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sav08:using_automata_to_decide_ws1s [2012/05/15 12:46] vkuncak |
sav08:using_automata_to_decide_ws1s [2012/05/15 15:38] vkuncak |
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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 $w \models F$. | ||
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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 1:** Compute automaton for formula $\exists X. \lnot (X \subseteq Y)$. | + | **Example 1:** Compute automaton for formula $\exists X. \lnot (X \subseteq Y)$. MONA syntax: |
+ | var2 Y; | ||
+ | ex2 X: ~(X sub Y); | ||
+ | Command to produce dot file: | ||
+ | mona -gw $1 | dot -Tps > output.ps | ||
+ | |||
+ | **Example 2:** Compute automaton for formula $\exists Y. (X < Y)$ where $<$ is interpreted treating $X,Y$ as digits of natural numbers. Also compute the automaton for the formula $\exists X. (X < Y)$. | ||
- | **Example 2:** Compute automaton for formula $\exists Y. (X < Y)$ where $<$ is interpreted treating $X,Y$ as digits of natural numbers. | + | Define less-than relation in MONA and encode this example. |
===== References ===== | ===== References ===== |