A language for multiplicative-additive linear logic

A term calculus for the proofs in multiplicative-additive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive categories with additives. It is also shown that proof equivalence is decidable by showing that the cut elimination rewrites supply a confluent rewriting system modulo equations.Comment: 16 pages without appendices, 30 with appendice

Complete Positivity for Mixed Unitary Categories

In this article we generalize the \CP^\infty-construction of dagger monoidal categories to mixed unitary categories. Mixed unitary categories provide a setting, which generalizes (compact) dagger monoidal categories and in which one may study quantum processes of arbitrary (infinite) dimensions. We show that the existing results for the \CP^\infty-construction hold in this more general setting. In particular, we generalize the notion of environment structures to mixed unitary categories and show that the \CP^\infty-construction on mixed unitary categories is characterized by this generalized environment structure.Comment: Lots of figure

Tangent Categories from the Coalgebras of Differential Categories

Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result applies when there are no negatives and thus encompasses examples arising from combinatorics and computer science

There Is Only One Notion of Differentiation

Differential linear logic was introduced as a syntactic proof-theoretic approach to the analysis of differential calculus. Differential categories were subsequently introduce to provide a categorical model theory for differential linear logic. Differential categories used two different approaches for defining differentiation abstractly: a deriving transformation and a coderiliction. While it was thought that these notions could give rise to distinct notions of differentiation, we show here that these notions, in the presence of a monoidal coalgebra modality, are completely equivalent

Components as processes: an exercise in coalgebraic modeling

IFIP TC6/WG6.1. Fourth International Conference on Formal Methods for Open Object-Based Distributed Systems (FMOODS 2000) September 6–8, 2000, Stanford, California, USASoftware components, arising, typically, in systems ’ analysis and design, are characterized by a public interface and a private encapsulated state. They persist (and evolve) in time, according to some behavioural patterns. This paper is an exercise in modeling such components as coalgebras for some kinds of endofunctors on , capturing both (interface) types and behavioural aspects. The construction of component categories, cofibred over the interface space, emerges by generalizing the usual notion of a coalgebra morphism. A collection of composition operators as well as a generic notion of bisimilarity, are discussed