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Modularity of Strong Normalization and Confluence in the algebraiclambdacube
, 1994
"... In this paper we present the algebraiccube, an extension of Barendregt's cube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraiccube, provided that the firstorder rewrite rules are nonduplicating and the hig ..."
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Cited by 25 (7 self)
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In this paper we present the algebraiccube, an extension of Barendregt's cube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraiccube, provided that the firstorder rewrite rules are nonduplicating and the higherorder rules satisfy the general schema of Jouannaud and Okada. This result is proven for the algebraic extension of the Calculus of Constructions, which contains all the systems of the algebraiccube. We also prove that local confluence is a modular property of all the systems in the algebraiccube, provided that the higherorder rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence. 1 Introduction Many different computational models have been developed and studied by theoretical computer scientists. One of the main motivations for the development This research was partially supported by ESPRIT Basic Research Act...
The Computability Path Ordering: the End of a Quest
"... Abstract. In this paper, we first briefly survey automated termination proof methods for higherorder calculi. We then concentrate on the higherorder recursive path ordering, for which we provide an improved definition, the Computability Path Ordering. This new definition appears indeed to capture ..."
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Cited by 13 (2 self)
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Abstract. In this paper, we first briefly survey automated termination proof methods for higherorder calculi. We then concentrate on the higherorder recursive path ordering, for which we provide an improved definition, the Computability Path Ordering. This new definition appears indeed to capture the essence of computability arguments à la Tait and Girard, therefore explaining the name of the improved ordering. 1
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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Cited by 9 (3 self)
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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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Cited by 9 (1 self)
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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...
Modularity of Strong Normalization in the Algebraicλcube
, 1996
"... In this paper we present the algebraicλcube, an extension of Barendregt's λcube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraicλcube, provided that the firstorder rewrite rules are nonduplicating and the ..."
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Cited by 8 (2 self)
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In this paper we present the algebraicλcube, an extension of Barendregt's λcube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraicλcube, provided that the firstorder rewrite rules are nonduplicating and the higherorder rules satisfy the general schema of Jouannaud and Okada. This result is proven for the algebraic extension of the Calculus of Constructions, which contains all the systems of the algebraicλcube. We also prove that local confluence is a modular property of all the systems in the algebraicλcube, provided that the higherorder rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence.
Higherorder termination: From kruskal to computability
 In 13th International Conf. on Logic for Programming, Artificial Intelligence, and Reasoning. Lecture Notes in Computer Science
"... Termination is a major question in both logic and computer science. In logic, termination is at the heart of proof theory where it is usually called strong normalization (of cut elimination). In computer science, termination has always been an important issue for showing programs correct. ..."
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Cited by 4 (2 self)
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Termination is a major question in both logic and computer science. In logic, termination is at the heart of proof theory where it is usually called strong normalization (of cut elimination). In computer science, termination has always been an important issue for showing programs correct.
Towards Rewriting in Coq
"... Equational reasoning in Coq is not straightforward. For a few years now there has been an ongoing research process towards adding rewriting to Coq. However, there are many research problems on this way. In this paper we give a coherent view of rewriting in Coq, we describe what is already done and w ..."
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Cited by 3 (0 self)
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Equational reasoning in Coq is not straightforward. For a few years now there has been an ongoing research process towards adding rewriting to Coq. However, there are many research problems on this way. In this paper we give a coherent view of rewriting in Coq, we describe what is already done and what remains to be done. We discuss such issues as strong normalization, confluence, logical consistency, completeness, modularity and extraction.
Variants of the Basic Calculus of Constructions
, 2004
"... In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version i ..."
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Cited by 1 (0 self)
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In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version in early papers on the subject by Seldin. None of these results is very deep, but it seems useful to collect them in one place.