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sav08:qbf_and_quantifier_elimination [2008/03/10 11:16] vkuncak |
sav08:qbf_and_quantifier_elimination [2008/03/11 16:14] vkuncak |
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| ====== QBF and Quantifier Elimination ====== | ====== QBF and Quantifier Elimination ====== | ||
| - | === Quantified Propositional Formula Syntax === | + | ===== Quantified Propositional Formula Syntax ===== |
| We extend the definition of formulas with quantifiers $\forall p.F$ and $\exists p.F$ where $p \in V$: | We extend the definition of formulas with quantifiers $\forall p.F$ and $\exists p.F$ where $p \in V$: | ||
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| - | === Eliminating quantifiers by expansion === | + | ===== Eliminating quantifiers by expansion ===== |
| We can apply the following rules to eliminate propositional quantifiers: | We can apply the following rules to eliminate propositional quantifiers: | ||
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| Notion of quantifier elimination applies to other logic as well. | Notion of quantifier elimination applies to other logic as well. | ||
| - | Definition: A logic has //quantifier elimination// if for every formula in the logic, there exists an equivalent formula without quantifiers. | + | **Definition:** A logic has //quantifier elimination// if for every formula in the logic, there exists an equivalent formula without quantifiers. |
| - | Definition: A quantifier elimination algorithm is an algorithm that takes a formula in a logic and converts it into an equivalent formula without quantifiers. | + | **Definition:** A quantifier elimination algorithm is an algorithm that takes a formula in a logic and converts it into an equivalent formula without quantifiers. |
| - | Note: Formula $F$ is valid iff $\forall p_1,\ldots,p_n. F$ is true. | + | **Observation:** Formula $F$ is valid iff $\forall p_1,\ldots,p_n. F$ is true. |
| - | Note: Formula $F$ is satisfiable iff $\exists p_1,\ldots,p_n. F$ is true. | + | **Observation:** Formula $F$ is satisfiable iff $\exists p_1,\ldots,p_n. F$ is true. |
| In general QBF formulas can have alternations between $\forall$ and $\exists$ quantifiers: | In general QBF formulas can have alternations between $\forall$ and $\exists$ quantifiers: | ||
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| - | === Quantified Propositional Formula Semantics === | + | ===== Quantified Propositional Formula Semantics ===== |
| - | We can similarly provide semantic function definition for propositional logic. | + | We can similarly provide recursive semantic function definition for propositional logic. |
| - | === Notes on Computational Complexity === | + | ===== Summary of Computational Complexity of Problems ===== |
| * checking satisfiability of propositional formula $F$ is NP-complete | * checking satisfiability of propositional formula $F$ is NP-complete | ||