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Interpolation for Propositional Logic
Given two propositional formulas and , an interpolant for is a propositional formula such that:
1)
2)
3)
denotes the set of free variables in the given propositional formula (see Propositional Logic Syntax).
Note that by transitivity of . The interpolant extracts what makes imply .
Here are two easy ways to create an interpolant. Let denote the operator that eliminates propositional quantifiers (see QBF and Quantifier Elimination):
- We can quantify existentially all variables in that are not in .
where
- We can quantify universally all variables in that are not in .
where
More precisely, let denote the set of all interpolants for , that is, \[
{\cal I}(F,G) = \{ H \mid H \mbox{ is interpolant for $(F,G)$ \}
\] Then the following properties hold:
We can also derive interpolants from resolution proofs, as we will see in lecture07.