An Abstract Approach to Stratification in Linear Logic

We study the notion of stratification, as used in subsystems of linear logic with low complexity bounds on the cut-elimination procedure (the so-called light logics), from an abstract point of view, introducing a logical system in which stratification is handled by a separate modality. This modality, which is a generalization of the paragraph modality of Girard's light linear logic, arises from a general categorical construction applicable to all models of linear logic. We thus learn that stratification may be formulated independently of exponential modalities; when it is forced to be connected to exponential modalities, it yields interesting complexity properties. In particular, from our analysis stem three alternative reformulations of Baillot and Mazza's linear logic by levels: one geometric, one interactive, and one semantic

The dagger lambda calculus

We present a novel lambda calculus that casts the categorical approach to the study of quantum protocols into the rich and well established tradition of type theory. Our construction extends the linear typed lambda calculus with a linear negation of "trivialised" De Morgan duality. Reduction is realised through explicit substitution, based on a symmetric notion of binding of global scope, with rules acting on the entire typing judgement instead of on a specific subterm. Proofs of subject reduction, confluence, strong normalisation and consistency are provided, and the language is shown to be an internal language for dagger compact categories.Comment: In Proceedings QPL 2014, arXiv:1412.810

Fibred Fibration Categories

We introduce fibred type-theoretic fibration categories which are fibred categories between categorical models of Martin-L\"{o}f type theory. Fibred type-theoretic fibration categories give a categorical description of logical predicates for identity types. As an application, we show a relational parametricity result for homotopy type theory. As a corollary, it follows that every closed term of type of polymorphic endofunctions on a loop space is homotopic to some iterated concatenation of a loop

Adjunctions for exceptions

An algebraic method is used to study the semantics of exceptions in computer languages. The exceptions form a computational effect, in the sense that there is an apparent mismatch between the syntax of exceptions and their intended semantics. We solve this apparent contradiction by efining a logic for exceptions with a proof system which is close to their syntax and where their intended semantics can be seen as a model. This requires a robust framework for logics and their morphisms, which is provided by categorical tools relying on adjunctions, fractions and limit sketches.Comment: In this Version 2, minor improvements are made to Version