a validator must initiate its state and sync it with other active validators. In particular, considering that voting is happening in different distinct rounds where each round of voting is assigned a unique sequential round number ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"r"),(0,n.kt)("mi",{parentName:"msub"},"v"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}_{{v}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r"))))))," equal to the voting round ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"r"),(0,n.kt)("mi",{parentName:"msub"},"n"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}_{{n}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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currently undergoing in the network. The mandated initialization procedure for the GRANDPA protocol for a joining validator is described in detail in ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-initiate-grandpa"},"Initiate-Grandpa"),"."),(0,n.kt)("p",null,"The process of joining a new voter set is different from the one of rejoining the current voter set after a network disconnect. The details of this distinction are described further in this section."),(0,n.kt)("h3",{id:"id-voter-set-changes"},"6.2.1. Voter Set Changes"),(0,n.kt)("p",null,"A GRANDPA voter node which is initiating GRANDPA protocol as part of joining a new authority set is required to execute ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-initiate-grandpa"},"Initiate-Grandpa"),". The algorithm mandates the initialization procedure for GRANDPA protocol."),(0,n.kt)("admonition",{type:"info"},(0,n.kt)("p",{parentName:"admonition"},"The GRANDPA round number reset to 0 for every authority set change.")),(0,n.kt)("p",null,"Voter set changes are signaled by Runtime via a consensus engine message (",(0,n.kt)("a",{parentName:"p",href:"chap-sync#sect-consensus-message-digest"},"Section 3.3.2."),"). When Authorities process such messages they must not vote on any block with a higher number than the block at which the change is supposed to happen. The new authority set should reinitiate GRANDPA protocol by executing ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-initiate-grandpa"},"Initiate-Grandpa"),"."),(0,n.kt)("h6",{id:"algo-initiate-grandpa"},"Algorithm 13. Initiate Grandpa"),(0,n.kt)("admonition",{type:"algorithm"},(0,n.kt)(m.Z,{content:p,algID:"initiateGrandpa",options:{lineNumber:!0},mdxType:"Pseudocode"}),(0,n.kt)("p",{parentName:"admonition"},"where ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mtext",{parentName:"msub"},"last"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{\\text{last}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"last"))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," is the last block which has been finalized on the chain (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-finalized-block"},"Definition 94"),"). ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"r"),(0,n.kt)("mtext",{parentName:"msub"},"last"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}_{{\\text{last}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"last"))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," is equal to the latest round the voter has observed that other voters are voting on. The voter obtains this information through various gossiped messages including those mentioned in ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-finalized-block"},"Definition 94"),". ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"r"),(0,n.kt)("mtext",{parentName:"msub"},"last"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}_{{\\text{last}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord text mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"last"))))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," is set to ",(0,n.kt)("em",{parentName:"p"},"0")," if the GRANDPA node is initiating the GRANDPA voting process as a part of a new authority set. 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We might even not finalize anything in a round (although ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-grandpa-round"},"Play-Grandpa-Round")," prevents us from moving to the round ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mo",{parentName:"mrow"},"+"),(0,n.kt)("mn",{parentName:"mrow"},"1")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}+{1}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"+"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"1"))))))," before finalizing the best final candidate of round ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mo",{parentName:"mrow"},"\u2212"),(0,n.kt)("mn",{parentName:"mrow"},"1")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{r}-{1}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"\u2212"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"1")))))),") The example in ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-unfinalized-candidate"},"Definition 92")," serves to demonstrate a situation where the best final candidate of a round cannot be finalized during its own round:"),(0,n.kt)("h6",{id:"defn-unfinalized-candidate"},"Definition 92. 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At round 1, we get 67 pre-votes for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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one pre-vote for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," which means that ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mtext",{parentName:"mrow"},"GRANDPA-GHOST"),(0,n.kt)("mrow",{parentName:"mrow"},(0,n.kt)("mo",{parentName:"mrow",fence:"true"},"("),(0,n.kt)("mn",{parentName:"mrow"},"1"),(0,n.kt)("mo",{parentName:"mrow",fence:"true"},")")),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\text{GRANDPA-GHOST}{\\left({1}\\right)}={B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord text"},(0,n.kt)("span",{parentName:"span",className:"mord"},"GRANDPA-GHOST")),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"minner"},(0,n.kt)("span",{parentName:"span",className:"mopen delimcenter",style:{top:"0em"}},"("),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"1")),(0,n.kt)("span",{parentName:"span",className:"mclose delimcenter",style:{top:"0em"}},")"))),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),"."),(0,n.kt)("p",{parentName:"admonition"},"Subsequently, potentially honest voters who could claim not seeing all the pre-votes for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," but receiving the pre-votes for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," would pre-commit to ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),". In this way, we receive 66 pre-commits for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," and 1 pre-commit for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),". Henceforth, we finalize ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," since we have a threshold commit (67 votes) for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"})))))))))),"."),(0,n.kt)("p",{parentName:"admonition"},"At this point, though, we have ",(0,n.kt)("span",{parentName:"p",className:"math 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Therefore, the condition of ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-grandpa-round"},"Play-Grandpa-Round")," is satisfied and we must proceed to round 2."),(0,n.kt)("p",{parentName:"admonition"},"Nonetheless, we must continue to attempt to finalize round ",(0,n.kt)("em",{parentName:"p"},"1")," in the background as the condition of ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#algo-attempt-to%E2%80%93finalize"},"Attempt-To-Finalize-At-Round")," has not been fulfilled."),(0,n.kt)("p",{parentName:"admonition"},"This prevents us from proceeding to round 3 until either:"),(0,n.kt)("ul",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"We finalize ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"2")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," in round 2, or")),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"We receive an extra pre-commit vote for ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05017em"}},"B")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},"1")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," in round 1. This will make it impossible to finalize ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"2"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{B}_{{2}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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matter to whom the remaining pre-commits are going to be cast for (even with considering the possibility of 1/3 of voter equivocating) and therefore we have ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mtext",{parentName:"mrow"},"Best-Final-Candidate"),(0,n.kt)("mrow",{parentName:"mrow"},(0,n.kt)("mo",{parentName:"mrow",fence:"true"},"("),(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mo",{parentName:"mrow",fence:"true"},")")),(0,n.kt)("mo",{parentName:"mrow"},"="),(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"B"),(0,n.kt)("mn",{parentName:"msub"},"1"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\mathtt{\\text{Best-Final-Candidate}}}{\\left({r}\\right)}={B}_{{1}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord 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",(0,n.kt)("span",{parentName:"p",className:"math 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math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mtext",{parentName:"mrow"},"Last-Finalized-Block")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\mathtt{\\text{Last-Finalized-Block}}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord text"},(0,n.kt)("span",{parentName:"span",className:"mord"},"Last-Finalized-Block")))))))," and the latter with decreasing ",(0,n.kt)("span",{parentName:"p",className:"math 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Forced Authority Set Changes"),(0,n.kt)("p",null,"In a case of emergency where the Polkadot network is unable to finalize blocks, such as in the event of a mass validator outage, the Polkadot governance mechanism must enact a forced change, which the Host must handle in a specific manner. Given that in such a case, finality cannot be relied on, the Host must detect the forced change (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-consensus-message-grandpa"},"Definition 91"),") in a (valid) block and apply it to all forks."),(0,n.kt)("p",null,"The ",(0,n.kt)("span",{parentName:"p",className:"math 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math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"n")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{n}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"n"))))))," is a GRANDPA voter and has already caught up to its peers according to the process described in Section ",(0,n.kt)("a",{parentName:"p",href:"sect-finality#sect-grandpa-catchup"},"Section 6.6.1."),"."))),(0,n.kt)("p",{parentName:"admonition"},"Note that all Polkadot relay chain nodes are supposed to process GRANDPA commit messages regardless of their GRANDPA voter status.")),(0,n.kt)("h3",{id:"sect-grandpa-catchup"},"6.6.1. 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Sending the catch-up requests"),(0,n.kt)("p",null,"When a Polkadot voter node has the same authority list as a peer voter node who is reporting a higher number for the ",(0,n.kt)("em",{parentName:"p"},"finalized round")," field, it should send a catch-up request message (",(0,n.kt)("a",{parentName:"p",href:"chap-networking#defn-grandpa-catchup-request-msg"},"Definition 53"),") to the reporting peer. This will allow the node to to catch up to the more advanced finalized round, provided that the following criteria hold:"),(0,n.kt)("ul",null,(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"The peer node is a GRANDPA voter, and:")),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"The last known finalized round for the Polkadot node is at least two rounds behind the finalized round for the peer."))),(0,n.kt)("h4",{id:"id-processing-the-catch-up-requests"},"6.6.1.2. 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Bridge design (BEEFY)"),(0,n.kt)("admonition",{type:"caution"},(0,n.kt)("p",{parentName:"admonition"},"The BEEFY protocol specifications are Work-In-Progress. It is currently being tested on test-nets like Rococco and has not yet been deployed on Polkadot. ")),(0,n.kt)("p",null,"The BEEFY (Bridge Efficiency Enabling Finality Yielder) is a secondary protocol to GRANDPA to support efficient bridging between the Polkadot network (relay chain) and remote, segregated blockchains, such as Ethereum, which were not built with the Polkadot interchain operability in mind. BEEFY\u2019s aim is to enable clients to efficiently follow a chain that has GRANDPA finality, a finality gadget created for Substrate/Polkadot ecosystem. This is useful for bridges (e.g., Polkadot->Ethereum), where a chain can follow another chain and light clients suitable for low storage devices such as mobile phones."),(0,n.kt)("p",null,"The protocol allows participants of the remote network to verify finality proofs created by the Polkadot relay chain validators. In other words: clients in the target network (e.g., Ethereum) can verify that the Polkadot network is at a specific state."),(0,n.kt)("p",null,"Storing all the information necessary to verify the state of the remote chain, such as the block headers, is too expensive. BEEFY stores the information in a space-efficient way, and clients can request additional information over the protocol."),(0,n.kt)("h3",{id:"id-motivation-beefy-1"},"6.7.1. Motivation"),(0,n.kt)("p",null,"A client could just follow GRANDPA using GRANDPA justifications, sets of signatures from validators. This is used for some substrate-substrate bridges and in light clients such as the Substrate Connect browser extension. GRANDPA was designed for fast and secure finality. Certain design decisions, like validators voting for different blocks in a justification and using ED25519 signatures, allow fast finality. However, GRANDPA justifications are large and are expensive to verify on other chains like Ethereum that do not support some cryptographic signature schemes. Thus, BEEFY adds an extra layer of finality that allows lighter bridges and clients for Polkadot."),(0,n.kt)("p",null,"To summarise, the goals of BEEFY are:"),(0,n.kt)("ul",null,(0,n.kt)("li",{parentName:"ul"},"Allow customization of signature schemes to adapt to different target chains. "),(0,n.kt)("li",{parentName:"ul"},"Minimize the size of the finality proof and the effort required by a light client to follow finality. "),(0,n.kt)("li",{parentName:"ul"},"Unify data types and use backward-compatible versioning so that the protocol can be extended (additional payload, different signature schemes) without breaking existing light clients.")),(0,n.kt)("h3",{id:"id-overview-beefy-1"},"6.7.2. Protocol Overview"),(0,n.kt)("p",null,"Since BEEFY runs on top of GRANDPA, similarly to how GRANDPA is lagging behind the best produced (non-finalized) block, BEEFY finalized block lags behind the best GRANDPA (finalized) block. "),(0,n.kt)("ul",null,(0,n.kt)("li",{parentName:"ul"},"The BEEFY validator set is the same as GRANDPA's. However, they might be using different types of session keys to sign BEEFY messages. "),(0,n.kt)("li",{parentName:"ul"},"From a single validator perspective, BEEFY has at most one active voting round. "),(0,n.kt)("li",{parentName:"ul"},"Since GRANDPA validators are reaching finality, we assume they are online and well-connected and have a similar view of the state of the blockchain.")),(0,n.kt)("p",null,"BEEFY consists of two components:"),(0,n.kt)("p",null,"a. ",(0,n.kt)("strong",{parentName:"p"},"Consensus Extension")," on GRANDPA finalization that is a voting round. "),(0,n.kt)("p",null,"The consensus extension serves to have a smaller consensus justification than GRANDPA and alternative cryptography, which helps the light client side of the BEEFY protocol described below. "),(0,n.kt)("p",null,"b. ",(0,n.kt)("strong",{parentName:"p"},"Light client")," protocol for convincing the other chain/device efficiently about the finality vote."),(0,n.kt)("p",null,"In the BEEFY light client, a prover, a full node, or a bridge relayer, wants to convince a verifier, a light client that may be a bridge implementation on the target chain, of the outcome of a BEEFY vote. The prover has access to all voting data from the BEEFY voting round. In the light client part, the prover may generate a proof and send it to the verifier or they may engage in an interactive protocol with several rounds of communication."),(0,n.kt)("h3",{id:"id-preliminaries-2"},"6.7.3. Preliminaries"),(0,n.kt)("h6",{id:"defn-beefy-key"},"Definition 95. BEEFY Session Keys"),(0,n.kt)("p",null,"Validators use an ",(0,n.kt)("inlineCode",{parentName:"p"},"ECDSA")," key scheme for signing Beefy messages. This is different from schemes like ",(0,n.kt)("inlineCode",{parentName:"p"},"sr25519")," and ",(0,n.kt)("inlineCode",{parentName:"p"},"ed25519"),", which are commonly used in Substrate for other components like BABE, and GRANDPA. The most noticeable difference is that an ",(0,n.kt)("inlineCode",{parentName:"p"},"ecdsa"),"\npublic key is ",(0,n.kt)("inlineCode",{parentName:"p"},"33")," bytes long, instead of ",(0,n.kt)("inlineCode",{parentName:"p"},"32")," bytes for a ",(0,n.kt)("inlineCode",{parentName:"p"},"sr25519")," based public key. As a consequence, the ",(0,n.kt)("inlineCode",{parentName:"p"},"AccountId")," (32-bytes) matches the ",(0,n.kt)("inlineCode",{parentName:"p"},"PublicKey")," for other session keys, but note that it's not the case for BEEFY."),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"BEEFY session key pair")," ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mo",{parentName:"mrow",fence:"true"},"("),(0,n.kt)("mi",{parentName:"mrow"},"s"),(0,n.kt)("msubsup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msubsup"},"k"),(0,n.kt)("mi",{parentName:"msubsup"},"j"),(0,n.kt)("mi",{parentName:"msubsup"},"B")),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("mi",{parentName:"mrow"},"p"),(0,n.kt)("msubsup",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msubsup"},"k"),(0,n.kt)("mi",{parentName:"msubsup"},"j"),(0,n.kt)("mi",{parentName:"msubsup"},"B")),(0,n.kt)("mo",{parentName:"mrow",fence:"true"},")")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\left({s}{{k}_{{j}}^{B}},{p}{{k}_{{j}}^{B}}\\right)}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1.2448em",verticalAlign:"-0.3948em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"minner"},(0,n.kt)("span",{parentName:"span",className:"mopen delimcenter",style:{top:"0em"}},(0,n.kt)("span",{parentName:"span",className:"delimsizing size1"},"(")),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"s")),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03148em"}},"k")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.8413em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.4413em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.05724em"}},"j"))))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.05017em"}},"B"))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3948em"}},(0,n.kt)("span",{parentName:"span"}))))))),(0,n.kt)("span",{parentName:"span",className:"mpunct"},","),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"p")),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03148em"}},"k")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.8413em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.4413em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.05724em"}},"j"))))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.05017em"}},"B"))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3948em"}},(0,n.kt)("span",{parentName:"span"}))))))),(0,n.kt)("span",{parentName:"span",className:"mclose delimcenter",style:{top:"0em"}},(0,n.kt)("span",{parentName:"span",className:"delimsizing size1"},")"))))))))," is a ",(0,n.kt)("inlineCode",{parentName:"p"},"secp256k1")," key pair which the BEEFY authority node ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub",mathvariant:"script"},"P"),(0,n.kt)("mi",{parentName:"msub"},"j"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{\\mathcal{{P}}}_{{j}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.9694em",verticalAlign:"-0.2861em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathcal",style:{marginRight:"0.08222em"}},"P"))),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3117em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.05724em"}},"j")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.2861em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," uses to sign the BEEFY signed commitments (justifications).\n")),(0,n.kt)("h3",{id:"id-beefy-merkle-mountain-ranges"},"6.7.4. Merkle Mountain Ranges"),(0,n.kt)("h6",{id:"defn-mmr"},"Definition 96. Merkle Mountain Ranges"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"Merkle Mountain Ranges, ",(0,n.kt)("strong",{parentName:"p"},"MMR"),", are used as an efficient way to send block headers and signatures to light clients. Merkle Mountain Ranges (",(0,n.kt)("inlineCode",{parentName:"p"},"MMR"),") is an improvement of the traditional Merkle tree data structure. Just like a Merkle tree, an ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR")," is a binary tree where each leaf node represents a data element and each non-leaf node is the hash of its child nodes. The key difference between a traditional Merkle tree and a MMR lies in the way nodes are organized. In traditional Merkle trees, whenever a leaf node is appended or removed, the tree must be rebuilt and the hashes of the non-leaf nodes must be recalculated. The overhead of recomputing the hashes up to the root makes traditional Merkle unsuitable for handling dynamic data. The ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR")," is designed to optimize the appending and removal of elements without requiring a complete rebuild of the tree, which makes it more efficient to handle growing lists of leaf nodes."),(0,n.kt)("p",{parentName:"admonition"},"MMR structure "),(0,n.kt)("p",{parentName:"admonition"},"A ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR")," structure can be seen as a list of perfectly balanced binary sub-trees in descending order of height. It is a strictly append-only structure where nodes are added from left to right, such that a parent node is added as soon as two children exist. The following representation shows a ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR")," with 11 elements, 7 leaf nodes and 4 non-leaf nodes, where the value of each node corresponds to the order in which it was inserted into the tree."),(0,n.kt)("pre",{parentName:"admonition"},(0,n.kt)("code",{parentName:"pre"},"       7\n      / \\\n     /   \\\n    /     \\\n   3       6        10\n  / \\     / \\       / \\   \n /   \\   /   \\     /   \\   \n1     2 4     5   8     9    11\n"))),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("h6",{parentName:"admonition",id:"defn-mmr-root"},"Definition 97. Merkle Mountain Ranges root (MMR-root)"),(0,n.kt)("p",{parentName:"admonition"},"An ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR")," does not have a single ",(0,n.kt)("inlineCode",{parentName:"p"},"root")," by design, as a conventional Merkle tree. Every sub-tree has a separate sub-root, which we refer to as the ",(0,n.kt)("inlineCode",{parentName:"p"},"peak")," of the sub-tree.\nBagging the peaks is the process used for hashing the peaks in order to compute the ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR-root"),". It is important to define the order in which the peaks are hashed to ensure that a given sub-set of peaks will always derive a unique ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR-root"),". Here we state that peaks are merged from right to left and bagged via ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"h"),(0,n.kt)("mi",{parentName:"mrow"},"a"),(0,n.kt)("mi",{parentName:"mrow"},"s"),(0,n.kt)("mi",{parentName:"mrow"},"h"),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},"("),(0,n.kt)("mi",{parentName:"mrow"},"r"),(0,n.kt)("mi",{parentName:"mrow"},"i"),(0,n.kt)("mi",{parentName:"mrow"},"g"),(0,n.kt)("mi",{parentName:"mrow"},"h"),(0,n.kt)("mi",{parentName:"mrow"},"t"),(0,n.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,n.kt)("mi",{parentName:"mrow"},"l"),(0,n.kt)("mi",{parentName:"mrow"},"e"),(0,n.kt)("mi",{parentName:"mrow"},"f"),(0,n.kt)("mi",{parentName:"mrow"},"t"),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},")")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"hash(right, left)")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"ha"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"s"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"h"),(0,n.kt)("span",{parentName:"span",className:"mopen"},"("),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"i"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"g"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"h"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"t"),(0,n.kt)("span",{parentName:"span",className:"mpunct"},","),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.01968em"}},"l"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"e"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10764em"}},"f"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"t"),(0,n.kt)("span",{parentName:"span",className:"mclose"},")"))))),"."),(0,n.kt)("p",{parentName:"admonition"},"Therefore, given an MMR tree with n peaks ordered in decreasing order of height, the ",(0,n.kt)("inlineCode",{parentName:"p"},"MMR-root")," of the tree is calculated as follows."),(0,n.kt)("div",{parentName:"admonition",className:"math 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",(0,n.kt)("a",{parentName:"li",href:"chap-host-api#sect-finality#id-ext_hashing_keccak_256"},"Keccak")," hash function used to merge the peaks."))),(0,n.kt)("p",null,"A distinguished feature of this process is that whenever new leaf nodes are added to the tree, the earlier hash computations of peaks are reused, making new leaf nodes less expensive to insert and to prove (i.e., to verify the integrity of leaf data). "),(0,n.kt)("h6",{id:"defn-mmr-operations"},"Definition 98. MMR operations"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"Here are the basic operations we should be able to perform on the MMR:"),(0,n.kt)("ul",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Append Leaf Node (appendData):"),(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"Signature: append(data: T) -> None"),(0,n.kt)("li",{parentName:"ul"},"Description: appends a new leaf element with the provided data to the MMR.\n"))),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Create MMR root (bagPeaks):"),(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"Signature: baggingPeaks(peaksIndexes: List","[int]",") -> str"),(0,n.kt)("li",{parentName:"ul"},"Description: creates the single MMR root based on the list of peaks, and returns the hash string of the  MMR root corresponding to the current state of the tree. "))),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Verify Node (verifyProof):"),(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"Signature: verifyNode(nodeHash: str, requiredProofNodes: List","[str]",", MMRroot: str) -> bool"),(0,n.kt)("li",{parentName:"ul"},"Description: verifies if the given node hash can be proved based on the list of required proof nodes and the MMR root hash."))))),(0,n.kt)("h6",{id:"defn-beefy-payload"},"Definition 99. Payload"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"Payload:")," is the Merkle root of the MMR generated where the leaf data contains the following fields for each block:"),(0,n.kt)("ul",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("em",{parentName:"li"},"LeafVersion"),": a byte indicating the current version number of the Leaf Format. The first 3 bits are for major versions and the last 5 bits are for minor versions. "),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("em",{parentName:"li"},"BeefyNextAuthoritySetInfo"),": It is a tuple consisting of:",(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"ValidatorSetID"),(0,n.kt)("li",{parentName:"ul"},"Len (u32): length of the validator set"),(0,n.kt)("li",{parentName:"ul"},"Merkle Root of the list of Next Beefy Authority Set (ECDSA public keys). The exact format depends on the implementation.  "))),(0,n.kt)("li",{parentName:"ul"},"Parent Block number and Parent Block Hash."),(0,n.kt)("li",{parentName:"ul"},"Extra Leaf Data: Currently the Merkle root of the list of (ParaID, ParaHeads)"))),(0,n.kt)("h6",{id:"defn-beefy-Commitment"},"Definition 100. Signed Commitment"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"Signed Commitment:"),"\n",(0,n.kt)("inlineCode",{parentName:"p"},"commitment")," is a tuple of ",(0,n.kt)("inlineCode",{parentName:"p"},"(payload, Block Number, ValidatorSetID)"),". A ",(0,n.kt)("inlineCode",{parentName:"p"},"signed commitment")," is a tuple ",(0,n.kt)("inlineCode",{parentName:"p"},"(commitment, signatures)"),", where ",(0,n.kt)("inlineCode",{parentName:"p"},"signatures")," is a list of optional signatures of the validator set on the SCALE encoded ",(0,n.kt)("inlineCode",{parentName:"p"},"commitment"),". Note that the number of signatures in ",(0,n.kt)("inlineCode",{parentName:"p"},"signatures")," may be less than the length of the Validator Set. ")),(0,n.kt)("h6",{id:"defn-beefy-witness-data"},"Definition 101. Witness Data"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"Signed Commitment Witnesses")," contains the commitment and an array indicating which validator of the Polkadot network voted for the payload (but not the signatures themselves). The indicators of which validator voted for the payload are just claims and provide no proof. It also contains the signature of one validator on the commitment, which is used only by the subsampling-based Light Clients. The network message is defined in ",(0,n.kt)("a",{parentName:"p",href:"chap-networking#defn-grandpa-beefy-signed-commitment-witness"},"Definition 58")," and the relayer saves it on the chain of the remote network.")),(0,n.kt)("h6",{id:"defn-beefy-light-client"},"Definition 102. Light Client"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"A ",(0,n.kt)("strong",{parentName:"p"},"light client")," is an abstract entity in a remote network such as Ethereum. It can be a node or a smart contract with the intent of requesting finality proofs from the Polkadot network. A light client reads the witness data (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-witness-data"},"Definition 101")," from the chain, then requests the signatures directly from the relayer in order to verify those.")),(0,n.kt)("h6",{id:"defn-beefy-relayer"},"Definition 103. Relayer"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"A ",(0,n.kt)("strong",{parentName:"p"},"relayer"),' (or "prover") is an abstract entity that takes finality proofs from the Polkadot network and makes those available to the light clients. The relayer attempts to convince the light clients that the finality proofs have been voted for by the Polkadot relay chain validators. The relayer operates off-chain and can for example be a node or a collection of nodes.')),(0,n.kt)("h3",{id:"id-voting-on-payloads"},"6.7.5. Voting on Payloads"),(0,n.kt)("p",null,"The Polkadot Host signs the MMR payload (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-payload"},"Definition 99"),") and gossips it as part of a vote (",(0,n.kt)("a",{parentName:"p",href:"chap-networking#defn-msg-beefy-gossip"},"Definition 56"),") to its peers on every new finalized block. The Polkadot Host uses ECDSA for signing the payload since Ethereum has better compatibility for it compared to SR25519 or ED25519."),(0,n.kt)("h3",{id:"sect-beefy-committing-witnesses"},"6.7.6. Committing Witnesses"),(0,n.kt)("p",null,"The relayer (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-relayer"},"Definition 103"),") participates in the Polkadot network by collecting the gossiped votes (",(0,n.kt)("a",{parentName:"p",href:"chap-networking#defn-msg-beefy-gossip"},"Definition 56"),"). Those votes are converted into the witness data structure (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-witness-data"},"Definition 101"),"). The relayer saves the data on the chain of the remote network. The occurrence of saving witnesses on remote networks is undefined."),(0,n.kt)("h3",{id:"id-requesting-signed-commitments"},"6.7.7. Requesting Signed Commitments"),(0,n.kt)("p",null,"A light client (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-light-client"},"Definition 102"),") fetches the Signed Commitment Witness (",(0,n.kt)("a",{parentName:"p",href:"sect-finality#defn-beefy-witness-data"},"Definition 101"),") from the chain. Once the light client knows which validators apparently voted for the specified payload, it needs to request the signatures from the relayer to verify whether the claims are actually true. This is achieved by requesting signed commitments (",(0,n.kt)("a",{parentName:"p",href:"chap-networking#defn-grandpa-beefy-signed-commitment"},"Definition 57"),")."),(0,n.kt)("p",null,"How those signed commitments are requested by the light client and delivered by the relayer varies among networks or implementations."),(0,n.kt)("h6",{id:"defn-consensus-message-beefy"},"Definition 104. 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",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"V"),(0,n.kt)("mi",{parentName:"msub"},"B"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{V}_{{B}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.22222em"}},"V")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3283em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.05017em"}},"B")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," is the array of the new BEEFY authorities\u2019s public keys and ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"V"),(0,n.kt)("mi",{parentName:"msub"},"i"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{V}_{{i}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.22222em"}},"V")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3117em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight"},"i")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," is the identifier of the remote validator set.")),(0,n.kt)("tr",{parentName:"tbody"},(0,n.kt)("td",{parentName:"tr",align:null},"2"),(0,n.kt)("td",{parentName:"tr",align:null},"implies ",(0,n.kt)("strong",{parentName:"td"},"on disabled"),": an index to the individual authority in ",(0,n.kt)("span",{parentName:"td",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("msub",{parentName:"mrow"},(0,n.kt)("mi",{parentName:"msub"},"V"),(0,n.kt)("mi",{parentName:"msub"},"B"))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"{V}_{{B}}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.22222em"}},"V")),(0,n.kt)("span",{parentName:"span",className:"msupsub"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3283em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.55em",marginRight:"0.05em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,n.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.05017em"}},"B")))))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,n.kt)("span",{parentName:"span"}))))))))))," that should be immediately disabled until the next authority change.")),(0,n.kt)("tr",{parentName:"tbody"},(0,n.kt)("td",{parentName:"tr",align:null},"3"),(0,n.kt)("td",{parentName:"tr",align:null},"implies ",(0,n.kt)("strong",{parentName:"td"},"MMR root"),": a 32-byte array containing the MMR root payload."))))),(0,n.kt)("h3",{id:"id-consensus-mechanism-beefy-1"},"6.7.8. Consensus Mechanism"),(0,n.kt)("p",null,"Role of various Actors in BEEFY:"),(0,n.kt)("ul",null,(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Validators are expected to additionally:"),(0,n.kt)("ol",{parentName:"li"},(0,n.kt)("li",{parentName:"ol"},"Produce & broadcast vote for the current round."))),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Regular nodes perform the following tasks:"),(0,n.kt)("ol",{parentName:"li"},(0,n.kt)("li",{parentName:"ol"},"Receive & validate votes for the current round and broadcast them to their peers."),(0,n.kt)("li",{parentName:"ol"},"Receive & validate BEEFY Justifications and broadcast them to their peers."),(0,n.kt)("li",{parentName:"ol"},"Return BEEFY Justifications for ",(0,n.kt)("strong",{parentName:"li"},"Mandatory Blocks")," on demand."),(0,n.kt)("li",{parentName:"ol"},"Optionally return BEEFY Justifications for non-mandatory blocks on demand.")))),(0,n.kt)("p",null,"A ",(0,n.kt)("strong",{parentName:"p"},"round")," is an attempt by BEEFY validators to produce a BEEFY Justification. ",(0,n.kt)("strong",{parentName:"p"},"Round number")," is simply defined as a block number the validators are voting for, or to be more precise, the Commitment for that block number. Round ends when the next round is started, which may happen when one of the events occur:"),(0,n.kt)("ol",null,(0,n.kt)("li",{parentName:"ol"},"Either the node collects ",(0,n.kt)("inlineCode",{parentName:"li"},"2/3rd + 1")," valid votes for that round."),(0,n.kt)("li",{parentName:"ol"},"Or the node receives a BEEFY Justification for a block greater than the current best BEEFY block.")),(0,n.kt)("p",null,'In both cases the node proceeds to determine the new round number using "Round Selection"\nprocedure.'),(0,n.kt)("p",null,"Both kinds of actors are expected to fully participate in the protocol ONLY IF they believe they are up-to-date with the rest of the network, i.e. they are fully synced. Before this happens, the node should continue processing imported BEEFY Justifications and votes without actively voting themselves."),(0,n.kt)("p",null,(0,n.kt)("strong",{parentName:"p"},"Round Selection")),(0,n.kt)("p",null,"Every node (both regular nodes and validators) determines locally what it believes\nthe current round number is. The choice is based on their knowledge of:"),(0,n.kt)("ol",null,(0,n.kt)("li",{parentName:"ol"},"Best GRANDPA finalized block number (",(0,n.kt)("inlineCode",{parentName:"li"},"best_grandpa"),")."),(0,n.kt)("li",{parentName:"ol"},"Best BEEFY finalized block number (",(0,n.kt)("inlineCode",{parentName:"li"},"best_beefy"),")."),(0,n.kt)("li",{parentName:"ol"},"Starting block of the current session (",(0,n.kt)("inlineCode",{parentName:"li"},"session_start"),").")),(0,n.kt)("p",null,(0,n.kt)("strong",{parentName:"p"},"Session")," means a period of time (or rather a number of blocks) where the validator set (keys) does not change. Session are synonymous to epochs (",(0,n.kt)("a",{parentName:"p",href:"sect-block-production#defn-epoch-slot"},"Definition 59"),"). Since the BEEFY authority set is the same as the GRANDPA authority set for any GRANDPA finalized block, the session boundaries for BEEFY are exactly the same as the ones for GRANDPA."),(0,n.kt)("p",null,"We define two kinds of blocks from the perspective of the BEEFY protocol:"),(0,n.kt)("ol",null,(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("strong",{parentName:"li"},"Mandatory Blocks")),(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("strong",{parentName:"li"},"Non-mandatory Blocks"))),(0,n.kt)("p",null,"Mandatory blocks are the ones that MUST have BEEFY justification. That means that the validators\nwill always start and conclude a round at mandatory blocks. For non-mandatory blocks, there may\nor may not be a justification and validators may never choose these blocks to start a round."),(0,n.kt)("p",null,"Every ",(0,n.kt)("strong",{parentName:"p"},"first block in")," each ",(0,n.kt)("strong",{parentName:"p"},"session")," is considered a ",(0,n.kt)("strong",{parentName:"p"},"mandatory block"),". All other blocks\nin the session are non-mandatory, however validators are encouraged to finalize as many blocks as\npossible to enable lower latency for light clients and hence end users. Since GRANDPA is\nconsidering session boundary blocks as mandatory as well, ",(0,n.kt)("inlineCode",{parentName:"p"},"session_start")," block will always have\nboth GRANDPA and BEEFY Justification."),(0,n.kt)("h6",{id:"defn-beefy-round-number"},"Definition 105. BEEFY Round NUmber"),(0,n.kt)("admonition",{type:"definition"},(0,n.kt)("p",{parentName:"admonition"},"The formula for determining the current round number is defined as:"),(0,n.kt)("pre",{parentName:"admonition"},(0,n.kt)("code",{parentName:"pre"},"round_number =\n      (1 - M) * session_start\n   +        M * Minimum(next_session_start, (best_beefy + NEXT_POWER_OF_TWO((best_grandpa - best_beefy + 1) / 2)))\n")),(0,n.kt)("p",{parentName:"admonition"},"where:"),(0,n.kt)("ul",{parentName:"admonition"},(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("inlineCode",{parentName:"li"},"M")," is ",(0,n.kt)("inlineCode",{parentName:"li"},"1")," if the mandatory block in the current session is already finalized and ",(0,n.kt)("inlineCode",{parentName:"li"},"0")," otherwise."),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("inlineCode",{parentName:"li"},"NEXT_POWER_OF_TWO(x)")," returns the smallest number greater or equal to ",(0,n.kt)("inlineCode",{parentName:"li"},"x")," that is a power of two."))),(0,n.kt)("p",null,"Intuitively, the next round number should be the oldest mandatory block without a justification,\nor the highest GRANDPA-finalized block, whose block number difference with ",(0,n.kt)("inlineCode",{parentName:"p"},"best_beefy")," block is\na power of two. The mental model for round selection is to first finalize the mandatory block and\nthen to attempt to pick a block taking into account how fast BEEFY catches up with GRANDPA.