A Mathematical Game Semantics of Concurrency and Nondeterminism

  • Julian Gutierrez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9399)


Concurrent games as event structures form a partial order model of concurrency where concurrent behaviour is captured by nondeterministic concurrent strategies—a class of maps of event structures. Extended with winning conditions, the model is also able to give semantics to logics of various kinds. An interesting subclass of this game model is the one considering deterministic strategies only, where the induced model of strategies can be fully characterised by closure operators. The model based on closure operators exposes many interesting mathematical properties and allows one to define connections with many other semantic models where closure operators are also used. However, such a closure operator semantics has not been investigated in the more general nondeterministic case. Here we do so, and show that some nondeterministic concurrent strategies can be characterised by a new definition of nondeterministic closure operators which agrees with the standard game model for event structures and with its extension with winning conditions.


Concurrent games Event structures Closure operators 



I thank Paul Harrenstein, Glynn Winskel, and Michael Wooldridge for their comments and support. Also, I acknowledge with gratitude the support of the ERC Advanced Research Grant 291528 (“RACE”) at Oxford.


  1. 1.
    Abramsky, S.: Sequentiality vs. concurrency in games and logic. Math. Struct. Comput. Sci. 13(4), 531–565 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abramsky, S., Jagadeesan, R., Malacaria, P.: Full abstraction for PCF. Inf. Comput. 163(2), 409–470 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Abramsky, S., Melliès, P.: Concurrent games and full completeness. In: LICS, pp. 431–442. IEEE Computer Society (1999)Google Scholar
  4. 4.
    Berry, G.: Modèles complètement adéquats et stables des lambda-calculs typés. Ph.D. thesis, University of Paris VII (1979)Google Scholar
  5. 5.
    Clairambault, P., Gutierrez, J., Winskel, G.: The winning ways of concurrent games. In: LICS, pp. 235–244. IEEE Computer Society (2012)Google Scholar
  6. 6.
    Desel, J., Esparza, J.: ree Choice Petri Nets, Cambridge Tracts in Theoretical Computer Science, vol. 40. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  7. 7.
    Gutierrez, J.: Concurrent logic games on partial orders. In: Beklemishev, L.D., de Queiroz, R. (eds.) WoLLIC 2011. LNCS, vol. 6642, pp. 146–160. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  8. 8.
    Harmer, R., Hyland, M., Melliès, P.: Categorical combinatorics for innocent strategies. In: LICS, pp. 379–388. IEEE Computer Society (2007)Google Scholar
  9. 9.
    Hyland, J.M.E., Ong, C.L.: On full abstraction for PCF: I, II, and III. Inf. Comput. 163(2), 285–408 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kahn, G., Plotkin, G.: Concrete domains. Technical report, INRIA (1993)Google Scholar
  11. 11.
    Melliès, P.-A., Mimram, S.: Asynchronous games: innocence without alternation. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 395–411. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  12. 12.
    Nielsen, M., Palamidessi, C., Valencia, F.D.: Temporal concurrent constraint programming: Denotation, logic and applications. Nord. J. Comput. 9(1), 145–188 (2002)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Nielsen, M., Winskel, G.: Models for concurrency. In: Handbook of Logic in Computer Science, pp. 1–148. Oxford University Press (1995)Google Scholar
  14. 14.
    Olarte, C., Valencia, F.D.: Universal concurrent constraint programing: symbolic semantics and applications to security. In: SAC, pp. 145–150. ACM (2008)Google Scholar
  15. 15.
    Plotkin, G.D.: A powerdomain construction. SIAM J. Comput. 5(3), 452–487 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rideau, S., Winskel, G.: Concurrent strategies. In: LICS, pp. 409–418. IEEE Computer Society (2011)Google Scholar
  17. 17.
    Saraswat, V.A., Rinard, M.C., Panangaden, P.: Semantic foundations of concurrent constraint programming. In: POPL, pp. 333–352. ACM Press (1991)Google Scholar
  18. 18.
    Saunders-Evans, L., Winskel, G.: Event structure spans for nondeterministic dataflow. Electron. Notes Theoret. Comput. Sci. 175(3), 109–129 (2007)CrossRefzbMATHGoogle Scholar
  19. 19.
    Winskel, G.: Deterministic concurrent strategies. Formal Aspects Comput. 24(4–6), 647–660 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Winskel, G.: Strategies as profunctors. In: Pfenning, F. (ed.) FOSSACS 2013 (ETAPS 2013). LNCS, vol. 7794, pp. 418–433. Springer, Heidelberg (2013) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK

Personalised recommendations