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--- expressing_finite_automata_in_msol_over_strings [2007/05/06 21:25]
vkuncak
+++ expressing_finite_automata_in_msol_over_strings [2009/04/29 11:01]
vkuncak
@@ Line 7: / Line 7: @@
   \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
+\begin{equation*}
+  \exists Q_0,\ldots, Q_m.\ \mbox{Init} \land (\forall p. p \leq k \rightarrow \mbox{Trans}) \land \mbox{Final}
 \end{equation*}
+Note: we need to take care of the end of the string as well, by introducing the set of all positions in the input string, often denoted \$.