N. Dershowitz and G. Sivakumar. Solving Goals in Equational Languages. In S. Kaplan and J. Joaunnaud, editors, Proc. First Int'l Workshop on Conditional Term Rewriting, volume 308 of Lecture Notes in Computer Science, pages 45--55. Springer-Verlag, Berlin, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....yields more determinism in narrowing derivations. Since the rules are required to be confluent and terminating, normal forms are unique and can be computed by any rewriting strategy. Therefore rewriting can be implemented as a deterministic computation process like reductions in functional lan 8 [37] describes an extension of the rejection rule where the requirement for different constructors is weakened to incomparable function symbols. 11 guages whereas narrowing needs a nondeterministic implementation as in logic languages, i.e. normalizing narrowing unifies the operational principles ....

N. Dershowitz and G. Sivakumar. Solving Goals in Equational Languages. In Proc. 1st Int. Workshop on Conditional Term Rewriting Systems, pp. 45--55. Springer LNCS 308, 1987.

.... s # 1 , s m = s # m , L 1 , L n , #) Mutate: f(s 1 , s m ) t, L 1 , L n , #) # (s 1 = l 1 , s m = l m , r = t, #) where f(l 1 , l m ) r is a renamed rule in R. 6 The above rules are a subset of the transformation rules used in [DS87, MMR89, Mit94] for semantic unification. They yield a complete forwarddecomposition calculus for solving goals of the form s = N , where N is in ground normal form, provided that the underlying TRS is convergent and either variable preserving or left linear (see [DMS92] In the next section, we present a ....

....is introduced next. 4.3 Operator derivability In order to further reduce rule nondeterminism, we provide the RR calculus with some knowledge about the forward derivation paths. In an attempt to combine a forward strategy with a reverse strategy, we adopt the technique of operator rewriting (see [DS87]) or operator derivability. Definition 19 The TRS ROP is derived from a TRS R by the following rules. For each defined function f in R, we add a rule f # f to ROP . For each rule g(l 1 , l q ) # f(r 1 , r m ) # R, where f is a defined function, we add a rule g # f ....

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N. Dershowitz and G. Sivakumar. Solving goals in equational languages. In Proceedings of the 1st International Workshop on Conditional Term Rewriting Systems, volume 308 of Lecture Notes in Computer Science, pages 45--55, Berlin, 1987. SpringerVerlag. 28

....: s m = s 0 m ; L 1 ; L n ; oe) Mutate: f(s 1 ; s m ) t; L 1 ; L n ; oe) s 1 = l 1 ; s m = l m ; r = t; oe) where f(l 1 ; l m ) r is a renamed rule in R. The above rules are a subset of the transformation rules which were used in [DS87, MMR89, Mit94] for semantic unification. They yield a complete forward decomposition calculus for solving goals of the form s = N , where N is in ground normal form, provided that the underlying TRS is convergent and either variable preserving or left linear [DMS92] In the next section, we will present a ....

....Derivability In order to further reduce rule nondeterminism, we provide the RR calculus with some knowledge about the forward derivation paths. This is a first approach to combine kind of a forward strategy with a reverse strategy. We will do this by adopting the technique of operator rewriting [DS87] or operator derivability. Definition 4.6 The TRS ROP is derived from a TRS R by the following rules. For each defined function f in R, we add a rule f f to ROP . For each rule g(l 1 ; l q ) f(r 1 ; r m ) 2 R, where f is a defined function, we add a rule g f to ROP . ....

[Article contains additional citation context not shown here]

N. Dershowitz and G. Sivakumar. Solving goals in equational languages. In Proceedings of the 1st International Workshop on Conditional Term Rewriting Systems, number 308 in Lecture Notes in Computer Science, pages 45--55. Springer Verlag, 1987.

