Fitting Semantics for Conditional Term Rewriting

This paper investigates the semantics of conditional term rewriting systems with negation, which may not satisfy desirable properties like termination. It is shown that the approach used by Fitting [5] for Prolog-style logic programs is applicable in this context. A monotone operator is developed, whose fixpoints describe the semantics of conditional rewriting. Several examples illustrate this semantics for non-terminating rewrite systems which could not be easily handled by previous approaches. 1 Introduction Conditional term rewriting systems (CTRS) have attracted much attention in the recent past as a useful generalization of the simpler formalism of term rewriting systems (TRS). But CTRS have not been unconditionally accepted, due to the absence of well defined semantics for conditional rewriting mechanisms. This paper suggests one remedy, following the approach of Melvin Fitting, who suggested similar semantics for Prolog-style logic programs [5]. Past work on the semantics of conditional term rewriting has followed three directions: 1. Impose restrictions on the syntax of the CTRS formalism to ensure termination and the existence of a unique precongruence which is considered to describe the meaning of the rewrite relation [8]. This approach does not define the meaning of rewriting when the CTRS does not satisfy the relevant termination criterion. Also, the termination criterion itself is undecidable, and is not a necessary condition for each rewrite step and all rewrite sequences to terminate finitely. 2. Give logical semantics for a CTRS R as a set of conditional equations £(R) together with a set of "default" negative equality literals [13]. This approach is useful if all rewrite sequences terminate or if the CTRS is intended to describe a specification based on a set of free constructor functions. 3. Transform CTRS into "equivalent" TRS, and identify the semantics of the CTRS with that of the transformed systems [1]. Assign an "initial algebra" semantics for TRS. The drawback of this approach is that it does not adequately describe the operational use of CTRS with negative literals in the antecedents of rules. This paper attempts to fill the lacuna using an elegant approach of Fitting, following Kripke[10] who brought together Kleene's multivalued logics [9], and Tarski's lattice-theoretical fixpoint theorem [16]. Fitting [5] uses this approach to present an alternative to the semantics of logic programming given by Apt and Van Emden [2]. The main contribution of this paper is to show that this approach can also successfully explain the meaning of conditional rewriting …