Simon Castellan

Symmetry in concurrent games

Behavioural symmetry is introduced into concurrent games. It expresses when plays are essentially the same. A characterization of strategies on games with symmetry is provided. This leads to a bi-category of strategies on games with symmetry. Symmetry helps allay the perhaps overly-concrete nature of games and strategies, and shares many mathematical features with homotopy. In the presence of symmetry we can consider monads for which the monad laws do not hold on the nose but do hold up to symmetry. This broadening of the concept of monad has a dramatic effect on the types concurrent games can support and allows us, for example, to recover the replication needed to express and extend traditional game semantics.

The Parallel Intensionally Fully Abstract Games Model of PCF

We describe a framework for truly concurrent game semantics of programming languages, based on Rideau and Winskels concurrent games on event structures. The model supports a notion of innocent strategy that permits concurrent and non-deterministic behaviour, but which coincides with traditional Hyland-Ong innocent strategies if one restricts to the deterministic sequential case. In this framework we give an alternative interpretation of Plot kins PCF, that takes advantage of the concurrent nature of strategies and formalizes the idea that although PCF is a sequential language, certain sub-computations are independent and can be computed in a parallel fashion. We show that just as Hyland and Ongs sequential interpretation of PCF, our parallel interpretation yields a model that is intensionally fully abstract for PCF.

Causality vs. interleavings in concurrent game semantics

We investigate relationships between interleaving and causal notions of game semantics for concurrent programming languages, focusing on the existence of canonical compact causal representations of the interleaving game semantics of programs. We perform our study on an affine variant of Idealized Parallel Algol (IPA), for which we present two games model: and interleaving model (an adaptation of Ghica and Murawski’s fully abstract games model for IPA up to may-testing), and a causal model (a variant of Rideau and Winskel’s games on event structures). Both models are sound and adequate for affine IPA. Then, we relate the two models. First we give a causality-forgetting operation mapping functorially the causal model to the interleaving one. We show that from an interleaving strategy we can reconstruct a causal strategy, from which it follows that the interleaving model is the observational quotient of the causal one. Then, we investigate several reconstructions of causal strategies from interleaving ones, showing finally that there are programs which are inherently causally ambiguous, with several distinct minimal causal representations.

Undecidability of Equality in the Free Locally Cartesian Closed Category

We show that a version of Martin-Lof type theory with extensional identity, a unit type N1, Sigma, Pi, and a base type is a free category with families (supporting these type formers) both in a 1- and a 2-categorical sense. It follows that the underlying category of contexts is a free locally cartesian closed category in a 2-categorical sense because of a previously proved biequivalence. We then show that equality in this category is undecidable by reducing it to the undecidability of convertibility in combinatory logic.

The concurrent game semantics of Probabilistic PCF

We define a new games model of Probabilistic PCF (PPCF) by enriching thin concurrent games with symmetry, recently introduced by Castellan et al, with probability. This model supports two interpretations of PPCF, one sequential and one parallel. We make the case for this model by exploiting the causal structure of probabilistic concurrent strategies. First, we show that the strategies obtained from PPCF programs have a deadlock-free interaction, and therefore deduce that there is an interpretation-preserving functor from our games to the probabilistic relational model recently proved fully abstract by Ehrhard et al. It follows that our model is intensionally fully abstract. Finally, we propose a definition of probabilistic innocence and prove a finite definability result, leading to a second (independent) proof of full abstraction.

Non-angelic Concurrent Game Semantics.

The hiding operation, crucial in the compositional aspect of game semantics, removes computation paths not leading to observable results. Accordingly, games models are usually biased towards angelic non-determinism: diverging branches are forgotten.

Observably Deterministic Concurrent Strategies and Intensional Full Abstraction for Parallel-or.

Although Plotkins parallel-or is inherently deterministic, it has a non-deterministic interpretation in games based on (prime) event structures-in which an event has a unique causal history-because they do not directly support disjunctive causality. General event structures can express disjunctive causality and have a more permissive notion of determinism, but do not support hiding. We show that (structures equivalent to) deterministic general event structures do support hiding, and construct a new category of games based on them with a deterministic interpretation of aPCFpor, an affine variant of PCF extended with parallel-or. We then exploit this deterministic interpretation to give a relaxed notion of determinism (observable determinism) on the plain event structures model. Putting this together with our previously introduced concurrent notions of well-bracketing and innocence, we obtain an intensionally fully abstract model of aPCFpor.

Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended Version)

We show that a version of Martin-Lof type theory with an extensional identity type former I, a unit type N-1, Sigma-types, Pi-types, and a base type is a free category with families (supporting these type formers) both in a 1- and a 2-categorical sense. It follows that the underlying category of contexts is a free locally cartesian closed category in a 2-categorical sense because of a previously proved biequivalence. We show that equality in this category is undecidable by reducing it to the undecidability of convertibility in combinatory logic. Essentially the same construction also shows a slightly strengthened form of the result that equality in extensional Martin-Lof type theory with one universe is undecidable.