Innocent strategies as presheaves and interactive equivalences for CCS

Seeking a general framework for reasoning about and comparing programming languages, we derive a new view of Milner's CCS. We construct a category E of plays, and a subcategory V of views. We argue that presheaves on V adequately represent innocent strategies, in the sense of game semantics. We then equip innocent strategies with a simple notion of interaction. This results in an interpretation of CCS. Based on this, we propose a notion of interactive equivalence for innocent strategies, which is close in spirit to Beffara's interpretation of testing equivalences in concurrency theory. In this framework we prove that the analogues of fair and must testing equivalences coincide, while they differ in the standard setting.Comment: In Proceedings ICE 2011, arXiv:1108.014

Full abstraction for fair testing in CCS

In previous work with Pous, we defined a semantics for CCS which may both be viewed as an innocent presheaf semantics and as a concurrent game semantics. It is here proved that a behavioural equivalence induced by this semantics on CCS processes is fully abstract for fair testing equivalence. The proof relies on a new algebraic notion called playground, which represents the 'rule of the game'. From any playground, two languages, equipped with labelled transition systems, are derived, as well as a strong, functional bisimulation between them.Comment: 15 pages, to appear in CALCO '13. To appear Lecture notes in computer science (2013

Justified Sequences in String Diagrams: a Comparison Between Two Approaches to Concurrent Game Semantics

Recent developments of game semantics have given rise to new models of concurrent languages. On the one hand, an approach based on string diagrams has given models of CCS and the pi-calculus, and on the other hand, Tsukada and Ong have designed a games model for a non-deterministic lambda-calculus. There is an obvious, shallow relationship between the two approaches, as they both define innocent strategies as sheaves for a Grothendieck topology embedding "views" into "plays". However, the notions of views and plays differ greatly between the approaches: Tsukada and Ong use notions from standard game semantics, while the authors of this paper use string diagrams. We here aim to bridge this gap by showing that even though the notions of plays, views, and innocent strategies differ, it is mostly a matter of presentation

An intensionally fully-abstract sheaf model for π (expanded version)

International audienceFollowing previous work on CCS, we propose a compositional model for the π-calculus in which processes are interpreted as sheaves on certain simple sites. Such sheaves are a concurrent form of innocent strategies, in the sense of Hyland-Ong/Nickau game semantics. We define an analogue of fair testing equivalence in the model and show that our interpretation is intensionally fully abstract for it. That is, the interpretation preserves and reflects fair testing equivalence; and furthermore, any innocent strategy is fair testing equivalent to the interpretation of some process. The central part of our work is the construction of our sites, relying on a combinatorial presentation of π-calculus traces in the spirit of string diagrams

Bisimulation maps in presheaf categories

The category of presheaves on a (small) category is a suitable semantic universe to study behaviour of various dynamical systems. In particular, presheaves can be used to record the executions of a system and their morphisms correspond to simulation maps for various kinds of state-based systems. In this paper, we introduce a notion of bisimulation maps between presheaves (or executions) to capture well known behavioural equivalences in an abstract way. We demonstrate the versatility of this framework by working out the characterisations for standard bisimulation, ∀-fair bisimulation, and branching bisimulation

Full abstraction for fair testing in CCS (expanded version)

In previous work with Pous, we defined a semantics for CCS which may both be viewed as an innocent form of presheaf semantics and as a concurrent form of game semantics. We define in this setting an analogue of fair testing equivalence, which we prove fully abstract w.r.t. standard fair testing equivalence. The proof relies on a new algebraic notion called playground, which represents the `rule of the game'. From any playground, we derive two languages equipped with labelled transition systems, as well as a strong, functional bisimulation between them.Comment: 80 page

Constructing Condensed Memories in Functorial Time

If episodic memory is constructive, experienced time is also a construct. We develop an event-based formalism that replaces the traditional objective, agent-independent notion of time with a constructive, agent-dependent notion of time. We show how to make this agent-dependent time entropic and hence well-defined. We use sheaf-theoretic techniques to render agent-dependent time functorial and to construct episodic memories as sequences of observed and constructed events with well-defined limits that maximize the consistency of categorizations assigned to objects appearing in memories. We then develop a condensed formalism that represents episodic memories as pure constructs from single events. We formulate an empirical hypothesis that human episodic memory implements a particular time-symmetric constructive functor, and discuss possible experimental tests

Probabilistic concurrent game semantics

This thesis presents a variety of models for probabilistic programming languages in the framework of concurrent games. Our starting point is the model of concurrent games with symmetry of Castellan, Clairambault and Winskel. We show that they form a symmetric monoidal closed bicategory, and that this can be turned into a cartesian closed bicategory using a linear exponential pseudo-comonad inspired by linear logic. Then, we enrich this with probability, relying heavily on Winskel's model of probabilistic concurrent strategies. We see that the bicategorical structure is not perturbed by the addition of probability. We apply this model to two probabilistic languages: a probabilistic untyped λ-calculus, and Probabilistic PCF. For the former, we relate the semantics to the probabilistic Nakajima trees of Leventis, thus obtaining a characterisation of observational equivalence for programs in terms of strategies. For the latter, we show a definability result in the spirit of the game semantics tradition. This solves an open problem, as it is notoriously difficult to model Probabilistic PCF with sequential game semantics. Finally, we introduce a model for measurable game semantics, in which games and strategies come equipped with measure-theoretic structure allowing for an accurate description of computation with continuous data types. The objective of this model is to support computation with arbitrary probability measures on the reals. In the last part of this thesis we see how this can be done by equipping strategies with parametrised families of probability measures (also known as stochastic kernels), and we construct a bicategory of measurable concurrent games and probabilistic measurable strategies