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sav08:using_automata_to_decide_ws1s [2010/10/15 15:07] vkuncak |
sav08:using_automata_to_decide_ws1s [2012/05/15 12:46] vkuncak |
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**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 ===== |