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Comparing and Implementing Calculi of Explicit Substitutions with Eta Reduction
 Annals of Pure and Applied Logic
, 2005
"... The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that e ..."
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Cited by 10 (8 self)
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The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that eta reduction is useful for adapting substitution calculi for practical problems like higher order uni cation. This paper concentrates on rewrite rules for eta reduction in three dierent styles of explicit substitution calculi: , se and the suspension calculus. Both and se when extended with eta reduction, have proved useful for solving higher order uni cation. We enlarge the suspension calculus with an adequate etareduction which we show to preserve termination and conuence of the associated substitution calculus and to correspond to the etareductions of the other two calculi. We prove that and se as well as and the suspension calculus are non comparable while se is more adequate than the suspension calculus in simulating one step of betacontraction.
Revisiting the Notion of Function
"... Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both pro ..."
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Cited by 6 (5 self)
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Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both processes were implemented in Frege's Begriffschrift [17], Russell's Ramified Type Theory [42] and the lambdacalculus (originally introduced by Church [12, 13]) showing that the lambdacalculus misses a crucial aspect of functionalisation. We then pay attention to some special forms of function abstraction that do not exist in the lambdacalculus and we show that various logical constructs (e.g., let expressions and definitions and the use of parameters in mathematics), can be seen as forms of the missing part of functionalisation. Our study of the function concept leads...
Explicit Substitutions and All That
, 2000
"... Explicit substitution calculi are extensions of the lambdacalculus where the substitution mechanism is internalized into the theory. This feature makes them suitable for implementation and theoretical study of logic based tools as strongly typed programming languages and proof assistant systems. In ..."
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Cited by 3 (3 self)
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Explicit substitution calculi are extensions of the lambdacalculus where the substitution mechanism is internalized into the theory. This feature makes them suitable for implementation and theoretical study of logic based tools as strongly typed programming languages and proof assistant systems. In this paper we explore new developments on two of the most successful styles of explicit substitution calculi: the lambdasigma and lambda_secalculi.
Unification via the ...Style of Explicit Substitutions
, 2001
"... A unication method based on the se style of explicit substitution is proposed. This method together with appropriate translations, provide a Higher Order Unication (HOU) procedure for the pure calculus. Our method is inuenced by the treatment introduced by Dowek, Hardin and Kirchner using the sty ..."
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Cited by 2 (2 self)
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A unication method based on the se style of explicit substitution is proposed. This method together with appropriate translations, provide a Higher Order Unication (HOU) procedure for the pure calculus. Our method is inuenced by the treatment introduced by Dowek, Hardin and Kirchner using the style of explicit substitution. Correctness and completeness properties of the proposed seunication method are shown and its advantages, inherited from the qualities of the se calculus, are pointed out. Our method needs only one sort of objects: terms. And in contrast to the HOU approach based on the calculus, it avoids the use of substitution objects. This makes our method closer to the syntax of the calculus. Furthermore, detection of redices depends on the search for solutions of simple arithmetic constraints which makes our method more operational than the one based on the style of explicit substitution. Keywords: Higher order unication, explicit substitution, lambdacalculi. 1
Higher Order Unification via ...Style of Explicit Substitution
"... A higher order unification (HOU) method based on the ...style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...style of explicit substitution. Correctness and completeness properties of the proposed approach ..."
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A higher order unification (HOU) method based on the ...style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...style of explicit substitution. Correctness and completeness properties of the proposed approach are shown and advantages of the method, inherited from the qualities of the ... calculus, are pointed out.
Explicit Substitutions a la de Bruijn: the local . . .
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 85 NO. 7 (2003)
, 2003
"... Kamareddine and Nederpelt [9], resp. Kamareddine and Ríos [11] gave two calculi of explicit of substitutions highly inuenced by de Bruijn's notation of the calculus. These calculi added to the explosive pool of work on explicit substitution in the past 15 years. As far as we know, calculi of explic ..."
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Kamareddine and Nederpelt [9], resp. Kamareddine and Ríos [11] gave two calculi of explicit of substitutions highly inuenced by de Bruijn's notation of the calculus. These calculi added to the explosive pool of work on explicit substitution in the past 15 years. As far as we know, calculi of explicit substitutions: a) are unable to handle local substitutions, and b) have answered (positively or negatively) the question of the termination of the underlying calculus of substitutions. The exception to a) is the calculus of [9] where substitution is handled both locally and globally. However, the calculus of [9] does not satisfy properties like conuence and termination. The exception to b) is the s e calculus [11] for which termination of the s e calculus, the underlying calculus of substitutions, remains unsolved. This paper has two aims: (i) To provide a calculus a la de Bruijn which deals with local substitution and whose underlying calculus of substitutions is terminating and conuent.
Comparing Calculi of Explicit . . .
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 67 (2002)
, 2002
"... The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. Three styles of explicit substi ..."
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The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. Three styles of explicit substitutions are treated in this paper: the and the s e which have proved useful for solving practical problems like higher order uni cation, and the suspension calculus related to the implementation of the language Prolog. We enlarge the suspension calculus with an adequate etareduction which we show to preserve termination and conuence of the associated substitution calculus and to correspond to the etareductions of the other two calculi. Additionally, we prove that and s e as well as and the suspension calculus are non comparable while s e is more adequate than the suspension calculus.