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.

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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 higher-order logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higher-order logic subsumes full higher-order resolution.

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 level-confluent and terminating conditional systems with extra variables in the conditions to systems that may also have extra variables in the right-hand 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...

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

We present a completion-based method for handling a new version of E-unification, called “mixed” E-unification, that is a combination of the classical “universal” E-unification and “rigid” E-unification. Rigid E-unification is an important method for handling equality in Gentzen-type first-order calculi, such as free-variable semantic tableaux or matings. The performance of provers using E-unification can be increased considerably, if mixed E-unification is used instead of the purely rigid version. We state soundness and completeness results, and describe experiments with an implementation of our method.

We present the problem of integrating functional languages and logic languages. We explain why the narrowing-based 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 SLD-resolution 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 first-order logic with equality [26], that is, first-order 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...

The paper presents a framework to describe, experiment and study the combination of different computational systems including the constraint solving paradigm. Computational systems are interpreted in a first-order setting thanks to an evaluator that rewrites formulas.

. 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 ...