Skip to main content
SpringerLink
Log in

Max-plus Algebraic Tools for Discrete Event Systems, Static Analysis, and Zero-Sum Games

  • Conference paper
Book cover Formal Modeling and Analysis of Timed Systems (FORMATS 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5813))

  • 488 Accesses

Abstract

The max-plus algebraic approach of timed discrete event systems emerged in the eighties, after the discovery that synchronization phenomena can be modeled in a linear way in the max-plus setting. This led to a number of results, like the determination of long term characteristics (throughput, stationary regime) by spectral theory methods or the representation of the input-output behavior by rational series.

Since these early developments, the max-plus scene has considerably evolved. Many analytical results appeared to carry over to a larger class of dynamical systems, involving monotone or nonexpansiveness operators. For instance, discrete dynamics in which the operations of maximum, minimum, positive linear combinations or log-exp type combinations simultaneously appear fall into this class.

Such generalizations are based on the study of non-linear fixed point problems by methods of Perron-Frobenius theory. They keep, however, a combinatorial flavor reminiscent of the max-plus case. Then, the same monotone fixed point problems were seen to arise in other fields, including zero-sum games and static analysis by abstract interpretation, leading to the design of algorithms inspired by control and game theory (policy iteration) in static analysis.

Finally, the recent flourishing of tropical geometry, in which max-plus objects are thought of as projections of classical objects by some valuations, has motivated new theoretical works, in particular on max-plus polyhedra. The latter were initially used to represent some invariant spaces (like the reachable sets of discrete event systems), they have arisen more recently in relation with game or static analysis problems. They now appear to be mathematical objects of an intrinsic interest, to which the arsenal of algorithms from computational geometry can be adapted.

This survey will give a unified perspective on these developments, shedding light on recent results concerning zero-sum games, static analysis, non-linear Perron-Frobenius theory, and polyhedra.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baccelli, F., Cohen, G., Olsder, G., Quadrat, J.: Synchronization and Linearity. Wiley, Chichester (1992)

    MATH  Google Scholar 

  2. Cohen, G., Gaubert, S., Quadrat, J.: Max-plus algebra and system theory: where we are and where to go now. Annual Reviews in Control 23, 207–219 (1999)

    Article  Google Scholar 

  3. Heidergott, B., Olsder, G.J., van der Woude, J.: Max Plus at Work: Modeling and Analysis of Synchronized Systems, A Course on Max-Plus Algebra and Its Applications. Princeton University Press, Princeton (2005)

    Book  Google Scholar 

  4. Cohen, G., Gaubert, S., Quadrat, J.: Asymptotic throughput of continuous timed petri nets. In: Proceedings of the 34th Conference on Decision and Control, New Orleans (December 1995)

    Google Scholar 

  5. Olsder, G.: Eigenvalues of dynamic min-max systems. J. of Discrete Event Dynamic Systems 1, 177–207 (1991)

    Article  MATH  Google Scholar 

  6. Gunawardena, J.: Min-max functions. Discrete Event Dynamic Systems 4, 377–406 (1994)

    Article  MATH  Google Scholar 

  7. Gaubert, S., Gunawardena, J.: A non-linear hierarchy for discrete event dynamical systems. In: Proc. of the Fourth Workshop on Discrete Event Systems (WODES 1998), Cagliari, Italy, IEE (1998)

    Google Scholar 

  8. Gunawardena, J.: From max-plus algebra to nonexpansive maps: a nonlinear theory for discrete event systems. Theoretical Computer Science 293, 141–167 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gaubert, S., Gunawardena, J.: The Perron-Frobenius theorem for homogeneous, monotone functions. Trans. of AMS 356(12), 4931–4950 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gaubert, S.: Nonlinear Perron-Frobenius theory and discrete event systems. JESA 39, 175–190 (2005)

    Article  Google Scholar 

  11. Bousch, T., Mairesse, J.: Finite-range topical functions and uniformly topical functions. Dyn. Syst. 21(1), 73–114 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Akian, M., Gaubert, S., Lemmens, B., Nussbaum, R.: Iteration of order preserving subhomogeneous maps on a cone. Math. Proc. Cambridge Philos. Soc. 140(1), 157–176 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nussbaum, R.D.: Hilbert’s projective metric and iterated nonlinear maps. Memoirs of the AMS 75(391) (1988)

