Categorical combinatorics of scheduling and synchronization in game semantics

• Published in: Proceedings of ACM on programming languages Vol. 3; no. POPL; pp. 1 - 30
• Main Author: Mellies, Paul-André
• Format: Journal Article
• Language: English
• Published: ACM 01.01.2019
• Subjects: Algebraic Geometry; Algebraic Topology; Category Theory; Computation and Language; Computer Science; Computer Science and Game Theory; Formal Languages and Automata Theory; Logic; Logic in Computer Science; Mathematical Software; Mathematics; Programming Languages; Quantum Algebra; Theory of computation; Categorical semantics; Concurrency; Game semantics; Denotational semantics; Sequential alternating games; Synchronization algebras; Bicategorical models of linear logic; Concurrent non-alternating games; Template games
• ISSN: 2475-1421; 2475-1421
• DOI: 10.1145/3290336

Game semantics is the art of interpreting types as games and programs as strategies interacting in space and time with their environment. In order to reflect the interactive behavior of programs, strategies are required to follow specific scheduling policies. Typically, in the case of a purely sequential programming language, the program (Player) and its environment (Opponent) will play one after the other, in a strictly alternating way. On the other hand, in the case of a concurrent language, Player and Opponent will be allowed to play several moves in a row, in a non-alternating way. In both cases, the scheduling policy is designed very carefully in order to ensure that the strategies synchronize properly and compose well when plugged together. A longstanding conceptual problem has been to understand when and why a given scheduling policy works and is compositional in that sense. In this paper, we exhibit a number of simple and fundamental combinatorial structures which ensure that a given scheduling policy encoded as synchronization template defines a symmetric monoidal closed (and in fact star-autonomous) bicategory of games, strategies and simulations. To that purpose, we choose to work at a very general level, and illustrate our method by constructing two template game models of linear logic with different flavors (alternating and non-alternating) using the same categorical combinatorics, performed in the category of small categories. As a whole, the paper may be seen as a hymn in praise of synchronization, building on the notion of synchronization algebra in process calculi and adapting it smoothly to programming language semantics, using a combination of ideas at the converging point of game semantics and of categorical algebra.