# Interpolation for Propositional Logic

Definition of Interpolant: Given two propositional formulas and , such that , an interpolant for is a propositional formula such that:
1)
2)
3)

What is FV?

Note that by transitivity of . The interpolant extracts the essential part of formula which makes imply .

Let 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 that are not in .

where

• We can quantify universally all variables in that are not in .

where

Definition: denote the set of all interpolants for , that is,

Theorem: The following properties hold for , , defined above:

We can also derive interpolants from resolution proofs, as we will see in lecture07.