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.
F i t t i n g Semantics for C o n d i t i o n a l T e r m R e w r i t i n g Chilukuri K. Mohan School of Computer and Information Science Syracuse University, Syracuse, NY 13244-4100
Abstract This paper investigates the semantics of conditional term rewriting systems w i t h negation, which may not satisfy desirable properties like termination. It is shown that the approach used by F i t t i n g [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 ( C T R S ) have attracted much attention in the recent past as a useful generalization of the simpler formalism of term rewriting systems (TRS). B u t C T R S 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 F i t t i n g , who suggested similar semantics for Prolog-style logic programs [5]. Past work on the semantics of conditional t e r m rewriting has followed three directions: 1. Impose restrictions on the syntax of the C T R S 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 C T R S does not satisfy the relevant termination criterion. Also, the termination criterion itself is undecidable, and is not a necessary condit i o n for each rewrite step and all rewrite sequences to terminate finitely. 2. Give logical semantics for a C T R S R as a set of conditional equations £(R) £(R) together w i t h a set of "default" negative equality literals [13]. This approach is useful if all rewrite sequences terminate or if the C T R S is intended to describe a specification based on a set of free constructor functions. 3. Transform C T R S into "equivalent" T R S , and ident i f y the semantics of the C T R S w i t h that of the
transformed systems [1]. Assign an " i n i t i a l algebra" semantics for T R S . The drawback of this approach is that it does not adequately describe the operational use of C T R S w i t h negative literals in the antecedents of rules. This paper attempts to fill the lacuna using an elegant approach of F i t t i n g , following Kripke[10] who brought together Kleene's multivalued logics [9], and Tarski's lattice-theoretical fixpoint theorem [16]. F i t t i n g [5] uses this approach to present an alternative to the semantics of logic programming given by A p t 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 systems w i t h negation, including the problematic C T R S whose semantics have eluded the grasp of previous approaches In the next section, we introduce C T R S and point out the deficiencies of a two-valued fixpoint semantics. In section 3, following some mathematical preliminaries, we describe the new semantics for conditional rewriting. Several examples are then given in section 4 to illustrate the semantics. References follow concluding remarks.
2 2.1
Preliminaries Conditional Rewriting
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 f o r m
where Ihs and rhs are two terms, and the antecedent is a conjunction of zero or more equations s i = t i and negated equality literals Every variable occurring in each and rhs must also occur in Ihs. Following the notation of refers to the subterm o f p a t position j , and ' p [ q ] j ' refers t o the result of replacing by q in p. For instance, when positions are described in Dewey deci m a l notation, and
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Besides m a t c h i n g a n d r e p l a c e m e n t , c o n d i t i o n a l r e w r i t i n g requires c h e c k i n g t h a t t h e a n t e c e d e n t o f a r u l e h o l d s . T h e basic i d e a u n d e r l y i n g t h e d e f i n i t i o n o f E l - r e w r i t i n g i s t o c o n c l u d e t h a t t w o t e r m s are e q u a l i f t h e y have c o n v e r g i n g r e d u c t i o n sequences, a n d n o t e q u a l o t h e r w i s e (for g r o u n d t e r m s ) . N o t surprisingly, the a t t e m p t t o f i n d a v a l l e y - p r o o f f o r a n e q u a l i t y does n o t a l w a y s t e r m i n a t e . B u t i f r e d u c t i o n sequences f r o m t w o g r o u n d t e r m s p , q d o t e r m i n a t e w i t h o u t c o n v e r g i n g , t h e n w e can assert t h a t p = q has no ' v a l l e y - p r o o f u s i n g t h e g i v e n r e w r i t i n g system. V a r i a b l e s in d i f f e r e n t rules are f i r s t r e n a m e d to be dist i n c t f r o m each o t h e r a n d f r o m those i n t h e t e r m t o b e El-reduced. D e f i n i t i o n 2 : A t e r m m E I - R e d u c e s ( o r EI-Rewrites) to n using an E I - C T R S R ( w r i t t e n if R contains a rule a n d t h e r e is a s u b s t i t u t i o n a m a t c h i n g Ihs w i t h a s u b t e r such t h a t each of the f o l l o w i n g conditions h o l d :
