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1 Contextsensitive Conditional Reduction Systems
"... We introduce Contextsensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems ( ..."
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We introduce Contextsensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems (finiteness of developments, confluence, permutation equivalence) carry over immediately. This can be used e.g. to infer confluence from the subject reduction property in several typed λcalculi possibly enriched with patternmatching definitions. Second, we express several proof and transition systems as CERSs. In particular, we give encodings of Hilbertstyle proof systems, Gentzenstyle sequentcalculi, rewrite systems with rule priorities, and the πcalculus into CERSs. This last encoding is an important example of real contextsensitive rewriting. ○c
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...
A Bargain for Intersection Types: A Simple Strong Normalization Proof
"... This pearl gives a discount proof of the folklore theorem that every strongly #normalizing #term is typable with an intersection type. (We consider typings that do not use the empty intersection # which can type any term.) The proof uses the perpetual reduction strategy which finds a longest path. ..."
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This pearl gives a discount proof of the folklore theorem that every strongly #normalizing #term is typable with an intersection type. (We consider typings that do not use the empty intersection # which can type any term.) The proof uses the perpetual reduction strategy which finds a longest path. This is a simplification over existing proofs that consider any longest reduction path. The choice of reduction strategy avoids the need for weakening or strengthening of type derivations. The proof becomes a bargain because it works for more intersection type systems, while being simpler than existing proofs.
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 ..."
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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.