Skip to main content
SpringerLink
Log in

On Probabilistic Distributed Strategies

  • Conference paper
  • First Online:
Theoretical Aspects of Computing - ICTAC 2015 (ICTAC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9399))

Included in the following conference series:

  • 708 Accesses

Abstract

In a distributed game we imagine a team Player engaging a team Opponent in a distributed fashion. No longer can we assume that moves of Player and Opponent alternate. Rather the history of a play more naturally takes the form of a partial order of dependency between occurrences of moves. How are we to define strategies within such a game, and how are we to adjoin probability to such a broad class of strategies? The answer yields a surprisingly rich language of probabilistic distributed strategies and the possibility of programming (optimal) probabilistic strategies. Along the way we shall encounter solutions to: the need to mix probability and nondeterminism; the problem of parallel causes in which members of the same team can race to make the same move, and why this leads us to invent a new model for the semantics of distributed systems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We consider two strategies \(\sigma :S\rightarrow A\) and \(\sigma ':S'\rightarrow A\) to be essentially the same if there is an isomorphism \(f:S\cong S'\) of event structures respecting polarity such that \(\sigma = \sigma 'f\).

  2. 2.

    A Scott-open subset of configurations is upwards-closed w.r.t. inclusion and such that if it contains the union of a directed subset S of configurations then it contains an element of S. A continuous valuation is a function w from the Scott-open subsets of \(\,\!\mathcal{C}^\infty (E)\) to [0, 1] which is (normalized) \(w(\,\!\mathcal{C}^\infty (E)) = 1\); (strict) \(w(\emptyset ) = 0 \); (monotone) \(U \subseteq V \implies w(U)\le w(V)\); (modular) \(w(U \cup V) + w(U\cap V) = w(U) + w(V)\); and

    (continuous) \(w(\bigcup _{i\in I} U_i) = \mathrm{sup}_{i\in I} w(U_i)\), for directed unions. The idea: w(U) is the probability of a result in open set U.

  3. 3.

    Samy Abbes has pointed out that the same “drop condition”  appears in early work of the Russian mathematician V.A.Rohlin [14](as relation (6) of Sect. 3, p.7). Its rediscovery in the context of event structures was motivated by the need to tie probability to the occurrences of events; it is sufficient to check the ‘drop condition’ for , in which the configurations \(x_i\) extend y with a single event.

  4. 4.

    The use of “schedulers to resolve the probability or nondeterminism” in earlier work is subsumed by that of probabilistic and deterministic counter-strategies. Deterministic strategies coincide with those with assignment one to each finite configuration.

  5. 5.

    Their treatment in [15] is slapdash.

  6. 6.

    One way to define a rigid embedding is as a rigid map whose function is injective and reflects consistency.

References

  1. Winskel, G., Nielsen, M.: Handbook of Logic in Computer Science. Oxford University Press, New York (1995). ch. Models for Concurrency

    Google Scholar 

  2. Winskel, G.: Event structure semantics for CCS and related languages. In: Schmidt, E.M., Nielsen, M. (eds.) Automata, Languages and Programming. LNCS, vol. 140, pp. 561–576. Springer, Heidelberg (1982)

    Chapter  Google Scholar 

  3. —, “Event structures,” in Advances in Petri Nets, ser. Lecture Notes in Computer Science, vol. 255. Springer, pp. 325–392 (1986)

    Google Scholar 

  4. Rideau, S., Winskel, G.: Concurrent strategies. In: LICS 2011. IEEE Computer Society (2011)

    Google Scholar 

  5. Nielsen, M., Plotkin, G., Winskel, G.: Petri nets, event structures and domains. Theor. Comput. Sci. 13, 85–108 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Conway, J.: On Numbers and Games. A K Peters, Wellesley (2000)

    MATH  Google Scholar 

  7. Joyal, A.: Remarques sur la théorie des jeux à deux personnes. Gaz. des Sci. Mathématiques du Que 1(4), 46–52 (1997)

    Google Scholar 

  8. Winskel, G.: Deterministic concurrent strategies. Formal Asp. Comput. 24(4–6), 647–660 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Chapter  Google Scholar 