    Google Scholar 

  14. Bewley, T., Kohlberg, E.: The asymptotic solution of a recursion equation occurring in stochastic games. Math. Oper. Res. 1(4), 321–336 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kohlberg, E.: Invariant half-lines of nonexpansive piecewise-linear transformations. Math. Oper. Res. 5(3), 366–372 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  16. Neyman, A.: Stochastic games and nonexpansive maps. In: Stochastic games and applications (Stony Brook, NY, 1999). NATO Sci. Ser. C Math. Phys. Sci., vol. 570, pp. 397–415. Kluwer Acad. Publ., Dordrecht (2003)

    Chapter  Google Scholar 

  17. Rosenberg, D., Sorin, S.: An operator approach to zero-sum repeated games. Israel J. Math. 121, 221–246 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Vigeral, G.: Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces. In: ESAIM:COCV (to appear, 2009)

    Google Scholar 

  19. Hoffman, A.J., Karp, R.M.: On nonterminating stochastic games. Management sciences 12(5), 359–370 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cochet-Terrasson, J., Gaubert, S., Gunawardena, J.: A constructive fixed point theorem for min-max functions. Dynamics and Stability of Systems 14(4), 407–433 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gaubert, S., Gunawardena, J.: The duality theorem for min-max functions. C.R. Acad. Sci. 326, 43–48 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vöge, J., Jurdziński, M.: A discrete strategy improvement algorithm for solving parity games. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  23. Dhingra, V., Gaubert, S.: How to solve large scale deterministic games with mean payoff by policy iteration. In: Valuetools 2006: Proceedings of the 1st international conference on Performance evaluation methodologies and tools, p. 12. ACM Press, New York (2006)

    Google Scholar 

  24. Cochet-Terrasson, J., Gaubert, S.: A policy iteration algorithm for zero-sum stochastic games with mean payoff. C. R. Math. Acad. Sci. Paris 343(5), 377–382 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jurdziński, M., Paterson, M., Zwick, U.: A deterministic subexponential algorithm for solving parity games. SIAM J. Comput. 38(4), 1519–1532 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Friedmann, O.: An exponential lower bound for the parity game strategy improvement algorithm as we know it. Accepted for publication in the proceedings of LICS, also arXiv:0901.2731 (2009)

    Google Scholar 

  27. Costan, A., Gaubert, S., Goubault, É., Martel, M., Putot, S.: A policy iteration algorithm for computing fixed points in static analysis of programs. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 462–475. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  28. Gaubert, S., Goubault, É., Taly, A., Zennou, S.: Static analysis by policy iteration on relational domains. In: De Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 237–252. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  29. Gawlitza, T., Seidl, H.: Precise fixpoint computation through strategy iteration. In: De Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 300–315. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  30. Adje, A., Gaubert, S., Goubault, E.: Computing the smallest fixed point of nonexpansive mappings arising in game theory and static analysis of programs. In: Proceedings of the Eighteenth International Symposium on Mathematical Theory of Networks and Systems (MTNS 2008), Blacksburg, Virginia (July 2008) arXiv:0806.1160

    Google Scholar 

  31. Zimmermann, K.: A general separation theorem in extremal algebras. Ekonom.-Mat. Obzor 13(2), 179–201 (1977)

    MathSciNet  MATH  Google Scholar 

  32. Litvinov, G., Maslov, V., Shpiz, G.: Idempotent functional analysis: an algebraic approach. Math. Notes 69(5), 696–729 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Cohen, G., Gaubert, S., Quadrat, J.: Hahn-Banach separation theorem for max-plus semimodules. In: Menaldi, J., Rofman, E., Sulem, A. (eds.) Optimal Control and Partial Differential Equations, pp. 325–334. IOS Press, Amsterdam (2001)

    Google Scholar 

  34. Cohen, G., Gaubert, S., Quadrat, J.P.: Duality and separation theorem in idempotent semimodules. Linear Algebra and Appl. 379, 395–422 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Cohen, G., Gaubert, S., Quadrat, J.P., Singer, I.: Max-plus convex sets and functions. In: Litvinov, G.L., Maslov, V.P. (eds.) Idempotent Mathematics and Mathematical Physics. Contemporary Mathematics. Contemporary Mathematics, pp. 105–129. American Mathematical Society (2005)

    Google Scholar 

  36. Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1–27 (2004) (electronic)

    MathSciNet  MATH  Google Scholar 

  37. Joswig, M.: Tropical halfspaces. In: Combinatorial and computational geometry. Math. Sci. Res. Inst. Publ., vol. 52, pp. 409–431. Cambridge Univ. Press, Cambridge (2005)