\nIn case GRANDPA makes progress, but BEEFY seems to be lagging behind, validators are changing\nrounds less often to increase the chance of concluding them."),(0,n.kt)("p",null,"As mentioned earlier, every time the node picks a new ",(0,n.kt)("inlineCode",{parentName:"p"},"round_number")," (and the validator casts a vote)\nit ends the previous one, no matter if finality was reached (i.e. the round concluded) or not.\nVotes for an inactive round should not be propagated."),(0,n.kt)("p",null,"Note that since BEEFY only votes for GRANDPA-finalized blocks, ",(0,n.kt)("inlineCode",{parentName:"p"},"session_start"),' here actually means:\n"the latest session for which the start of is GRANDPA-finalized", i.e. block production might\nhave already progressed, but BEEFY needs to first finalize the mandatory block of the older\nsession.'),(0,n.kt)("p",null,"While it is useful to finalize non-mandatory blocks frequently, in good networking conditions BEEFY may end up finalizing each and every block GRANDPA finalized block. Practically, with short block times, it's going to be rare and might be excessive, so\nit's suggested for implementations to introduce a ",(0,n.kt)("inlineCode",{parentName:"p"},"min_delta")," parameter which will limit the frequency with which new rounds are started. The affected component of the formula would be:\n",(0,n.kt)("inlineCode",{parentName:"p"},"best_beefy + MAX(min_delta, NEXT_POWER_OF_TWO(...))"),", so we start a new round only if the\nthe power-of-two component is greater than the min delta. Note that if ",(0,n.kt)("inlineCode",{parentName:"p"},"round_number > best_grandpa")," the validators are not expected to start any round."),(0,n.kt)("h3",{id:"id-light-client-beefy-1"},"6.7.9. BEEFY Light Client"),(0,n.kt)("p",null,"A light client following BEEFY could request ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"N"),(0,n.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"/"),(0,n.kt)("mn",{parentName:"mrow"},"3"),(0,n.kt)("mo",{parentName:"mrow"},"+"),(0,n.kt)("mn",{parentName:"mrow"},"1")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"N/3 +1")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10903em"}},"N"),(0,n.kt)("span",{parentName:"span",className:"mord"},"/3"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"+"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"1")))))," signatures to be checked, where ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"N")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"N")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10903em"}},"N")))))," is the number of validators on Polkadot. 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diff --git a/assets/js/runtime~main.a2453290.js b/assets/js/runtime~main.23d931e1.js
similarity index 97%
rename from assets/js/runtime~main.a2453290.js
rename to assets/js/runtime~main.23d931e1.js
index 477945bf9..b17b7bbe4 100644
--- a/assets/js/runtime~main.a2453290.js
+++ b/assets/js/runtime~main.23d931e1.js
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diff --git a/chap-host-api.html b/chap-host-api.html
index d27541232..31607b974 100644
--- a/chap-host-api.html
+++ b/chap-host-api.html
@@ -6,7 +6,7 @@
 <title data-rh="true">Appendix B: Host API | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/chap-host-api"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="Appendix B: Host API | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="Description of the expected environment available for import by the Polkadot Runtime"><meta data-rh="true" property="og:description" content="Description of the expected environment available for import by the Polkadot Runtime"><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/chap-host-api"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-host-api" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-host-api" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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diff --git a/chap-networking.html b/chap-networking.html
index 6286bff9c..0246e9e05 100644
--- a/chap-networking.html
+++ b/chap-networking.html
@@ -6,7 +6,7 @@
 <title data-rh="true">4. Networking | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/chap-networking"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="4. Networking | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="This chapter, in its current form, is incomplete and considered work in progress. Authors appreciate receiving requests for clarification or any reports regarding deviation from the current Polkadot network protocol. This can be done by filing an issue in Polkadot Specification repository."><meta data-rh="true" property="og:description" content="This chapter, in its current form, is incomplete and considered work in progress. Authors appreciate receiving requests for clarification or any reports regarding deviation from the current Polkadot network protocol. This can be done by filing an issue in Polkadot Specification repository."><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/chap-networking"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-networking" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-networking" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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diff --git a/chap-overview.html b/chap-overview.html
index c5d331921..a636c51b9 100644
--- a/chap-overview.html
+++ b/chap-overview.html
@@ -6,7 +6,7 @@
 <title data-rh="true">1. Overview | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/chap-overview"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="1. Overview | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="The Polkadot Protocol differentiates between different classes of Polkadot Hosts. Each class differs in its trust roots and how active or passively they interact with the network."><meta data-rh="true" property="og:description" content="The Polkadot Protocol differentiates between different classes of Polkadot Hosts. Each class differs in its trust roots and how active or passively they interact with the network."><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/chap-overview"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-overview" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-overview" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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+<script src="/assets/js/runtime~main.23d931e1.js" defer="defer"></script>
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 <body class="navigation-with-keyboard">
diff --git a/chap-runtime-api.html b/chap-runtime-api.html
index 96d0b0797..bc07b9ddd 100644
--- a/chap-runtime-api.html
+++ b/chap-runtime-api.html
@@ -6,7 +6,7 @@
 <title data-rh="true">Appendix C: Runtime API | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/chap-runtime-api"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="Appendix C: Runtime API | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="Description of how to interact with the Runtime through its exported functions"><meta data-rh="true" property="og:description" content="Description of how to interact with the Runtime through its exported functions"><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/chap-runtime-api"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-runtime-api" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-runtime-api" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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-<script src="/assets/js/runtime~main.a2453290.js" defer="defer"></script>
+<script src="/assets/js/runtime~main.23d931e1.js" defer="defer"></script>
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 <body class="navigation-with-keyboard">
diff --git a/chap-state.html b/chap-state.html
index 48e51d7b1..ea3e2e1f3 100644
--- a/chap-state.html
+++ b/chap-state.html
@@ -6,7 +6,7 @@
 <title data-rh="true">2. States and Transitions | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/chap-state"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="2. States and Transitions | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="2.1. Introduction"><meta data-rh="true" property="og:description" content="2.1. Introduction"><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/chap-state"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-state" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-state" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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diff --git a/chap-sync.html b/chap-sync.html
index 152bfc9d8..35b0be037 100644
--- a/chap-sync.html
+++ b/chap-sync.html
@@ -6,7 +6,7 @@
 <title data-rh="true">3. Synchronization | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/chap-sync"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="3. Synchronization | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="Many applications that interact with the Polkadot network, to some extent, must be able to retrieve certain information about the network. Depending on the utility, this includes validators that interact with Polkadot’s consensus and need access to the full state, either from the past or just the most up-to-date state, or light clients that are only interested in the minimum information required in order to verify some claims about the state of the network, such as the balance of a specific account. To allow implementations to quickly retrieve the required information, different types of synchronization protocols are available, respectively Full, Fast, and Warp sync suited for different needs."><meta data-rh="true" property="og:description" content="Many applications that interact with the Polkadot network, to some extent, must be able to retrieve certain information about the network. Depending on the utility, this includes validators that interact with Polkadot’s consensus and need access to the full state, either from the past or just the most up-to-date state, or light clients that are only interested in the minimum information required in order to verify some claims about the state of the network, such as the balance of a specific account. To allow implementations to quickly retrieve the required information, different types of synchronization protocols are available, respectively Full, Fast, and Warp sync suited for different needs."><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/chap-sync"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-sync" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chap-sync" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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diff --git a/chapter-anv.html b/chapter-anv.html
index 6b2e7851d..86445ca15 100644
--- a/chapter-anv.html
+++ b/chapter-anv.html
@@ -6,7 +6,7 @@
 <title data-rh="true">8. Availability &amp; Validity | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/chapter-anv"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="8. Availability &amp; Validity | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="Polkadot serves as a replicated shared-state machine designed to resolve scalability issues and interoperability among blockchains. The validators of Polkadot execute transactions and participate in the consensus of Polkadots primary chain, the so-called relay chain. Parachains are independent networks that maintain their own state and are connected to the relay chain. Those parachains can take advantage of the relay chain consensus mechanism, including sending and receiving messages to and from other parachains. Parachain nodes that send parachain blocks, known as candidates, to the validators in order to be included in relay chain are referred to as collators."><meta data-rh="true" property="og:description" content="Polkadot serves as a replicated shared-state machine designed to resolve scalability issues and interoperability among blockchains. The validators of Polkadot execute transactions and participate in the consensus of Polkadots primary chain, the so-called relay chain. Parachains are independent networks that maintain their own state and are connected to the relay chain. Those parachains can take advantage of the relay chain consensus mechanism, including sending and receiving messages to and from other parachains. Parachain nodes that send parachain blocks, known as candidates, to the validators in order to be included in relay chain are referred to as collators."><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/chapter-anv"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chapter-anv" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/chapter-anv" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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diff --git a/id-consensus.html b/id-consensus.html
index 20ea633db..2864c7781 100644
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diff --git a/id-cryptography-encoding.html b/id-cryptography-encoding.html
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 <title data-rh="true">Appendix A: Cryptography &amp; Encoding | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/id-cryptography-encoding"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="Appendix A: Cryptography &amp; Encoding | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="The appendix chapter contains various protocol details."><meta data-rh="true" property="og:description" content="The appendix chapter contains various protocol details."><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/id-cryptography-encoding"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/id-cryptography-encoding" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/id-cryptography-encoding" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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index 82a83256a..b00ef5b54 100644
