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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 &lambda ..."
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Higherorder unification is equational unification for &beta;&eta;conversion. 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 &lambda;&sigma;calculus of explicit substitutions.
COMPARING APPROACHES TO RESOLUTION BASED HIGHERORDER THEOREM PROVING
, 2002
"... We investigate several approaches to resolution based automated theorem proving in classical higherorder logic (based on Church’s simply typed λcalculus) and discuss their requirements with respect to Henkin completeness and full extensionality. In particular we focus on Andrews’ higherorder res ..."
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Cited by 17 (11 self)
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We investigate several approaches to resolution based automated theorem proving in classical higherorder logic (based on Church’s simply typed λcalculus) and discuss their requirements with respect to Henkin completeness and full extensionality. In particular we focus on Andrews’ higherorder resolution (Andrews 1971), Huet’s constrained resolution (Huet 1972), higherorder Eresolution, and extensional higherorder resolution (Benzmüller and Kohlhase 1997). With the help of examples we illustrate the parallels and differences of the extensionality treatment of these approaches and demonstrate that extensional higherorder resolution is the sole approach that can completely avoid additional extensionality axioms.
Unification in an Extensional Lambda Calculus with Ordered Function Sorts and Constant Overloading
, 1994
"... We develop an ordersorted higherorder calculus suitable for automatic theorem proving applications by extending the extensional simply typed lambda calculus with a higherorder ordered sort concept and constant overloading. Huet's wellknown techniques for unifying simply typed lambda ter ..."
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Cited by 4 (1 self)
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We develop an ordersorted higherorder calculus suitable for automatic theorem proving applications by extending the extensional simply typed lambda calculus with a higherorder ordered sort concept and constant overloading. Huet's wellknown techniques for unifying simply typed lambda terms are generalized to arrive at a complete transformationbased unification algorithm for this sorted calculus. Consideration of an ordersorted logic with functional base sorts and arbitrary term declarations was originally proposed by the second author in a 1991 paper; we give here a corrected calculus which supports constant rather than arbitrary term declarations, as well as a corrected unification algorithm, and prove in this setting results corresponding to those claimed there.
Theory and Practice of Minimal Modular HigherOrder EUnification
 PROCEEDINGS OF THE 12TH CONFERENCE ON AUTOMATED DEDUCTION, NUMBER 814 IN LNAI
, 1994
"... Modular higherorder Eunification, as described in [6], produces numerous redundant solutions in many practical cases. We present a refined algorithm for finitary theories with a finite number of nonfree constants that avoids most redundant solutions, analyse it theoretically, and give first exper ..."
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Modular higherorder Eunification, as described in [6], produces numerous redundant solutions in many practical cases. We present a refined algorithm for finitary theories with a finite number of nonfree constants that avoids most redundant solutions, analyse it theoretically, and give first experimental results. In comparison to [6], the description of the Eunification algorithm is enriched by a definition of the translation process between first and higherorder terms and an explicit handling of new variables. This explicit handling gives deeper insight into the reasons for redundant solutions and thus provides a method for their avoidance. In order to study the efficiency of this method and the performance of the modular approach in general, some benchmark examples are presented and an interpretation of their empirical evaluation is given.
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
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...
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.
HigherOrder and Semantic Unification
"... We provide a complete system of transformation rules for semantic unification with respect to theories defined by convergent rewrite systems. We show that this standard unification procedure, with slight modifications, can be used to solve the satisfiability problem in combinatory logic with a conve ..."
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We provide a complete system of transformation rules for semantic unification with respect to theories defined by convergent rewrite systems. We show that this standard unification procedure, with slight modifications, can be used to solve the satisfiability problem in combinatory logic with a convergent set of algebraic axioms R, thus resulting in a complete higherorder unification procedure for R. Furthermore, we use the system of transformation rules to provide a syntactic characterization for R which results in decidability of semantic unification.