Untyped lambda-calculus

No abstract

Distilling abstract machines

It is well-known that many environment-based abstract machines can be seen as strategies in lambda calculi with explicit substitutions (ES). Recently, graphical syntaxes and linear logic led to the linear substitution calculus (LSC), a new approach to ES that is halfway between small-step calculi and traditional calculi with ES. This paper studies the relationship between the LSC and environment-based abstract machines. While traditional calculi with ES simulate abstract machines, the LSC rather distills them: some transitions are simulated while others vanish, as they map to a notion of structural congruence. The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic. We show that such a pattern applies uniformly in call-by-name, call-by-value, and call-by-need, catching many machines in the literature. We start by distilling the KAM, the CEK, and a sketch of the ZINC, and then provide simplified versions of the SECD, the lazy KAM, and Sestoft's machine. Along the way we also introduce some new machines with global environments. Moreover, we show that distillation preserves the time complexity of the executions, i.e. The LSC is a complexity-preserving abstraction of abstract machines.Fil: Accattoli, Beniamino. Universidad de Bologna; Italia. University of Carnegie Mellon; Estados UnidosFil: Barenbaum, Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires; ArgentinaFil: Mazza, Damiano. Centre National de la Recherche Scientifique; Francia. Universite de Paris 13-Nord; Franci

Are Types needed for Natural Language?

Logic, due to the paradoxes, is absent from the type free -calculus. This makes such a calculus an unsuitable device for Natural Language Semantics. Moreover, the problems that arise from mixing the type free -calculus with logic lead to type theory and hence formalisations of Natural Language were carried out in a strictly typed framework. It was shown however, that strict type theory cannot capture the self-referential nature of language ([Parsons 79], [Chierchia, Turner 88] and [Kamareddine, Klein 93]) and hence other approaches were needed. For example, the approach carried out by Parsons is based on creating a notion of floating types which can be instantiated to particular instances of types whereas the approaches of Chierchia, Turner and Kamareddine, Klein are based on a type free framework. In this paper, we will embed the typing system of [Parsons 79] into a version of the one proposed in [Kamareddine, Klein 93] giving an interpretation of Parsons' system in a type free theory..