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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 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...
Polarized HigherOrder Subtyping
, 1997
"... The calculus of higher order subtyping, known as F ω ≤ , a higherorder polymorphic λcalculus with subtyping, is expressive enough to serve as core calculus for typed objectoriented languages. The versions considered in the literature usually support only pointwise subtyping of type operators, whe ..."
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Cited by 32 (1 self)
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The calculus of higher order subtyping, known as F ω ≤ , a higherorder polymorphic λcalculus with subtyping, is expressive enough to serve as core calculus for typed objectoriented languages. The versions considered in the literature usually support only pointwise subtyping of type operators, where two types S U and T U are in subtype relation, if S and T are. In the widely cited, unpublished note [Car90], Cardelli presents F ω ≤ in a more general form going beyond pointwise subtyping of type applications in distinguishing between monotone and antimonotone operators. Thus, for instance, T U1 is a subtype of T U2, if U1 ≤ U2 and T is a monotone operator. My thesis extends F ω ≤ by polarized application, it explores its proof theory, establishing decidability of polarized F ω ≤. The inclusion of polarized application rules leads to an interdependence of the subtyping and the kinding system. This contrasts with pure F ω ≤ , where subtyping depends on kinding but not vice versa. To retain decidability of the system, the equalbounds subtyping rule for alltypes is rephrased in the polarized setting as a mutualsubtype requirement of the upper bounds.
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 4 (2 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...
The LambdaCube With Classes Which Approximate Reductional Equivalence
, 1995
"... We study lambda calculus and refine the notions of fireduction and fiequality. In particular, we define the operation TS (term reshuffling) on terms which reshuffles a term in such a way that more redexes become visible. Two terms are called shuffleequivalent if they have syntactically equivalent ..."
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We study lambda calculus and refine the notions of fireduction and fiequality. In particular, we define the operation TS (term reshuffling) on terms which reshuffles a term in such a way that more redexes become visible. Two terms are called shuffleequivalent if they have syntactically equivalent TSimages. The shuffleequivalence classes are shown to divide the classes of fiequal terms into smaller classes consisting of terms with similar reduction behaviour. The refinement of fireduction from a relation on terms to a relation on shuffle classes, called shufflereduction, allows one to make more redexes visible and to contract these newly visible redexes. This enables one to have more freedom in choosing the reduction path of a term, which can result in smaller terms along the reduction path if a clever reduction strategy is used. Moreover, this gain in reductional breadth is not at the expense of reductional length. We show that the cube of [Barendregt 92] extended with shuffle...
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 redexes, but classic ..."
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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