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--- expressing_finite_automata_in_msol_over_strings [2007/05/06 21:25]
vkuncak
+++ expressing_finite_automata_in_msol_over_strings [2015/04/21 17:32] (current)
@@ Line 1: / Line 1: @@
-====== Expressing finite automaton in MSOL over strings ======
+====== Expressing finite automaton in WS1S ======
 Consider a finite automaton with alphabet $\Sigma=\{0,1\}^n$, states $Q = \{q_0,q_1,\ldots,q_m\}$, transition relation $\Delta$, initial state $q_0$ and final states $F$.
+Let the sets $v_i$ denote the encoding of input string of the automaton.
 There exists an execution of an automaton iff there exist the intermediate states given by transition relation $\Delta$.  We describe intermediate states using sets $Q_0,\ldots,Q_m$, one for each state of the in $Q$.  The state $Q_i$ is the set of positions in the execution at which the automaton is in state $q_i$.  We encode the transition relation $\Delta$ using the formula
@@ Line 7: / Line 9: @@
   \mbox{Tran} = \bigwedge (p+1) \in Q_w \leftrightarrow \bigvee \{ (p \in Q_u \land \bigwedge_{i=1}^n (p \in v_i)^{a_{ij}}) \mid (q_u,(a_1,\ldots,a_n),q_w) \in \Delta \}
 \end{equation*}
-Initially, the autonaton is in the initial state:
+Initially, the automaton is in the initial state:
 \begin{equation*}
   \mbox{Init} = 0 \in Q_0 \land \bigwedge_{u=1}^m (0 \notin Q_u)
+\end{equation*}
+and at some final position $k$ it should be in final state:
+\begin{equation*}
+  \mbox{Final} = \bigvee_{u=0}^m (k \in Q_u)
+\end{equation*}
+To express the property that an automaton accepts a string whose length is given by free variable $k$ we then use formula $F$:
+\begin{equation*}
+  \exists Q_0,\ldots, Q_m.\ \mbox{Init} \land (\forall p. p \leq k \rightarrow \mbox{Trans}) \land \mbox{Final}
+\end{equation*}
+Then the word is accepted if the above formula holds for the entire input
+\begin{equation*}
+   \exists k. (F \land \forall p. k < p \rightarrow \bigwedge_i p \notin v_i)
 \end{equation*}