Metaconfluence of Calculi with Explicit Substitutions at a Distance

Confluence is a key property of rewriting calculi that guarantees uniqueness of normal-forms when they exist. Metaconfluence is even more general, and guarantees confluence on open/meta terms, i.e. terms with holes, called metavariables that can be filled up with other (open/meta) terms. The difficulty to deal with open terms comes from the fact that the structure of metaterms is only partially known, so that some reduction rules became blocked by the metavariables. In this work, we establish metaconfluence for a family of calculi with explicit substitutions (ES) that enjoy preservation of strong-normalization (PSN) and that act at a distance. For that, we first extend the notion of reduction on metaterms in such a way that explicit substitutions are never structurally moved, i.e. they also act at a distance on metaterms. The resulting reduction relations are still rewriting systems, i.e. they do not include equational axioms, thus providing for the first time an interesting family of lambda-calculi with explicit substitutions that enjoy both PSN and metaconfluence without requiring sophisticated notions of reduction modulo a set of equations

Encoding Tight Typing in a Unified Framework

This paper explores how the intersection type theories of call-by-name (CBN) and call-by-value (CBV) can be unified in a more general framework provided by call-by-push-value (CBPV). Indeed, we propose tight type systems for CBN and CBV that can be both encoded in a unique tight type system for CBPV. All such systems are quantitative, i.e. they provide exact information about the length of normalization sequences to normal form as well as the size of these normal forms. Moreover, the length of reduction sequences are discriminated according to their multiplicative and exponential nature, a concept inherited from linear logic. Last but not least, it is possible to extract quantitative measures for CBN and CBV from their corresponding encodings in CBPV

Inhabitation for Non-idempotent Intersection Types

The inhabitation problem for intersection types in the lambda-calculus is known to be undecidable. We study the problem in the case of non-idempotent intersection, considering several type assignment systems, which characterize the solvable or the strongly normalizing lambda-terms. We prove the decidability of the inhabitation problem for all the systems considered, by providing sound and complete inhabitation algorithms for them

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

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

Quantitative Types for Intuitionistic Calculi

We define quantitative type systems for two intuitionistic term languages. While the first language in natural deduction style is already known in the literature, the second one is one of the contributions of the paper, and turns out to be a natural computational interpretation of sequent calculus style by means of a non-idempotent type discipline. The type systems are able to characterize linear-head, weak and strong normalization sets of terms. All such characterizations are given by means of combinatorial arguments, i.e. there is a measure based on type derivations which is decreasing with respect to the different reduction relations considered in the paper

Types as Resources for Classical Natural Deduction

We define two resource aware typing systems for the lambda-mu-calculus based on non-idempotent intersection and union types. The non-idempotent approach provides very simple combinatorial arguments - based on decreasing measures of type derivations - to characterize head and strongly normalizing terms. Moreover, typability provides upper bounds for the length of head-reduction sequences and maximal reduction sequences

Strong Normalization through Intersection Types and Memory

AbstractWe characterize β-strongly normalizing λ-terms by means of a non-idempotent intersection type system. More precisely, we first define a memory calculus K together with a non-idempotent intersection type system K, and we show that a K-term t is typable in K if and only if t is K-strongly normalizing. We then show that β-strong normalization is equivalent to K-strong normalization. We conclude since λ-terms are strictly included in K-terms