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An Implementation of Narrowing Strategies
 Journal of the ACM
, 2001
"... This paper describes an implementation of narrowing, an essential component of implementations of modern functional logic languages. These implementations rely on narrowing, in particular on some optimal narrowing strategies, to execute functional logic programs. We translate functional logic progra ..."
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Cited by 291 (121 self)
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This paper describes an implementation of narrowing, an essential component of implementations of modern functional logic languages. These implementations rely on narrowing, in particular on some optimal narrowing strategies, to execute functional logic programs. We translate functional logic programs into imperative (Java) programs without an intermediate abstract machine. A central idea of our approach is the explicit representation and processing of narrowing computations as data objects. This enables the implementation of operationally complete strategies (i.e., without backtracking) or techniques for search control (e.g., encapsulated search). Thanks to the use of an intermediate and portable representation of programs, our implementation is general enough to be used as a common back end for a wide variety of functional logic languages.
Theorem Proving Modulo
 Journal of Automated Reasoning
"... Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first ..."
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Cited by 78 (14 self)
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Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higherorder logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higherorder logic subsumes full higherorder resolution.
Completeness Results for Basic Narrowing
, 1994
"... In this paper we analyze completeness results for basic narrowing. We show that basic narrowing is not complete with respect to normalizable solutions for equational theories defined by confluent term rewriting systems, contrary to what has been conjectured. By imposing syntactic restrictions on the ..."
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Cited by 44 (2 self)
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In this paper we analyze completeness results for basic narrowing. We show that basic narrowing is not complete with respect to normalizable solutions for equational theories defined by confluent term rewriting systems, contrary to what has been conjectured. By imposing syntactic restrictions on the rewrite rules we recover completeness. We refute a result of Holldobler which states the completeness of basic conditional narrowing for complete (i.e. confluent and terminating) conditional term rewriting systems without extra variables in the conditions of the rewrite rules. In the last part of the paper we extend the completeness result of Giovannetti and Moiso for levelconfluent and terminating conditional systems with extra variables in the conditions to systems that may also have extra variables in the righthand sides of the rules. 1985 Mathematics Subject Classification: 68Q50 1987 CR Categories: F.4.1, F.4.2 Key Words and Phrases: narrowing, basic narrowing, conditional narrowin...
The Integration of Functions into Logic Programming: A Survey
, 1994
"... Functional and logic programming are the most important declarative programming paradigms, and interest in combining them has grown over the last decade. Early research concentrated on the definition and improvement of execution principles for such integrated languages, while more recently efficient ..."
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Cited by 35 (0 self)
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Functional and logic programming are the most important declarative programming paradigms, and interest in combining them has grown over the last decade. Early research concentrated on the definition and improvement of execution principles for such integrated languages, while more recently efficient implementations of these execution principles have been developed so that these languages became relevant for practical applications. In this paper we survey the development of the operational semantics as well as
A completionbased method for mixed universal and rigid Eunification
 PROC. 12TH CONFERENCE ON AUTOMATED DEDUCTION CADE, NANCY/FRANCE, LNAI 814
, 1994
"... We present a completionbased method for handling a new version of Eunification, called “mixed” Eunification, that is a combination of the classical “universal” Eunification and “rigid” Eunification. Rigid Eunification is an important method for handling equality in Gentzentype firstorder cal ..."
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Cited by 27 (8 self)
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We present a completionbased method for handling a new version of Eunification, called “mixed” Eunification, that is a combination of the classical “universal” Eunification and “rigid” Eunification. Rigid Eunification is an important method for handling equality in Gentzentype firstorder calculi, such as freevariable semantic tableaux or matings. The performance of provers using Eunification can be increased considerably, if mixed Eunification is used instead of the purely rigid version. We state soundness and completeness results, and describe experiments with an implementation of our method.
Implementation of Narrowing: The PrologBased Approach
 Logic programming languages: constraints, functions, and objects
, 1993
"... We present the problem of integrating functional languages and logic languages. We explain why the narrowingbased techniques have so far prevailed as operational mechanisms for the functional logic interpreters. We then discuss various strategies of narrowing. Finally we explain how to simulate the ..."
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Cited by 18 (0 self)
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We present the problem of integrating functional languages and logic languages. We explain why the narrowingbased techniques have so far prevailed as operational mechanisms for the functional logic interpreters. We then discuss various strategies of narrowing. Finally we explain how to simulate these strategies of narrowing using the leftmost SLDresolution rule of Prolog, and compare some experimental results with those obtained with direct narrowing implementations. 1. Introduction There has been a flurry of research on the integration of functional programming (FP) and logic programming (LP). A natural framework would be to consider the union of a set H of Horn clauses with a set E of conditional equations as a program. The declarative semantics of a program is then given by firstorder logic with equality [26], that is, firstorder logic extended with an equality symbol and the standard equality axioms. The operational semantics of a program is usually given by a system of infere...
Combining Lazy Narrowing and Simplification
 In Proc. of the 6th International Symposium on Programming Language Implementation and Logic Programming
, 1994
"... . Languages that integrate functional and logic programming styles with a complete operational semantics are based on narrowing. In order to avoid useless computations and to deal with infinite data structures, lazy narrowing strategies have been proposed in the past. This paper presents an impo ..."
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Cited by 14 (5 self)
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. Languages that integrate functional and logic programming styles with a complete operational semantics are based on narrowing. In order to avoid useless computations and to deal with infinite data structures, lazy narrowing strategies have been proposed in the past. This paper presents an important improvement of lazy narrowing by incorporating deterministic simplification steps into lazy narrowing derivations. These simplification steps reduce the search space so that in some cases infinite search spaces are reduced to finite ones. We show that the completeness of lazy narrowing is not destroyed by the simplification process and demonstrate the improved operational behavior by means of several examples. 1 Introduction In recent years, a lot of proposals have been made to amalgamate functional and logic programming languages [19]. Functional logic languages with a sound and complete operational semantics are based on narrowing, a combination of the reduction principle of ...
Lazy Unification with Simplification
 In Proc. 5th European Symposium on Programming
, 1994
"... . Unification in the presence of an equational theory is an important problem in theoremproving and in the integration of functional and logic programming languages. This paper presents an improvement of the proposed lazy unification methods by incorporating simplification into the unification ..."
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Cited by 13 (3 self)
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. Unification in the presence of an equational theory is an important problem in theoremproving and in the integration of functional and logic programming languages. This paper presents an improvement of the proposed lazy unification methods by incorporating simplification into the unification process. Since simplification is a deterministic computation process, more efficient unification algorithms can be achieved. Moreover, simplification reduces the search space so that in some case infinite search spaces are reduced to finite ones. We show soundness and completeness of our method for equational theories represented by ground confluent and terminating rewrite systems which is a reasonable class w.r.t. functional logic programming. 1 Introduction Unification is not only an important operation in theorem provers but also the most important operation in logic programming systems. Unification in the presence of an equational theory, also known as Eunification, is necessary...