Equality Reasoning in Clause Graphics

A method to control the application of equality derivation rules in an automatic theorem proving system is presented. When handling equality in the search for a proof of a theorem two main problems arise: 1) to obtain a control mechanism for the search and application of useful equality derivation steps in order to support global strategies which plan and control the whole proof, thus conducing to an efficient and complete proof procedure. 2) to find proper equations rendering two terms unifiable. These problems are solved by combining the clause graph method and the Mparamodulation-if-needed" idea by introducing Morris' E-resolution into the clause graph proof procedure. The necessary equations to form possible E-resolvents are searched for in the initial graph and are inherited after-wards. The search space for possible E-resolutions will be reduced by exploiting constraints using the information in the clause graph. 0. INTRODUCTION The use of equality axioms in a theorem prover based on the resolution principle has turned out to be very inefficient because too many additional resolution operations involving the equality axioms are possible. This problem is well recognized in A way out is the direct incorporation of equality into the proof procedure. One of the various methods proposed with this aim in mind is paramodulation [RW69]: with one additional rule of inference, the paramodulation rule, the equality axioms become superfluous except for the reflexivity axion-. But paramodulation, i.e. the replacement of terms by equal terms, can be applied almost everywhere in a clause set and therefore paramodulation alone still does not solve the problem of "how to handle equality in an automatic theorem proving system" (ATP). Strategies or methods are required to control the enormous amount of potential steps and to make sensible use of the paramodulation rule. A promising control mechanism may result from the "paramodulation if needed" idea, which states that the paramodulation rule should only be used to reduce differences between potentially complementary unifiable literals, such that an inference step by resolution becomes possible. Two literals (in diffe-IN CLAUSE GRAPHS Blasius University of Karlsruhe rent clauses) are called potentially complementary unifiable, if they have the same predicate symbol and opposite sign. There are several methods known to realize the "if needed" idea (e.g. [Sh78], [HR78], [Di79]) the most explicit realization of which is Morris' E-resolution [Mo69]. An E-resolution step can be regarded as a sequence of paramodulation steps such that two potentially complementary …