Jan A. Bergstra and Jan Willem Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Sciences 32, 323--362, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....same logical strength as the underlying conditional equational system. In this paper we summarize known logicality results and we present new sufficient conditions for logicality of the important class of oriented conditional term rewriting systems. 1 Introduction Conditional term rewriting ([4, 6, 8]) provides a useful framework for the study of a wide range of problems in computation and programming. In this paper we investigate the logical strength of conditional rewrite systems. A conditional rewrite system is called logical if it has the same logical strength as the underlying conditional ....

....for s 1 oe t 1 oe; s n oe t n oe. If we replace coe by coe #Rn we obtain the rewrite relation of a join CTRS and if we replace coe by coe we obtain the rewrite relation of an oriented CTRS. This classification of CTRSs goes back to Bergstra and Klop [4] who use the terminology type I, II, and III. Natural CTRSs are also called semiequational in the literature and join CTRSs are sometimes called standard. Note that we don t put any restrictions on the distribution of variables among the different parts of conditional rewrite rules. In particular, ....

J.A. Bergstra and J.W. Klop, Conditional Rewrite Rules: Confluence and Termination, JCSS 32 (1986) 323--362.

....narrowing and reflection is complete for canonical conditional term rewriting systems and rewriting can be applied as a simplification rule, where a goal clause G rewrites to G iff G Gamma p(oe) G and oe does not bind a variable in G. Note, this definition differs from the one given in e. g [Bergstra and Klop, 1986] or [Kaplan, 1984] The reason is that we are mainly interested in equation solving and the conditions of a rewrite rule applied are simply added to the new goal clause. Recalling the CREDIT example we find that (credibility(y) high can be rewritten to (high = high paid(y) yes using (c1) ....

J. A. Bergstra and J. W. Klop. Conditional rewrite rules: Confluence and termination. Computer and System Sciences, 32:323--362, 1986.

....left linear right ground TRSs. Furthermore, we believe that this approach gives better understanding of expressiveness of CTRSs subjecting to various restrictions. One of the best known undecidability results in conditional term rewriting is that one step reduction of CTRSs is no longer decidable[2][12] We will see this fact is quite severe in a sense that even in a limited conditional extension of a well behaved class of TRSs the situation is the same. Several criteria for decidability of one step reduction, which also imply termination, are known: e.g. decreasing CTRSs[7] deterministic ....

....decidability of one step reduction, which also imply termination, are known: e.g. decreasing CTRSs[7] deterministic quasi reductive CTRSs[8] These criteria are, however, undecidable in general, and tools to detect these criteria have been investigated. More similar to our approach is a result in [2], which says that whether a given term is normal is decidable for orthogonal normal oriented CTRSs with subterm property. They proves in fact decidability of innermost one step reduction for these CTRSs this property is, however, out of scope of this paper. The paper is organized as follows. ....

[Article contains additional citation context not shown here]

J. A. Bergstra and J. W. Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Sciences, 32:323--362, 1986. 17

....the discussion to first order computations and the focus to strategies. Figure 1 summarizes the state of the art in this field. All the TRSs in the figure are constructor based (follow the constructor discipline [21] and left linear. For the time being, we ignore whether they are conditional [8]. We will see later that this is not a substantial characterization for our discussion. The inductively sequential TRSs are the first order component of functional languages, such as ML and Haskell, and the constructorbased restriction of the strongly sequential TRSs [14, 15] A sound and ....

....rule l r in R, a substitution and a context C such that s = C[l ] and t = C[r ] A conditional rewrite rule is a triple l r ( c, where l and r are defined as in the unconditional case and c is a sequence of conditions. Various options have been considered for the form of the conditions [8]. Since we are interested in narrowing computations in possibly non terminating constructor based TRSs, it is appropriate to consider conditions consisting of sequences of elementary equational constraints, i.e. pairs u v, where the symbol is an ordinary (infix, overloaded) operation, ....

[Article contains additional citation context not shown here]

J. A. Bergstra and J. W. Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Sciences, 32(3):323--362, 1986.

....the unconditional case and c is a sequence of elementary equational constraints, i.e. pairs of terms of the form t = u. The definition of the rewrite relation for conditional TRSs is fairly more complicated than for unconditional TRSs. The classic approach to conditional rewriting is discussed in [9]. A left linear, conditional, constructor based TRS is a good model for a functional or a logic program. Computations are (expressed by) operation rooted terms ultimately applied to values. Example 2. In programming languages, values are introduced by data type declarations such as: data bool = ....