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 <title data-rh="true">9. Extrinsics | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/id-extrinsics"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="9. Extrinsics | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="9.1. Introduction"><meta data-rh="true" property="og:description" content="9.1. Introduction"><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/id-extrinsics"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/id-extrinsics" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/id-extrinsics" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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index 56ca72060..c6b62492d 100644
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0-.727.3l-.254 1.557c-.117.061-.233.124-.35.188l-1.434-.654a.46.46 0 0 0-.654.436l.055 1.58a14.34 14.34 0 0 0-.303.25l-1.538-.363a.464.464 0 0 0-.557.556l.363 1.539c-.085.1-.168.2-.25.302l-1.579-.055a.462.462 0 0 0-.437.654l.654 1.436c-.063.115-.126.231-.187.35l-1.556.252a.462.462 0 0 0-.301.727l.92 1.279c-.04.126-.078.253-.115.38l-1.476.553a.462.462 0 0 0-.153.771l1.155 1.077c-.015.13-.028.26-.039.391l-1.343.832a.462.462 0 0 0 0 .786l1.343.831c.011.131.024.262.039.392l-1.155 1.077a.462.462 0 0 0 .153.771l1.476.552c.037.128.076.255.116.38l-.921 1.28a.462.462 0 0 0 .301.726l1.556.253c.061.118.123.235.188.35l-.655 1.435a.462.462 0 0 0 .437.654l1.579-.055c.082.103.165.203.25.303l-.363 1.539a.46.46 0 0 0 .557.555l1.538-.362c.1.085.201.167.303.249l-.055 1.58a.461.461 0 0 0 .654.436l1.435-.654c.115.064.232.127.35.188l.253 1.555a.461.461 0 0 0 .727.302l1.279-.922c.126.04.253.08.38.116l.552 1.476a.46.46 0 0 0 .771.153l1.078-1.155c.13.015.26.028.391.04l.832 1.343a.463.463 0 0 0 .786 0l.831-1.344c.131-.011.262-.024.392-.039l1.077 1.155a.46.46 0 0 0 .77-.153l.553-1.476c.127-.036.254-.076.38-.116l1.28.922a.463.463 0 0 0 .726-.302l.254-1.556c.117-.06.233-.124.349-.187l1.435.654a.461.461 0 0 0 .654-.437l-.055-1.58c.102-.08.203-.163.303-.248l1.538.362a.46.46 0 0 0 .556-.555l-.362-1.539c.084-.1.167-.2.249-.303l1.58.055a.46.46 0 0 0 .436-.654l-.654-1.435c.064-.115.126-.232.187-.35l1.556-.253a.46.46 0 0 0 .301-.726l-.92-1.28a17.5 17.5 0 0 0 .115-.38l1.476-.552a.46.46 0 0 0 .153-.771l-1.155-1.077c.014-.13.027-.261.039-.392l1.343-.831a.462.462 0 0 0 0-.786zM38.79 42.752a.952.952 0 0 1 .399-1.861.952.952 0 0 1-.4 1.861zm-.457-3.087a.866.866 0 0 0-1.028.666l-.477 2.226A11.649 11.649 0 0 1 32 43.597c-1.76 0-3.43-.39-4.929-1.087l-.477-2.225a.866.866 0 0 0-1.028-.667l-1.965.422a11.68 11.68 0 0 1-1.016-1.197h9.561c.108 0 .18-.02.18-.118v-3.382c0-.099-.072-.118-.18-.118H29.35V33.08h3.024c.276 0 1.476.079 1.86 1.613.12.471.384 2.006.564 2.497.18.551.912 1.652 1.692 1.652h4.764a.977.977 0 0 0 .173-.017c-.33.449-.693.874-1.083 1.27l-2.01-.431zm-13.223 3.04a.952.952 0 0 1-.399-1.861.95.95 0 0 1 .398 1.862zm-3.627-14.707a.95.95 0 1 1-1.737.771.95.95 0 1 1 1.737-.771zm-1.115 2.643 2.047-.91a.868.868 0 0 0 .44-1.145l-.421-.953h1.658v7.474h-3.345a11.714 11.714 0 0 1-.38-4.466zm8.983-.726v-2.203h3.948c.204 0 1.44.236 1.44 1.16 0 .767-.948 1.043-1.728 1.043h-3.66zM43.7 31.898c0 .292-.011.581-.033.868h-1.2c-.12 0-.168.08-.168.197v.551c0 1.298-.732 1.58-1.373 1.652-.61.068-1.288-.256-1.371-.63-.36-2.025-.96-2.458-1.908-3.206 1.176-.746 2.4-1.848 2.4-3.323 0-1.593-1.092-2.596-1.836-3.088-1.044-.688-2.2-.826-2.512-.826H23.285a11.684 11.684 0 0 1 6.545-3.694l1.463 1.535c.331.346.88.36 1.225.028l1.638-1.566a11.71 11.71 0 0 1 8.009 5.704l-1.121 2.532a.869.869 0 0 0 .44 1.145l2.159.958c.037.383.056.77.056 1.163zM31.294 19.093a.95.95 0 0 1 1.344.031.952.952 0 0 1-.032 1.346.949.949 0 0 1-1.343-.032.953.953 0 0 1 .031-1.345zm11.123 8.951a.95.95 0 1 1 1.737.772.95.95 0 1 1-1.737-.772z" fill="#fff"></path></svg></div></div><div class="text--center padding-horiz--md"><h3>Rust</h3><p>See <code><a href="https://github.com/paritytech/polkadot">Node Implementation by Parity</a></code> and <code><a href="https://github.com/smol-dot/smoldot">smoldot</a></code></p></div></div><div class="col col--4"><div class="text--center"><div class="fillWhite"><svg width="800" height="800" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg" class="featureSvg_GfXr" role="img"><path d="M1.811 10.231c-.047 0-.058-.023-.035-.059l.246-.315c.023-.035.081-.058.128-.058h4.172c.046 0 .058.035.035.07l-.199.303c-.023.036-.082.07-.117.07zM.047 11.306c-.047 0-.059-.023-.035-.058l.245-.316c.023-.035.082-.058.129-.058h5.328c.047 0 .07.035.058.07l-.093.28c-.012.047-.058.07-.105.07zm2.828 1.075c-.047 0-.059-.035-.035-.07l.163-.292c.023-.035.07-.07.117-.07h2.337c.047 0 .07.035.07.082l-.023.28c0 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1.495-.315.06-.63.07-.934.106zm2.78-4.72c-.011-.153-.011-.27-.034-.387-.21-1.157-1.274-1.81-2.384-1.554-1.087.245-1.788.935-2.045 2.033-.21.912.234 1.835 1.075 2.21.643.28 1.285.244 1.905-.07.923-.48 1.425-1.228 1.484-2.233z"></path></svg></div></div><div class="text--center padding-horiz--md"><h3>Go</h3><p>See <code><a href="https://github.com/ChainSafe/gossamer">Gossamer by Chainsafe</a></code></p></div></div><div class="col col--4"><div class="text--center"><div class="fillWhite"><svg height="24" viewBox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg" class="featureSvg_GfXr kagome_eyY2" role="img"><path d="M20.66 7a1.51 1.51 0 0 0-.55-.57l-7.34-4.24a1.67 1.67 0 0 0-1.54 0L3.89 6.43a1.68 1.68 0 0 0-.77 1.33v8.48a1.57 1.57 0 0 0 .22.76 1.51 1.51 0 0 0 .55.57l7.34 4.24a1.67 1.67 0 0 0 1.54 0l7.34-4.24a1.51 1.51 0 0 0 .55-.57 1.57 1.57 0 0 0 .22-.76V7.76a1.57 1.57 0 0 0-.22-.76zM12 17.92A5.92 5.92 0 1 1 17.13 9L16 9.71l-.36.2-1 .61A3 3 0 0 0 9 12a2.88 2.88 0 0 0 .4 1.48 3 3 0 0 0 5.13 0l2.6 1.52A5.94 5.94 0 0 1 12 17.92zm5.92-5.59h-.66V13h-.65v-.66H16v-.66h.66V11h.65v.66h.66zm2.47 0h-.66V13h-.66v-.66h-.65v-.66h.65V11h.66v.66h.66z"></path></svg></div></div><div class="text--center padding-horiz--md"><h3>C++</h3><p>See <code><a href="https://github.com/soramitsu/kagome">Kagome by Soramitsu</a></code></p></div></div></div></div></section></main></div><footer class="footer footer--dark"><div class="container container-fluid"><div class="row footer__links"><div class="col footer__col"><div class="footer__title">More Info</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://github.com/w3f/polkadot-spec" target="_blank" rel="noopener noreferrer" class="footer__link-item">GitHub<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 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0l.831-1.344c.131-.011.262-.024.392-.039l1.077 1.155a.46.46 0 0 0 .77-.153l.553-1.476c.127-.036.254-.076.38-.116l1.28.922a.463.463 0 0 0 .726-.302l.254-1.556c.117-.06.233-.124.349-.187l1.435.654a.461.461 0 0 0 .654-.437l-.055-1.58c.102-.08.203-.163.303-.248l1.538.362a.46.46 0 0 0 .556-.555l-.362-1.539c.084-.1.167-.2.249-.303l1.58.055a.46.46 0 0 0 .436-.654l-.654-1.435c.064-.115.126-.232.187-.35l1.556-.253a.46.46 0 0 0 .301-.726l-.92-1.28a17.5 17.5 0 0 0 .115-.38l1.476-.552a.46.46 0 0 0 .153-.771l-1.155-1.077c.014-.13.027-.261.039-.392l1.343-.831a.462.462 0 0 0 0-.786zM38.79 42.752a.952.952 0 0 1 .399-1.861.952.952 0 0 1-.4 1.861zm-.457-3.087a.866.866 0 0 0-1.028.666l-.477 2.226A11.649 11.649 0 0 1 32 43.597c-1.76 0-3.43-.39-4.929-1.087l-.477-2.225a.866.866 0 0 0-1.028-.667l-1.965.422a11.68 11.68 0 0 1-1.016-1.197h9.561c.108 0 .18-.02.18-.118v-3.382c0-.099-.072-.118-.18-.118H29.35V33.08h3.024c.276 0 1.476.079 1.86 1.613.12.471.384 2.006.564 2.497.18.551.912 1.652 1.692 1.652h4.764a.977.977 0 0 0 .173-.017c-.33.449-.693.874-1.083 1.27l-2.01-.431zm-13.223 3.04a.952.952 0 0 1-.399-1.861.95.95 0 0 1 .398 1.862zm-3.627-14.707a.95.95 0 1 1-1.737.771.95.95 0 1 1 1.737-.771zm-1.115 2.643 2.047-.91a.868.868 0 0 0 .44-1.145l-.421-.953h1.658v7.474h-3.345a11.714 11.714 0 0 1-.38-4.466zm8.983-.726v-2.203h3.948c.204 0 1.44.236 1.44 1.16 0 .767-.948 1.043-1.728 1.043h-3.66zM43.7 31.898c0 .292-.011.581-.033.868h-1.2c-.12 0-.168.08-.168.197v.551c0 1.298-.732 1.58-1.373 1.652-.61.068-1.288-.256-1.371-.63-.36-2.025-.96-2.458-1.908-3.206 1.176-.746 2.4-1.848 2.4-3.323 0-1.593-1.092-2.596-1.836-3.088-1.044-.688-2.2-.826-2.512-.826H23.285a11.684 11.684 0 0 1 6.545-3.694l1.463 1.535c.331.346.88.36 1.225.028l1.638-1.566a11.71 11.71 0 0 1 8.009 5.704l-1.121 2.532a.869.869 0 0 0 .44 1.145l2.159.958c.037.383.056.77.056 1.163zM31.294 19.093a.95.95 0 0 1 1.344.031.952.952 0 0 1-.032 1.346.949.949 0 0 1-1.343-.032.953.953 0 0 1 .031-1.345zm11.123 8.951a.95.95 0 1 1 1.737.772.95.95 0 1 1-1.737-.772z" fill="#fff"></path></svg></div></div><div class="text--center padding-horiz--md"><h3>Rust</h3><p>See <code><a href="https://github.com/paritytech/polkadot">Node Implementation by Parity</a></code> and <code><a href="https://github.com/smol-dot/smoldot">smoldot</a></code></p></div></div><div class="col col--4"><div class="text--center"><div class="fillWhite"><svg width="800" height="800" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg" class="featureSvg_GfXr" role="img"><path d="M1.811 10.231c-.047 0-.058-.023-.035-.059l.246-.315c.023-.035.081-.058.128-.058h4.172c.046 0 .058.035.035.07l-.199.303c-.023.036-.082.07-.117.07zM.047 11.306c-.047 0-.059-.023-.035-.058l.245-.316c.023-.035.082-.058.129-.058h5.328c.047 0 .07.035.058.07l-.093.28c-.012.047-.058.07-.105.07zm2.828 1.075c-.047 0-.059-.035-.035-.07l.163-.292c.023-.035.07-.07.117-.07h2.337c.047 0 .07.035.07.082l-.023.28c0 .047-.047.082-.082.082zm12.129-2.36c-.736.187-1.239.327-1.963.514-.176.046-.187.058-.34-.117-.174-.199-.303-.327-.548-.444-.737-.362-1.45-.257-2.115.175-.795.514-1.204 1.274-1.192 2.22.011.935.654 1.706 1.577 1.835.795.105 1.46-.175 1.987-.77.105-.13.198-.27.315-.434H10.47c-.245 0-.304-.152-.222-.35.152-.362.432-.97.596-1.274a.315.315 0 0 1 .292-.187h4.253c-.023.316-.023.631-.07.947a4.983 4.983 0 0 1-.958 2.29c-.841 1.11-1.94 1.8-3.33 1.986-1.145.152-2.209-.07-3.143-.77-.865-.655-1.356-1.52-1.484-2.595-.152-1.274.222-2.419.993-3.424.83-1.086 1.928-1.776 3.272-2.02 1.098-.2 2.15-.07 3.096.571.62.41 1.063.97 1.356 1.648.07.105.023.164-.117.2m3.868 6.461c-1.064-.024-2.034-.328-2.852-1.029a3.665 3.665 0 0 1-1.262-2.255c-.21-1.32.152-2.489.947-3.529.853-1.122 1.881-1.706 3.272-1.95 1.192-.21 2.314-.095 3.33.595.923.63 1.496 1.484 1.648 2.605.198 1.578-.257 2.863-1.344 3.962-.771.783-1.718 1.273-2.805 1.495-.315.06-.63.07-.934.106zm2.78-4.72c-.011-.153-.011-.27-.034-.387-.21-1.157-1.274-1.81-2.384-1.554-1.087.245-1.788.935-2.045 2.033-.21.912.234 1.835 1.075 2.21.643.28 1.285.244 1.905-.07.923-.48 1.425-1.228 1.484-2.233z"></path></svg></div></div><div class="text--center padding-horiz--md"><h3>Go</h3><p>See <code><a href="https://github.com/ChainSafe/gossamer">Gossamer by Chainsafe</a></code></p></div></div><div class="col col--4"><div class="text--center"><div class="fillWhite"><svg height="24" viewBox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg" class="featureSvg_GfXr kagome_eyY2" role="img"><path d="M20.66 7a1.51 1.51 0 0 0-.55-.57l-7.34-4.24a1.67 1.67 0 0 0-1.54 0L3.89 6.43a1.68 1.68 0 0 0-.77 1.33v8.48a1.57 1.57 0 0 0 .22.76 1.51 1.51 0 0 0 .55.57l7.34 4.24a1.67 1.67 0 0 0 1.54 0l7.34-4.24a1.51 1.51 0 0 0 .55-.57 1.57 1.57 0 0 0 .22-.76V7.76a1.57 1.57 0 0 0-.22-.76zM12 17.92A5.92 5.92 0 1 1 17.13 9L16 9.71l-.36.2-1 .61A3 3 0 0 0 9 12a2.88 2.88 0 0 0 .4 1.48 3 3 0 0 0 5.13 0l2.6 1.52A5.94 5.94 0 0 1 12 17.92zm5.92-5.59h-.66V13h-.65v-.66H16v-.66h.66V11h.65v.66h.66zm2.47 0h-.66V13h-.66v-.66h-.65v-.66h.65V11h.66v.66h.66z"></path></svg></div></div><div class="text--center padding-horiz--md"><h3>C++</h3><p>See <code><a href="https://github.com/soramitsu/kagome">Kagome by Soramitsu</a></code></p></div></div></div></div></section></main></div><footer class="footer footer--dark"><div class="container container-fluid"><div class="row footer__links"><div class="col footer__col"><div class="footer__title">More Info</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://github.com/w3f/polkadot-spec" target="_blank" rel="noopener noreferrer" class="footer__link-item">GitHub<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 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diff --git a/part-polkadot-host.html b/part-polkadot-host.html
index 21a48c14d..3abcb4858 100644
--- a/part-polkadot-host.html
+++ b/part-polkadot-host.html
@@ -6,7 +6,7 @@
