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HigherOrder Narrowing with Convergent Systems
 In 4th Int. Conf. Algebraic Methodology and Software Technology, AMAST '95. Springer LNCS 936
, 1995
"... . Higherorder narrowing is a general method for higherorder equational reasoning and serves for instance as the foundation for the integration of functional and logic programming. We present several refinements of higherorder lazy narrowing for convergent (terminating and confluent) term rewr ..."
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. Higherorder narrowing is a general method for higherorder equational reasoning and serves for instance as the foundation for the integration of functional and logic programming. We present several refinements of higherorder lazy narrowing for convergent (terminating and confluent) term rewrite systems and their application to program transformation. The improvements of narrowing include a restriction of narrowing at variables, generalizing the firstorder case. Furthermore, functional evaluation via normalization is shown to be complete and a partial answer to the eager variable elimination problem is presented. 1 Introduction and Overview Higherorder narrowing is a method for solving higherorder equations modulo a set of rewrite rules. It forms the basis of functionallogic programming and has been extensively studied in the firstorder case, for a survey see [10]. Motivated by functional programming, there exist several higherorder extensions for such languages [7, ...
Development Closed Critical Pairs
, 1996
"... . The class of orthogonal rewriting systems (rewriting systems where rewrite steps cannot depend on one another) is the main class of notnecessarilyterminating rewriting systems for which confluence is known to hold. Huet and Toyama have shown that for leftlinear firstorder term rewriting sys ..."
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. The class of orthogonal rewriting systems (rewriting systems where rewrite steps cannot depend on one another) is the main class of notnecessarilyterminating rewriting systems for which confluence is known to hold. Huet and Toyama have shown that for leftlinear firstorder term rewriting systems (TRSs) the orthogonality restriction can be relaxed somewhat by allowing critical pairs (arising from maximally general ways of dependence between steps), but requiring them to be parallel closed. We extend these results by replacing the parallel closed condition by a development closed condition. This also permits to generalise them to higherorder term rewriting, yielding a confluence criterion for Klop's combinatory reduction systems (CRSs), Khasidashvili's expression reduction systems (ERSs), and Nipkow's higherorder pattern rewriting systems (PRSs). 1 Introduction This paper is concerned with a method to prove confluence of rewriting systems. It's an extension of some co...
Observations About Using Logic as a Specification Language
, 1995
"... This extended abstract contains some nontechnical observations about the roles that logic can play in the specification of computational systems. In particular, computationasdeduction, metaprogramming, and higherorder abstract syntax are briefly discussed. 1 Two approaches to specifications In ..."
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This extended abstract contains some nontechnical observations about the roles that logic can play in the specification of computational systems. In particular, computationasdeduction, metaprogramming, and higherorder abstract syntax are briefly discussed. 1 Two approaches to specifications In the specification of computational systems, logics are generally used in one of two approaches. In one approach, computations are mathematical structures, containing such items as nodes, transitions, and state, and logic is used in an external sense to make statements about those structures. That is, computations are used as models for logical expressions. Intensional operators, such as the modals of temporal and dynamic logics or the triples of Hoare logic, are often employed to express propositions about the change in state. For example, nexttime modal operators are used to describe the possible evolution of state; expressions in the HennesseyMilner are evaluated against the transitions...
Some Applications of FunctionalLogic Programming
, 1996
"... We show examples for higherorder functional logic programming in which both programming concepts nicely complement each other. These include modeling distributed systems, parsing and hardware synthesis. ..."
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We show examples for higherorder functional logic programming in which both programming concepts nicely complement each other. These include modeling distributed systems, parsing and hardware synthesis.
Higherorder Lazy Narrowing Calculus: a Solver for HigherOrder Equations
 in Proc. 8th Int’l Conf. Computer Aided Systems (EuroCAST 2001), LNCS
, 2001
"... This paper introduces a higherorder lazy narrowing calculus (HOLN for short) that solves higherorder equations over the domain of simply typed #terms. HOLN is an extension and refinement of Prehofer's higherorder narrowing calculus LN using the techniques developed in the refinement of ..."
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This paper introduces a higherorder lazy narrowing calculus (HOLN for short) that solves higherorder equations over the domain of simply typed #terms. HOLN is an extension and refinement of Prehofer's higherorder narrowing calculus LN using the techniques developed in the refinement of a firstorder lazy narrowing calculus LNC.
Refinements of lazy narrowing for leftlinear fully extened pattern rewrite systems
, 2001
"... Abstract. Lazy narrowing is a general Eunification procedure for equational theories presented by confluent term rewriting systems. It has been deeply studied in the first order case and various higherorder extensions have been proposed in an attempt to improve its expressive power. Such extension ..."
