A needed narrowing strategy

Abstract

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.

Trends in Functional Programming

“…Forward Reach uses lazy narrowing in order to generate inputs e ciently. Lazy narrowing is a concept from functional logic programming [2,7] and can be described as the natural extension of a non-strict semantics to a language with free variables. Free variables are only bound when their value is required for evaluation to proceed.…”

“…We introduce and compare two particularly relevant theories to ours. In their seminal work, Antoy et al [2] established the soundness and completeness of the related notion of needed narrowing, and the optimality of needed narrowing within a restricted domain. However, whereas our formalisation is based on extending a small-step semantics, theirs is based on classical rewrite systems.…”

A Virtual Machine for Functional Logic Computations

“…We compile each definitional tree into a sequence of virtual machine instructions. Because a definitional tree is a high-level abstraction for the definition of a sound, complete and theoretically efficient narrowing strategy [6], mapping this strategy into virtual machine instructions increases our confidence in both the correctness and the efficiency of the execution. The notation for the variant of definitional trees we use is summarized in Figure 3.…”

Functional Logic Design Patterns

“…An appropriate notion of "need" is a subtle point in the presence of a nondeterministic choice, e.g., a typical computation step, since the need of an argument might depend on the choice itself. [4] proposes a needed narrowing strategy to solve this problem. This strategy is optimal w.r.t.…”

“…Since it is based on an optimal evaluation strategy [4], Curry can be considered a generalization of concurrent constraint programming [36] with a lazy (optimal) evaluation strategy. Furthermore, Curry also offers features for application programming like modules, monadic I/O, encapsulated search [27], ports for distributed programming [22], libraries for GUI [23] and HTML programming [24] etc.…”