1 - Completion Without Failure

Abstract

This chapter discusses completion without failure. The design of efficient methods for dealing with the equality predicate is one of the major goals in automated theorem proving. Just adding equality axioms almost invariably leads to unacceptable inefficiencies. A number of special methods have been devised for reasoning about equality. Within resolution-based provers, demodulation, that is, using equations in only one direction to rewrite terms to a simpler form, is frequently employed. A complete method for handling equations is paramodulation in which equational consequences are generated by using all equations in both directions. Paramodulation is difficult to control and may produce hosts of irrelevant or redundant formulas. The chapter discusses the purely equational case in which a theory is presented as a set of equations and one is interested in proving a given equation to be valid in that equational theory. In important special cases, validity can be decided using canonical rewrite systems that have the property that all equal terms simplify to an identical form. Deciding validity in theories for which canonical systems are known is thus easy and reasonably efficient.