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...

Partial evaluation is a method for program specialization based on fold/unfold transformations [8, 25]. Partial evaluation of pure functional programs uses mainly static values of given data to specialize the program [15, 44]. In logic programming, the so-called static/dynamic distinction is hardly present, whereas considerations of determinacy and choice points are far more important for control [12]. We discuss these issues in the context of a (lazy) functional logic language. We formalize a two-phase specialization method for a non-strict, first order, integrated language which makes use of lazy narrowing to specialize the program w.r.t. a goal. The basic algorithm (first phase) is formalized as an instance of the framework for the partial evaluation of functional logic programs of [2, 3], using lazy narrowing. However, the results inherited by [2, 3] mainly regard the termination of the PE method, while the (strong) soundness and completeness results must be restated for the lazy strategy. A post-processing renaming scheme (second phase) is necessary which we describe and illustrate on the well-known matching example. This phase is essential also for other non-lazy narrowing strategies, like innermost narrowing, and our method can be easily extended to these strategies. We show that our method preserves the lazy narrowing semantics and that the inclusion of simplification steps in narrowing derivations can improve control during specialization.

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

. 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 ...

We describe decision procedures for certain classes of semantic matching problems, where the equational theory with respect to which the semantic matching is performed has a convergent rewrite system. We give counterexamples to show that semantic matching becomes undecidable (as it generally is) when the conditions wegive are weakened.

. Unification in the presence of an equational theory is an important problem in theorem-proving 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 E-unification, is necessary...

We present a number of new results on inductive theorem proving for design specifications based on Horn logic with equality. Induction is explicit here because induction orderings are supposed to be part of the specification. We show how the automatic support for program verification is enhanced if the specification satisfies a bunch of rewrite properties, summarized under the notion of canonicity. The enhancement is due to inference rules and corresponding strategies whose soundness is implied by the specification's canonicity. The second main result of the paper provides a method for proving canonicity by using the same rules, which are applied in proofs of conjectures about the specification and the functional-logic programs it contains. Contents 1 Introduction 2 1.1 Expander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Proof by term rewriting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

Languages that integrate functional and logic programming styles with a complete operational semantics are based on narrowing. In order to avoid useless computations, lazy narrowing strategies have been proposed in the past. This paper presents an 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 consider two classes of programs where this strategy can be applied. Firstly, we show soundness and completeness of our strategy for functional logic programs based on ground confluent and terminating rewrite systems. Then, we show similar results for constructor-based weakly orthogonal (not necessarily terminating) rewrite systems. Finally, we demonstrate the improved operational behavior by means of several examples. Since most functional logic languages are based on programs belonging to one of these classes, our result is a significant step to improve the operational semantics of existing functional logic languages.

Let E be a first-order equational theory. A translation of typed higher-order E-unification problems into a typed combinatory logic framework is presented and justified. The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modification of ordinary narrowing is shown to be a complete method for enumerating higher-order E-unifiers. In fact, we treat a more general problem, in which the types of terms contain type variables. 1 Introduction Investigation of the interaction between first-order and higher-order equational reasoning has emerged as an active line of research. The collective import of a recent series of papers, originating with [Bre88] and including (among others) [Bar90], [BG91a], [BG91b], [Dou92], [JO91] and [Oka89], is that when various typed -calculi are enriched by first-order equational theories, the validity problem is well-behaved, and furthermore that the respective computational approaches to ...

We present an algorithm for unification in the simply typed lambda calculus which enumerates complete sets of unifiers using a finitely branching search space. In fact, the types of terms may contain type-variables, so that a solution may involve typesubstitution as well as term-substitution. the problem is first translated into the problem of unification with respect to extensional equality in combinatory logic, and the algorithm is defined in terms of transformations on systems of combinatory terms. These transformations are based on a new method (itself based on systems) for deciding extensional equality between typed combinatory logic terms. 1 Introduction This paper develops a new algorithm for higher-order unification. A higher-order unification problem is specified by two terms F and G of the explicitly simply typed lambda calculus LC; a solution is a substitution oe such that oeF = fij oeG. We will always assume the extensionality axiom j in this paper. In fact we tre...