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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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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.
Characterizing LambdaTerms With Equal Reduction Behavior
"... We define an equivalence relation on lambdaterms called shuffleequivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The aim of reductional equivalence is to characterize the evaluation behavior of programs. The shuffleequivalence classes are s ..."
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We define an equivalence relation on lambdaterms called shuffleequivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The aim of reductional equivalence is to characterize the evaluation behavior of programs. The shuffleequivalence classes are shown to divide the classes of betaequal strongly normalising terms (programs which lead to the same final value/output) into smaller ones consisting of terms with similar evaluation behavior. We refine betareduction from a relation on terms to a relation on shuffleequivalence classes, called shufflereduction, and show that this refinement captures existing generalisations of lambdareduction. Shufflereduction allows one to make more redexes visible and to contract these newly visible redexes. Moreover, it allows 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. This can benefit both programming language...
De Bruijn's syntax and reductional behaviour of λterms
"... In this paper, a notation influenced by de Bruijn's syntax of the λcalculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly ..."
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In this paper, a notation influenced by de Bruijn's syntax of the λcalculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly than the classical notation. We de ne reduction modulo equivalence classes of terms up to the permutation of redexes in canonical forms and show that this reduction contains other notions of reductions in the literature including the reduction of Regnier. We establish all the desirable properties of our reduction modulo equivalence classes. Then, we give two extensions of the Barendregt cube, one with the  reduction of Regnier and the other with our class reduction and show that the subject reduction property fails in each of these extensions. We then show that adding de nitions in the contexts of type derivations, enables each of these extensions to satisfy all the desirable properties of type systems, including subject reduction and strong normalisation.
De Bruijn's syntax and reductional equivalence of λterms
, 2001
"... In this paper, a notation inuenced by de Bruijn's syntax of the λcalculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly than ..."
Abstract
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In this paper, a notation inuenced by de Bruijn's syntax of the λcalculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly than the classical notation and we establish the desirable properties of our reduction modulo equivalence classes rather than single terms. Finally, we extend the cube consisting of eight type systems with class reduction and show that this extension satis es all the desirable properties of type systems.