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Perpetual Reductions in λCalculus
, 1999
"... This paper surveys a part of the theory of fireduction in λcalculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λterms (when possible), and with perpetual r ..."
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This paper surveys a part of the theory of fireduction in λcalculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λterms (when possible), and with perpetual redexes, i.e., redexes whose contraction in λterms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in λcalculus and type theory.
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.
Conservation and Uniform Normalization in Lambda Calculi With Erasing Reductions
, 2002
"... For a notion of reduction in a #calculus one can ask whether a term satises conservation and uniform normalization. Conservation means that singlestep reductions of the term preserve innite reduction paths from the term. Uniform normalization means that either the term will have no reduction path ..."
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For a notion of reduction in a #calculus one can ask whether a term satises conservation and uniform normalization. Conservation means that singlestep reductions of the term preserve innite reduction paths from the term. Uniform normalization means that either the term will have no reduction paths leading to a normal form, or all reduction paths will lead to a normal form.
Weak and Strong Beta Normalisations in Typed λCalculi
 IN: PROC. OF THE 3 RD INTERNATIONAL CONFERENCE ON TYPED LAMBDA CALCULUS AND APPLICATIONS, TLCA'97
, 1997
"... We present a technique to study relations between weak and strong finormalisations in various typed calculi. We first introduce a translation which translates a λterm into a λIterm, and show that a λterm is strongly finormalisable if and only if its translation is weakly finormalisable. We t ..."
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We present a technique to study relations between weak and strong finormalisations in various typed calculi. We first introduce a translation which translates a λterm into a λIterm, and show that a λterm is strongly finormalisable if and only if its translation is weakly finormalisable. We then prove that the translation preserves typability of λterms in various typed λcalculi. This enables us to establish the equivalence between weak and strong finormalisations in these typed λcalculi. This translation can deal with Curry typing as well as Church typing, strengthening some recent closely related results. This may bring some insights into answering whether weak and strong finormalisations in all pure type systems are equivalent.
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 eleg ..."
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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 ..."
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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.
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...
Normalization in λCalculus and Type Theory
, 1997
"... and HigherOrder Rewriting. PhD thesis, Vrije Universiteit Amsterdam, 1994. [95] V. van Oostrom. Take five. IR406, Vrije Universiteit Amsterdam, 1996. [96] V. van Oostrom. Finite family developments. In H. Comon, editor, Rewriting Techniques and Applications, volume 1232 of Lecture Notes in Compu ..."
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and HigherOrder Rewriting. PhD thesis, Vrije Universiteit Amsterdam, 1994. [95] V. van Oostrom. Take five. IR406, Vrije Universiteit Amsterdam, 1996. [96] V. van Oostrom. Finite family developments. In H. Comon, editor, Rewriting Techniques and Applications, volume 1232 of Lecture Notes in Computer Science, pages 308322. SpringerVerlag, 1997. [97] V. van Oostrom and F. van Raamsdonk. Comparing combinatory reduction systems and higherorder rewrite systems. In J. Heering, K. Meinke, B. Moller, and T. Nipkow, editors, Higher Order Algebra, Bibliography 157 Logic and Term Rewriting, volume 816 of Lecture Notes in Computer Science. SpringerVerlag, 1994. [98] M. Parigot. Internal labellings in lambdacalculus. In B. Rovan, editor, Symposium on Mathematical Foundations of Computer Science, volume 452 of Lecture Notes in Computer Science, pages 439445. SpringerVerlag, 1990. [99] D.A. Plaisted. Polynomial time termination and constraint satisfaction tests. In C. Kirchner, editor...
An Approximation of Reductional Equivalence
"... We define an equivalence relation on terms called shuffleequivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The shuffleequivalence classes are shown to divide the classes of fiequal terms into smaller ones consisting of terms with similar re ..."
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We define an equivalence relation on terms called shuffleequivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The shuffleequivalence classes are shown to divide the classes of fiequal terms into smaller ones consisting of terms with similar reduction behaviour. We refine fireduction from a relation on terms to a relation on shuffleequivalence classes, called shufflereduction, and show that this refinement captures existing generalisations of fireduction. Shufflereduction moreover, apart from allowing one to make more redexes visible and to contract these newly visible redexes, 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. This can benefit both programming languages and theorem provers since this flexibility and freedom in chosing reduction paths can be exploited to produce the shortest program evalu...