....in this section. The focus, as in the rest of this paper, is on strategies. The classes of TRSs discussed earlier are all unconditional. The well known outermost fair rewrite strategy, which is normalizing for almost orthogonal TRSs [20] is also normalizing for conditional almost orthogonal TRSs [9]. For the constructor based TRSs, the results presented earlier about evaluation strategies are extended to the conditional case with little effort. The strategies discussed in Section 3 are based, either directly or indirectly, on definitional trees. Definitional trees are concerned with the ....

J. A. Bergstra and J. W. Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Sciences, 32(3):323--362, 1986.

....when studying reappearances of redexes in orthogonal TRSs. The origin relation restricted to the common variables rule and the context rule, and applied only to orthogonal, non conditional TRSs is equivalent to the descendants relation. Sometimes the notion of quasi descendant is used [BK86] Here the redex and contractum are also related. Thus, the origin relation without the common subterms rule (again restricted to orthogonal, non conditional TRSs) is equivalent to the quasidescendants relation. 3.4 Primitive Recursive Schemes A Primitive Recursive Scheme is a program scheme ....

J.A. Bergstra and J.W. Klop. Conditional rewrite rules: confluence and termination. Journal of Computer and System Sciences, 32(3):323--362, 1986.

....system (CTRS) is a set of conditional rewrite rules. For instance, Example 1.1 is a CTRS. We consider an equational logic program as a CTRS. 3. 1 Basic Definitions In order to give a precise definition of the computation with CTRS, we recall basic notions of (conditional) term rewriting [BK86, DJ90] Substitutions and most general unifiers are defined as in logic programming [Llo87] A position p in a term t is represented by a sequence of natural numbers (where denotes the root position) tj p denotes the subterm of t at position p, and t[s] p denotes the result of replacing the ....

....3.2 Equational Logic Programs The computation mechanism of unconditional term rewrite systems was defined by the rewrite relation R in the previous section. If we want to define the computation with a CTRS, we have to explain the evaluation of the condition in a rewrite step. Due to [BK86, DO90] we can distinguish the following possibilities. A condition s 1 = t 1 ; s k = t k is satisfied if (i) semi equational systems) s 1 t 1 ; s k t k (i.e. the left hand side of each condition can be converted into the right hand side by equational reasoning) ii) ....

[Article contains additional citation context not shown here]

J.A. Bergstra and J.W. Klop. Conditional Rewrite Rules: Confluence and Termination. Journal of Computer and System Sciences, Vol. 32, No. 3, pp. 323--362, 1986. 30

....the cycle is completed by replacing t by the instantiated right hand side oe(r) A term for which no rule is applicable to any of its subterms is called a normal form; the process of rewriting a term to its normal form (if it exists) is referred to as normalizing. A conditional rewrite rule [3] (such as [Er4] and [Er5] in Figure 4) is only applicable if all its conditions succeed; this is determined by instantiating and normalizing the left hand side and the right hand side of each condition. Positive (equality) conditions (of the form t 1 = t 2 ) succeed iff the resulting normal forms ....

.... TENV LIST type of(TENV LIST, EXPR) TYPE variables [ C TENV [ D [ DECL ; D[ DECL [ D [ DECL ; Tenv[ TENV [ Tenv [ TENV [ Tenv [ TENV [ TenvList[ TENV LIST equations [1] IntConst = INTEGER [2] RealConst = REAL [3] BoolConst = BOOLEAN [4] ARRAY[IntConst . IntConst ] OF Type) INTEGER ] Type Figure 10: Module TcTenv of the ASF SDF specification of the CLaX type checker. ffl Type correct statements (e.g. assignments whose left hand side and right hand side are both rewritten to INTEGER ) are are ....

[Article contains additional citation context not shown here]

Bergstra, J., and Klop, J. Conditional rewrite rules: confluence and termination. Journal of Computer and System Sciences 32, 3 (1986), 323--362.

....model which we use in our extension. Other aspects of the formalism are out of the scope of this presentation. 2.1 Conditional Rewrite Rules Conditional equations are used to specify language semantics and occur very frequently in realistic specifications. Conditional rewrite rules [BK86] are used to execute conditional equations. A conditional rewrite rule takes the form s 1 = t 1 ; Delta Delta Delta ; s n = t n s 0 = t 0 with n 0, and s i ; t i (0 i n) term. There are well definedness constraints imposed on the variables of the conditions [Wal91, p.16] in order to ....

J.A. Bergstra and J.W. Klop. Conditional rewrite rules: confluence and termination. Journal of Computer and System Sciences, 32(3):323--362, 1986.

....term rewriting as one of the basic implementation paradigms [29] This research has resulted in the design of the Asf Sdf formalism and its support environment, the Asf Sdf Meta Environment. ASF stands for Algebraic Specification Formalism [6] It is an implementation of conditional term rewriting [36,8] possibly containing negative premises [37,45,46,26] It also uses a form of prioritized rewriting [3,44] there are default rewrite rules that apply if other rewrite rules do not match. SDF stands for Syntax Definition Formalism [28] In fact, SDF is the part to conveniently define the signature ....

J.A. Bergstra and J.W. Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer System Science, 32(3):323--362, 1986.

....does not normalize when applied to a looping expression. Conditional equations are executed by first trying to normalize the various condition sides. Only if the condition sides of each condition yield the same normal form, the conclusion is applied as a rewrite rule. In the terminology of [BK86], conditional rewrite rules are executed as join systems. Finally, as seen in the last condition of equation [6] in module Substitute (Module 2.3.3) the Asf Sdf system can deal with conditions where at most one side introduces variables not yet occurring in the left hand side of the conclusion. ....

J.A. Bergstra and J.W. Klop. Conditional rewrite rules: confluence and termination. Journal of Computer and System Sciences, 32(3):323--362, 1986.

....to true if and only if s and t are syntactically equal. This equality function can be incorporated in the rewrite rules, to eliminate multiple occurrences of the same variable in the left hand side of a rewrite rule; see [25, page 28] for an elaborate example. We do not consider conditional TRSs [8], where rewrite rules are allowed to carry conditions. A sensible way to compile a conditional TRS properly, is to eliminate the conditions in its rewrite rules first. For example, conditions of the form s # t, i.e. s and t have the same normal form, can be expressed by means of an equality ....

J.A. Bergstra and J.W. Klop. Conditional rewrite rules: confluence and termination. Journal of Computer and System Sciences, 32(3):323--362, 1986.

....the right hand side of the head of a clause in also occurs in the left hand side of the head. The equational theory is said to be canonical if the binary one step rewriting relation defined by is noetherian and confluent. For syntactical characterizations of confluent conditional theories refer to [7, 11, 50]. We need the following: 0 = 1 = 7 denotes the set of occurrences (sequences of positive integers denoting an access path in a term, with the empty sequence represented by ) of the term t and is partially ordered by the prefix ordering: iff . We use to denote the subterm of t at ....

J.A. Bergstra and J.W. Klop. Conditional Rewrite Rules: confluence and termination. , 32:323--362, 1986.

....by the outermost fair rewrite sequence f(g(a) f(g(a) f(g(a) The proof presented in [27] is abstract in nature and applies also to the case of higher order rewriting. It makes use of ideas that are also present in [34, 20, 8] For proofs according to the sketch given above, see [24, 4, 32]. Theorem 6. Outermost fair rewriting is normalising for higher order rewriting systems that are weakly orthogonal and fully extended. ....

J.A. Bergstra and J.W. Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Sciences, 32:323--362, 1986.

....extendedness. As in the first order case, an immediate corollary of the main result is that the parallel outermost strategy is normalising for orthogonal higher order rewriting systems that are fully extended. Our result extends and corrects a result by Bergstra and Klop, proved in the appendix of [BK86], which states that outermost fair rewriting is normalising for orthogonal Combinatory Reduction Systems. Unfortunately, the proof presented in [BK86] is not entirely correct. The remainder of this paper is organised as follows. The next section is concerned with the preliminaries. In Section 3 ....

....higher order rewriting systems that are fully extended. Our result extends and corrects a result by Bergstra and Klop, proved in the appendix of [BK86] which states that outermost fair rewriting is normalising for orthogonal Combinatory Reduction Systems. Unfortunately, the proof presented in [BK86] is not entirely correct. The remainder of this paper is organised as follows. The next section is concerned with the preliminaries. In Section 3 the notion of outermost fair rewriting is explained. In Section 4 the main result of this paper is proved, namely that outermost fair rewriting is ....

J.A. Bergstra and J.W. Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Sciences, 32:323--362, 1986.

....to be finite, CCP (T) is finite as well. 4.3 Level Joinability of Critical Pairs Now we define a property which is related to the notion of shallow joinability of critical pairs in conditional TRS. In contrast to the latter, which does not guarantee the confluence (at least for join systems; cf. [2]) our condition characterizes the level coherence of an ORT and, thus, assures the completeness of the implementation. Definition 11. Let (s, t, C) # CCP (T) and n # N. A substitution # # Sub is called (C, n) feasible i# c# ER AC # T i n # ER AC # c # # for every c ....

J.A. Bergstra and J.W. Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Sciences, 32(3):323--326, 1986.