1. Match and Replace: 2 . D e m o n s t r a b l e Convergence: For each e q u a l i t y cond, t h e r e is a c o m m o n t e r m to w h i c h and can b e E l - r e d u c e d i n a f i n i t e n u m b e r o f steps. 3. D e m o n s t r a b l e Non-Convergence: For every n e g a t e d equality literal cond, i t can b e d e m o n s t r a t e d b y f i n i t e E l - r e w r i t i n g sequences t h a t and are g r o u n d t e r m s w i t h n o c o m m o n r e d u c t . M o r e precisely, t h e f o l l o w i n g c o n d i t i o n s m u s t h o l d : (i) and are g r o u n d t e r m s ; ( i i ) a l l E l - r e w r i t i n g sequences f r o m and terminate; and ( i i i ) t h e set o f a l l r e d u c t s o f is disjoint f r o m that of For a m o r e d e t a i l e d d e s c r i p t i o n o f E l - r e w r i t i n g w i t h e x a m p l e s , see [14]. 2.2
Fixpoint Semantics
D e f i n i t i o n 3 If / is a f u n c t i o n ( o f one a r g u m e n t ) whose d o m a i n a n d r a n g e are t h e s a m e , t h e n S is a f i x p o i n t ( o r fixed p o i n t ) of / w h e n e v e r f(S) = 5. If t h e d o m a i n e l e m e n t s are p a r t i a l l y o r d e r e d , / m a y have zero o r m o r e p a r t i a l l y ordered fixpoints: - a f i x p o i n t S is m i n i m a l if t h e r e is no o t h e r f i x p o i n t T such in the ordering; - a m i n i m a l f i x p o i n t S is t h e least f i x p o i n t if , for every fixpoint; - a f i x p o i n t S is m a x i m a l if t h e r e is no o t h e r f i x p o i n t T such t h a t in the ordering; - t w o f i x p o i n t s S, T are c o m p a t i b l e if t h e y have a c o m m o n upper b o u n d which is a fixpoint, - a f i x p o i n t is intrinsic [10] (or o p t i m a l [12]) i f f it is c o m p a t i b l e w i t h every f i x p o i n t o f / . In the f i x p o i n t semantics approach, the 'meaning* of a p r o g r a m is considered to be t h e least fixpoint of a f u n c t i o n / r e l a t i o n w h i c h represents t h e b e h a v i o r o f t h e p r o g r a m on some i n p u t . T h e following fixpoint semantics can be suggested f o r c o n d i t i o n a l r e w r i t i n g , as in [8]. A function R is associated w i t h each C T R S R such t h a t if S is a b i n a r y r e l a t i o n , a n d is i t s reflexive t r a n s i t i v e
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K e y O b s e r v a t i o n : The fixpoints of describe the semantics of conditional rewriting with an EI-CTRS R; particularly important are the least fixpoint and the largest intrinsic fixpoint. We now investigate the fixpoints of the monotone relation corresponding to each rewrite system R.
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5
Conclusions
W e have i n v e s t i g a t e d t h e f i x p o i n t s e m a n t i c s o f c o n d i tional t e r m r e w r i t i n g systems w i t h negation. Twov a l u e d s e m a n t i c s does n o t ascribe a m e a n i n g t o C T R S ' s t h a t d o n o t s a t i s f y useful p r o p e r t i e s such a s t e r m i n a t i o n . W e have s h o w n t h a t a t h r e e - v a l u e d a p p r o a c h used b y F i t t i n g [5] f o r P r o l o g - s t y l e l o g i c p r o g r a m s i s a p p l i c a b l e i n t h i s c o n t e x t . A m o n o t o n e o p e r a t o r is d e v e l o p e d , whose f i x p o i n t s describe t h e s e m a n t i c s o f c o n d i t i o n a l r e w r i t i n g . Several e x a m p l e s i l l u s t r a t e t h i s s e m a n t i c s for ' t r o u b l e s o m e ' r e w r i t e s y s t e m s w h i c h c o u l d n o t b e h a n d l e d easily by previous approaches. T h i s w o r k supports the cont e n t i o n t h a t r e s u l t s achieved i n research o n P r o l o g - s t y l e logic p r o g r a m m i n g can b e u s e f u l i n t h e c o n t e x t o f c o n ditional term rewriting. W e have h e s i t a t e d t o say w h e t h e r i t i s t h e least f i x p o i n t o r t h e greatest i n t r i n s i c f i x p o i n t w h i c h b e t t e r describes t h e s e m a n t i c s o f t h e E I - C T R S . T h e e x a m p l e s m a y m o t i v a t e a preference for one o r t h e o t h e r . T h e o p e r a t i o n a l m e c h a n i s m described i n [13] a n d [8] c o m p u t e d m e m b e r s o f t h e least f i x p o i n t . T o c o m p u t e t h e greatest i n t r i n s i c f i x p o i n t , we need a d i f f e r e n t o p e r a t i o n a l m e c h a n i s m w h i c h uses " d i s j u n c t i v e r e w r i t i n g " [15] (cf. exa m p l e 9) as w e l l as a m e c h a n i s m w h i c h r e t u r n s f a i l u r e i n some cases w h e n n a i v e e v a l u a t i o n o f t h e antecedent leads t o n o n - t e r m i n a t i o n (cf. e x a m p l e 5 ) . T h e f o r m u l a t i o n o f such a r e w r i t i n g m e c h a n i s m , w h i c h c o m p u t e s precisely t h e greatest i n t r i n s i c fixpoint, is an issue f o r f u t u r e w o r k . I n n o n - c o n t r o v e r s i a l cases, w h e n t e r m i n a t i o n r e q u i r e m e n t s are s a t i s f i e d , t h e least f i x p o i n t a n d t h e greatest i n t r i n s i c f i x p o i n t c o i n c i d e (cf. e x a m p l e s 3 , 4 , 7 ) . a n d are e s s e n t i a l l y e q u i v a l e n t t o t h e s e m a n t i c s g i v e n i n p r e v i o u s w o r k for such w e l l - b e h a v e d r e w r i t e s y s t e m s .
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