 <title data-rh="true">Polkadot Host | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/part-polkadot-host"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="Polkadot Host | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="With the current document, we aim to specify the Polkadot Host part of the Polkadot protocol as a replicated state machine. After defining the different types of hosts in Chapter 1, we proceed to specify the representation of a valid state of the Protocol in Chapter 2. We also identify the protocol states by explaining the Polkadot state transition and discussing the detail based on which the Polkadot Host interacts with the state transition function, i.e., Runtime, in the same chapter. Following, we specify the input messages triggering the state transition and the system behavior. In Chapter 4, we specify the communication protocols and network messages required for the Polkadot Host to communicate with other nodes in the network, such as exchanging blocks and consensus messages. In Chapter 5 and Chapter 6, we specify the consensus protocol, which is responsible for keeping all the replicas in the same state. Finally, the initial state of the machine is identified and discussed in Section A.3.3.. A Polkadot Host implementation that conforms with this part of the specification should successfully be able to sync its states with the Polkadot network."><meta data-rh="true" property="og:description" content="With the current document, we aim to specify the Polkadot Host part of the Polkadot protocol as a replicated state machine. After defining the different types of hosts in Chapter 1, we proceed to specify the representation of a valid state of the Protocol in Chapter 2. We also identify the protocol states by explaining the Polkadot state transition and discussing the detail based on which the Polkadot Host interacts with the state transition function, i.e., Runtime, in the same chapter. Following, we specify the input messages triggering the state transition and the system behavior. In Chapter 4, we specify the communication protocols and network messages required for the Polkadot Host to communicate with other nodes in the network, such as exchanging blocks and consensus messages. In Chapter 5 and Chapter 6, we specify the consensus protocol, which is responsible for keeping all the replicas in the same state. Finally, the initial state of the machine is identified and discussed in Section A.3.3.. A Polkadot Host implementation that conforms with this part of the specification should successfully be able to sync its states with the Polkadot network."><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/part-polkadot-host"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/part-polkadot-host" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/part-polkadot-host" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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diff --git a/part-polkadot-runtime.html b/part-polkadot-runtime.html
index 5111b494d..d33dbdcb9 100644
--- a/part-polkadot-runtime.html
+++ b/part-polkadot-runtime.html
@@ -6,7 +6,7 @@
 <title data-rh="true">Polkadot Runtime | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/part-polkadot-runtime"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="Polkadot Runtime | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="Description of various useful Runtime internals"><meta data-rh="true" property="og:description" content="Description of various useful Runtime internals"><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/part-polkadot-runtime"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/part-polkadot-runtime" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/part-polkadot-runtime" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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diff --git a/sect-block-production.html b/sect-block-production.html
index 62c942dca..d436c73e1 100644
--- a/sect-block-production.html
+++ b/sect-block-production.html
@@ -6,7 +6,7 @@
 <title data-rh="true">5. Block Production | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/sect-block-production"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="5. Block Production | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="5.1. Introduction"><meta data-rh="true" property="og:description" content="5.1. Introduction"><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/sect-block-production"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/sect-block-production" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/sect-block-production" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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diff --git a/sect-finality.html b/sect-finality.html
index 8d2ff7562..f97baf0b0 100644
--- a/sect-finality.html
+++ b/sect-finality.html
@@ -6,7 +6,7 @@
 <title data-rh="true">6. Finality | Polkadot Protocol Specification</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://spec.polkadot.network/sect-finality"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-default-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-rh="true" property="og:title" content="6. Finality | Polkadot Protocol Specification"><meta data-rh="true" name="description" content="6.1. Introduction"><meta data-rh="true" property="og:description" content="6.1. Introduction"><link data-rh="true" rel="icon" href="/img/polkadot-logo.png"><link data-rh="true" rel="canonical" href="https://spec.polkadot.network/sect-finality"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/sect-finality" hreflang="en"><link data-rh="true" rel="alternate" href="https://spec.polkadot.network/sect-finality" hreflang="x-default"><link data-rh="true" rel="preconnect" href="https://59CYO0AR6V-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css" integrity="sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM" crossorigin="anonymous">
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@@ -190,7 +190,7 @@
 public key is <code>33</code> bytes long, instead of <code>32</code> bytes for a <code>sr25519</code> based public key. As a consequence, the <code>AccountId</code> (32-bytes) matches the <code>PublicKey</code> for other session keys, but note that it&#x27;s not the case for BEEFY.</p><div class="alert alert--definition"><h5></h5><div><p><strong>BEEFY session key pair</strong> <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mi>s</mi><msubsup><mi>k</mi><mi>j</mi><mi>B</mi></msubsup><mo separator="true">,</mo><mi>p</mi><msubsup><mi>k</mi><mi>j</mi><mi>B</mi></msubsup><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">{\left({s}{{k}_{{j}}^{B}},{p}{{k}_{{j}}^{B}}\right)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2448em;vertical-align:-0.3948em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">s</span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em">k</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em"><span style="top:-2.4413em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em">B</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em"><span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">p</span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em">k</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em"><span style="top:-2.4413em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em">B</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span></span> is a <code>secp256k1</code> key pair which the BEEFY authority node <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">P</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">{\mathcal{{P}}}_{{j}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em">P</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span></span></span></span></span> uses to sign the BEEFY signed commitments (justifications).
 </p></div></div><h3 class="anchor anchorWithStickyNavbar_LWe7" id="id-beefy-merkle-mountain-ranges">6.7.4. Merkle Mountain Ranges<a href="#id-beefy-merkle-mountain-ranges" class="hash-link" aria-label="Direct link to 6.7.4. Merkle Mountain Ranges" title="Direct link to 6.7.4. Merkle Mountain Ranges">​</a></h3><h6 class="anchor anchorWithStickyNavbar_LWe7" id="defn-mmr">Definition 96. Merkle Mountain Ranges<a href="#defn-mmr" class="hash-link" aria-label="Direct link to Definition 96. Merkle Mountain Ranges" title="Direct link to Definition 96. Merkle Mountain Ranges">​</a></h6><div class="alert alert--definition"><h5></h5><div><p>Merkle Mountain Ranges, <strong>MMR</strong>, are used as an efficient way to send block headers and signatures to light clients. Merkle Mountain Ranges (<code>MMR</code>) is an improvement of the traditional Merkle tree data structure. Just like a Merkle tree, an <code>MMR</code> is a binary tree where each leaf node represents a data element and each non-leaf node is the hash of its child nodes. The key difference between a traditional Merkle tree and a MMR lies in the way nodes are organized. In traditional Merkle trees, whenever a leaf node is appended or removed, the tree must be rebuilt and the hashes of the non-leaf nodes must be recalculated. The overhead of recomputing the hashes up to the root makes traditional Merkle unsuitable for handling dynamic data. The <code>MMR</code> is designed to optimize the appending and removal of elements without requiring a complete rebuild of the tree, which makes it more efficient to handle growing lists of leaf nodes.</p><p>MMR structure </p><p>A <code>MMR</code> structure can be seen as a list of perfectly balanced binary sub-trees in descending order of height. It is a strictly append-only structure where nodes are added from left to right, such that a parent node is added as soon as two children exist. The following representation shows a <code>MMR</code> with 11 elements, 7 leaf nodes and 4 non-leaf nodes, where the value of each node corresponds to the order in which it was inserted into the tree.</p><div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token plain">       7</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">      / \</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">     /   \</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">    /     \</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">   3       6        10</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">  / \     / \       / \   </span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> /   \   /   \     /   \   </span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">1     2 4     5   8     9    11</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div></div></div><div class="alert alert--definition"><h5></h5><div><h6 class="anchor anchorWithStickyNavbar_LWe7" id="defn-mmr-root">Definition 97. Merkle Mountain Ranges root (MMR-root)<a href="#defn-mmr-root" class="hash-link" aria-label="Direct link to Definition 97. Merkle Mountain Ranges root (MMR-root)" title="Direct link to Definition 97. Merkle Mountain Ranges root (MMR-root)">​</a></h6><p>An <code>MMR</code> does not have a single <code>root</code> by design, as a conventional Merkle tree. Every sub-tree has a separate sub-root, which we refer to as the <code>peak</code> of the sub-tree.
 Bagging the peaks is the process used for hashing the peaks in order to compute the <code>MMR-root</code>. It is important to define the order in which the peaks are hashed to ensure that a given sub-set of peaks will always derive a unique <code>MMR-root</code>. Here we state that peaks are merged from right to left and bagged via <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mi>a</mi><mi>s</mi><mi>h</mi><mo stretchy="false">(</mo><mi>r</mi><mi>i</mi><mi>g</mi><mi>h</mi><mi>t</mi><mo separator="true">,</mo><mi>l</mi><mi>e</mi><mi>f</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hash(right, left)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">ha</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em">r</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.03588em">g</span><span class="mord mathnormal">h</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.01968em">l</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span></span>.</p><p>Therefore, given an MMR tree with n peaks ordered in decreasing order of height, the <code>MMR-root</code> of the tree is calculated as follows.</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext mathvariant="monospace">MMR</mtext><mtext> </mtext><mtext mathvariant="monospace">root</mtext></mrow><mo>=</mo><mi>h</mi><mi>a</mi><mi>s</mi><mi>h</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>+</mo><mi>h</mi><mi>a</mi><mi>s</mi><mi>h</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mn>2</mn></msub><mo>+</mo><mi>h</mi><mi>a</mi><mi>s</mi><mi>h</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mn>3</mn></msub><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><mi>h</mi><mi>a</mi><mi>s</mi><mi>h</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mi>n</mi></msub><mo>−</mo><mn>1</mn><mo>+</mo><msub><mi>p</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\texttt{MMR root} = hash (p_1 + hash( p_2 + hash( p_3 +... + hash ( p_n-1 + p_n ))))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em"></span><span class="mord text"><span class="mord texttt">MMR root</span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">ha</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">ha</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">ha</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em"></span><span class="mord">...</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">ha</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">))))</span></span></span></span></span></div><p><strong>where</strong></p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>p</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>p</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">p_1, p_2,...,p_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>: represents the list of current peaks in decreasing order of heights.</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mi>a</mi><mi>s</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">hash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal">ha</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span></span></span></span></span>: corresponds to the 256-bit <a href="/chap-host-api#sect-finality#id-ext_hashing_keccak_256">Keccak</a> hash function used to merge the peaks.</li></ul></div></div><p>A distinguished feature of this process is that whenever new leaf nodes are added to the tree, the earlier hash computations of peaks are reused, making new leaf nodes less expensive to insert and to prove (i.e., to verify the integrity of leaf data). </p><h6 class="anchor anchorWithStickyNavbar_LWe7" id="defn-mmr-operations">Definition 98. MMR operations<a href="#defn-mmr-operations" class="hash-link" aria-label="Direct link to Definition 98. MMR operations" title="Direct link to Definition 98. MMR operations">​</a></h6><div class="alert alert--definition"><h5></h5><div><p>Here are the basic operations we should be able to perform on the MMR:</p><ul><li><p>Append Leaf Node (appendData):</p><ul><li>Signature: append(data: T) -&gt; None</li><li>Description: appends a new leaf element with the provided data to the MMR.
