Uniform Normalisation beyond Orthogonality

  title={Uniform Normalisation beyond Orthogonality},
  author={Zurab Khasidashvili and Mizuhito Ogawa and Vincent van Oostrom},
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 well-known fact is uniform normalisation of orthogonal non-erasing term rewrite systems, e.g. the λ/-calculus. In the present paper both restrictions are analysed. Orthogonality is seen to pertain to the linear part and non-erasingness to the non-linear part… 

Vicious circles in rewriting systems

textabstractWe continue our study of the difference between Weak Normalisation (WN) and Strong Normalisation (SN). We extend our earlier result that orthogonal TRSs with the property WN do not admit

Equivalence of Reductions in Higher-Order Rewriting

The main result of this dissertation is that for local, orthogonal HRSs the three notions of equivalence coincide, and the proof that H RSs enjoy finite family developments, meaning that in infinite reductions, there is no bound on the number of steps which were involved in creating a given symbol.

Perpetuality for Full and Safe Composition (in a Constructive Setting)

The strong normalisation proof is based on implicit substitution rather than explicit substitution, so that it turns out to be modular w.r.t. the well-known proofs for typed lambda-calculus.

The Theory of Calculi with Explicit Substitutions Revisited

Very simple technology is used to establish a general theory of explicit substitutions for the lambda-calculus which enjoys fundamental properties such as simulation of one-step beta-reduction, confluence on metaterms, preservation of beta-strong normalisation, strong normalisation of typed terms and full composition.

Weak Convergence and Uniform Normalization in Infinitary Rewriting

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Extending the Explicit Substitution Paradigm

We present a simple term language with explicit operators for erasure, duplication and substitution enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic's

Delimiting diagrams

We propose a way in which a strategy may exceed another one. In particular, we say that a strategy uniformly exceeds another one, if every maximal reduction for the former is as least as long as

Metamathematics in Coq

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Resource operators for lambda-calculus

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A characterization of the class of higher-order rewriting systems which can be encoded by first- order rewriting modulo an empty theory (that is, Ɛ = θ), which includes of course the λ-calculus.



Development Closed Critical Pairs

This work extends results from Huet and Toyama on left-linear first-order term rewriting systems by replacing the parallel closed condition by a development closed condition, yielding a confluence criterion for Klop's combinatory reduction systems), Khasidashvili's expression reduction systems, and Nipkow's higher-order pattern rewriting systems.

Perpetuality and Uniform Normalization in Orthogonal Rewrite Systems

We study perpetuality of reduction steps, as well as perpetuality of redexes, in orthogonal rewrite systems. A perpetual step is a reduction step which retains the possibility of infinite reductions.

Strong sequentiality of left-linear overlapping term rewriting systems

  • Y. Toyama
  • Mathematics
    [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science
  • 1992
It is shown that index reduction is normalizing for the class of strongly sequential left-linear term rewriting systems in which every critical pair can be joined with root balanced reductions.

Normalisation in Weakly Orthogonal Rewriting

(Infinitary) normalisation is established and a counterexample against head-normalisation is given for the larger class of weakly orthogonal rewrite systems.

The Longest Perpetual Reductions in Orthogonal Expression Reduction Systems

A strategy is designed that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise, and the Conservation Theorem for OERSs follows easily from the properties of the strategy.

Conflunt reductions: Abstract properties and applications to term rewriting systems

  • G. Huet
  • Computer Science
    18th Annual Symposium on Foundations of Computer Science (sfcs 1977)
  • 1977
This paper gives new results, and presents old ones in a unified formalism, concerning Church-Rosser theorems for rewriting systems, and shows how these results yield efficient methods for the mechanization of equational theories.

lambda-nu, A Calculus of Explicit Substitutions which Preserves Strong Normalisation

Abstract Explicit substitutions were proposed by Abadi, Cardelli, Curien, Hardin and Lévy to internalise substitutions into λ-calculus and to propose a mechanism for computing on substitutions. λν is

Perpetuality in a named lambda calculus with explicit substitutions

  • E. Bonelli
  • Mathematics
    Mathematical Structures in Computer Science
  • 2001
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