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Abstract. Lazy narrowing is a general Eunification procedure for equational theories presented by confluent term rewriting systems. It has been deeply studied in the first order case and various higherorder extensions have been proposed in an attempt to improve its expressive power. Such extensions suffer from huge search space in guessing the solutions of variables of functional type. For practical purposes, the need to reduce the search space of solutions is of paramount importance. In this paper we introduce HOLN, a higherorder lazy narrowing calculus for Eunification in theories presented by pattern rewrite systems. The calculus is designed to deal with both oriented and unoriented equations, and keeps track of the variables which are to be bound to normalized solutions. We discuss the operating principle of HOLN, its main properties, and propose refinements to reduce its search space for solutions. Our refinements are defined for classes of leftlinear fullyextended pattern rewrite systems which are widely used in higherorder functional logic programming. 1
Refinements of Lazy Narrowing for LeftLinear FullyExtended Pattern Rewrite Systems Mircea Marin
, 2001
"... Lazy narrowing is a general Eunification procedure for equational theories presented by confluent term rewriting systems. It has been deeply studied in the first order case and various higherorder extensions have been proposed in an attempt to improve its expressive power. Such extensions suffer f ..."
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Cited by 1 (1 self)
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Lazy narrowing is a general Eunification procedure for equational theories presented by confluent term rewriting systems. It has been deeply studied in the first order case and various higherorder extensions have been proposed in an attempt to improve its expressive power. Such extensions suffer from huge search space in guessing the solutions of variables of functional type. For practical purposes, the need to reduce the search space of solutions is of paramount importance. In this paper we introduce HOLN, a higherorder lazy narrowing calculus for Eunification in theories presented by pattern rewrite systems. The calculus is designed to deal with both oriented and unoriented equations, and keeps track of the variables which are to be bound to normalized solutions. We discuss the operating principle of HOLN, its main properties, and propose refinements to reduce its search space for solutions. Our refinements are defined for classes of leftlinear fullyextended pattern rewrite systems which are widely used in higherorder functional logic programming.
A Distributed System for Solving Equational Constraints Based on Lazy Narrowing Calculi
, 1999
"... In this paper we describe the architecture and implementation of a system that aims at extending in a consistent way a functional logic programming language with solving techniques of various constraint solving systems. The system is called CFLP (Constrained Functional Logic Programming language), a ..."
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In this paper we describe the architecture and implementation of a system that aims at extending in a consistent way a functional logic programming language with solving techniques of various constraint solving systems. The system is called CFLP (Constrained Functional Logic Programming language), and consists of a lazy functional logic interpreter extended in two directions: the possibility to specify constraints, and the possibility to specify AND and ORparallelism. For solving the constraints, a distributed constraint solving system was implemented.
HigherOrder FunctionalLogic Programming: A Systematic Development
 In Proc. of the Second Fuji International Workshop on Functional and Logic Programming. World Scientific
, 1997
"... We develop an effective model for higherorder functionallogic programming by refining higherorder narrowing calculi. The refinements reduce the high degree of nondeterminism in narrowing calculi, utilizing properties of functional(logic) programs. These include convergent and leftlinear rewrit ..."
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We develop an effective model for higherorder functionallogic programming by refining higherorder narrowing calculi. The refinements reduce the high degree of nondeterminism in narrowing calculi, utilizing properties of functional(logic) programs. These include convergent and leftlinear rewrite rules. All refinements can be combined to a narrowing strategy which generalizes callbyneed as in functional programming. Furthermore, we consider conditional rewrite rules which are often convenient for programming applications. 1. Introduction We present a systematic development of a calculus which integrates higherorder functional and logic programming, based on narrowing. Narrowing is a general method for solving equations modulo a set of rewrite rules. Functionallogic languages with a sound and complete operational semantics are mainly based on narrowing. For a survey on the topic we refer to [9]. In our higherorder equational logic we use a rewrite relation due to Nipkow [18],...
An Optimized Decision Algorithm for Stratified Context Unification
, 2000
"... Context unification is a variant of second order unification. It can also be seen as a generalization of string unification to tree unification. Currently it is not known whether context unification is decidable. A specialization of context unification is stratified context unification, which is ..."
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Context unification is a variant of second order unification. It can also be seen as a generalization of string unification to tree unification. Currently it is not known whether context unification is decidable. A specialization of context unification is stratified context unification, which is decidable. However, the previous algorithm has a very bad worst case complexity. Recently it turned out that stratified context unification is equivalent to satisfiability of onestep rewrite constraints. This paper contains an optimized algorithm for stratified context unification exploiting sharing and power expressions. We prove that the complexity is determined mainly by the maximal depth of SOcycles. Two observations are used: i. For every ambiguous SOcycle, there is a context variable that can be instantiated with a ground context of main depth O(c d), where c is the number of context variables, and d is the depth of the SOcycle. ii. the exponent of periodicity is O(2 ...