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sav08:using_automata_to_decide_ws1s [2010/05/21 01:45] vkuncak |
sav08:using_automata_to_decide_ws1s [2012/05/15 12:48] vkuncak |
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| ====== Using Automata to Decide WS1S ====== | ====== Using Automata to Decide WS1S ====== | ||
| - | Consider a formula $F$ of [[weak_monadic_logic_of_one_successor|WS1S]] without quantifiers. Let $V$ be a finite set of all variables in $F$. We construct an automaton $A(F)$ in the finite alphabet | + | Consider a formula $F$ of [[weak_monadic_logic_of_one_successor|WS1S]]. Let $V$ be a finite set of all variables in $F$. We construct an automaton $A(F)$ in the finite alphabet |
| \[ | \[ | ||
| \Sigma = \Sigma_1^V | \Sigma = \Sigma_1^V | ||
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| \] | \] | ||
| - | Instead of $e(F)(D,\alpha(w))={\it true}$ we write for short $w \models F$. | + | Instead of $e(F)(D,\alpha(w))={\it true}$ we write for short $w \models F$. So, we design automata so that: |
| + | \[ | ||
| + | w \in L(A(F)) \ \ \ \iff \ \ \ w \models F | ||
| + | \] | ||
| The following lemma follows from the definition of semantic evaluation function 'e' and the shorthand $w \models F$. | The following lemma follows from the definition of semantic evaluation function 'e' and the shorthand $w \models F$. | ||
| **Lemma**: Let $F,F_i$ denote formulas, $w \in \Sigma^*$. Then | **Lemma**: Let $F,F_i$ denote formulas, $w \in \Sigma^*$. Then | ||
| - | * $w \models (F_1 \lor F_2)$ iff $w \models F_1$ and $w \models F_2$ | + | * $w \models (F_1 \lor F_2)$ iff $w \models F_1$ or $w \models F_2$ |
| * $w \models \lnot F$ iff $\lnot (w \models F)$ | * $w \models \lnot F$ iff $\lnot (w \models F)$ | ||
| * $w \models \exists x.F$ iff $\exists b. patch(w,x,b) \models F$ | * $w \models \exists x.F$ iff $\exists b. patch(w,x,b) \models F$ | ||
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| * if $q$ is a final state and $zero_x \in \Sigma$ is such that $zero_x(v)=0$ for all $x \neq v$, and if $\delta(q',zero_x)=q$, then set $q'$ also to be final | * if $q$ is a final state and $zero_x \in \Sigma$ is such that $zero_x(v)=0$ for all $x \neq v$, and if $\delta(q',zero_x)=q$, then set $q'$ also to be final | ||
| - | **Example:** Compute automaton for formula $\exists X. \lnot (X \subseteq Y)$. | + | **Example 1:** Compute automaton for formula $\exists X. \lnot (X \subseteq Y)$. |
| + | |||
| + | **Example 2:** Compute automaton for formula $\exists Y. (X < Y)$ where $<$ is interpreted treating $X,Y$ as digits of natural numbers. | ||
| ===== References ===== | ===== References ===== | ||