  10. Saunders-Evans, L., Winskel, G.: Event structure spans for nondeterministic dataflow. Electr. Notes Theor. Comput. Sci. 175(3), 109–129 (2007)

    Article  MATH  Google Scholar 

  11. Berry, G.: Stable models of typed lambda-calculi. In: ICALP, ser. Lecture Notes in Computer Science, vol. 62. Springer, pp. 72–89 (1978)

    Google Scholar 

  12. Winskel, G.: Probabilistic and quantum event structures. In: van Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds.) Horizons of the Mind. LNCS, vol. 8464, pp. 476–497. Springer, Heidelberg (2014)

    Google Scholar 

  13. Jones, C., Plotkin, G.: A probabilistic powerdomain of valuations. In: LICS 1989. IEEE Computer Society (1989)

    Google Scholar 

  14. Rohlin, V.A.: On the fundamental ideas of measure theory. Amer. Math. Soc. Translation. 71, 55 (1952)

    MathSciNet  Google Scholar 

  15. Winskel, G.: Distributed probabilistic and quantum strategies. Electr. Notes Theor. Comput. Sci. 298, 403–425 (2013)

    Article  MathSciNet  Google Scholar 

  16. Clairambault, P., Winskel, G.: On concurrent games with payoff. Electr. Notes Theor. Comput. Sci. 298, 71–92 (2013)

    Article  MathSciNet  Google Scholar 

  17. Winskel, G.: Winning, losing and drawing in concurrent games with perfect or imperfect information. In: Constable, R.L., Silva, A. (eds.) Logic and Program Semantics, Kozen Festschrift. LNCS, vol. 7230, pp. 298–317. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  18. Peters, J.F.: In: Peters, J.F. (ed.). ISRL, vol. 63, pp. 1–76. Springer, Heidelberg (2014)

    Google Scholar 

  19. Castellan, S., Hayman, J., Lasson, M., Winskel, G.: Strategies as concurrent processes. Electr. Notes Theor. Comput. Sci. 308, 87–107 (2014)

    Article  MathSciNet  Google Scholar 

  20. de Visme, M., Winskel, G.: Strategies with Parallel Causes. Draft (2015)

    Google Scholar 

Download references

Acknowledgements

Thanks to Samy Abbes, Nathan Bowler, Simon Castellan, Pierre Clairambault, Marcelo Fiore, Mai Gehrke, Julian Gutierrez, Jonathan Hayman, Martin Hyland, Marc Lasson, Silvain Rideau, Daniele Varacca and Marc de Visme for helpful discussions. The concluding section is based on recent joint work with Marc de Visme while on his internship from ENS Paris. The support of Advanced Grant ECSYM of the European Research Council is acknowledged with gratitude.

Author information

Authors and Affiliations

  1. Computer Laboratory, University of Cambridge, Cambridge, UK

    Glynn Winskel

Authors
  1. Glynn Winskel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Glynn Winskel .

Editor information

Editors and Affiliations

  1. University of Lübeck, Lübeck, Schleswig-Holstein, Germany

    Martin Leucker

  2. Pontificia Universidad Javeriana-Cali, Cali, Colombia

    Camilo Rueda

  3. Pontificia Universidad Javeriana-Cali, CNRS LIX École Polytechnique de Paris, Palaiseau Cedex, France

    Frank D. Valencia

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Winskel, G. (2015). On Probabilistic Distributed Strategies. In: Leucker, M., Rueda, C., Valencia, F. (eds) Theoretical Aspects of Computing - ICTAC 2015. ICTAC 2015. Lecture Notes in Computer Science(), vol 9399. Springer, Cham. https://doi.org/10.1007/978-3-319-25150-9_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-25150-9_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25149-3

  • Online ISBN: 978-3-319-25150-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation