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12
On Stepwise Explicit Substitution
, 1993
"... This paper starts by setting the ground for a lambda calculus notation that strongly mirrors the two fundamental operations of term construction, namely abstraction and application. In particular, we single out those parts of a term, called items in the paper, that are added during abstraction and a ..."
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Cited by 41 (30 self)
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This paper starts by setting the ground for a lambda calculus notation that strongly mirrors the two fundamental operations of term construction, namely abstraction and application. In particular, we single out those parts of a term, called items in the paper, that are added during abstraction and application. This item notation proves to be a powerful device for the representation of basic substitution steps, giving rise to different versions of fireduction including local and global fi reduction. In other words substitution, thanks to the new notation, can be easily formalised as an object language notion rather than remaining a meta language one. Such formalisation will have advantages with respect to various areas including functional application and the partial unfolding of definitions. Moreover our substitution is, we believe, the most general to date. This is shown by the fact that our framework can accommodate most of the known reduction strategies, which range from local to...
The Barendregt Cube with Definitions and Generalised Reduction
, 1997
"... In this paper, we propose to extend the Barendregt Cube by generalising reduction and by adding definition mechanisms. We show that this extension satisfies all the original properties of the Cube including Church Rosser, Subject Reduction and Strong Normalisation. Keywords: Generalised Reduction, ..."
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Cited by 37 (17 self)
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In this paper, we propose to extend the Barendregt Cube by generalising reduction and by adding definition mechanisms. We show that this extension satisfies all the original properties of the Cube including Church Rosser, Subject Reduction and Strong Normalisation. Keywords: Generalised Reduction, Definitions, Barendregt Cube, Church Rosser, Subject Reduction, Strong Normalisation. Contents 1 Introduction 3 1.1 Why generalised reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Why definition mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The item notation for definitions and generalised reduction . . . . . . . . . . 4 2 The item notation 7 3 The ordinary typing relation and its properties 10 3.1 The typing relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Properties of the ordinary typing relation . . . . . . . . . . . . . . . . . . . . 13 4 Generalising reduction in the Cube 15 4.1 The generalised...
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.
Canonical typing and Πconversion in the Barendregt Cube
, 1996
"... In this article, we extend the Barendregt Cube with \Piconversion (which is the analogue of betaconversion, on product type level) and study its properties. We use this extension to separate the problem of whether a term is typable from the problem of what is the type of a term. ..."
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Cited by 4 (3 self)
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In this article, we extend the Barendregt Cube with \Piconversion (which is the analogue of betaconversion, on product type level) and study its properties. We use this extension to separate the problem of whether a term is typable from the problem of what is the type of a term.
Canonical typing and Πconversion
, 1997
"... In usual type theory, if a function f is of type oe ! oe and an argument a is of type oe, then the type of fa is immediately given to be oe and no mention is made of the fact that what has happened is a form of ficonversion. A similar observation holds for the generalized Cartesian product typ ..."
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Cited by 3 (3 self)
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In usual type theory, if a function f is of type oe ! oe and an argument a is of type oe, then the type of fa is immediately given to be oe and no mention is made of the fact that what has happened is a form of ficonversion. A similar observation holds for the generalized Cartesian product types, \Pi x:oe : . In fact, many versions of type theory assume that fi holds of both types and terms, yet only a few attempt to study the theory where terms and types are really treated equally and where ficonversion is used for both. A unified treatment however, of types and terms is becoming indispensible especially in the approaches which try to generalise many systems under a unique one. For example, [Barendregt 91] provides the Barendregt cube and the Pure Type Systems (PTSs) which are a generalisation of many type theories. Yet even such a generalisation does not use ficonversion for both types and terms. This is unattractive, in a calculus where types have the same syntax as terms (such as the calculi of the cube or the PTSs). For example, in those systems, even though compatibility holds for the typing of abstraction, it does not hold for the typing of application. That is, even though M : N ) y:P :M : \Pi y:P :N holds, the following does not hold: Based on this observation, we present a calculus in which the conversion rules apply to types as well as terms. Abstraction and application, moreover, range over both types and terms. We extend the calculus with a canonical type operator in order to associate types to terms. The type of fa will then be Fa, where F is the type of f and the statement \Gamma ` t : oe from usual type theory is split in two statements in our system: \Gamma ` t and (\Gamma; t) = oe. Such a splitting enables us to discuss the two questio...
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.
Are Types needed for Natural Language?
, 1996
"... Logic, due to the paradoxes, is absent from the type free calculus. This makes such a calculus an unsuitable device for Natural Language Semantics. Moreover, the problems that arise from mixing the type free calculus with logic lead to type theory and hence formalisations of Natural Language were ..."
