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--- sav08:qbf_and_quantifier_elimination [2008/03/05 21:34]
vkuncak created
+++ sav08:qbf_and_quantifier_elimination [2008/03/10 11:16]
vkuncak
@@ Line 1: / Line 1: @@
 ====== QBF and Quantifier Elimination ======
-Quantified Propositional Formulas
+=== Quantified Propositional Formula Syntax ===
-Satisfiability is:
+We extend the definition of formulas with quantifiers $\forall p.F$ and $\exists p.F$ where $p \in V$:
+\[\begin{array}{rcl}
+   F  & ::= & V \mid {\it false} \mid {\it true} \mid (F \land F) \mid (F \lor F) \mid (\lnot F) \mid (F \rightarrow F) \mid (F \leftrightarrow F) \\
+      & \mid & \forall V. F\ \mid\ \exists V. F
+\end{array}
+\]
-Validity is:
+=== Eliminating quantifiers by expansion ===
-[[Quantifier Elimination]]
+We can apply the following rules to eliminate propositional quantifiers:
+\[
+    \exists p. F \ \leadsto subst(\{p \mapsto {\it false}\},F) \lor subst(\{p \mapsto {\it true}\},F)
+\]
+\[
+    \forall p. F \ \leadsto subst(\{p \mapsto {\it false}\},F) \land subst(\{p \mapsto {\it true}\},F)
+\]
+Note that the expansion can produce exponentially larger formula.
-Notion of PSPACE-completeness.
+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 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.
+Note: 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:
+\[
+   \forall p. \exists q. ((p \rightarrow q) \land (q \rightarrow r))
+\]
+=== Quantified Propositional Formula Semantics ===
+We can similarly provide semantic function definition for propositional logic.
+=== Notes on Computational Complexity ===
+  * checking satisfiability of propositional formula $F$ is NP-complete
+  * checking validity of propositional formula $F$ is coNP-complete
+  * checking truth value of arbitrary QBF formula is PSPACE-complete