The Parallel Intensionally Fully Abstract Games Model of PCF

Abstract

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

Resource-Tracking Concurrent Games

“…As in [5] we chose to interpret an affine language: this lets us focus on the key phenomena which are already at play, avoiding the technical hindrance caused by replication. As suggested by recent experience with concurrent games [6,10], we expect the developments presented here to extend transparently in the presence of symmetry [8,9]; this would allow us to move to the general (non-affine) setting.…”

Game semantics for quantum programming

“…Recently, this formulation has been actively developed (e.g. [Castellan et al 2015;Clairambault et al 2012]) and proved to be a powerful and versatile basis for semantics; including with probabilities [Castellan et al 2018]. See [Castellan et al 2017] for an introduction.…”

“…But one must then express that these copies are indistinguishable. In concurrent games, this was achieved by concurrent games with symmetry [Castellan et al 2014[Castellan et al , 2015, later generalized to probabilistic games [Castellan et al 2018], simply requiring probability valuations to be invariant under symmetry. We now extend it to the quantum case, and complete our interpretation.…”

“…We base our semantics on concurrent games, a framework for games based around ideas from concurrency theory. Initiated by [Abramsky and Melliès 1999;Melliès 2005], this family of game semantics has been actively developed recently, prompted by new foundations introduced in [Rideau and Winskel 2011] ś milestones relevant to the present contribution include fully abstract models of higher-order [Castellan et al 2015], concurrent and probabilistic [Castellan et al 2018;Winskel 2013] programs. A notion of quantum strategies was already proposed in this setting [Winskel 2013] but it was not broad enough for our purposes so we had to develop a new notion from scratch.…”

“…In Section 5, we build on QCG the further structure required to interpret the affine fragment, and prove adequacy. Finally, in Section 6 we mix quantum annotations with symmetry from [Castellan et al 2015], and extend our adequacy result to the full calculus.…”