Implementing Term Rewriting by Graph Reduction: Termination of Combined Systems

It is well known that the disjoint union of terminating term rewriting systems does not yield a terminating system in general. We show that this undesirable phenomenon vanishes if one implements term rewriting by graph reduction: given two terminating term rewrite systems 7~o and 7~1, the graph reduction system implementing 7~0 -F T~1 is terminating. In fact, we prove the stronger result that the graph reduction system for the union ~o U T~ 1 is terminating provided that the left-hand sides of ~i have no common function symbols with the right-hand sides of 7~1-i (i = O, 1). The implementation is complete in the sense that it computes a normal form for each term over the signature of R0 u 7~1. 1 I n t r o d u c t i o n The operational semantics of algebraic specification languages are usually based on term rewriting (see, e.g., [BCV 85], [BHK 89], [EM 85], [FG3M 85], [GH 86]). In this context, confluence and termination are particularly relevant properties of term rewriting systems. Hence, when specifications are structured as combinations of smaller subspecifications, the question arises whether these properties are preserved by a given combination mechanism. Recently, research in this direction has been started by,considering the disjoint union T£0 +7~1 of two term rewriting systems 7"£o and 7~1: the rule set of 7£o ~-7~1 is the union of the rules of T~ and T£1 where the function symbols occurring in T~ and 7~1 are disjoint (or are made disjoint by renaming). Toyama [Toy 87a] proves that 7~o + 7~1 is confluent *Work supported by ESPRIT project #390, PROSPECTRA, and by ESPRIT Basic Research Working Group #3264, COMPASS.