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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...
Normal Forms in Combinatory Logic
 Wesleyan University
, 1992
"... Abstract Let R be a convergent term rewriting system, and let CRequality on (simply typed) combinatory logic terms be the equality induced by βηRequality on terms of the (simply typed) lambda calculus under any of the standard translations between these two frameworks for higherorder reasoning. We ..."
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Abstract Let R be a convergent term rewriting system, and let CRequality on (simply typed) combinatory logic terms be the equality induced by βηRequality on terms of the (simply typed) lambda calculus under any of the standard translations between these two frameworks for higherorder reasoning. We generalize the classical notion of strong reduction to a reduction relation which generates CRequality and whose irreducibles are exactly the translates of long βRnormal forms. The classical notion of strong normal form in combinatory logic is also generalized, yielding yet another description of these translates. Their resulting tripartite characterization extends to the combined firstorder algebraic and higherorder setting the classical combinatory logic descriptions of the translates of long βnormal forms in the lambda calculus. As a consequence, the translates of long βRnormal forms are easily seen to serve as canonical representatives for CRequivalence classes of combinatory logic terms for nonempty, as well as for empty, R. 573
A Combinatorbased Ordersorted Higherorder Unification Algorithm
, 1993
"... This paper develops a sound and complete transformationbased algorithm for unification in an extensional ordersorted combinatory logic supporting constant overloading and a higherorder sort concept. Appropriate notions of ordersorted weak equality and extensionality  reflecting ordersorted f ..."
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This paper develops a sound and complete transformationbased algorithm for unification in an extensional ordersorted combinatory logic supporting constant overloading and a higherorder sort concept. Appropriate notions of ordersorted weak equality and extensionality  reflecting ordersorted fijequality in the corresponding lambda calculus given by Johann and Kohlhase  are defined, and the typed combinatorbased higherorder unification techniques of Dougherty are modified to accommodate unification with respect to the theory they generate. The algorithm presented here can thus be viewed as a combinatory logic counterpart to that of Johann and Kohlhase, as well as a refinement of that of Dougherty, and provides evidence that combinatory logic is wellsuited to serve as a framework for incorporating ordersorted higherorder reasoning into deduction systems aiming to capitalize on both the expressiveness of extensional higherorder logic and the efficiency of ordersorted calculi.