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Oostrom, Uniform normalisation beyond orthogonality
 Proceedings of the Twelfth International Conference on Rewriting Techniques and Applications (RTA ’01), Lecture Notes in Computer Science (2001
, 2001
"... Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A wellknown fact is uniform normalisation of orthogonal ..."
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Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A wellknown fact is uniform normalisation of orthogonal nonerasing term rewrite systems, e.g. the λIcalculus. In the present paper both restrictions are analysed. Orthogonality is seen to pertain to the linear part and nonerasingness to the nonlinear part of rewrite steps. Based on this analysis, a modular proof method for uniform normalisation is presented which allows to go beyond orthogonality. The method is shown applicable to biclosed first and secondorder term rewrite systems as well as to a λcalculus with explicit substitutions. 1
A NOTE ON SHORTEST DEVELOPMENTS
, 708
"... Abstract. De Vrijer has presented a proof of the finite developments theorem which, in addition to showing that all developments are finite, gives an effective reduction strategy computing longest developments as well as a simple formula computing their length. We show that by applying a rather simp ..."
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Abstract. De Vrijer has presented a proof of the finite developments theorem which, in addition to showing that all developments are finite, gives an effective reduction strategy computing longest developments as well as a simple formula computing their length. We show that by applying a rather simple and intuitive principle of duality to de Vrijer’s approach one arrives at a proof that some developments are finite which in addition yields an effective reduction strategy computing shortest developments as well as a simple formula computing their length. The duality fails for general βreduction. Our results simplify previous work by Khasidashvili. 1.
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.
0 Least Upper Bounds on the Size of Confluence and ChurchRosser Diagrams in Term Rewriting and λCalculus 1
"... We study confluence and the ChurchRosser property in term rewriting and λcalculus with explicit bounds on term sizes and reduction lengths. Given a system R, we are interested in the lengths of the reductions in the smallest valleys t → ∗ s ′ ∗ ← t ′ expressed as a function: — for confluence a ..."
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We study confluence and the ChurchRosser property in term rewriting and λcalculus with explicit bounds on term sizes and reduction lengths. Given a system R, we are interested in the lengths of the reductions in the smallest valleys t → ∗ s ′ ∗ ← t ′ expressed as a function: — for confluence a function vsR(m, n) where the valleys are for peaks t ∗ ← s → ∗ t ′ with s of size at most m and the reductions of maximum length n, and — for the ChurchRosser property a function cvsR(m, n) where the valleys are for conversions t ↔ ∗ t ′ with t and t ′ of size at most m and the conversion of maximum length n. For confluent term rewriting systems (TRSs), we prove that vsR is a total computable function, and for linear such systems that cvsR is a total computable function. Conversely, we show that every total computable function is the lower bound on the functions vsR(m, n) and cvsR(m, n) for some TRS R: In particular, we show that for every total computable function ϕ: N − → N there is a TRS R with a single term s such that vsR(s, n) ≥ ϕ(n) and cvsR(n, n) ≥ ϕ(n) for all n. For orthogonal TRSs R we prove that there is a constant k such that (a) vsR(m, n) is bounded from above by a function exponential in k and (b) cvsR(m, n) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy. Similarly, for λcalculus, we show that vsR(m, n) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy.