Hierachical Termination

From a practical perspective, it is important for programs to have modular correctness properties. Some (largely syntactic) sufficient conditions are given here for the union of terminating rewrite systems to be terminating, particularly in the hierarchical case, when one of the systems makes no reference to functions defined by the other. 1 I n t r o d u c t i o n A rewrite rule is an equation between first-order terms used to replace equals-byequals in one direction only. A rewrite system, that is a set of rewrite rules, is a form of applicative program that computes by reducing (that is, repeatedly rewriting) a variable-free term to its normal form (an unrewritable term), where the order in which rules are applied and the choice of subterm to which to apply a rule is arbitrary. Rewrite systems have other important applications in programming language semantics and automated deduction. For recent surveys of rewriting, see [Dershowitz and Jouannaud, 1990; Avenhaus and Madlener, 1990; Klop, 1992; Plaisted, 1993]. When no infinite sequences of rewrites are possible, a rewrite system is said to have the (strong, or uniform) termination property. In practice, one usually guarantees termination by devising a well-founded partial ordering for which a rewritten term is always smaller than the original. For a survey of methods of proving termination, see [Dershowitz, t987]; for examples of these methods, see [Dershowitz, 1995]. Rewrite systems provide a simple, intuitive, nondeterministic functional language. As such, it would be quite valuable to be able to combine systems possessing desirable properties. In particular, we look for sufficient conditions under which the union of two terminating systems would be terminating. The conditions given here are based on syntactic restrictions of the systems in question. The first to consider modularity issues in rewriting was Bidoit [1981] with his "gracious" conditions. A rule 1 ~ r is used to rewrite a term s containing an instance la of its left-hand side l at some position p to 8[r~]p, the same term, except that the subterm at p has * This research was supported in part by the U. S. National Science Foundation under Grants CCR-90-07195 and CCR-90-24271, by a Lady Davis fellowship at the Hebrew University of Jerusalem, Israel, and by a Meyerhoff Visiting Professorship at the Weizmann Institute of Science, Rehovot, Israel.