....the considered goal instead of enumerating an infinite set of solutions. Obviously, we can also apply a same kind of reasoning as sketched above in order to improve the resolution of goals of the form t 1 6= t 2 . Several authors have defined techniques to prune the search tree of narrowing. Works [8], 5] are based on the inspection of concrete rewrite rules. In [8] the idea is to analyze how the operators at the top of the left hand sides are replaced by operators on the top of the right hand sides ; and then compute the set of possible head constructors of a term. In [5] Chabin and ....

....Obviously, we can also apply a same kind of reasoning as sketched above in order to improve the resolution of goals of the form t 1 6= t 2 . Several authors have defined techniques to prune the search tree of narrowing. Works [8] 5] are based on the inspection of concrete rewrite rules. In [8], the idea is to analyze how the operators at the top of the left hand sides are replaced by operators on the top of the right hand sides ; and then compute the set of possible head constructors of a term. In [5] Chabin and R ety improve the previous method by taking into account terms and ....

Nachum Dershowitz and G. Sivakumar. Solving goals in equational languages. In Proceedings of the 1st International Conference on Conditional Term Rewriting Systems, CTRS'87, Orsay, France, number 308 in Lecture Notes in Computer Science, pages 45--55. Springer-Verlag, July 1987.

....recognized as a basis of E unification problems (which are sometimes called semantic unification problems) where an E unification problem is the one to decide unifiability of goal terms with respect to axioms. Actually, in some E unification procedures such as narrowing[7] 13] decomposition[5] and transformation method[9] the E unification problem is solved by reducing a given instance of the problem into a set of syntactic unification problems. Usually, the substitution oe used to unify goal terms is a free substitution in the sense that an arbitrary Manuscript received August 31, ....

N. Dershowitz and G. Sivakumar, "Solving goals in equational languages," Proc. of the First Workshop of CTRS, LNCS 308, pp.45--55, Springer-Verlag, Berlin, 1987.

....is satisfied ffl R contains a type clause g( u) h( t) C ; ffl R contains a type clause x : h( t) C with x : g( u) 2 C . The relation ae defines a dependency graph (see also [15] This relation can also been considered as a rewrite system R F for function symbols (see [4]) Let ae denote the transitive closure of ae . Lemma 1 Let h; h 1 ; h 3 2 F t be type constructors and let f 2 F o be a object constructor. ffl If h ae f does not hold, then 9f( t) h( u) cannot follow from R . ffl If there is not any g 2 F o such that h 1 ae g and h 2 ae g ....

N. Dershowitz and G. Sivakumar. Solving goals in equational languages. In S. Kaplan and J.-P. Jouannaud, editors, Conditional Term Rewriting Systems, volume 308 of Lecture Notes in Computer Science, pages 45--55. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1988.

....and a set of constraints representing goal solutions. After the compilation halts, the procedure starts with the goal solutions and words backwards from the goal to the initial equations, solving constraints along the way. If our technique is restricted to narrowing, we could consider the work of [5, 2] to be special cases. We consider our work to be an extension of [8] where it was rst shown how to decide the word problem for ground terms in polynomial time, with a congruence closure algorithm. We show that it is possible to perform congruence closure in polynomial time, even for non ground ....

N. Dershowitz and G. Sivakumar. Solving goals in equational languages. In Proc. of the First Intl. Workshop on Conditional and Typed Rewriting Systems, Lect. Notes in Comput. Sci., vol. 308, (1987) pp. 4555.

.... rule in R This forward decomposition calculus is a complete procedure for solving goals s = N , where N is in ground normal form, and the underlying TRS is canonical and variable preserving or left linear [DMS92] The above rules are a subset of the transformation rules which were used in [DS87, MMR89, Mit90, Mit94] for semantic unification. However, this approach does not explicitly exploit the fact that the right hand side of an initial goal has to be in ground normal form. In the next section we will present a calculus which takes more advantage of this since it starts with the result on the right hand ....

