ABSTRACT
Game semantics is a rich and successful class of denotational models for programming languages. Most game models feature a rather intuitive setup, yet surprisingly difficult proofs of such basic results as associativity of composition of strategies. We seek to unify these models into a basic abstract framework for game semantics, game settings. Our main contribution is the generic construction, for any game setting, of a category of games and strategies. Furthermore, we extend the framework to deal with innocence, and prove that innocent strategies form a subcategory. We finally show that our constructions cover many concrete cases, mainly among the early models [5, 23] and the recent, sheaf-based ones [40].
- IEEE 1997. Proc. 12th Symposium on Logic in Computer Science IEEE.Google Scholar
- IEEE 2015. Proc. 30th Symposium on Logic in Computer Science IEEE.Google Scholar
- S. Abramsky. 2003. Sequentiality vs. Concurrency In Games And Logic. Mathematical Structures in Computer Science 13, 4 (2003), 531--565. Google ScholarDigital Library
- S. Abramsky, K. Honda, and G. McCusker. 1998. A Fully Abstract Game Semantics for General References. In Proc. 13th Symposium on Logic in Computer Science IEEE, 334--344. Google ScholarDigital Library
- S. Abramsky, R. Jagadeesan, and P. Malacaria. 2000. Full Abstraction for PCF. Information and Computation 163, 2 (2000), 409--470. Google ScholarDigital Library
- P. Baillot, V. Danos, T. Ehrhard, and L. Regnier. 1997. Believe it or not, AJM's Games Model is a Model of Classical Linear Logic, See {1}, 68--75. Google ScholarDigital Library
- A. Blass. 1972. Degrees of indeterminacy in games. Fundamenta Mathematica LXXVII (1972), 151--162.Google Scholar
- A. Blass. 1992. A game semantics for linear logic. Annals of Pure and Applied Logic 56, 1-3 (1992), 183--220.Google ScholarCross Ref
- N.J. Bowler. 2011. A unified approach to the construction of categories of games. Ph.D. Dissertation. University of Cambridge.Google Scholar
- S. Castellan, P. Clairambault, and G. Winskel. 2015. The parallel intensionally fully abstract games model of PCF, See {2}.Google Scholar
- C. Eberhart and T. Hirschowitz. 2017. Game semantics as a singular functor, and definability as geometric realisation. (2017). Preprint hal-01527171.Google Scholar
- C. Eberhart and T. Hirschowitz. 2017. What's in a game? A theory of game models. (2017). Preprint hal-01634162.Google Scholar
- M. P. Fiore. 2012. Discrete Generalised Polynomial Functors - (Extended Abstract). In Proc. 39th International Colloquium on Automata, Languages and Programming (LNCS), Vol. 7392. Springer, 214--226. Google ScholarDigital Library
- R. H. G. Garner and T. Hirschowitz. 2018. Shapely monads and analytic functors. Journal of Logic and Computation 28, 1 (2018), 33--83.Google ScholarCross Ref
- R. Guitart. 1980. Relations et carrés exacts. Annales des Sciences Mathématiques du Québec 4, 2 (1980), 103--125.Google Scholar
- R. Harmer. 1999. Games and Full Abstraction for Nondeterministic Languages. Ph.D. Dissertation. Imperial College, University of London.Google Scholar
- R. Harmer, J. M. E. Hyland, and P.-A. Melliès. 2007. Categorical Combinatorics for Innocent Strategies. In Proc. 22nd Symposium on Logic in Computer Science IEEE, 379--388. Google ScholarDigital Library
- R. Harmer and G. McCusker. 1999. A Fully Abstract Game Semantics for Finite Nondeterminism. In LICS. IEEE, 422--430. Google ScholarDigital Library
- F. Hatat. 2013. Graphical games and proof theory. Ph.D. Dissertation. Université de Grenoble.Google Scholar
- M. Hirschowitz. 2004. Jeux abstraits et composition catégorique. Ph.D. Dissertation. Université Paris-Diderot - Paris VII.Google Scholar
