The direct sum of two term rewriting systems is the union of systems having disjoint sets of function symbols. It is shown that if two term rewriting systems both have the Chruch-Rosser property, then the direct sum of these systems also has this property.
IntroductionWe consider properties of the direct sum system RI @ R2 obtained from two term rewriting systems R1 and RZ [3]. The first study on the direct sum system was conducted by Klop [3] in order to consider the Church-Rosser property for combinatory reduction systems having nonlinear rewriting rules, which contain term rewriting systems as a special case. He showed that if R, is a regular (i.e., linear and nonambiguous) system and R2 consists of the single nonlinear rule D(x, x) D x, then the direct sum R1 @ RZ has the Church-Rosser property. He also showed in the same manner that, if RZ consists of the nonlinear rules
The direct sum of two term rewriting systems is the union of systems having disjoint sets of function symbols. It is shown that the direct sum of two term rewriting systems is not terminating, even if these systems are both terminating.
A term rewriting system is called growing if each variable occurring on both the left-hand side and the right-hand side of a rewrite rule occurs at depth zero or one in the left-hand side. Jacquemard showed that the reachability and the sequentiality of linear (i.e., left-right-linear) growing term rewriting systems are decidable. In this paper we show that Jacquemard's result can be extended to left-linear growing rewriting systems that may have right-nonlinear rewrite rules. This implies that the reachability and the joinability of some class of right-linear term rewriting systems are decidable, which improves the results for right-ground term rewriting systems by Oyamaguchi. Our result extends the class of left-linear term rewriting systems having a decidable call-by-need normalizing strategy. Moreover, we prove that the termination property is decidable for almost orthogonal growing term rewriting systems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.