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26
Calculi of Generalised βReduction and Explicit Substitutions: The TypeFree and Simply Typed Versions
, 1998
"... Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substit ..."
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Cited by 16 (8 self)
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Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the  calculus, because it allows postponement of work in two different but complementary ways. Moreover, gs (and also s) satisfies properties desirable for calculi of explicit substitutions and generalized reductions. In particular, we show that gs preserves strong normalization, is a conservative extension of g, and simulates fireduction of g and the classical calculus. Furthermore, we study the simply typed versions of s and gs, and show that welltyped terms are strongly normalizing and that other properties,...
Bridging de Bruijn indices and variable names in explicit substitutions calculi
 Logic Journal of the Interest Group of Pure and Applied Logic (IGPL
, 1996
"... Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding ffconversion and being as close to machines as possible. De Bruijn indices however, though very suitable for the machine, are difficult to human users. This is the reason for a renew ..."
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Cited by 11 (7 self)
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Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding ffconversion and being as close to machines as possible. De Bruijn indices however, though very suitable for the machine, are difficult to human users. This is the reason for a renewed interest in systems of explicit substitutions using variable names. Formal systems of explicit substitutions using variable names is a new area however and we believe, it should not develop without being welltied to existing work on explicit substitutions. The aim of this paper is to establish a bridge between explicit substitutions using de Bruijn indices and using variable names. In our aim to do so, we provide the tcalculus: a calculus `a la de Bruijn which can be translated into a calculus with explicit substitutions written with variables names. We present explicitly this translation and use it to obtain preservation of strong normalisation for t. Moreover, we show several prope...
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 7 (6 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...
Strong normalization from weak normalization in typed λcalculi
 Information and Computation
, 1997
"... For some typed λcalculi it is easier to prove weak normalization than strong normalization. Techniques to infer the latter from the former have been invented over the last twenty years by Nederpelt, Klop, Khasidashvili, Karr, de Groote, and Kfoury and Wells. However, these techniques infer strong n ..."
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Cited by 4 (1 self)
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For some typed λcalculi it is easier to prove weak normalization than strong normalization. Techniques to infer the latter from the former have been invented over the last twenty years by Nederpelt, Klop, Khasidashvili, Karr, de Groote, and Kfoury and Wells. However, these techniques infer strong normalization of one notion of reduction from weak normalization of a more complicated notion of reduction. This paper presents a new technique to infer strong normalization of a notion of reduction in a typed λcalculus from weak normalization of the same notion of reduction. The technique is demonstrated to work on some wellknown systems including secondorder λcalculus and the system of positive, recursive types. It gives hope for a positive answer to the BarendregtGeuvers conjecture stating that every pure type system which is weakly normalizing is also strongly normalizing. The paper also analyzes the relationship between the techniques mentioned above, and reviews, in less detail, other techniques for proving strong normalization.
Reviewing the classical and the de Bruijn notation for λcalculus and pure type systems
 Logic and Computation
, 2001
"... This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentat ..."
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Cited by 3 (0 self)
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This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λcalculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λcalculus is introduced and some of its advantages are outlined.
Calculi of generalised #reduction and explicit substitutions: The type free and simply typed versions
 J. Funct. Logic Programming
, 1998
"... Extending the calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently and still has many open problems. This paper is the rst investigation into the properties of a calculus combining both generalised reduction and explicit substitutio ..."
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Cited by 3 (3 self)
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Extending the calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently and still has many open problems. This paper is the rst investigation into the properties of a calculus combining both generalised reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the calculus because it allows postponment of work in two dierent but complementary ways. Moreover, gs (and also s) satises desirable properties of calculi of explicit substitutions and generalised reductions. In particular, we show that gs preserves strong normalisation, is a conservative extension of g, and simulates reduction of g and the classical calculus. Furthermore, we study the simply typed versions of s and gs and show that well typed terms are strongly normalising and that other properties such as typing of subterms and subject reduction hold. Our proof of the preservation of strong normalisation (PSN) is based on the minimal derivation method. It is however much simpler because we prove the commutation of arbitrary internal and external reductions. Moreover, we use one proof to show both the preservation of strong normalisation in s and the preservation of gstrong normalisation in gs. We remark that the technique of these proofs is not suitable for calculi without explicit substitutions (e.g. the preservation of strong normalisation in g requires a dierent technique). 1
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
Explicit substitutions for control operators
, 1997
"... Abstract. The calculus is a calculus with a local operator closely related to normalisation procedures in classical logic and control operators in functional programming. We introduce exp, an explicit substitution calculus for , show it preserves strong normalisation and that its simply typed ve ..."
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Cited by 1 (0 self)
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Abstract. The calculus is a calculus with a local operator closely related to normalisation procedures in classical logic and control operators in functional programming. We introduce exp, an explicit substitution calculus for , show it preserves strong normalisation and that its simply typed version is strongly normalising. Interestingly, exp is the rst example for which the decency method of showing preservation of strong normalisation (PSN) works whereas the structure preserving method which is based on the decency method does not. In particular, exp is a very simple calculus yet is not structure preserving. This shows that the structure preserving notion intended to give a general description of calculi of explicit substitution that satisfy PSN, is restrictive. To our knowledge, exp is the rst calculus of explicit substitution that is not structure preserving. 5 1
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 e ..."
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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.