lec1.tex

\section{Deterministic Finite Automaton (DFA)}
% ==================================================

A machine consists of different states drawn in circles with names. Often a state drawn as
a double circle is an ``acceptive state,'' and a plain circle indicates a ``rejective
state.'' A machine receives a string consisted of `1's and `0's as input and the states
change as the machine reads through input digits. An arrow is used to indicate which state
is to start with. See \autoref{exa:a_DFA} for detailed information.

\subsection{Expressions of DFA's}
% --------------------------------------------------

\begin{example}[A DFA]
    \label{exa:a_DFA}

    Let's first look at the DFA below which starts at state $A$.
    \centgraph{mp/dfa-0.pdf}

    If the string ``010110'' is input to the machine, will it result in true or false?
    Will the state be acceptive or rejective?
    \[
        \xrightarrow{010110}
        \fbox{M}
        \xrightarrow{1/0 \text{ (True / False, Accept / Reject)}}
    \]

    There are two arrows leaving state $A$: one with a label reading `1' which points to
    state $B$ and one reading `0' which goes back to state $A$ itself. That means, if an
    input digit reads `1,' the state changes to $B$, and if `0' the state stays in $A$.

    Now step through the procedure:
    \begin{compactenum}
    \item
        The machine starts off at state $A$ with input `0,' which, as explained above,
        changes the state to $A$ itself.
    \item
        Next, the second digit `1' is read so the state is changed to $B$.
    \item
        The next digit `0' makes state $B$ to switch to state $C$.
    \item
        Then state $C$ reads `1' so no state change occurs.
    \item
        The next digit is `1' again so the state remains still on $C$.
    \item
        Last, the digit `0' switches the state from $C$ to $B$.
    \end{compactenum}
    Thus the input string ``010110'' changes the machine to state $B$, which is an
    acceptive state.

\end{example}

\begin{definition}[DFA]
    \label{def:DFA}

    A DFA is a $5$-tuple
    \[ \DFA \]
    where
    \begin{compactdesc}
    \item[$Q$]      is a finite set,    for states
    \item[$\Sigma$] is a finite set,    for input alphabet
    \item[$s$]      $\in Q$,            for start states
    \item[$F$]      $\subseteq Q$,      for accepting states
    \item[$\delta$]
        $Q \times \Sigma \mapsto Q$,
        a function that specifies the transition between states
    \end{compactdesc}
\end{definition}

\begin{example}[Denoting machine in \autoref{exa:a_DFA}]

    According to definition \autoref{def:DFA}, 
    the machine in \autoref{exa:a_DFA} can be denoted by 
    \[ \DFA \]
    where
    \begin{compactitem}
    \item $Q        = \set{ A,B,C }$
    \item $\Sigma   = \set{ 0,1 }$
    \item $s        = \set{ A }$
    \item $F        = \set{ B,C }$
    \end{compactitem}
    And function $\delta$ can be described by the table below.
    \begin{center} \begin{tabular}{Mc Mc Mc}
        \hline
        \delta & 0 & 1  \\
        \hline
        A      & A & B  \\
        B      & C & A  \\
        C      & B & C  \\
        \hline
    \end{tabular} \end{center}

\end{example}

\begin{definition}[$f_M$]
    For any DFA $\DFA$,
    let 
    \[
        f_M : \Sigma^* \mapsto \set{ \true, \false }
    \]
    where $\Sigma^*$ is a set of strings over $\Sigma$.

    \[
        f_M(w)
        = \begin{cases}
            \true,   &\delta^*(s,w) \in F  \\
            \false,  &\text{else}
        \end{cases}
    \]
\end{definition}

\begin{definition}[$\delta^*$]
    \[
        \delta^* : Q \times \Sigma^* \mapsto Q
    \]
    which is an inductive function defined as
    \[
        \begin{cases}
            \delta^*(q, \varepsilon) = q \\
            \delta^*(q, aw) = \delta^*\( \delta(q,a), w \)
        \end{cases}
    \]
    where $varepsilon$ is an empty string and $q \in Q, a \in \Sigma, w \in \Sigma^*)$.
\end{definition}