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expressing_finite_automata_in_msol_over_strings [2009/04/29 11:08] vkuncak |
expressing_finite_automata_in_msol_over_strings [2010/05/21 02:31] vkuncak |
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| 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 $F$. | 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 $F$. | ||
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| + | 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 | 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 16: | Line 18: | ||
| \end{equation*} | \end{equation*} | ||
| - | To express the property that an automaton accepts a string whose length is given by free variable $k$ we then use formula | + | 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*} | \begin{equation*} | ||
| \exists Q_0,\ldots, Q_m.\ \mbox{Init} \land (\forall p. p \leq k \rightarrow \mbox{Trans}) \land \mbox{Final} | \exists Q_0,\ldots, Q_m.\ \mbox{Init} \land (\forall p. p \leq k \rightarrow \mbox{Trans}) \land \mbox{Final} | ||
| \end{equation*} | \end{equation*} | ||
| + | Then the word is accepted if the above formula holds for the entire input | ||
| + | \[ | ||
| + | \exists k. (F \land \forall p. k < p \rightarrow \bidwedge_i p \notin v_i) | ||
| + | \] | ||