Constructing Craig Interpolation Formulas

1 Background and Introduction Let 27 and H be two inconsistent first order theories. Then by Craig's Interpolation Theorem, there is a sentence 8, called a Craig interpolant, such that 8 is t rue in 27 and false in H and every nonlogical symbol occurring in 8 occurs in bo th 27 and H. Craig interpolants can be used to solve the problem of learning a first order concept by letting 27 and H be the lists of positive and negative examples of the concept to be learned. The s tandard nonconstructive model-theoretic proof of Craig's Theorem is in [3]. Lyndon showed how to construct an interpolant from a special form of natural deduction (see [1]). We show how to construct an interpolant from a refutat ion proof which uses binary resolution, factoring and paramodulation. In our examples, we use O T T E R (the s tandard text on O T T E R is [4]) to generate such proofs. Craig interpolants can be used to find a sentence which distinguishes two nonisomorphic finite structures. Let 27 and H be the atomic diagrams of the two structures. Then they are inconsistent and any Craig interpolant for them is a sentence which is t rue in one s tructure and false in the other. 2 Constructing Interpolation Formulas from Refutations Let L~: and L ~ be two languages, 27 a theory in L~:, and H a theory in L ~ such that 27 t9 H is not consistent. In this paper we use <> to represent contradiction, use O to indicate the end of a proof, and suppose P is a refutation of E U / / ~ <> involving only binary resolutions, paramodulations, and factorings. The input clauses (clauses at the top of the refutation) are required to be instances of clauses from 27 and / / . For convenience, we will assume that different input clauses have disjoint sets of variables.