Chapter 27 - Combining Superposition, Sorts and Splitting

Abstract

This chapter is about the implementation of first-order saturation based clausal theorem provers. At the heart of such an implementation is a first-order calculus. It consists of inference rules that generate new clauses and reduction rules that reduce the number of clauses or transform clauses into simpler ones. The chapter introduces a great variety of clause set based inference and reduction rules that can be composed to various sound and complete first-order calculi. The clause store data structure together with such a calculus is the basis for most of today's theorem proving systems. The chapter goes one step further by introducing a splitting rule that supports explicit case analysis. This generalizes the standard clause store based approach to a clause store collection approach where different clause stores represent the different cases. Therefore, the splitting rule introduces a second dimension in automated theorem proving.