Termination

This survey describes methods for proving tha t systems of rewri te rules terminate . Illust ra t ions of the use of path orderings and other simplification orderings in terminat ion proofs are given. The effect of restrictions, such as l inearity, on the form of rules is considered. In general, though, te rminat ion is an undeeidable p roper ty of rewri te systems. 1. I N T R O D U C T I O N A term-rewriting (rewrite) system R over a set of terms T is a set of rewrite rules, each of the form l -> r , where I and r are terms in T or are terms containing variables ranging over T . Such a rule applies to a te rm t in T if a sub te rm s of t matches the left-hand side l with some subst i tut ion a of te rms in T for variables appearing in I (i.e. s--~la). The rule is applied by replacing the sub te rm s in t with the corresponding r ight -hand side ra of the rule, within which the same subst i tut ion a of terms for variables has been made. We write t~RU , or jus t t ~ u , to indicate tha t a t e rm u in T is derivable in this way from the t e rm t in T by a single application of some rule in R . If t=* • • • ~ u in zero or more steps, abbrevia ted t ~ * u , then we say tha t t reduces to u ; if no rule can be applied to t , we say tha t t is irreducible; when t reduces to an irreducible t e rm u, we say tha t u is a normal form of t . There are five propert ies involved in the verification of rewrite systems: 1) termination--no infinite derivat ions are possible, 2) confluence--each t e rm has at most one normal form, 3) soundness--terms are only rewri t ten to equal terms, 4) completeness--equal te rms have the same normal form, 5) correctness--all normal forms satisfy given requirements . *The preparation of this survey was supported in part by the National Science Foundation under Grant MCS 83-07755.