Lab for Automated Reasoning and Analysis LARA

Projection of Automata

Consider alphabet $\Sigma$ is $\{0,1\}^V$.

Given a language $L \subseteq \Sigma^*$, define

\begin{equation*}
   proj(x,L) = 
\begin{array}[t]{@{}l}
\{ a_1 \ldots a_n \mid \exists b_1,\ldots,b_n \in \{0,1\}. a_1[x:=b_1] \ldots a_n[x:=b_n] \in L \} \\
\{ a_1[x:=b_1] \ldots a_n[x:=b_n] \mid b_1,\ldots,b_n \in \{0,1\}, \ a_1 \ldots a_n \in L \}
\end{array}
\end{equation*}

Given a finite automaton $A = (\Sigma,Q,\Delta,q_0,F)$, define projection $proj(x,A)$ as $(\Sigma,Q,\Delta',q_0,F)$ where

\begin{equation*}
    \Delta' = \{(q,a,q') \mid \exists b \in \{0,1\}. (q,a[x:=b],q') \in \Delta \}
\end{equation*}

Lemma: $L(proj(x,A)) = proj(x,L(A))$

Given a regular expression $r$, define $proj(x,r)$ by

\begin{equation*}
\begin{array}{l}
proj(x,[F]) = [qe(\exists x.F)] \\
proj(x,r_1 r_2) = proj(x,r_1) proj(x,r_2) \\
proj(x,r^*) = proj(x,r)^* \\
proj(x,r_1 + r_2) = proj(x,r_1) + proj(x,r_2)
\end{array}
\end{equation*}

Lemma: $L(proj(x,r)) = proj(x,L(r))$

 
sav08/projection_of_automata.txt · Last modified: 2015/04/21 17:30 (external edit)