Equality Elimination for the Inverse Method and Extension Procedures

We demonstrate how to handle equality in the inverse method using equality elimination. In the equality elimination method, proofs consist of two parts. In the first part we try to solve equations obtaining so called solution clauses. Solution clauses are obtained by a very refined strategy — basic superposition with selection function. In the second part, we perform the usual sequent proof search by the inverse method. Our approach is called equality elimination because we eliminate all occurrences of equality in the first part of the proof. Unlike the previous approach proposed by Maslov, our method uses most general substitutions, ordering restrictions and selection functions. We also note that this technique is directly applicable to extension procedures, like the connection method. Unlike other approaches, we do not require the use of rigid or mixed E-unification. 1 The inverse method The inverse method of theorem proving in sequent cal-culi has been proposed by Maslov in the 1960s. The method is based on the bottom-up1 search in sequent calculi. The inverse method is completely local [Maslov and Mints, 1983; Degtyarev and Voronkov, 1994b] and can be efficiently implemented both for classical and non-classical logics [Voronkov, 1992]. In terms of efficiency, it is competitive with resolution2. The inverse method requires no normal forms which can be an advantage for interactive provers. The introduction of equality in the inverse method has not yet received proper attention. In this paper we * Supported by a grant from the Swedish Institute. *On leave from Kiev University. t Supported by a TFR grant. 1 Bottom-up search means the search from axioms to the goal. 2In fact, the inverse method can be simulated by resolution using structure-preserving clause-form translation [Maslov, 1983; Boy de la Tour, 1990]. consider the inverse method with equality. Our approach is based on equality elimination. The structure of this paper is the following. In this section we briefly discuss introduction of equality in the inverse method in general. In Section 2 we introduce main definitions and notation. In Section 3 we define the equality elimination method and give several examples. In Section 4 we show how one can generalize our technique to the connection method. The first natural generalization of the inverse method to include equality has been made by Maslov [Maslov, 1971]. It was based on theorems proven in [Kanger, 1983; Lifschitz, 1968] about the specialization of proofs in the sequent calculus with …