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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 ..."
Abstract

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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 ...
Chew's Theorem Revisited  Uniquely Normalizing Property of Nonlinear Term Rewriting Systems 
, 1992
"... . This paper gives a purely syntactical proof, based on proof normalization techniques, of an extension of Chew's theorem. The main theorem is that a weakly compatible TRS is uniquely normalizing. Roughly speaking, the weakly compatible condition allows possibly nonlinear TRSs to have nonroot o ..."
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. This paper gives a purely syntactical proof, based on proof normalization techniques, of an extension of Chew's theorem. The main theorem is that a weakly compatible TRS is uniquely normalizing. Roughly speaking, the weakly compatible condition allows possibly nonlinear TRSs to have nonroot overlapping rules that return the same results. This result implies the consistency of CLpc which is an extension of the combinatory logic CL with parallelif rules. 1 Introduction The ChurchRosser (CR) property is one of the most important properties for term rewriting systems (TRSs). When a TRS is nonterminating, a wellknown condition for CR is Rosen's theorem, which states that a leftlinear weakly nonoverlapping TRS is CR  or, simply, that a leftlinear nonoverlapping TRS is CR[11, 13]. A pair of reduction rules is said to be overlapping if their applications interfere with each other (i.e., they are unified at some nonvariable position), and a TRS is said to be nonoverlapping if none of ...