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

$\begin{equation*} \mbox{Tran} = \bigwedge (p+1) \in Q_w \leftrightarrow \bigvee \{ (p \in Q_u \land \bigwedge_{i=1}^n (p \in v_i)^{a_{ij}}) \mid (q_u,(a_1,\ldots,a_n),q_w) \in \Delta \} \end{equation*}$

Initially, the 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

$\begin{equation*} \exists k. (F \land \forall p. k < p \rightarrow \bigwedge_i p \notin v_i) \end{equation*}$