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123
Term Rewriting Systems
, 1992
"... Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstract Re ..."
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Cited by 567 (16 self)
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Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstract Reduction Systems
Termination of Term Rewriting Using Dependency Pairs
 Comput. Sci
, 2000
"... We present techniques to prove termination and innermost termination of term rewriting systems automatically. In contrast to previous approaches, we do not compare left and righthand sides of rewrite rules, but introduce the notion of dependency pairs to compare lefthand sides with special subter ..."
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Cited by 210 (47 self)
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We present techniques to prove termination and innermost termination of term rewriting systems automatically. In contrast to previous approaches, we do not compare left and righthand sides of rewrite rules, but introduce the notion of dependency pairs to compare lefthand sides with special subterms of the righthand sides. This results in a technique which allows to apply existing methods for automated termination proofs to term rewriting systems where they failed up to now. In particular, there are numerous term rewriting systems where a direct termination proof with simplification orderings is not possible, but in combination with our technique, wellknown simplification orderings (such as the recursive path ordering, polynomial orderings, or the KnuthBendix ordering) can now be used to prove termination automatically. Unlike previous methods, our technique for proving innermost termination automatically can also be applied to prove innermost termination of term rewriting systems that are not terminating. Moreover, as innermost termination implies termination for certain classes of term rewriting systems, this technique can also be used for termination proofs of such systems.
Solution of the Robbins Problem
 Journal of Automated Reasoning
, 1997
"... . In this article we show that the three equations known as commutativity, associativity, and the Robbins equation are a basis for the variety of Boolean algebras. The problem was posed by Herbert Robbins in the 1930s. The proof was found automatically by EQP, a theoremproving program for equationa ..."
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Cited by 130 (3 self)
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. In this article we show that the three equations known as commutativity, associativity, and the Robbins equation are a basis for the variety of Boolean algebras. The problem was posed by Herbert Robbins in the 1930s. The proof was found automatically by EQP, a theoremproving program for equational logic. We present the proof and the search strategies that enabled the program to find the proof. Key words: Associativecommutative unification, Boolean algebra, EQP, equational logic, paramodulation, Robbins algebra, Robbins problem. 1. Introduction This article contains the answer to the Robbins question of whether all Robbins algebras are Boolean. The answer is yes, all Robbins algebras are Boolean. The proof that answers the question was found by EQP, an automated theoremproving program for equational logic. In 1933, E. V. Huntington presented the following three equations as a basis for Boolean algebra [6, 5]: x + y = y + x, (commutativity) (x + y) + z = x + (y + z), (associativit...
Higherorder Unification via Explicit Substitutions (Extended Abstract)
 Proceedings of LICS'95
, 1995
"... Higherorder unification is equational unification for βηconversion. But it is not firstorder equational unification, as substitution has to avoid capture. In this paper higherorder unification is reduced to firstorder equational unification in a suitable theory: the λσcal ..."
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Cited by 103 (13 self)
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Higherorder unification is equational unification for βηconversion. But it is not firstorder equational unification, as substitution has to avoid capture. In this paper higherorder unification is reduced to firstorder equational unification in a suitable theory: the λσcalculus of explicit substitutions.
Natural termination
 Theoretical Computer Science
"... Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1 ..."
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Cited by 83 (11 self)
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Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1
Narrowingdriven Partial Evaluation of Functional Logic Programs
 ACM TRANSACTIONS ON PROGRAMMING LANGUAGES AND SYSTEMS
, 1996
"... Languages that integrate functional and logic programming with a complete operational semantics are based on narrowing, a unificationbased goalsolving mechanism which subsumes the reduction principle of functional languages and the resolution principle of logic languages. Formal methods of transfo ..."
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Cited by 82 (38 self)
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Languages that integrate functional and logic programming with a complete operational semantics are based on narrowing, a unificationbased goalsolving mechanism which subsumes the reduction principle of functional languages and the resolution principle of logic languages. Formal methods of transformation of functional logic programs can be based on this wellestablished operational semantics. In this paper, we present a partial evaluation scheme for functional logic languages based on an automatic unfolding algorithm which builds narrowing trees. We study the semantic properties of the transformation and the conditions under which the technique terminates, is sound and complete, and is also generally applicable to a wide class of programs. We illustrate our method with several examples and discuss the relation with Supercompilation and Partial Evaluation. To the best of our knowledge this is the first formal approach to partial evaluation of functional logic programs.
Higherorder narrowing with definitional trees
 Neural Computation
, 1996
"... Functional logic languages with a sound and complete operational semantics are mainly based on narrowing. Due to the huge search space of simple narrowing, steadily improved narrowing strategies have been developed in the past. Needed narrowing is currently the best narrowing strategy for firstord ..."
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Cited by 78 (23 self)
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Functional logic languages with a sound and complete operational semantics are mainly based on narrowing. Due to the huge search space of simple narrowing, steadily improved narrowing strategies have been developed in the past. Needed narrowing is currently the best narrowing strategy for firstorder functional logic programs due to its optimality properties w.r.t. the length of derivations and the number of computed solutions. In this paper, we extend the needed narrowing strategy to higherorder functions and λterms as data structures. By the use of definitional trees, our strategy computes only incomparable solutions. Thus, it is the first calculus for higherorder functional logic programming which provides for such an optimality result. Since we allow higherorder logical variables denoting λterms, applications go beyond current functional and logic programming languages.
Theorem Proving Modulo
 Journal of Automated Reasoning
"... Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first ..."
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Cited by 75 (14 self)
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Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higherorder logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higherorder logic subsumes full higherorder resolution.
ContextSensitive Computations in Functional and Functional Logic Programs
 JOURNAL OF FUNCTIONAL AND LOGIC PROGRAMMING
, 1998
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