-</li></ul></li><li><p>Create MMR root (bagPeaks):</p><ul><li>Signature: baggingPeaks(peaksIndexes: List<!-- -->[int]<!-- -->) -&gt; str</li><li>Description: creates the single MMR root based on the list of peaks, and returns the hash string of the  MMR root corresponding to the current state of the tree. </li></ul></li><li><p>Verify Node (verifyProof):</p><div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token plain">- Signature: verifyNode(nodeHash: str, requiredProofNodes: List[str], MMRroot: str) -&gt; bool</span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">- Description: verifies if the given node hash can be proved based on the list of required proof nodes and the MMR root hash.</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div><p>:::</p></li></ul><h6 class="anchor anchorWithStickyNavbar_LWe7" id="defn-beefy-payload">Definition 99. Payload<a href="#defn-beefy-payload" class="hash-link" aria-label="Direct link to Definition 99. Payload" title="Direct link to Definition 99. Payload">​</a></h6><div class="alert alert--definition"><h5></h5><div><p><strong>Payload:</strong> is the Merkle root of the MMR generated where the leaf data contains the following fields for each block:</p><ul><li><em>LeafVersion</em>: a byte indicating the current version number of the Leaf Format. The first 3 bits are for major versions and the last 5 bits are for minor versions. </li><li><em>BeefyNextAuthoritySetInfo</em>: It is a tuple consisting of:<ul><li>ValidatorSetID</li><li>Len (u32): length of the validator set</li><li>Merkle Root of the list of Next Beefy Authority Set (ECDSA public keys). The exact format depends on the implementation.  </li></ul></li><li>Parent Block number and Parent Block Hash.</li><li>Extra Leaf Data: Currently the Merkle root of the list of (ParaID, ParaHeads)</li></ul></div></div><h6 class="anchor anchorWithStickyNavbar_LWe7" id="defn-beefy-Commitment">Definition 100. Signed Commitment<a href="#defn-beefy-Commitment" class="hash-link" aria-label="Direct link to Definition 100. Signed Commitment" title="Direct link to Definition 100. Signed Commitment">​</a></h6><div class="alert alert--definition"><h5></h5><div><p><strong>Signed Commitment:</strong>
+</li></ul></li><li><p>Create MMR root (bagPeaks):</p><ul><li>Signature: baggingPeaks(peaksIndexes: List<!-- -->[int]<!-- -->) -&gt; str</li><li>Description: creates the single MMR root based on the list of peaks, and returns the hash string of the  MMR root corresponding to the current state of the tree. </li></ul></li><li><p>Verify Node (verifyProof):</p><ul><li>Signature: verifyNode(nodeHash: str, requiredProofNodes: List<!-- -->[str]<!-- -->, MMRroot: str) -&gt; bool</li><li>Description: verifies if the given node hash can be proved based on the list of required proof nodes and the MMR root hash.</li></ul></li></ul></div></div><h6 class="anchor anchorWithStickyNavbar_LWe7" id="defn-beefy-payload">Definition 99. Payload<a href="#defn-beefy-payload" class="hash-link" aria-label="Direct link to Definition 99. Payload" title="Direct link to Definition 99. Payload">​</a></h6><div class="alert alert--definition"><h5></h5><div><p><strong>Payload:</strong> is the Merkle root of the MMR generated where the leaf data contains the following fields for each block:</p><ul><li><em>LeafVersion</em>: a byte indicating the current version number of the Leaf Format. The first 3 bits are for major versions and the last 5 bits are for minor versions. </li><li><em>BeefyNextAuthoritySetInfo</em>: It is a tuple consisting of:<ul><li>ValidatorSetID</li><li>Len (u32): length of the validator set</li><li>Merkle Root of the list of Next Beefy Authority Set (ECDSA public keys). The exact format depends on the implementation.  </li></ul></li><li>Parent Block number and Parent Block Hash.</li><li>Extra Leaf Data: Currently the Merkle root of the list of (ParaID, ParaHeads)</li></ul></div></div><h6 class="anchor anchorWithStickyNavbar_LWe7" id="defn-beefy-Commitment">Definition 100. Signed Commitment<a href="#defn-beefy-Commitment" class="hash-link" aria-label="Direct link to Definition 100. Signed Commitment" title="Direct link to Definition 100. Signed Commitment">​</a></h6><div class="alert alert--definition"><h5></h5><div><p><strong>Signed Commitment:</strong>
 <code>commitment</code> is a tuple of <code>(payload, Block Number, ValidatorSetID)</code>. A <code>signed commitment</code> is a tuple <code>(commitment, signatures)</code>, where <code>signatures</code> is a list of optional signatures of the validator set on the SCALE encoded <code>commitment</code>. Note that the number of signatures in <code>signatures</code> may be less than the length of the Validator Set. </p></div></div><h6 class="anchor anchorWithStickyNavbar_LWe7" id="defn-beefy-witness-data">Definition 101. Witness Data<a href="#defn-beefy-witness-data" class="hash-link" aria-label="Direct link to Definition 101. Witness Data" title="Direct link to Definition 101. Witness Data">​</a></h6><div class="alert alert--definition"><h5></h5><div><p><strong>Signed Commitment Witnesses</strong> contains the commitment and an array indicating which validator of the Polkadot network voted for the payload (but not the signatures themselves). The indicators of which validator voted for the payload are just claims and provide no proof. It also contains the signature of one validator on the commitment, which is used only by the subsampling-based Light Clients. The network message is defined in <a href="/chap-networking#defn-grandpa-beefy-signed-commitment-witness">Definition 58</a> and the relayer saves it on the chain of the remote network.</p></div></div><h6 class="anchor anchorWithStickyNavbar_LWe7" id="defn-beefy-light-client">Definition 102. Light Client<a href="#defn-beefy-light-client" class="hash-link" aria-label="Direct link to Definition 102. Light Client" title="Direct link to Definition 102. Light Client">​</a></h6><div class="alert alert--definition"><h5></h5><div><p>A <strong>light client</strong> is an abstract entity in a remote network such as Ethereum. It can be a node or a smart contract with the intent of requesting finality proofs from the Polkadot network. A light client reads the witness data (<a href="/sect-finality#defn-beefy-witness-data">Definition 101</a> from the chain, then requests the signatures directly from the relayer in order to verify those.</p></div></div><h6 class="anchor anchorWithStickyNavbar_LWe7" id="defn-beefy-relayer">Definition 103. Relayer<a href="#defn-beefy-relayer" class="hash-link" aria-label="Direct link to Definition 103. Relayer" title="Direct link to Definition 103. Relayer">​</a></h6><div class="alert alert--definition"><h5></h5><div><p>A <strong>relayer</strong> (or &quot;prover&quot;) is an abstract entity that takes finality proofs from the Polkadot network and makes those available to the light clients. The relayer attempts to convince the light clients that the finality proofs have been voted for by the Polkadot relay chain validators. The relayer operates off-chain and can for example be a node or a collection of nodes.</p></div></div><h3 class="anchor anchorWithStickyNavbar_LWe7" id="id-voting-on-payloads">6.7.5. Voting on Payloads<a href="#id-voting-on-payloads" class="hash-link" aria-label="Direct link to 6.7.5. Voting on Payloads" title="Direct link to 6.7.5. Voting on Payloads">​</a></h3><p>The Polkadot Host signs the MMR payload (<a href="/sect-finality#defn-beefy-payload">Definition 99</a>) and gossips it as part of a vote (<a href="/chap-networking#defn-msg-beefy-gossip">Definition 56</a>) to its peers on every new finalized block. The Polkadot Host uses ECDSA for signing the payload since Ethereum has better compatibility for it compared to SR25519 or ED25519.</p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="sect-beefy-committing-witnesses">6.7.6. Committing Witnesses<a href="#sect-beefy-committing-witnesses" class="hash-link" aria-label="Direct link to 6.7.6. Committing Witnesses" title="Direct link to 6.7.6. Committing Witnesses">​</a></h3><p>The relayer (<a href="/sect-finality#defn-beefy-relayer">Definition 103</a>) participates in the Polkadot network by collecting the gossiped votes (<a href="/chap-networking#defn-msg-beefy-gossip">Definition 56</a>). Those votes are converted into the witness data structure (<a href="/sect-finality#defn-beefy-witness-data">Definition 101</a>). The relayer saves the data on the chain of the remote network. The occurrence of saving witnesses on remote networks is undefined.</p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="id-requesting-signed-commitments">6.7.7. Requesting Signed Commitments<a href="#id-requesting-signed-commitments" class="hash-link" aria-label="Direct link to 6.7.7. Requesting Signed Commitments" title="Direct link to 6.7.7. Requesting Signed Commitments">​</a></h3><p>A light client (<a href="/sect-finality#defn-beefy-light-client">Definition 102</a>) fetches the Signed Commitment Witness (<a href="/sect-finality#defn-beefy-witness-data">Definition 101</a>) from the chain. Once the light client knows which validators apparently voted for the specified payload, it needs to request the signatures from the relayer to verify whether the claims are actually true. This is achieved by requesting signed commitments (<a href="/chap-networking#defn-grandpa-beefy-signed-commitment">Definition 57</a>).</p><p>How those signed commitments are requested by the light client and delivered by the relayer varies among networks or implementations.</p><h6 class="anchor anchorWithStickyNavbar_LWe7" id="defn-consensus-message-beefy">Definition 104. BEEFY Consensus Message<a href="#defn-consensus-message-beefy" class="hash-link" aria-label="Direct link to Definition 104. BEEFY Consensus Message" title="Direct link to Definition 104. BEEFY Consensus Message">​</a></h6><div class="alert alert--definition"><h5></h5><div><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>CM</mtext><mrow><mi>b</mi><mi>e</mi><mi>e</mi><mi>f</mi><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\text{CM}_{{beefy}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord text"><span class="mord">CM</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em"><span style="top:-2.55em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mord mathnormal mtight">ee</span><span class="mord mathnormal mtight" style="margin-right:0.10764em">f</span><span class="mord mathnormal mtight" style="margin-right:0.03588em">y</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span></span></span></span></span>, the consensus message for BEEFY, is of the following format:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mtext>CM</mtext><mrow><mi>b</mi><mi>e</mi><mi>e</mi><mi>f</mi><mi>y</mi></mrow></msub><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>V</mi><mi>B</mi></msub><mo separator="true">,</mo><msub><mi>V</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>A</mi><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>R</mi></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\text{CM}_{{beefy}}={\left\lbrace\begin{matrix}{1}&amp;{\left({V}_{{B}},{V}_{{i}}\right)}\\{2}&amp;{A}_{{i}}\\{3}&amp;{R}\end{matrix}\right.}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord text"><span class="mord">CM</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em"><span style="top:-2.55em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mord mathnormal mtight">ee</span><span class="mord mathnormal mtight" style="margin-right:0.10764em">f</span><span class="mord mathnormal mtight" style="margin-right:0.03588em">y</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em"></span><span class="mord"><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-2.5em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.492em"><span class="pstrut" style="height:3.15em"></span><span style="height:0.016em;width:0.8889em"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.016em" style="width:0.8889em" viewBox="0 0 888.89 16" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-3.15em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292em"><span class="pstrut" style="height:3.15em"></span><span style="height:0.016em;width:0.8889em"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.016em" style="width:0.8889em" viewBox="0 0 888.89 16" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-4.3em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord">1</span></span></span></span><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord">2</span></span></span></span><span style="top:-1.81em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em">V</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em"><span style="top:-2.55em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em">B</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em">V</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathnormal">A</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.81em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em">R</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></div><p><strong>where</strong></p><table><thead><tr><th></th><th></th></tr></thead><tbody><tr><td>1</td><td>implies that the remote <strong>authorities have changed</strong>. <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">{V}_{{B}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em">V</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em"><span style="top:-2.55em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em">B</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> is the array of the new BEEFY authorities’s public keys and <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">{V}_{{i}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em">V</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> is the identifier of the remote validator set.</td></tr><tr><td>2</td><td>implies <strong>on disabled</strong>: an index to the individual authority in <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">{V}_{{B}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em">V</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em"><span style="top:-2.55em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em">B</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> that should be immediately disabled until the next authority change.</td></tr><tr><td>3</td><td>implies <strong>MMR root</strong>: a 32-byte array containing the MMR root payload.</td></tr></tbody></table></div></div><h3 class="anchor anchorWithStickyNavbar_LWe7" id="id-consensus-mechanism-beefy-1">6.7.8. Consensus Mechanism<a href="#id-consensus-mechanism-beefy-1" class="hash-link" aria-label="Direct link to 6.7.8. Consensus Mechanism" title="Direct link to 6.7.8. Consensus Mechanism">​</a></h3><p>Role of various Actors in BEEFY:</p><ul><li><p>Validators are expected to additionally:</p><ol><li>Produce &amp; broadcast vote for the current round.</li></ol></li><li><p>Regular nodes perform the following tasks:</p><ol><li>Receive &amp; validate votes for the current round and broadcast them to their peers.</li><li>Receive &amp; validate BEEFY Justifications and broadcast them to their peers.</li><li>Return BEEFY Justifications for <strong>Mandatory Blocks</strong> on demand.</li><li>Optionally return BEEFY Justifications for non-mandatory blocks on demand.</li></ol></li></ul><p>A <strong>round</strong> is an attempt by BEEFY validators to produce a BEEFY Justification. <strong>Round number</strong> is simply defined as a block number the validators are voting for, or to be more precise, the Commitment for that block number. Round ends when the next round is started, which may happen when one of the events occur:</p><ol><li>Either the node collects <code>2/3rd + 1</code> valid votes for that round.</li><li>Or the node receives a BEEFY Justification for a block greater than the current best BEEFY block.</li></ol><p>In both cases the node proceeds to determine the new round number using &quot;Round Selection&quot;
 procedure.</p><p>Both kinds of actors are expected to fully participate in the protocol ONLY IF they believe they are up-to-date with the rest of the network, i.e. they are fully synced. Before this happens, the node should continue processing imported BEEFY Justifications and votes without actively voting themselves.</p><p><strong>Round Selection</strong></p><p>Every node (both regular nodes and validators) determines locally what it believes
 the current round number is. The choice is based on their knowledge of:</p><ol><li>Best GRANDPA finalized block number (<code>best_grandpa</code>).</li><li>Best BEEFY finalized block number (<code>best_beefy</code>).</li><li>Starting block of the current session (<code>session_start</code>).</li></ol><p><strong>Session</strong> means a period of time (or rather a number of blocks) where the validator set (keys) does not change. Session are synonymous to epochs (<a href="/sect-block-production#defn-epoch-slot">Definition 59</a>). Since the BEEFY authority set is the same as the GRANDPA authority set for any GRANDPA finalized block, the session boundaries for BEEFY are exactly the same as the ones for GRANDPA.</p><p>We define two kinds of blocks from the perspective of the BEEFY protocol:</p><ol><li><strong>Mandatory Blocks</strong></li><li><strong>Non-mandatory Blocks</strong></li></ol><p>Mandatory blocks are the ones that MUST have BEEFY justification. That means that the validators
@@ -212,7 +212,7 @@
 session.</p><p>While it is useful to finalize non-mandatory blocks frequently, in good networking conditions BEEFY may end up finalizing each and every block GRANDPA finalized block. Practically, with short block times, it&#x27;s going to be rare and might be excessive, so
 it&#x27;s suggested for implementations to introduce a <code>min_delta</code> parameter which will limit the frequency with which new rounds are started. The affected component of the formula would be:
 <code>best_beefy + MAX(min_delta, NEXT_POWER_OF_TWO(...))</code>, so we start a new round only if the
-the power-of-two component is greater than the min delta. Note that if <code>round_number &gt; best_grandpa</code> the validators are not expected to start any round.</p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="id-light-client-beefy-1">6.7.9. BEEFY Light Client<a href="#id-light-client-beefy-1" class="hash-link" aria-label="Direct link to 6.7.9. BEEFY Light Client" title="Direct link to 6.7.9. BEEFY Light Client">​</a></h3><p>A light client following BEEFY could request <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mi mathvariant="normal">/</mi><mn>3</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N/3 +1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10903em">N</span><span class="mord">/3</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span></span> signatures to be checked, where <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">N</span></span></span></span></span> is the number of validators on Polkadot. Assuming a maximum of <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mi mathvariant="normal">/</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">N/3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10903em">N</span><span class="mord">/3</span></span></span></span></span> malicious validators, the light client can be certain of the payloads finality if all the signatures it requested are valid. </p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="id-subsampling-light-client-beefy-1">6.7.10. Subsampling Light Client<a href="#id-subsampling-light-client-beefy-1" class="hash-link" aria-label="Direct link to 6.7.10. Subsampling Light Client" title="Direct link to 6.7.10. Subsampling Light Client">​</a></h3><p>It is an interactive protocol between the light-client (verifier) and the relayer (prover) to convince the Light Client with a high probability that the payload sent by the prover is signed by honest Polkadot validators. The protocol prioritizes efficiency and tries to minimize the number (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo><mo>&lt;</mo><mi>N</mi><mi mathvariant="normal">/</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">&lt;&lt; N/3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em"></span><span class="mrel">&lt;&lt;</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10903em">N</span><span class="mord">/3</span></span></span></span></span>) of signature checks (computationally expensive operations) on the light client side. </p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="id-APK-light-client-beefy-1">6.7.11. APK Proof based Light Clients<a href="#id-APK-light-client-beefy-1" class="hash-link" aria-label="Direct link to 6.7.11. APK Proof based Light Clients" title="Direct link to 6.7.11. APK Proof based Light Clients">​</a></h3><p>TODO: Section on using Aggregatable Signatures for efficient verification on light clients.</p></div></div></div><footer class="theme-doc-footer docusaurus-mt-lg"><div class="theme-doc-footer-edit-meta-row row"><div class="col"><a href="https://github.com/w3f/polkadot-spec/blob/main/docs/sect-finality.md" target="_blank" rel="noopener noreferrer" class="theme-edit-this-page"><svg fill="currentColor" height="20" width="20" viewBox="0 0 40 40" class="iconEdit_Z9Sw" aria-hidden="true"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="col lastUpdated_vwxv"></div></div></footer></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Docs pages"><a class="pagination-nav__link pagination-nav__link--prev" href="/sect-block-production"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">5. Block Production</div></a><a class="pagination-nav__link pagination-nav__link--next" href="/sect-lightclient"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">7. Light Clients</div></a></nav></div></div><div class="col col--3"><div class="tableOfContents_bqdL thin-scrollbar theme-doc-toc-desktop"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#id-introduction-4" class="table-of-contents__link toc-highlight">6.1. Introduction</a></li><li><a href="#id-initiating-the-grandpa-state" class="table-of-contents__link toc-highlight">6.2. Initiating the GRANDPA State</a><ul><li><a href="#id-voter-set-changes" class="table-of-contents__link toc-highlight">6.2.1. Voter Set Changes</a></li></ul></li><li><a href="#id-rejoining-the-same-voter-set" class="table-of-contents__link toc-highlight">6.3. Rejoining the Same Voter Set</a></li><li><a href="#id-voting-process-in-round-r" class="table-of-contents__link toc-highlight">6.4. Voting Process in Round {r}</a></li><li><a href="#sect-finality-forced-changes" class="table-of-contents__link toc-highlight">6.5. Forced Authority Set Changes</a></li><li><a href="#sect-block-finalization" class="table-of-contents__link toc-highlight">6.6. Block Finalization</a><ul><li><a href="#sect-grandpa-catchup" class="table-of-contents__link toc-highlight">6.6.1. Catching up</a></li></ul></li><li><a href="#sect-grandpa-beefy" class="table-of-contents__link toc-highlight">6.7. Bridge design (BEEFY)</a><ul><li><a href="#id-motivation-beefy-1" class="table-of-contents__link toc-highlight">6.7.1. Motivation</a></li><li><a href="#id-overview-beefy-1" class="table-of-contents__link toc-highlight">6.7.2. Protocol Overview</a></li><li><a href="#id-preliminaries-2" class="table-of-contents__link toc-highlight">6.7.3. Preliminaries</a></li><li><a href="#id-beefy-merkle-mountain-ranges" class="table-of-contents__link toc-highlight">6.7.4. Merkle Mountain Ranges</a></li></ul></li></ul></div></div></div></div></main></div></div></div><footer class="footer footer--dark"><div class="container container-fluid"><div class="row footer__links"><div class="col footer__col"><div class="footer__title">More Info</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://github.com/w3f/polkadot-spec" target="_blank" rel="noopener noreferrer" class="footer__link-item">GitHub<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a href="https://web3.foundation/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Website<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a class="footer__link-item" href="/Support Docs/privacy_policy">Privacy Policy</a></li><li class="footer__item"><a href="https://web3.foundation/legal-disclosures/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Legal Disclosures</a></li></ul></div><div class="col footer__col"><div class="footer__title">Connect</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://twitter.com/Web3foundation" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">© 2023 Web3 Foundation</div></div></div></footer></div>
+the power-of-two component is greater than the min delta. Note that if <code>round_number &gt; best_grandpa</code> the validators are not expected to start any round.</p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="id-light-client-beefy-1">6.7.9. BEEFY Light Client<a href="#id-light-client-beefy-1" class="hash-link" aria-label="Direct link to 6.7.9. BEEFY Light Client" title="Direct link to 6.7.9. BEEFY Light Client">​</a></h3><p>A light client following BEEFY could request <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mi mathvariant="normal">/</mi><mn>3</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N/3 +1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10903em">N</span><span class="mord">/3</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span></span> signatures to be checked, where <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">N</span></span></span></span></span> is the number of validators on Polkadot. Assuming a maximum of <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mi mathvariant="normal">/</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">N/3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10903em">N</span><span class="mord">/3</span></span></span></span></span> malicious validators, the light client can be certain of the payloads finality if all the signatures it requested are valid. </p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="id-subsampling-light-client-beefy-1">6.7.10. Subsampling Light Client<a href="#id-subsampling-light-client-beefy-1" class="hash-link" aria-label="Direct link to 6.7.10. Subsampling Light Client" title="Direct link to 6.7.10. Subsampling Light Client">​</a></h3><p>It is an interactive protocol between the light-client (verifier) and the relayer (prover) to convince the Light Client with a high probability that the payload sent by the prover is signed by honest Polkadot validators. The protocol prioritizes efficiency and tries to minimize the number (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo><mo>&lt;</mo><mi>N</mi><mi mathvariant="normal">/</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">&lt;&lt; N/3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em"></span><span class="mrel">&lt;&lt;</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10903em">N</span><span class="mord">/3</span></span></span></span></span>) of signature checks (computationally expensive operations) on the light client side. </p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="id-APK-light-client-beefy-1">6.7.11. APK Proof based Light Clients<a href="#id-APK-light-client-beefy-1" class="hash-link" aria-label="Direct link to 6.7.11. APK Proof based Light Clients" title="Direct link to 6.7.11. APK Proof based Light Clients">​</a></h3><p>TODO: Section on using Aggregatable Signatures for efficient verification on light clients.</p></div><footer class="theme-doc-footer docusaurus-mt-lg"><div class="theme-doc-footer-edit-meta-row row"><div class="col"><a href="https://github.com/w3f/polkadot-spec/blob/main/docs/sect-finality.md" target="_blank" rel="noopener noreferrer" class="theme-edit-this-page"><svg fill="currentColor" height="20" width="20" viewBox="0 0 40 40" class="iconEdit_Z9Sw" aria-hidden="true"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="col lastUpdated_vwxv"></div></div></footer></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Docs pages"><a class="pagination-nav__link pagination-nav__link--prev" href="/sect-block-production"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">5. Block Production</div></a><a class="pagination-nav__link pagination-nav__link--next" href="/sect-lightclient"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">7. Light Clients</div></a></nav></div></div><div class="col col--3"><div class="tableOfContents_bqdL thin-scrollbar theme-doc-toc-desktop"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#id-introduction-4" class="table-of-contents__link toc-highlight">6.1. Introduction</a></li><li><a href="#id-initiating-the-grandpa-state" class="table-of-contents__link toc-highlight">6.2. Initiating the GRANDPA State</a><ul><li><a href="#id-voter-set-changes" class="table-of-contents__link toc-highlight">6.2.1. Voter Set Changes</a></li></ul></li><li><a href="#id-rejoining-the-same-voter-set" class="table-of-contents__link toc-highlight">6.3. Rejoining the Same Voter Set</a></li><li><a href="#id-voting-process-in-round-r" class="table-of-contents__link toc-highlight">6.4. Voting Process in Round {r}</a></li><li><a href="#sect-finality-forced-changes" class="table-of-contents__link toc-highlight">6.5. Forced Authority Set Changes</a></li><li><a href="#sect-block-finalization" class="table-of-contents__link toc-highlight">6.6. Block Finalization</a><ul><li><a href="#sect-grandpa-catchup" class="table-of-contents__link toc-highlight">6.6.1. Catching up</a></li></ul></li><li><a href="#sect-grandpa-beefy" class="table-of-contents__link toc-highlight">6.7. Bridge design (BEEFY)</a><ul><li><a href="#id-motivation-beefy-1" class="table-of-contents__link toc-highlight">6.7.1. Motivation</a></li><li><a href="#id-overview-beefy-1" class="table-of-contents__link toc-highlight">6.7.2. Protocol Overview</a></li><li><a href="#id-preliminaries-2" class="table-of-contents__link toc-highlight">6.7.3. Preliminaries</a></li><li><a href="#id-beefy-merkle-mountain-ranges" class="table-of-contents__link toc-highlight">6.7.4. Merkle Mountain Ranges</a></li><li><a href="#id-voting-on-payloads" class="table-of-contents__link toc-highlight">6.7.5. Voting on Payloads</a></li><li><a href="#sect-beefy-committing-witnesses" class="table-of-contents__link toc-highlight">6.7.6. Committing Witnesses</a></li><li><a href="#id-requesting-signed-commitments" class="table-of-contents__link toc-highlight">6.7.7. Requesting Signed Commitments</a></li><li><a href="#id-consensus-mechanism-beefy-1" class="table-of-contents__link toc-highlight">6.7.8. Consensus Mechanism</a></li><li><a href="#id-light-client-beefy-1" class="table-of-contents__link toc-highlight">6.7.9. BEEFY Light Client</a></li><li><a href="#id-subsampling-light-client-beefy-1" class="table-of-contents__link toc-highlight">6.7.10. Subsampling Light Client</a></li><li><a href="#id-APK-light-client-beefy-1" class="table-of-contents__link toc-highlight">6.7.11. APK Proof based Light Clients</a></li></ul></li></ul></div></div></div></div></main></div></div></div><footer class="footer footer--dark"><div class="container container-fluid"><div class="row footer__links"><div class="col footer__col"><div class="footer__title">More Info</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://github.com/w3f/polkadot-spec" target="_blank" rel="noopener noreferrer" class="footer__link-item">GitHub<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a href="https://web3.foundation/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Website<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a class="footer__link-item" href="/Support Docs/privacy_policy">Privacy Policy</a></li><li class="footer__item"><a href="https://web3.foundation/legal-disclosures/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Legal Disclosures</a></li></ul></div><div class="col footer__col"><div class="footer__title">Connect</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://twitter.com/Web3foundation" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">© 2023 Web3 Foundation</div></div></div></footer></div>
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