Modalities, Cohesion, and Information Flow

It is informally understood that the purpose of modal type constructors in programming calculi is to control the flow of information between types. In order to lend rigorous support to this idea, we study the category of classified sets, a variant of a denotational semantics for information flow proposed by Abadi et al. We use classified sets to prove multiple noninterference theorems for modalities of a monadic and comonadic flavour. The common machinery behind our theorems stems from the the fact that classified sets are a (weak) model of Lawvere's theory of axiomatic cohesion. In the process, we show how cohesion can be used for reasoning about multi-modal settings. This leads to the conclusion that cohesion is a particularly useful setting for the study of both information flow, but also modalities in type theory and programming languages at large

Dual-Context Calculi for Modal Logic

We present natural deduction systems and associated modal lambda calculi for the necessity fragments of the normal modal logics K, T, K4, GL and S4. These systems are in the dual-context style: they feature two distinct zones of assumptions, one of which can be thought as modal, and the other as intuitionistic. We show that these calculi have their roots in in sequent calculi. We then investigate their metatheory, equip them with a confluent and strongly normalizing notion of reduction, and show that they coincide with the usual Hilbert systems up to provability. Finally, we investigate a categorical semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089

On the Semantics of Intensionality and Intensional Recursion

Intensionality is a phenomenon that occurs in logic and computation. In the most general sense, a function is intensional if it operates at a level finer than (extensional) equality. This is a familiar setting for computer scientists, who often study different programs or processes that are interchangeable, i.e. extensionally equal, even though they are not implemented in the same way, so intensionally distinct. Concomitant with intensionality is the phenomenon of intensional recursion, which refers to the ability of a program to have access to its own code. In computability theory, intensional recursion is enabled by Kleene's Second Recursion Theorem. This thesis is concerned with the crafting of a logical toolkit through which these phenomena can be studied. Our main contribution is a framework in which mathematical and computational constructions can be considered either extensionally, i.e. as abstract values, or intensionally, i.e. as fine-grained descriptions of their construction. Once this is achieved, it may be used to analyse intensional recursion.Comment: DPhil thesis, Department of Computer Science & St John's College, University of Oxfor

Syllepsis in Homotopy Type Theory

International audienceThe Eckmann-Hilton argument shows that any two monoid structures on the same set satisfying the interchange law are in fact the same operation, which is moreover commutative. When the monoids correspond to the vertical and horizontal composition of a sufficiently higher-dimensional category, the Eckmann-Hilton argument itself appears as a higher cell. This cell is often required to satisfy an additional piece of coherence, which is known as the syllepsis. We show that the syllepsis can be constructed from the elimination rule of intensional identity types in Martin-Löf type theory

Client-Server Sessions in Linear Logic

We introduce coexponentials, a new set of modalities for Classical Linear Logic. As duals to exponentials, the coexponentials codify a distributed form of the structural rules of weakening and contraction. This makes them a suitable logical device for encapsulating the pattern of a server receiving requests from an arbitrary number of clients on a single channel. Guided by this intuition we formulate a system of session types based on Classical Linear Logic with coexponentials, which is suited to modelling client-server interactions. We also present a session-typed functional programming language for server-client programming, which we translate to our system of coexponentials