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--- sav08:interpolation_for_propositional_logic [2008/03/11 16:18]
vkuncak
+++ sav08:interpolation_for_propositional_logic [2015/04/21 17:30]
@@ Line 1: / Line 1: @@
-====== Interpolation for Propositional Logic ======
-**Definition of Interpolant:**
-Given two propositional formulas $F$ and $G$, an interpolant for $(F,G)$ is a propositional formula $H$ such that: \\
-) $\models F \rightarrow H$ \\
-) $\models H \rightarrow G$ \\
-) $FV(H)\subseteq FV(F) \cap FV(G)$
-++++What is FV?|
-$FV$ denotes the set of free variables in the given propositional formula, see [[Propositional Logic Syntax]].
-++++
-Note that $F \rightarrow G$ by transitivity of $\rightarrow$. The interpolant extracts the essential part of formula $F$ which makes $F$ imply $G$.
-Let $elim$ denote the operator that eliminates propositional quantifiers (see [[QBF and Quantifier Elimination]]).
-Here are two **simple ways to construct an interpolant:**
-  * We can quantify existentially all variables in $F$ that are not in $G$.
-$H_{min} \equiv elim(\exists p_1, p_2, ..., p_n.\  F)$ where $\{p_1, p_2, ..., p_n \}\ = FV(F)\backslash FV(G)$ \\
-  * We can quantify universally all variables in $G$ that are not in $F$.
-$H_{max} \equiv elim(\forall q_1, q_2, ..., q_m.\ G)$ where $\{q_1, q_2, ..., q_m \}\ = FV(G)\backslash FV(F)$ \\
-**Definition:** ${\cal I}(F,G)$ denote the set of all interpolants for $(F,G)$, that is,
-\[
-   {\cal I}(F,G) = \{ H \mid H \mbox{ is interpolant for $(F,G)$ \}
-\]
-**Theorem:** The following properties hold for $H_{min}$, $H_{max}$, ${\cal I}(F,G)$ defined above:
-  - $H_{min} \in {\cal I}(F,G)$
-  - $\forall H \in {\cal I}(F,G).\ \models (H_{min} \rightarrow H)$
-  - $H_{max} \in {\cal I}(F,G)$
-  - $\forall H \in {\cal I}(F,G).\ \models (H \rightarrow H_{max})$
-We can also derive interpolants from resolution proofs, as we will see in [[lecture07]].