Abstract. Functional logic programming languages combine the most important declarative programming paradigms, and attempts to combine these paradigms have a long history. The declarative multi-paradigm language Curry is influenced by recent advances in the foundations and implementation of functional logic languages. The development of Curry is an international initiative intended to provide a common platform for the research, teaching, and application of integrated functional logic languages. This paper surveys the foundations of functional logic programming that are relevant for Curry, the main features of Curry, and extensions and applications of Curry and functional logic programming. 1

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.

We define a transformation from a left-linear constructor-based conditional rewrite system into an overlapping inductively sequential rewrite system. This transformation is sound and complete for the computations in the source system. Since there exists a sound and complete narrowing strategy for the target system, the combination of these results offers the first procedure for provably sound and complete narrowing computations for the whole class of the leftlinear constructor-based conditional rewrite systems. We address the differences between demand driven and lazy strategies and between narrowing strategies and narrowing calculi. In this context, we analyze the efficiency and practicality of using our transformation for the implementation of functional logic programming languages. The results of this paper complement, extend, and occasionally rectify, previously published results in this area. Categories and Subject Descriptors D.1.1 [Programming Techniques]: Applicative (Functional) Programming; D.1.6 [Programming Techniques]: Logic Programming; D.3.3 [Programming Languages]: Language Constructs and Features---Control structures; D.3.4 [Programming Languages ]: Processors---Optimization; F.4.2 [Mathematical Logic and Formal Languages]: Grammars and Other Rewriting Systems; I.1.1 [Algebraic Manipulation]: Expressions and Their Representation ---Simplification of expressions; I.2.2 [Automatic Programming ]: Program transformation General Terms Algorithms, Languages, Performance, Theory Keywords Functional Logic Programming Languages, Rewrite Systems, Narrowing Strategies, Call-By-Need This work has been supported in part by the National Science Foundation grant INT-9981317. 1.

1 Introduction Michael Hanus Salvador Lucas Aart Middeldorp Strongly sequential and inductively sequential term rewriting systems hanus@informatik.rwth-aachen.de slucas@dsic.upv.es ami@score.is.tsukuba.ac.jp first(0,x) [] first(s(x),y::z) y::first(x,z) first x 1 0 s(x) x 2 first x 1 x 2 first 0 x 2 first s x x 2 [] first s x y z y first x z first inductively sequential outermost-needed strategy needed rewriting needed Definitional trees inductively sequential inductively sequential index trees forward-branching index trees matching dags outermost-needed strategy index reduction Informatik II, RWTH Aachen, D-52056 Aachen, Germany, . Work partially supported by DFG (under grant Ha 2457/1-1) and Acci'on Integrada. DSIC, U.P. de Valencia, Camino de la Vera s/n, Apdo. 22012, E-46071 Valencia, Spain, . Work partially supported by EEC-HCM grant ERBCHRXCT940624, Bancaixa (Bancaja-Europa grant), Acci'on Integrada (HA19970073) and CICYT (under grant TIC 95-0433-C03-03). Institute of Informa...

Abstract. Declarative programming languages advocate a programming style expressing the properties of problems and their solutions rather than how to compute individual solutions. Depending on the underlying formalism to express such properties, one can distinguish different classes of declarative languages, like functional, logic, or constraint programming languages. This paper surveys approaches to combine these different classes into a single programming language. 1

In this paper we study the non-determinism between the inference rules of the lazy narrowing calculus lnc (Middeldorp et al., 1996). We show that all non-determinism can be removed without losing the important completeness property by restricting the underlying term rewriting systems to left-linear con uent constructor systems and interpreting equality as strict equality. For the subclass of orthogonal constructor systems the resulting narrowing calculus is shown to have the nice property that solutions computed by di erent derivations starting from the same goal are incomparable. 1.

Functional logic languages combine the operational principles of the most important declarative programming paradigms, namely functional and logic programming. Inductively sequential programs admit the definition of optimal computation strategies and are the basis of several recent (lazy) functional logic languages. In this paper, we define a partial evaluator for inductively sequential functional logic programs. We prove strong correctness of this partial evaluator and show that the nice properties of inductively sequential programs carry over to the specialization process and the specialized programs. In particular, the structure of the programs is preserved by the specialization process. This is in contrast to other partial evaluation methods for functional logic programs which can destroy the original program structure. Finally, we present some experiments which highlight the practical advantages of our approach. 1 Introduction Functional logic languages combine the operational p...

. Recent advances in the foundations and the development of functional logic programming languages originate from far-reaching results on narrowing evaluation strategies. Narrowing is a computation similar to rewriting which yields substitutions in addition to normal forms. In functional logic programming, the classes of rewrite systems to which narrowing is applied are, for the most part, subclasses of the constructor-based, possibly conditional, rewrite systems. Many interesting narrowing strategies, particularly for the smallest subclasses of the constructor-based rewrite systems, are generalizations of wellknown rewrite strategies. However, some strategies for larger non-confluents subclasses have been developed just for functional logic computations. In this paper, I will discuss the elements that play a relevant role in evaluation strategies for functional logic programming, describe some important classes of rewrite systems that model functional logic programs, show examples of the differences in expressiveness provided by these classes, and review the characteristics of narrowing strategies proposed for each class of rewrite systems. 1

Functional logic languages with a complete operational semantics are based on narrowing, a unification-based goal-solving mechanism which subsumes the reduction principle of functional languages and the resolution principle of logic languages. Needed narrowing is an optimal narrowing strategy and the basis of several recent functional logic languages. In this paper, we define a partial evaluator for functional logic programs based on needed narrowing. We prove strong correctness of this partial evaluator and show that the nice properties of needed narrowing carry over to the specialization process and the specialized programs. In particular, the structure of the specialized programs provides for the application of optimal evaluation strategies. This is in contrast to other partial evaluation methods for functional logic programs which can change the original program structure in a negative way. Finally, we present some experiments which highlight the practical advantages of our approach.

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