- M. Hirschowitz, A. Hirschowitz, and T. Hirschowitz. 2007. A theory for game theories. In Proc. 27th Foundations of Software Technology and Theoretical Computer Science (LNCS), Vol. 4855. Springer, 192--203. Google ScholarDigital Library
- T. Hirschowitz. 2014. Full abstraction for fair testing in CCS (expanded version). Logical Methods in Computer Science 10, 4 (2014).Google Scholar
- J. M. E. Hyland and C.-H. L. Ong. 2000. On Full Abstraction for PCF: I, II, and III. Information and Computation 163, 2 (2000), 285--408. Google ScholarDigital Library
- A. Joyal. 1977. Remarques sur la théorie des jeux à deux personnes. Gazette des Sciences Mathématiques du Québec 1, 4 (1977), 46--52.Google Scholar
- A. Joyal. 1986. Foncteurs analytiques et espèces de structure. In Combinatoire énumérative (Montréal 1985) (Lecture Notes in Mathematics), Vol. 1234. Springer, 126--159.Google Scholar
- J. Laird. 1997. Full Abstraction for Functional Languages with Control, See {1}, 58--67. Google ScholarDigital Library
- S. Mac Lane. 1998. Categories for the Working Mathematician (2nd ed.). Number 5 in Graduate Texts in Mathematics. Springer.Google Scholar
- G. McCusker. 1996. Games and Full Abstraction for a Functional Metalanguage with Recursive Types. Ph.D. Dissertation. Imperial College, University of London.Google Scholar
- G. McCusker. 2000. Games and Full Abstraction for FPC. Information and Computation 160, 1-2 (2000), 1--61.Google ScholarCross Ref
- G. McCusker, J. Power, and C. Wingfield. 2012. A Graphical Foundation for Schedules. In Proc. 28th Conf. on the Math. Found. of Prog. Semantics (Electronic Notes in Theoretical Computer Science), Vol. 286. Elsevier, 273--289. Google ScholarDigital Library
- P.-A. Melliès. 2004. Asynchronous games 2: the true concurrency of innocence. In Proc. 15th International Conference on Concurrency Theory (LNCS), Vol. 3170. Springer, 448--465.Google ScholarCross Ref
- P.-A. Melliès. 2005. Asynchronous Games 4: A Fully Complete Model of Propositional Linear Logic. In Proc. 20th Symposium on Logic in Computer Science IEEE, 386--395. Google ScholarDigital Library
- P.-A. Melliès. 2012. Game Semantics in String Diagrams. In Proc. 27th Symposium on Logic in Computer Science IEEE, 481--490. Google ScholarDigital Library
- P.-A. Melliès and S. Mimram. 2007. Asynchronous Games: Innocence Without Alternation. In Proc. 19th International Conference on Concurrency Theory (LNCS), Vol. 4703. Springer, 395--411. Google ScholarDigital Library
- A. S. Murawski and N. Tzevelekos. 2016. Nominal Game Semantics. Foundations and Trends in Programming Languages 2, 4 (2016), 191--269. Google ScholarDigital Library
- H. Nickau. 1994. Hereditarily Sequential Functionals. In Proc. Logical Foundations of Computer Science (LNCS), Vol. 813. Springer, 253--264. Google ScholarDigital Library
- S. Rideau and G. Winskel. 2011. Concurrent Strategies. In Proc. 26th Symposium on Logic in Computer Science IEEE, 409--418. Google ScholarDigital Library
- E. Riehl. 2014. Categorical Homotopy Theory. Number 24 in New Mathematical Monographs. Cambridge University Press.Google Scholar
- T. Tsukada and C.-H. L. Ong. 2014. Innocent Strategies are Sheaves over Plays---Deterministic, Non-deterministic and Probabilistic Innocence. (2014). arXiv:cs/1409.2764Google Scholar
- T. Tsukada and C.-H. L. Ong. 2015. Nondeterminism in Game Semantics via Sheaves, See {2}.Google Scholar
- M. Weber. 2004. Generic morphisms, parametric representations and weakly cartesian monads. Theory and Applications of Categories 13 (2004), 191--234.Google Scholar
- M. Weber. 2007. Familial 2-functors and parametric right adjoints. Theory and Applications of Categories 18, 22 (2007), 665--732.Google Scholar
Index Terms
- What's in a game?: A theory of game models
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