The dynamics of many calculi can be most clearly defined by a reduction semantics. To work with a calculus, however, an understanding of operational congruences is fundamental; these can often be given tractable definitions or characterisations using a labelled transition semantics. This paper considers calculi with arbitrary reduction semantics of three simple classes, firstly ground term rewriting, then left-linear term rewriting, and then a class which is essentially the action calculi lacking substantive name binding. General definitions of labelled transitions are given in each case, uniformly in the set of rewrite rules, and without requiring the prescription of additional notions of observation. They give rise to bisimulation congruences. As a test of the theory it is shown that bisimulation for a fragment of CCS is recovered. The transitions generated for a fragment of the Ambient Calculus of Cardelli and Gordon, and for SKI combinators, are also discussed briefly.

Abstract not found

This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag.

Nominal rewriting is based on the observation that if we add support for alphaequivalence to first-order syntax using the nominal-set approach, then systems with binding, including higher-order reduction schemes such as lambda-calculus betareduction, can be smoothly represented. Nominal rewriting maintains a strict distinction between variables of the objectlanguage (atoms) and of the meta-language (variables or unknowns). Atoms may be bound by a special abstraction operation, but variables cannot be bound, giving the framework a pronounced first-order character, since substitution of terms for variables is not capture-avoiding. We show how good properties of first-order rewriting survive the extension, by giving an efficient rewriting algorithm, a critical pair lemma, and a confluence theorem

Consider Adam and Eve. Count generations starting from them. Supposing that there will always be people, then it's true that for any generation X, eventually there will be people belonging to the next generation X + 1. In this paper the same result is established for the class of higher order pattern rewriting systems. 1 Introduction Consider a set of structures and a set of transformations on them specifying how a structure may be transformed into another one. Suppose the transformations are of the following form: first a structure is decomposed into substructures, next some substructure is replaced by another one, and finally the substructures are composed into a structure again. (destroy) The parts of the initial structure eliminated in the course of the transformation (i.e. the parts of the replaced substructure as well as the parts eliminated in the initial decomposition) can be thought of as being destroyed . (create) The parts of the final structure introduced in the cou...

. We prove a version of the Standardization Theorem and the Discrete Normalization (DN) Theorem in stable Deterministic Residual Structures, which are Abstract Reduction Systems with axiomatized residual relation, and model orthogonal rewrite systems. The latter theorem gives a strategy for construction of reductions L'evy-equivalent (or permutation-equivalent) to a given, finite or infinite, regular (or semi-linear) reduction, based on the neededness concept of Huet and L'evy. This and other results of this paper add to the understanding of L'evy-equivalence of reductions, and consequently, L'evy's reduction space. We demonstrate how elements of this space can be used to give denotational semantics to known functional languages in an abstract manner. 1 Introduction Long ago, Curry and Feys [CuFe58] proved that repeated contraction of leftmostoutermost redexes in any normalizable -term eventually yields its normal form, even if the term possesses infinite reductions as well. The reaso...

As a minimal environment for the study of permutative reductions an extension LambdaJ of the untyped lambda-calculus is considered. In this non-terminating system with non-trivial critical pairs, conuence is established by studying triangle properties that allow to treat permutative reductions modularly and could be extended to more complex term systems with permutations. Standardization is shown by means of an inductive definition of standard reduction that closely follows the inductive term structure and captures the intuitive notion of standardness even for permutative reductions.

. A rewrite sequence is said to be outermost-fair if every outermost redex occurrence is eventually eliminated. Outermost-fair rewriting is known to be (head-)normalising for almost orthogonal rewrite systems. In this paper we study (head-)normalisation for the larger class of weakly orthogonal rewrite systems. Normalisation is established and a counterexample against head-normalisation is provided. Nevertheless, infinitary normalisation, which is usually obtained as a corollary of headnormalisation, is shown to hold. 1 Introduction The term f(a) in the term rewrite system fa ! a; f(x) ! bg can be rewritten to normal form b, but is also the starting point of the infinite rewrite sequence f(a) ! f(a) ! : : :. It is then of interest to design a normalising strategy, i.e. a restriction on rewriting which guarantees to reach a normal form if one can be reached. How to design a normalising strategy? Observe that in the example the normal form b was reached by contracting the redex closest...

This paper surveys a part of the theory of fi-reduction in -calculus 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 -terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in -terms 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 -calculus and type theory. 1. Introduction Considerable attention has been devoted to classification of reduction strategies in type-free -calculus [4, 6, 7, 15, 38, 44, 81]---see also [2, Ch. 13]. We are concerned with strategies differing in the length of reduction paths. This paper draws on several sources. In late 1994, van Raamsdonk and Severi [59] and Srensen [66, 67] independently developed ...

We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the -calculus due to Bergstra and Klop, and others. To unify our results with those in the literature, we introduce Context-sensitive Conditional Expression Reduction Systems (CCERSs) and prove confluence for orthogonal CCERSs. We then define a perpetual one-step reduction strategy which enables one to construct minimal (w.r.t. Levy's permutation ordering on reductions) infinite reductions in orthogonal CCERSs. We then prove (1) perpetuality (in a specific context) of a redex whose contraction does not erase potentially infinite arguments, which are possibly finite (i.e., strongly normalizable) arguments that may become infinite after a number of outside steps, and (2) perpetuality (in every con...