S. Kaplan, Conditional Rewrite Rules, TCS 33 (1984) 175--193.  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 ....

....because the bidirectional use of rewrite rules in the conditions goes against the spirit of rewriting. In a join system the applicability of conditional rewrite rules is determined by joinability of the conditions. Most of the literature on conditional rewriting addresses join systems. Kaplan [8] showed that join systems are logical, provided they are confluent. Recently, oriented systems emerged as the most natural type of conditional rewriting when modeling logic and functional programming, especially when allowing extra variables in Partially supported by the Advanced Information ....

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S. Kaplan, Conditional Rewrite Rules, TCS 33 (1984) 175--193.

....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) and substitution ....

....goal can be solved by applying reflection and we obtain the answer that Mary s credibility is high. One should observe that the question of whether a goal clause can be rewritten is decidable, whereas the same question is undecidable if we consider, for example, Kaplan s rewrite relation (see [Kaplan, 1984]) Moreover, we may apply other simplifications rules such as removal of trivial equations, decomposition of decomposible equations [Kirchner, 1984] and elimination of variables if the goal clause contains an equation of the form x =t and no defined function symbol occurs in t. Despite these ....

S. Kaplan. Conditional rewrite rules. Journal of Theoretical Computer Science, 33:175--193, 1984.

....; 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, di erent rewrite relations can be associated with a CTRS, cf. e.g. [11,12,15,16,20,22,23,26,27,29,32]. In our veri cation example, we transformed the problem into an oriented CTRS [32] 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) 6 J urgen Giesl, Thomas Arts ....

Kaplan, S.: Conditional rewrite rules. TCS 33, 175-193 (1984)

....unit equations. For instance, for obtaining confluent sets for rewrite relations like conditional and clausal rewriting, completion procedures were designed for transforming sets of conditional equations (definite Horn clauses with equality, i.e. of the form s 1 t 1 : s n t n s t) [Kaplan 1984, Jouannaud and Waldmann 1986, Kounalis and Rusinowitch 1991, Ganzinger 1991] or restricted equality clauses [Nieuwenhuis and Orejas 1990] The generalization of this kind of completion procedure to full first order clauses with equality required the development of more powerful proof techniques ....

Kaplan S. [1984], `Conditional rewrite rules', Theoretical Computer Science 33, 175--193.

....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 ....

S. Kaplan, Conditional rewrite rules. TCS, 33:175--193, 1984.

....the 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 ....

....because the bidirectional use of rewrite rules in the conditions goes against the spirit of rewriting. In a join system the applicability of conditional rewrite rules is determined by joinability of the conditions. Most of the literature on conditional rewriting addresses join systems. Kaplan [8] showed that join systems are logical, provided they are confluent. Recently, oriented systems emerged as the most natural type of conditional rewriting when modeling logic and functional programming, especially when allowing extra variables in Partially supported by the Advanced Information ....

[Article contains additional citation context not shown here]

S. Kaplan, Conditional Rewrite Rules, TCS 33 (1984) 175--193.

....orderings, and for confluence by computing critical pairs. Note that the termination arguments for the rules in RD and R C may be different: nothing prevents the whole set of rules to be non terminating. 2. 5 Kaplan s Specification of Integers An adaptation of Kaplan s specification of integers [23] is shown in Figure 1, together with its corresponding automaton in Figure 2. It is paradigmatic in many respects: it has non free constructors, defined symbols, and the rules make essential use of Horn clauses for giving the semantics of the defined symbols. Unlike Kaplan s original unsorted ....

S. Kaplan. Conditional rewrite rules. Theoretical Comput. Sci., 33:175--193, 1984.

....jE on T (F ; V) such that soe jE toe whenever s 1 oe jE t 1 oe; s n oe jE t n oe, for all s 1 t 1 ; s n t n ) s t 2 E and substitutions oe. We call jE the congruence generated by E. Proof. We only sketch the idea of the proof. The missing details can be found in [Kaplan, 1984] and [Kaplan, 1983] Let X be the complete lattice of congruences on T (F ; V) partially ordered by inclusion . We de ne the function : X X; 1 7 2 by: i) s 1 t implies s 2 t, for all s, t 2 T (F ; V) ii) 2 is the smallest congruence relation such that soe 2 toe whenever s 1 ....

....11 If E consists only of unconditional equations it is quite clear that jE= E . Kaplan showed that this also holds for the case of conditional equations. Proposition 2.29. Given a set E of conditional equations. s jE t if and only if s E t, for all terms s; t 2 T (F ; V) Proof. [Kaplan, 1984] 12 2. TERMINOLOGY AND CLASSICAL RESULTS 4. Rewrite Systems As mentioned in the last section, reasoning in equational theories can be seen as replacement of an instance of one side of an equation by the corresponding instance of the other side of the equation. The central idea in rewriting is ....

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Kaplan, S. (1984). Conditional rewrite rules. Theoret. Comput. Sci., 33:175193.

.... hold in the underlying theory [16] This semantics admits an equivalent fixpoint characterization and, for programs that can be oriented as a confluent term rewriting system , it can be further characterized as the set of all pairs of (ground) terms which can be reduced to a common normal form [10,16,18]. E m n n n m ca Example 1 Example 2 1 2 1 2 1 2 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 E E E [ E E E E f ( g E f ( g M E M E f g ( f g fg E f g E E f ( g E f (g M E f g M E f g M E [ E f g M E [ E M E M E E E f f ; f x E ; f ; f ; f ; f f ; f f f x x= x= E E E a b b c b c a a; b b; c c b ....

S. Kaplan. Conditional Rewrite Rules. , 33:175--193, 1984.

....theorem from structural operational semantics to conditional term rewriting. The result is useful for the development of software renovation factories, and for modular specification of abstract data types. 1 Introduction There is a strong link between the worlds of conditional term rewriting [30, 8] and of structural operational semantics (SOS) 35] In fact, from a conceptual level they can be seen as identical. In both fields, terms are built from a set of function symbols. The binary relations on terms, rewrite steps and transitions, both are defined inductively by means of proof rules, ....

S. Kaplan. Conditional rewrite rules. TCS, 33(2):175--193, 1984.

....using the reduction relation again. This may lead to a non terminating evaluation process for the conditions, even for terminating systems. In fact, the rewrite relation (and reducibility) may be undecidable even for ( nite) CTRSs without extra variables which are terminating and conAEuent ([8], cf. also [13] 3 Note that s # n R t is to denote that the depth of s #R t is at most n. In order to ensure decidability of the basic notions like reducibility one has to impose a stronger condition than termination: A CTRS R is said to be decreasing ( 5] if there exists a well founded ....

S. Kaplan. Conditional rewrite rules. TCS, 33:175193, 1984.

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S. Kaplan. Conditional rewrite rules. TCS, 33:175--193, 1984.

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Kaplan, S. (1984). Conditional Rewrite Rules. Theoretical Computer Science 33(2), pp. 175--193.

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