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A Combinatory Logic Approach to Higherorder Eunification
 in Proceedings of the Eleventh International Conference on Automated Deduction, SpringerVerlag LNAI 607
, 1992
"... Let E be a firstorder equational theory. A translation of typed higherorder Eunification 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 modifi ..."
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Let E be a firstorder equational theory. A translation of typed higherorder Eunification 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 higherorder Eunifiers. In fact, we treat a more general problem, in which the types of terms contain type variables. 1 Introduction Investigation of the interaction between firstorder and higherorder 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 firstorder equational theories, the validity problem is wellbehaved, and furthermore that the respective computational approaches to ...
HigherOrder Unification via Combinators
 Theoretical Computer Science
, 1993
"... 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 typevariables, so that a solution may involve typesubstitution as well as termsubstitution. the ..."
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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 typevariables, so that a solution may involve typesubstitution as well as termsubstitution. 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 higherorder unification. A higherorder 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...
HigherOrder Equational Unification via Explicit Substitutions
 in Proceedings of the tenth UNIF Workshop
, 1996
"... . We show how to reduce the unification problem modulo fij conversion and a firstorder equational theory E, into a firstorder unification problem in a union of two nondisjoint equational theories including E and a calculus of explicit substitutions. A rulebased unification procedure in thi ..."
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Cited by 5 (3 self)
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. We show how to reduce the unification problem modulo fij conversion and a firstorder equational theory E, into a firstorder unification problem in a union of two nondisjoint equational theories including E and a calculus of explicit substitutions. A rulebased unification procedure in this combined theory is described and may be viewed as an extension of the one initially designed by G. Dowek, T. Hardin and C. Kirchner for performing unification of simply typed terms in a firstorder setting via the oecalculus of explicit substitutions. Additional rules are used to deal with the interaction between E and oe. 1 Introduction Unification modulo an equational theory plays an important role in automated deduction and in logic programming systems. For example, Prolog[NM88] is based on higherorder unification, ie. unification modulo the fijconversion. In order to design more expressive higherorder logic programming systems enhanced with a firstorder equational theory E,...
HigherOrder Rigid EUnification
 5th International Conference on Logic Programming and Automated Reasoning', number 822 in `Lecture Notes in Artificial Intelligence
"... . Higherorder Eunification, i.e. the problem of finding substitutions that make two simply typed terms equal modulo fi or fij equivalence and a given equational theory, is undecidable. We propose to rigidify it, to get a resourcebounded decidable unification problem (with arbitrary high bound ..."
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. Higherorder Eunification, i.e. the problem of finding substitutions that make two simply typed terms equal modulo fi or fij equivalence and a given equational theory, is undecidable. We propose to rigidify it, to get a resourcebounded decidable unification problem (with arbitrary high bounds), providing a complete higherorder Eunification procedure. The techniques are inspired from Gallier's rigid Eunification and from Dougherty and Johann's use of combinatory logic to solve higherorder Eunification problems. We improve their results by using general equational theories, and by defining optimizations such as higherorder rigid Epreunification, where flexible terms are used, gaining much efficiency, as in the nonequational case due to Huet. 1 Introduction Higherorder Eunification is the problem of finding complete sets of unifiers of two simply typed terms modulo fi or fijequivalence, and modulo an equational theory E . This problem has applications in higherorder a...