7 - Inductive Completion by Ground Proof Transformation

Abstract

This chapter presents the example of Prolog that has shown the possible success of high-level programming with some powerful mathematical notation as language and an automated theorem-prover as an interpreter for its execution. One of the few serious challenges to this combination of first-order predicate logic and resolution theorem proving is provided to date by the language of term-equations together with rewrite-rule machines and the Knuth–Bendix completion procedure. The latter approach has important applications in the area of data-type specification, where term equations provide a natural specification language, whose expressive power and usefulness is well known from modern Algebra and the theory of recursive functions. Software specification through term-equations has the advantage that, by interpreting the equations as directed rewrite-rules, an easy to implement rewrite-rule machine allows their efficient application and, therefore, some means of rapid prototyping. The category of all models forms an equational variety, for which the usual equational reasoning is a complete proof procedure. In the variety, there are also certain initial or standard models that are important in practice because they capture some standard intuition; all data items in these models are denoted by terms without variables formed from the operators in the equations and, hence, they are finitely constructible.