....Derivability In order to further reduce rule nondeterminism, we provide the RR calculus with some knowledge about the forward derivation paths. This is a first approach to combine kind of a forward strategy with a reverse strategy. We will do this by adopting the technique of operator rewriting [DS87] or operator derivability. Definition 4.5 : Operator TRS ROP ) The TRS ROP is derived from a TRS R by the following rules: For each defined function in R, we add a rule f f to ROP . For each rule g(l 1 ; l q ) f(r 1 ; r m ) 2 R where f is a defined function, we add a rule g ....

[Article contains additional citation context not shown here]

N. Dershowitz and G. Sivakumar. Solving goals in equational languages. In S. Kaplan and J.-P. Jouannaud, editors, Proceedings of the 1st International Workshop on Conditional Term Rewriting Systems, volume 308 of Lecture Notes in Computer Science, pages 45--55, Orsay, France, July 1987.

.... Narrowing Approximations as an Optimization for Equational Logic Programs abstract narrower Conselleria de Cultura, Educaci o i Ci encia de la Generalitat Valenciana Mar ia Alpuente , Moreno Falaschi , Mar ia Jos e Ramis and Germ an Vidal The recent interest in logic programming with equations [12,16,18] has promoted much work on equational unification [14,16,27] and narrowing [16,23] Equational unification ( unification) characterizes the problem of solving equations an equational theory . The narrowing mechanism is a powerful Departamento de Sistemas Inform aticos y Computaci on, Universidad ....

....to prune useless paths from the search tree and to save a lot of unnecessary computations. This problem is, in general, undecidable. 5] defines a scheme based on abstract interpretation for the (static) analysis of the unsatisfiability of equation sets, and shows how various analyses such as [2,4,12,13] can be seen as instances of the scheme. In this work, we are concerned with equation solving or unification with respect to a given set of equations. We define an optimization of (basic) conditional narrowing for the purpose of generating complete sets of unifiers in equational theories that can ....

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N. Dershowitz and G. Sivakumar. Solving Goals in Equational Languages. In S. Kaplan and J. Joaunnaud, editors, , volume 308 of , pages 45--55. Springer-Verlag, Berlin, 1987.

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Nachum Dershowitz and G. Sivakumar. Solving goals in equational languages. In S. Kaplan and J.-P. Jouannaud, editors, Proceedings of the First International Workshop on Conditional Term Rewriting Systems #Orsay, France#,volume 308 of Lecture Notes in Computer Science, pages 45#55, Berlin, July 1987. Springer-Verlag.

....provably terminating, thus implying that the semantic unification or semantic matching problems in the corresponding theories are decidable. In this paper, we consider only equational theories for which there is a finite convergent rewrite system. We specialize the unification procedure given in [DS87, Mit90, JK91] and study the effect of some syntactic and semantic restrictions on the rewrite system presenting a theory, which result in decidability. In the remainder of this section, we briefly review the relevant basic notions, terminology and results for equational theories and rewrite systems. For ....

....in the augmented theory. 2 The Matching Procedure We describe a method for semantic matching that is complete for the special cases of matchings that we will consider in Section 3, and later in Section 4. This is a simplified version of the generally complete system for unification appearing in [DS87, Mit90, JK91], which is a refinement of narrowing, as studied in [ Hul80, NRS89, Ret87] and others. We consider equational theories that are given as finite convergent rewrite systems. Convergent systems allow one to ignore reducible solutions to semantic unification and matching problems. For an equational ....

[Article contains additional citation context not shown here]

Nachum Dershowitz and G. Sivakumar. Solving Goals in Equational Languages. In Proceedings of the First International Workshop Conditional Term Rewriting System, Orsay, France, July 1987. Vol. 308, pages 45--55, of Lecture Notes in Computer Science, Springer Verlag (1987).

....and Christian [1992] Termination of a system of rewrite rules is important for using rewriting as a computational tool, and for simplification in theorem provers. Typically, termination proofs are done using path orderings, or by interpreting function symbols as multivariate polynomials; see [Dershowitz, 1987] for a survey of the area. In this thesis, we develop a precedence based binary relation for proving termination of extended rewriting, modulo associativity and commutativity. Our ordering was inspired by the one in [Kapur et al. 1990] Similar research has been reported in [Bachmair and ....

