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The narrowing relation over terms constitutes the basis of the most important operational semantics of languages that integrate functional and logic programming paradigms. It also plays an important role in the definition of some algorithms of unification modulo equational theories that are defined by confluent term rewriting systems. Due to the inefficiency of simple narrowing, many refined narrowing strategies have been proposed in the last decade. This paper presents a new narrowing strategy that is optimal in several respects. For this purpose, we propose a notion of a needed narrowing step that, for inductively sequential rewrite systems, extends the Huet and Lévy notion of a needed reduction step. We define a strategy, based on this notion, that computes only needed narrowing steps. Our strategy is sound and complete for a large class of rewrite systems, is optimal with respect to the cost measure that counts the number of distinct steps of a derivation, computes only incomparable and disjoint unifiers, and is efficiently implemented by unification.

Narrowingis the operational principle of languages that integrate functional and logic programming.

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Combining the paradigm features of both logic and functional programming makes for some powerful implementations.

Abstract. Functional logic languages extend purely functional languages with two features: operations defined by overlapping rules and logic variables in both defining rules and expressions to evaluate. In this paper, we show that only one of these features is sufficient in a core language. On the one hand, overlapping rules can be eliminated by introducing logic variables in rules. On the other hand, logic variables can be eliminated by introducing operations defined by overlapping rules. The proposed transformations between different classes of programs not only give a better understanding of the features of functional logic programs but also may simplify implementations of functional logic languages. MotivationFunctional logic languages [20] integrate the best features of functional and logic languages in order to provide a variety of programming concepts. For instance, the concepts of demand-driven evaluation and higher-order functions from functional programming can be combined with logic programming features like computing with partial information (logic variables), constraint solving, and nondeterministic search for solutions. In contrast to purely functional languages, functional logic languages allow computations with overlapping rules (i.e., more than one rule can be applied to evaluate a function call) and logic variables (i.e., unbound variables occurring in the initial expression and/or rules, also called extra variables). Operationally, these features are supported by nondeterministic computation steps.Functional logic languages are modeled by constructor-based term rewriting systems (TRS) with narrowing as the evaluation mechanism. A crucial choice in the design of a language, both at the source level and the implementation level, is the class of rewrite systems used to model the programs. Early languages (e.g., ) were modeled by weakly orthogonal, constructor-based TRSs. Larger classes are more expressive, i.e., programs in

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