Superdevelopments for Weak Reduction

We study superdevelopments in the weak lambda calculus of Cagman and Hindley, a confluent variant of the standard weak lambda calculus in which reduction below lambdas is forbidden. In contrast to developments, a superdevelopment from a term M allows not only residuals of redexes in M to be reduced but also some newly created ones. In the lambda calculus there are three ways new redexes may be created; in the weak lambda calculus a new form of redex creation is possible. We present labeled and simultaneous reduction formulations of superdevelopments for the weak lambda calculus and prove them equivalent

The structural lambda-calculus

Inspired by a recent graphical formalism for lambda-calculus based on Linear Logic technology, we introduce an untyped structural lambda-calculus, called lambda_j, which combines action at a distance with exponential rules decomposing the substitution by means of weakening, contraction and dereliction. Firstly, we prove fundamental properties such as confluence and preservation of beta-strong normalisation. Secondly, we use lambda_j to describe known notions of developments and superdevelopments, and introduce a more general one called XL-development. Then we show how to reformulate Regnier's sigma-equivalence in lambda_j so that it becomes a strong bisimulation. Finally, we prove that explicit composition or de-composition of substitutions can be added to lambda_j while still preserving beta-strong normalisation

A Theory of Explicit Substitutions with Safe and Full Composition

Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the first part of this paper. Then, very simple technology in named variable-style notation is used to establish a theory of explicit substitutions for the lambda-calculus which enjoys a whole set of useful properties such as full composition, simulation of one-step beta-reduction, preservation of beta-strong normalisation, strong normalisation of typed terms and confluence on metaterms. Normalisation of related calculi is also discussed.Comment: 29 pages Special Issue: Selected Papers of the Conference "International Colloquium on Automata, Languages and Programming 2008" edited by Giuseppe Castagna and Igor Walukiewic

Equational term graph rewriting

We present an equational framework for term graph rewriting with cycles. The usual notion of homomorphism is phrased in terms of the notion of bisimulation, which is well-known in process algebra and concurrency theory. Specifically, a homomorphism is a functional bisimulation. We prove that the bisimilarity class of a term graph, partially ordered by functional bisimulation, is a complete lattice. It is shown how Equational Logic induces a notion of copying and substitution on term graphs, or systems of recursion equations, and also suggests the introduction of hidden or nameless nodes in a term graph. Hidden nodes can be used only once. The general framework of term graphs with copying is compared with the more restricted copying facilities embodied in the $mu$-rule, and translations are given between term graphs and $mu$-expressions. Using these, a proo

Matching modulo superdevelopments application to second-order matching

To perform higher-order matching, we need to decide the βη-equivalence on λ-terms. The first way to do it is to use simply typed λ-calculus and this is the usual framework where higher-order matching is performed. Another approach consists in deciding a restricted equivalence based on finite superdevelopments. We consider higher-order matching modulo this equivalence over untyped λ-terms for which we propose a terminating, sound and complete matching algorithm. This is in particular of interest since all second-order β-matches are matches modulo superdevelopments. We further propose a restriction to second-order matching that gives exactly all second-order matches