    Google Scholar 

  38. Allamigeon, X., Gaubert, S., Katz, R.D.: The number of extreme points of tropical polyhedra. Eprint arXiv:math/0906.3492 (submitted, 2009)

    Google Scholar 

  39. Butkovič, P., Schneider, H., Sergeev, S.: Generators, extremals and bases of max cones. Linear Algebra Appl. 421(2-3), 394–406 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Gaubert, S., Katz, R.: Max-plus convex geometry. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 192–206. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  41. Gaubert, S., Katz, R.: The Minkowski theorem for max-plus convex sets. Linear Algebra and Appl. 421, 356–369 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Nitica, V., Singer, I.: Max-plus convex sets and max-plus semispaces. I. Optimization 56(1-2), 171–205 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  43. Joswig, M., Sturmfels, B., Yu, J.: Affine buildings and tropical convexity. Albanian J. Math. 1(4), 187–211 (2007)

    MathSciNet  MATH  Google Scholar 

  44. Joswig, M.: Tropical convex hull computations, Eprint arXiv:0809.4694. to appear in Contemporary Mathematics (November 2008)

    Google Scholar 

  45. Gaubert, S., Katz, R.: The tropical analogue of polar cones. Linear Algebra and Appl. 431, 608–625 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. Gaubert, S., Meunier, F.: Carathéodory, Helly and the others in the max-plus world. Discrete Computational Geometry, Published online, also arXiv:0804.1361v1 (2009)

    Google Scholar 

  47. Allamigeon, X., Gaubert, S., Goubault, E.: Computing the extreme points of tropical polyhedra. Eprint arXiv:0904.3436 (2009)

    Google Scholar 

  48. Katz, R.D.: Max-plus (A,B)-invariant spaces and control of timed discrete event systems. IEEE Trans. Aut. Control 52(2), 229–241 (2007)

    Article  MathSciNet  Google Scholar 

  49. Di Loreto, M., Gaubert, S., Katz, R.D., Loiseau, J.J.: Duality between invariant spaces for max-plus linear discrete event systems. Eprint arXiv:0901.2915 (2009)

    Google Scholar 

  50. Allamigeon, X., Gaubert, S., Goubault, É.: Inferring min and max invariants using max-plus polyhedra. In: Alpuente, M., Vidal, G. (eds.) SAS 2008. LNCS, vol. 5079, pp. 189–204. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  51. Gaubert, S., Sergeev, S.N.: Cyclic projectors and separation theorems in idempotent convex geometry. Journal of Mathematical Sciences 155(6), 815–829 (2008); Russian version published. Fundamentalnaya i prikladnaya matematika 13(4), pp. 33–52 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. Bezem, M., Nieuwenhuis, R., Rodríguez-Carbonell, E.: The max-atom problem and its relevance. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 47–61. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  53. Itenberg, I., Mikhalkin, G., Shustin, E.: Tropical algebraic geometry. Oberwolfach seminars. Birkhäuser, Basel (2007)

    MATH  Google Scholar 

  54. Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Idempotent mathematics and mathematical physics. Contemp. Math., vol. 377, pp. 289–317. Amer. Math. Soc., Providence (2005)

    Chapter  Google Scholar 

  55. Block, F., Yu, J.: Tropical convexity via cellular resolutions. J. Algebraic Combin. 24(1), 103–114 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  56. Cohen, G., Gaubert, S., McGettrick, M., Quadrat, J.P.: Maxplus toolbox of scilab, http://minimal.inria.fr/gaubert/maxplustoolbox/ , now integrated in ScicosLab, http://www.scicoslab.org

  57. Gawrilow, E., Joswig, M.: Polymake, http://www.math.tu-berlin.de/polymake/ , http://www.math.tu-berlin.de/polymake/

Download references

Author information

Authors and Affiliations

  1. INRIA Saclay – Île-de-France and CMAP, École Polytechnique, France

    Stéphane Gaubert

Authors
  1. Stéphane Gaubert
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computing Laboratory, Oxford University, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Joël Ouaknine

  2. Institute for Computing and Information Sciences , Radboud University Nijmegen , P.O. Box 9010, 6500, Nijmegen, GL, The Netherlands

    Frits W. Vaandrager

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gaubert, S. (2009). Max-plus Algebraic Tools for Discrete Event Systems, Static Analysis, and Zero-Sum Games. In: Ouaknine, J., Vaandrager, F.W. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2009. Lecture Notes in Computer Science, vol 5813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04368-0_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-04368-0_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04367-3

  • Online ISBN: 978-3-642-04368-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation