lec2.tex

\subsection{Configurations of DFAs}
% --------------------------------------------------

\begin{definition}[Configurations]
    \[
        \Conf = Q \times \Sigma^*
    \]
\end{definition}

\begin{definition}[Initial Configurations]
    The initial configuration of a machine $I_M(w) \in \Conf$
    \[
        I_M(w) = ( s,w )
    \]
\end{definition}

\begin{definition}[Final Configurations]
    The final configuration of a machine $H_M(w) \subseteq \Conf$
    \[
        H_M(w) = \set{ (q,u) \mid q \in Q, u = \varepsilon }
    \]
\end{definition}

\begin{definition}[Output]
    The output of a machine is a function that returns either ``$\true$'' or ``$\false$.''
    \[
        O_M \colon H_M \mapsto \set{ \true,\false }
    \]
    defined as
    \[
        O_M(q,\varepsilon)
        = \begin{cases}
            \true,   & q \in F  \\
            \false,  & \text{else}
        \end{cases}
    \]
\end{definition}

\begin{definition}[Transition Relations] %% ?
    $R_M \subseteq \Conf$

    \[
        R_M = \rightarrow_M
        = \{
            (q, aw) \rightarrow \( \delta(q,a),w\)
            \mid q \in Q, a \in \Sigma, w \in \Sigma^*
        \}
    \]
\end{definition}

\begin{example}[Configurations of machine in \autoref{exa:a_DFA}]
    With input ``10010'' write in mathematical language the configurations of machine in
    \autoref{exa:a_DFA}:
    \begin{align*}
        I_M (10010)
        =               (A, 10010)
        \rightarrow     (B, 0010)
        \rightarrow     (C, 010)
        \rightarrow     (B, 10)
        \rightarrow     (A, 0)
        \rightarrow     (A, \varepsilon)
        \in             H_M
    \end{align*}
    And thus the output
    \[
        O_F (A,\varepsilon) = \false
    \].

    The machine in fact will only accept integers that are \emph{not} multiples of 3.
\end{example}

\begin{definition}[]
    \[
        f'_n (w) = O_F (C_n)
    \]
\end{definition}

% \[
%     L(M) = \set{ w \in \Sigma^* \mid f_M(w) = \true }
%     L(M_1)  \Sigma^*
% \]