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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,...
Cut Rules and Explicit Substitutions
, 2000
"... this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the l ..."
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Cited by 15 (0 self)
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this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the logical connectives which operate on the right, i.e., they manipulate the succedent formula. Examples are Gentzen's NJ and NK (Gentzen 1935). Logical deduction systems are given via leftintroduction and rightintroduction rules for the logical connectives. Although others have called these systems "sequent calculi", we call them Lsystems to avoid confusion with other systems given in sequent style. Examples are Gentzen's LK and LJ (Gentzen 1935). In this paper we are primarily interested in Lsystems. The advantage of Nsystems is that they seem closer to actual reasoning, while Lsystems on the other hand seem to have an easier proof theory. Lsystems are often extended with a "cut" rule as part of showing that for a given Lsystem and Nsystem, the derivations of each system can be encoded in the other. For example, NK proves the same as LK + cut (Gentzen 1935). Proof Normalization. A system is consistent when it is impossible to prove false, i.e., derive absurdity from zero assumptions. A system is analytic (has the analycity property) when there is an e#ective method to decompose any conclusion sequent into simpler premise sequents from which the conclusion can be obtained by some rule in the system such that the conclusion is derivable i# the premises are derivable (Maenpaa 1993). To achieve the goals of consistency and analycity, it has been customary to consider
A reduction relation for which postponement of Kcontractions, Conservation and Preservation of Strong Normalisation hold
 Univ. of Glasgow, Glasgow
, 1996
"... Postponement of fi K contractions and the conservation theorem do not hold for ordinary fi but have been established by de Groote for a mixture of fi with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation fi e which generalises fi. ..."
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Cited by 10 (7 self)
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Postponement of fi K contractions and the conservation theorem do not hold for ordinary fi but have been established by de Groote for a mixture of fi with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation fi e which generalises fi. This then is used to solve an open problem of fi e : the Preservation of Strong Normalisation 1 . Keywords: Generalised fireduction, Postponement of Kcontractions, Generalised Conservation, Preservation of Strong Normalisation. 1 Introduction 1.1 Background and Motivation In the term (( x : y :N)P )Q, the function starting with x and the argument P result in the redex ( x : y :N)P . It is also the case that the function starting with y and the argument Q will result in another redex when the first redex is contracted. This idea has been exploited by many researchers and reduction has been extended so that the generalised redex based on the matching y and Q is given the same priority a...
On \Piconversion in the lambdacube and the combination with abbreviations
, 1997
"... Typed calculus uses two abstraction symbols ( and \Pi) which are usually treated in different ways: x: :x has as type the abstraction \Pi x: :, yet \Pi x: : has type 2 rather than an abstraction; moreover, ( x:A :B)C is allowed and fireduction evaluates it, but (\Pi x:A :B)C is rarely allowed. Fu ..."
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Cited by 6 (3 self)
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Typed calculus uses two abstraction symbols ( and \Pi) which are usually treated in different ways: x: :x has as type the abstraction \Pi x: :, yet \Pi x: : has type 2 rather than an abstraction; moreover, ( x:A :B)C is allowed and fireduction evaluates it, but (\Pi x:A :B)C is rarely allowed. Furthermore, there is a general consensus that and \Pi are different abstraction operators. While we agree with this general consensus, we find it nonetheless important to allow \Pi to act as an abstraction operator. Moreover, experience with AUTOMATH and the recent revivals of \Pireduction as in [KN 95b, PM 97], illustrate the elegance of giving \Piredexes a status similar to redexes. However, \Pireduction in the cube faces serious problems as shown in [KN 95b, PM 97]: it is not safe as regards subject reduction, it does not satisfy type correctness, it loses the property that the type of an expression is wellformed and it fails to make any expression that contains a \Piredex wellfor...
Absolute Explicit Unification
 in International Conference on Rewriting Techniques and Applications (RTA'2000
, 2000
"... . This paper presents a system for explicit substitutions in Pure Type Systems (PTS). The system allows to solve type checking, type inhabitation, higherorder unification, and type inference for PTS using purely firstorder machinery. A novel feature of our system is that it combines substituti ..."
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. This paper presents a system for explicit substitutions in Pure Type Systems (PTS). The system allows to solve type checking, type inhabitation, higherorder unification, and type inference for PTS using purely firstorder machinery. A novel feature of our system is that it combines substitutions and variable declarations. This allows as a sideeffect to type check letbindings. Our treatment of metavariables is also explicit, such that instantiations of metavariables is internalized in the calculus. This produces a confluent calculus with distinguished holes and explicit substitutions that is insensitive to ffconversion, and allows directly embedding the system into rewriting logic. 1 Introduction Explicit substitutions provide a convenient framework for encoding higherorder typed calculus using firstorder machinery. In particular, this allows to integrate higherorder unification with firstorder provers, rewriting logic, and to delay evaluation and resolve scoping...
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.
Dependent Types and Explicit Substitutions
, 1999
"... We present a dependenttype system for a #calculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization. ..."
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Cited by 3 (0 self)
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We present a dependenttype system for a #calculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.
The Soundness of Explicit Substitution with Nameless Variables
, 1995
"... We show the soundness of a λcalculus B where de Bruijn indices are used, substitution is explicit, and reduction is stepwise. This is done by interpreting B in the classical calculus where the explicit substitution becomes implicit and de Bruijn indices become named variables. This is the first fl ..."
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Cited by 2 (1 self)
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We show the soundness of a λcalculus B where de Bruijn indices are used, substitution is explicit, and reduction is stepwise. This is done by interpreting B in the classical calculus where the explicit substitution becomes implicit and de Bruijn indices become named variables. This is the first flat semantics of explicit substitution and stepwise reduction and the first clear account of exactly when ffreduction is needed.
AUTOMATH and Pure Type Systems
, 1996
"... We study the position of Automath systems within the framework of the Pure Type Systems as discussed in [3]. ..."
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We study the position of Automath systems within the framework of the Pure Type Systems as discussed in [3].