Knuth–Bendix completion algorithm

Abstract: Publisher Summary This chapter focuses on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained. As a formalism, rewrite systems have the full power of Turing machines and may be thought of as nondeterministic Markov algorithms over terms rather than strings. The theory of rewriting is in essence a theory of normal forms. To some extent, it is an outgrowth of the study of A. Church's Lambda Calculus and H. B. Curry's Combinatory Logic. The chapter discusses the syntax and semantics of equations from the algebraic, logical, and operational points of view. To use a rewrite system as a decision procedure, it must be convergent. The chapter describes this fundamental concept as an abstract property of binary relations. To use a rewrite system for computation or as a decision procedure for validity of identities, the termination property is crucial. The chapter presents the basic methods for proving termination. The chapter discusses the question of satisfiability of equations and the convergence property applied to rewriting.

Abstract: This paper gives new results, and presents old ones m a umfied formahsm, concerning ChurchRosser theorems for rewrmng systems Abstract confluence propentes, depending solely on axioms for a binary relatton called reduction, are first presented Results of Newman and others are presented m a unified formahsm The systemattc use of a powerful mductmn pnnciple permRs the generahzauon of results of Sethl on reduction modulo eqmvalence. Simphficatton systems operating on terms of a first-order logic are then considered. Results by Rosen and Knuth and Bendix are extended to give several new crtteria for confluence of these systems It ts then shown how these criteria yield new methods for the mechanizaUon of equattonal theories

Abstract: 1. Abstract reduction systems 2. First-order term rewriting systems 3. Examples of TRSs and special rewriting formats 4. Orthogonality 5. Properties of rewriting: decidability and modularity 6. Termination 7. Completion of equational specifications 8. Equivalence of reductions 9. Strategies 10. Lambda calculus 11. Higher order rewriting 12. Infinitary rewriting 13. Term graph rewriting 14. Advanced ARS theory 15. Rewriting based languages and systems 16. Mathematical background.