Abstract-We describe a framework for truly concurrent game semantics of programming languages, based on Rideau and Winskel's concurrent games on event structures. The model supports a notion of innocent strategies 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 Plotkin's 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 Ong's sequential interpretation of PCF, our parallel interpretation yields a model that is intensionally fully abstract for PCF. I. INTRODUCTIONRegardless of mathematical elegance, partial order models of concurrent computation are in principle more informative than their interleaving counterparts: they avoid the state explosion problem inherent to interleavings, and retain explicit information on causality. This can be useful for instance for the purposes of error diagnostics, or security analysis. However, although we have truly concurrent models for simple process languages such as CCS, extracting partial order models from source code remains a challenge, especially if one considers rich concurrent programming languages with complex computational features such as higher-order, state or exceptions.In order to construct compositionally a fine-grained representation of the execution of higher-order programs, game semantics is a powerful tool. Game semantics proposes to see computation as an interaction between agents (strategies) exchanging messages, hence reducing higher-order computation to the exchange of first-order tokens. Thanks to this methodology, game semantics has not only given intensionally fully abstract models of PCF [10, 1] but also pushed beyond the purely functional setting and given effectively presentable fully abstract models of higher-order programming languages with rich computational features such as control or state.Most games models for concurrent programming languages [7,12], however, are based on interleavings. Several truly concurrent frameworks for game semantics have been proposed [6,14,16,9], but have yet to be applied to the semantics of programming languages beyond CCS or linear logic -this is in part due to the fact that truly concurrent notions of strategies are mathematically more elaborate than their interleaved counterparts, and are more subtle to handle. Moreover, changing

Abstract-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 bicategory 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 of programming languages.

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.