....it is undecidable to even check if a conditional rule can be applied to a term. By restricting the terms in the condition to be smaller (in some well founded ordering) we obtain a class of decreasing systems for which we have methods for checking important properties like confluence, etc. [Dershowitz et al. 1987]. Definition 1 (Decreasing) A conditional rewrite rule is decreasing, if there is a well founded extension of the proper subterm ordering that contains , such that for each rule c : l r and any substitution oe, loe coe. Most useful functions (like factorial) can be defined using ....

[Article contains additional citation context not shown here]

N. Dershowitz and G. Sivakumar. Solving goals in equational languages. In Proceedings of the First International Workshop Conditional Term Rewriting System, Orsay, France, 1987. Volume 308, pages 45--55, of Lecture Notes in Computer Science, Springer Verlag.

....canonical systems. A similar approach is taken in [7] As an alternative, Hullot [11] limits narrowing to certain positions within goals, which he calls basic, and shows that no (irreducible) solutions are lost this way, for Church Rosser systems. A top down version of basic narrowing is given in [4]. The combination of the above two restrictions, basic normal narrowing, turns out to be incomplete even for canonical systems; that is, sometimes solutions are lost. R ety [16] adds certain positions for consideration to assure completeness. 2.3 Flat Concurrent Prolog We give here only the ....

Dershowitz, N., Sivakumar, G., "Solving goals in equational languages", Proceedings of the 1st International Workshop on Conditional Term Rewriting Systems (Orsay, France), Lecture Notes in Computer Science 308, Springer, Berlin, pp. 45--55, 1988.

No context found.

Nachum Dershowitz and G. Sivakumar. Solving goals in equational languages. In S. Kaplan and J.-P. Jouannaud, editors, Proceedings of the First International Workshop on Conditional Term Rewriting Systems (Orsay, France), volume 308 of Lecture Notes in Computer Science, pages 45--55, Berlin, July 1987. Springer-Verlag.

....to so called basic positions is that when one is looking for irreducible solutions, one can ignore paths that narrow within what was a variable of the original goal. There are more general semantic unification methods as well as refinements of narrowing (see [15] Top down methods (e.g. [9, 19]) are particularly appealing. We take the liberty henceforth of referring to all equation solving methods that make use of confluence by the generic term narrowing. Additional superfluous narrowing paths can be avoided by making a distinction between constructor symbols and defined ones ....

....under some assumption of the operational behavior. With (ground confluent and) terminating rules, there is no need for an lazy evaluation strategy to ensure that a value for a term will be reached. Just as an innermost evaluation is appropriate, a narrowing derivation that mimics it suffices [9]. This narrowing derivation, however, might itself not be innermost. Slog [10] always chooses the leftmost innermost narrowing path, and, hence, is complete only in certain situations. The terminating approach to equational programming suggests a normalizing narrowing strategy [7, 8, 16, 19] ....

[Article contains additional citation context not shown here]

Nachum Dershowitz and G. Sivakumar. Solving goals in equational languages. In S. Kaplan and J.-P. Jouannaud, editors, Proceedings of the First International Workshop on Conditional Term Rewriting Systems (Orsay, France), volume 308 of Lecture Notes in Computer Science, pages 45--55, Berlin, July 1987. Springer-Verlag.

No context found.

N. Dershowitz and G. Sivakumar. Solving Goals in Equational Languages. In S. Kaplan and J. Joaunnaud, editors, Proc. First Int'l Workshop on Conditional Term Rewriting, volume 308 of Lecture Notes in Computer Science, pages 45--55. Springer-Verlag, Berlin, 1987.

No context found.

N. Dershowitz and G. Sivakumar. Solving Goals in Equational Languages. In S. Kaplan and J. Joaunnaud, editors, Proc. First Int'l Workshop on Conditional Term Rewriting, 1987.

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