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Cited by 1 (0 self)
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Logic, due to the paradoxes, is absent from the type free calculus. This makes such a calculus an unsuitable device for Natural Language Semantics. Moreover, the problems that arise from mixing the type free calculus with logic lead to type theory and hence formalisations of Natural Language were carried out in a strictly typed framework. It was shown however, that strict type theory cannot capture the selfreferential nature of language ([Parsons 79], [Chierchia, Turner 88] and [Kamareddine, Klein 93]) and hence other approaches were needed. For example, the approach carried out by Parsons is based on creating a notion of floating types which can be instantiated to particular instances of types whereas the approaches of Chierchia, Turner and Kamareddine, Klein are based on a type free framework. In this paper, we will embed the typing system of [Parsons 79] into a version of the one proposed in [Kamareddine, Klein 93] giving an interpretation of Parsons' system in a type free theory...
A Useful λNotation
, 1996
"... In this article, we introduce a notation that is useful for many concepts of the  calculus. The new notation is a simple translation of the classical one. Yet, it provides many nice advantages. First, we show that definitions such as compatibility, the heart of a term and firedexes become simpl ..."
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Cited by 1 (0 self)
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In this article, we introduce a notation that is useful for many concepts of the  calculus. The new notation is a simple translation of the classical one. Yet, it provides many nice advantages. First, we show that definitions such as compatibility, the heart of a term and firedexes become simpler in item notation. Second, we show that with this item notation, reduction can be generalised in a nice way. We find a relation ; fi which extends ! fi , which is Church Rosser and Strongly Normalising. This reduction relation may be the way to new reduction strategies. In classical notation, it is much harder to present this generalised reduction in a convincing manner. Third, we show that the item notation enables one to represent in a very simple way the canonical type ø (\Gamma; A) of a term A in context \Gamma. This canonical type plays the role of a preference type and can be used to split \Gamma ` A : B in the two parts: \Gamma ` A and ø (\Gamma; A) = B. This means that the questio...
Refining Reduction in the lambda calculus
, 1996
"... We introduce a calculus notation which enables us to detect in a term, more fi redexes than in the usual notation. On this basis, we define an extended fireduction which is yet a subrelation of conversion. The Church Rosser property holds for this extended reduction. Moreover, we show that we c ..."
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We introduce a calculus notation which enables us to detect in a term, more fi redexes than in the usual notation. On this basis, we define an extended fireduction which is yet a subrelation of conversion. The Church Rosser property holds for this extended reduction. Moreover, we show that we can transform generalised redexes into usual ones by a process called "term reshuffling". Keywords: Item notation, Redexes, Church Rosser. Contents 1 Introduction 3 1.1 The item notation and visible redexes . . . . . . . . . . . . . . . . . . . . . . 4 1.2 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Generalising redexes and fireduction 7 2.1 Extending redexes from segments to couples . . . . . . . . . . . . . . . . . . . 7 2.2 Extending fireduction and the Church Rosser theorem . . . . . . . . . . . . . 8 3 Term reshuffling 10 3.1 Partitioning terms into bachelor and wellbalanced segments . . . . . . . . . . 11 3.2 The reshuffling procedure...
Beyond betaReduction in Church's ...
, 1996
"... In this paper, we shall write ! using a notation, item notation, which enables one to make more redexes visible, and shall extend fireduction to all visible redexes. We will prove that ! written in item notation and accommodated with extended reduction, satisfies all its original properties (such a ..."
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In this paper, we shall write ! using a notation, item notation, which enables one to make more redexes visible, and shall extend fireduction to all visible redexes. We will prove that ! written in item notation and accommodated with extended reduction, satisfies all its original properties (such as Church Rosser, Subject Reduction and Strong Normalisation). The notation itself is very simple: if I translates classical terms to our notation, then I(t 1 t 2 ) j (I(t 2 )ffi)I(t 1 ) and I( v:ae :t) j (ae v )I(t). For example, t j (( x7 :X4 :( x6 :X3 : x5 :X1!X2 :x 5 x 4 )x 3 )x 2 )x 1 , can be written in our item notation as I(t) j (x 1 ffi)(x 2 ffi)(X 4 x7 )(x 3 ffi)(X 3 x6 )((X 1 ! X 2 ) x5 )(x 4 ffi)x 5 where the visible redexes are based on all the matching fficouples. So here, the redexes are based on (x 2 ffi)(X 4 x7 ), (x 3 ffi)(X 3 x6 ) and (x 1 ffi)((X 1 ! X 2 ) x5 ). In classical notation however, only the redexes based on ( x7 :X4 : \Gamma \Gamma)x 2 and ( x6 :X3 : \Gamm...