Fitting Semantics for Conditional Term Rewriting

Abstract

This paper investigates the semantics of condit ional term rewri t ing systems wi th negation, which may not satisfy desirable properties like terminat ion. It is shown that the approach used by F i t t ing [5] for Prolog-style logic programs is applicable in this context. A monotone operator is developed, whose fixpoints describe the semantics of conditional rewri t ing. Several examples i l lustrate this semantics for non-terminating rewrite systems which could not be easily handled by previous approaches. 1 I n t r o d u c t i o n Condit ional term rewri t ing systems (CTRS) have attracted much attention in the recent past as a useful generalization of the simpler formalism of term rewrit ing systems (TRS). But CTRS have not been unconditionally accepted, due to the absence of well defined semantics for conditional rewri t ing mechanisms. This paper suggests one remedy, fol lowing the approach of Melvin F i t t ing , 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 terminat ion 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 rewri t ing when the CTRS does not satisfy the relevant terminat ion criterion. Also, the terminat ion criterion itself is undecidable, and is not a necessary condit ion for each rewrite step and all rewrite sequences to terminate f ini tely. 2. Give logical semantics for a CTRS R as a set of conditional equations £(R) together w i th 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 ident i fy the semantics of the CTRS wi th that of the transformed systems [1]. Assign an " in i t ia l algebra" semantics for TRS. The drawback of this approach is that it does not adequately describe the operational use of CTRS wi th negative literals in the antecedents of rules. This paper attempts to fill the lacuna using an elegant approach of F i t t ing , fol lowing Kripke[10] who brought together Kleene's mult ivalued logics [9], and Tarski's lattice-theoretical fixpoint theorem [16]. F i t t ing [5] uses this approach to present an alternative to the semantics of logic programming given by Ap t and Van Emden [2]. The main contr ibution of this paper is to show that this approach can also successfully explain the meaning of conditional rewri t ing systems w i th negation, including the problematic CTRS whose semantics have eluded the grasp of previous approaches In the next section, we introduce CTRS and point out the deficiencies of a two-valued f ixpoint semantics. In section 3, following some mathematical preliminaries, we describe the new semantics for conditional rewrit ing. Several examples are then given in section 4 to i l lustrate the semantics. References follow concluding remarks. 2 Pre l im ina r ies 2.1 C o n d i t i o n a l R e w r i t i n g We define the formalism and operational use of a language for expressing data type and function specifications [13, 8]. D e f i n i t i o n 1 Equational-lnequational-Conditional Term Rewriting Systems (EI-CTRS) are finite sets of rules of the general form where Ihs and rhs are two terms, and the antecedent is a conjunction of zero or more equations si = ti and negated equality literals Every variable occurr ing in each and rhs must also occur in Ihs. Following the notat ion of refers to the subterm of p at position j , and 'p[q] j ' refers to the result of replacing by q in p. For instance, when positions are described in Dewey decimal notat ion, and

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Cite this paper

@inproceedings{Mohan1991FittingSF, title={Fitting Semantics for Conditional Term Rewriting}, author={Chilukuri K. Mohan}, booktitle={IJCAI}, year={1991} }