....with left linearity and normality; see [6] p. 36, Example B) If we do not require termination, the situation is even more complicated: There are left linear positive conditional rewrite systems that do not have any critical pairs but lack confluence (see [6] p. 36, Example A; taken from [2]) Therefore, syntactic confluence criteria for non terminating rewrite systems need to require strengthened forms of joinability of critical pairs and syntactic restrictions on rewrite rules such as left linearity and (weakened forms of) normality. Another major problem is caused by the infinite ....

....C is a set of constructor rules and R D a set of defining rules. Assume that Gamma R C is confluent. If each critical pair in CP(R) of the form (0; 1) 1; 0) or (1; 1) is complementary, then Gamma R is confluent. Note that this confluence criterion is stronger than a similar theorem of [2] (also cited in [6] since instead of normality it only requires weak normality, a property that is less restrictive because we can always achieve it by adding definedness atoms to the condition literals. Moreover, instead of requiring orthogonality, our theorem can deal with critical pairs ....

J. A. Bergstra and J. W. Klop. Conditional rewrite rules: Confluence and termination. J. Computer and System Sci., 32:323--362, 1986.

....s n = t n may be added to rewrite rules l r. In this paper, we restrict ourselves to CTRSs where all variables in the conditions s i ; t i also occur in l. Depending on the interpretation of the equality sign in the conditions, different rewrite relations can be associated with a CTRS, cf. e.g. [Kap84,BK86,DOS88,BG89,DO90,Mid93,Gra94,SMI95,Gra96a, Gra96b]. In our verification example, we transformed the problem into an oriented CTRS [SMI95] where the equality signs in conditions of rewrite rules are interpreted as reachability ( Thus, we denote rewrite rules by s 1 t 1 ; s n t n j l r: 3) In fact, we even have a normal ....

J. A. Bergstra & J. W. Klop, Conditional rewrite rules: confluence and termination. JCSS, 32:323--362, 1986.

....in Example 9 is confluent because it is orthogonal. 5 Confluence of Orthogonal Higher Order CTRSs In this section, we generalize the confluence result presented in the previous section to the case of conditional rewriting. Bergstra and Klop proved the confluence of first order orthogonal CTRSs in [BK86]. Their proof depends on the notion of development and the fact that every development is finite. Our result in this section generalizes their result to the higher order case and also simplifies their confluence proof, based on the parallel moves property. Definition 27 (conditional rewrite ....

J.A. Bergstra and J.W. Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Science, 32:323-- 362, 1986.

.... rewrite rules to reduction (simplification of ground expressions) as opposed to narrowing 2 (simplification of expressions that may contain variables) Furthermore a distinction is drawn between ordinary rewrite rules and conditional rewrite rules (rules with constraints on their applicability) [5]. We have been intrigued by the question whether Prolog Technology can offer the same gains in term rewriting that it has in theorem proving. Prolog comes tantalizingly close to term rewriting itself, because the narrowing or reduction step required in rewriting is closely related with the ....

....execution and proof: eq( prove:p(b,a,c) Proof) printproof(Proof) trying proof depth 0 trying proof depth s(0) trying proof depth s(s(0) trying proof depth s(s(s(0) trying proof depth s(s(s(s(0) Rule Rule Step used instance 1. [ 5] p(b,a,c) p(a,c,b) p(c,a,b) p(a,b,c) 2. 5] p(a,c,b) p(e,a,a) p(a,c,b) p(e,b,b) 3. 1] p(e,a,a) 4. 6] p(a,c,b) p(a,a,e) p(a,b,c) p(e,b,b) 5. 3] p(a,a,e) 6. 4] p(a,b,c) 7. 1] p(e,b,b) 8. 1] p(e,b,b) 9. 6] p(c,a,b) p(c,c,e) p(c,b,a) p(e,b,b) 15 10. 3] p(c,c,e) 11. 5] ....

[Article contains additional citation context not shown here]

J.A. Bergstra and J.W. Klop. Conditional rewrite rules: Confluence and termination. J. Comput. System Sci., 32:323--362, 1986.

No context found.

Jan A. Bergstra and Jan Willem Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Sciences 32, 323--362, 1986.

No context found.

J.A. Bergstra and J.W. Klop. Conditional Rewrite Rules: confluence and termination. Journal of Computer and System Sciences, 32:323--362, 1986.

No context found.

J. A. Bergstra and J. W. Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Sciences, 32:323-- 362, 1986.

No context found.

J. A. Bergstra and J. W. Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Sciences, 32(3):323--362, 1986.

No context found.

J.A. Bergstra and J.W. Klop. Conditional rewrite rules: Confluence and termination. J. Comput. System Sci., 32:323--362, 1986.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC