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24
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, ..."
Abstract

Cited by 39 (18 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...
A unified approach to Type Theory through a refined λcalculus
, 1994
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. ..."
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Cited by 14 (13 self)
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In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic.
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...
and
, 1996
"... We are grateful for the discussions with Roel Bloo, Tijn Borghuis, Erik Poll and Phil Wadler and for the helpful remarks received from them. In particular, we are grateful to Phil Wadler who has recommended that we dispose of 's and 's as in (x)( y)y and write (x)[y]y instead. We are also ..."
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We are grateful for the discussions with Roel Bloo, Tijn Borghuis, Erik Poll and Phil Wadler and for the helpful remarks received from them. In particular, we are grateful to Phil Wadler who has recommended that we dispose of 's and 's as in (x)( y)y and write (x)[y]y instead. We are also grateful to Peter Peters for his help concerning Latex and email which enabled us to keep exchanging drafts and corrections of the present paper. Last but not least, we are grateful to the anonymous referees for their useful comments. y
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...
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...
Term Reshuffling in the Barendregt Cube
"... This paper will concentrate on a new feature related to reshuffling terms so that more redexes become visible. The idea is explained as follows: Assume a redex is a `[' next to a `]'. What will happen in a term of the form `[ [ ] [ ] ]' ? We know that the two internal `[ ]' are ..."
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This paper will concentrate on a new feature related to reshuffling terms so that more redexes become visible. The idea is explained as follows: Assume a redex is a `[' next to a `]'. What will happen in a term of the form `[ [ ] [ ] ]' ? We know that the two internal `[ ]' are redexes, but classical notation does not allow us to say that the outside `[' and `]' form a redex. In [BKN 9x], we generalised the notion of a redex from a pair of adjacent matching parentheses to simply a pair of matching parentheses. Hence, with generalised reduction all the three redexes are visible in `[ [ ] [ ] ]'. In this paper, we propose to reshuffle `[ [ ] [ ] ]' to `[ ] [ ] [ ]' where the first `[' has been moved next to the last `]'. The item notation enables us to see the matching parentheses and to reshuffle terms so that all matching paretheses become adjacent
and
, 1996
"... We are grateful for the discussions with Roel Bloo, Tijn Borghuis, Erik Poll and Phil Wadler and for the helpful remarks received from them. In particular, we are grateful to Phil Wadler who has recommended that we dispose of 's and 's as in (x)( y)y and write (x)[y]y instead. We are also ..."
Abstract
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We are grateful for the discussions with Roel Bloo, Tijn Borghuis, Erik Poll and Phil Wadler and for the helpful remarks received from them. In particular, we are grateful to Phil Wadler who has recommended that we dispose of 's and 's as in (x)( y)y and write (x)[y]y instead. We are also grateful to Peter Peters for his help concerning Latex and email which enabled us to keep exchanging drafts and corrections of the present paper. Last but not least, we are grateful to the anonymous referees for their useful comments. y
and
, 1996
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, ..."
Abstract
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In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, the three subjects of calculus, logic and type theory, got separated due to the appearence of the paradoxes. Moreover, the existence of various versions of each topic is due to the need to get back to the lost paradise which allowed a great freedom in mixing expressivity and logic. In any case, the presence of such a variety of systems calls for a framework to unify them all. Barendregt's cube for example, is an attempt to unify various type systems and his associated logic cube is an attempt to nd connections between type theories and logic. We devise a new notation which enables categorising most of the known systems in a unied way. More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from Automathlike systems to singlytyped Pure Type Systems. The system We are grateful for Erik Poll who has read the paper carefully and for his productive comments. We are also grateful for discussions with Henk Barendregt, Inge Bethke, Tijn Borghuis and for the helpful remarks received from them. Furthermore, we are indebted to the anonymous referee for his/her useful suggestions and remarks. y