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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 ...
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
Automated Equational Reasoning in Nondeterministic λCalculi Modulo Theories H*
, 2009
"... In this thesis I study four extensions of untyped λcalculi all under the maximally coarse semantics of the theory H ∗ (observable equality), and implement a system for reasoning about and storing abstract knowledge expressible in languages with these extensions. The extensions are: (1) a semilattic ..."
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In this thesis I study four extensions of untyped λcalculi all under the maximally coarse semantics of the theory H ∗ (observable equality), and implement a system for reasoning about and storing abstract knowledge expressible in languages with these extensions. The extensions are: (1) a semilattice operation J, the join w.r.t the Scott ordering; (2) a random mixture R for stochastic λcalculus; (3) a computational comonad 〈code,apply,eval,quote, {−}〉 for Gödel codes modulo provable equality; and (4) a Π 1 1complete oracle O. I develop three languages from combinations of these extensions. The syntax of these languages is always simple: each is a finitely generated combinatory algebra. The semantics of these languages are various fragments of Dana Scott’s D ∞ models. Although the languages use ideas