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--- sav08:qbf_and_quantifier_elimination [2008/03/10 11:17]
vkuncak
+++ sav08:qbf_and_quantifier_elimination [2015/04/21 17:30] (current)
@@ Line 1: / Line 1: @@
 ====== 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 $p \in V$:
-\[\begin{array}{rcl}
+\begin{equation*}\begin{array}{rcl}
    F  & ::= & V \mid {\it false} \mid {\it true} \mid (F \land F) \mid (F \lor F) \mid (\lnot F) \mid (F \rightarrow F) \mid (F \leftrightarrow F) \\
       & \mid & \forall V. F\ \mid\ \exists V. F
 \end{array}
-\]
+\end{equation*}
-=== Eliminating quantifiers by expansion ===
+===== Eliminating quantifiers by expansion =====
 We can apply the following rules to eliminate propositional quantifiers:
-\[
+\begin{equation*}
     \exists p. F \ \leadsto subst(\{p \mapsto {\it false}\},F) \lor subst(\{p \mapsto {\it true}\},F)
-\]
+\end{equation*}
-\[
+\begin{equation*}
     \forall p. F \ \leadsto subst(\{p \mapsto {\it false}\},F) \land subst(\{p \mapsto {\it true}\},F)
-\]
+\end{equation*}
 Note that the expansion can produce exponentially larger formula.
 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:
-\[
+\begin{equation*}
    \forall p. \exists q. ((p \rightarrow q) \land (q \rightarrow r))
-\]
+\end{equation*}
-=== Quantified Propositional Formula Semantics ===
+===== Quantified Propositional Formula Semantics =====
 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