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International Symposium on Mathematical Foundations of Computer Science

MFCS 2015: Mathematical Foundations of Computer Science 2015 pp 179–191Cite as

Relating Paths in Transition Systems: The Fall of the Modal Mu-Calculus

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9234))

Abstract

We revisit Janin and Walukiewicz’s classic result on the expressive completeness of the modal mu-calculus w.r.t. MSO, when transition systems are equipped with a binary relation over paths. We obtain two natural extensions of MSO and the mu-calculus: MSO with path relation and the jumping mu-calculus. While “bounded-memory” binary relations bring about no extra expressivity to either of the two logics, “unbounded-memory” binary relations make the bisimulation-invariant fragment of MSO with path relation more expressive than the jumping mu-calculus: the existence of winning strategies in games with imperfect-information inhabits the gap.

Catalin Dima acknowledges financial support from the ANR project EQINOCS. Bastien Maubert acknowledges financial support from the ERC project EPS 313360.

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Notes

  1. 1.

    i.e. finite sequences of states and actions that start in the initial state and follow the binary relations.

  2. 2.

    The notion of “associated state” is only used to define unfoldings and is left informal.

  3. 3.

    i.e. it progresses at the same pace on each tape.

  4. 4.

    i.e. has a winning strategy.

  5. 5.

    a one-state transducer that accepts it can easily be exhibited.

References

  1. Alur, R., Černý, P., Chaudhuri, S.: Model checking on trees with path equivalences. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 664–678. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  2. Apt, K.R., Grädel, E.: Lectures in Game Theory for Computer Scientists. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  3. Berstel, J.: Transductions and Context-free Languages, vol. 4. Teubner, Stuttgart (1979)

    Book  MATH  Google Scholar 

  4. Berwanger, D., Chatterjee, K., De Wulf, M., Doyen, L., Henzinger, T.A.: Strategy construction for parity games with imperfect information. Inf. Comput. 208(10), 1206–1220 (2010)

    Article  MATH  Google Scholar 

  5. Berwanger, D., Kaiser, L.: Information tracking in games on graphs. J. Logic Lang. Inf. 19(4), 395–412 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berwanger, D., Mathew, A.B.: Games with recurring certainty. In: Proceedings of SR Games with Recurring Certainty 2014, pp. 91–96 (2014)

    Google Scholar 

  7. Bozianu, R., Dima, C., Enea, C.: Model checking an epistemic mu-calculus with synchronous and perfect recall semantics. In: TARK 2013 (2013)

    Google Scholar 

  8. Caucal, D.: On infinite transition graphs having a decidable monadic theory. Theor. Comput. Sci. 290(1), 79–115 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chatterjee, K., Doyen, L.: The complexity of partial-observation parity games. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 1–14. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  10. Courcelle, B., Engelfriet, J.: Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  11. Elgot, C.C., Rabin, M.O.: Decidability and undecidability of extensions of second (first) order theory of (generalized) successor. J. Symb. Log. 31(2), 169–181 (1966)

    Article  MATH  Google Scholar 

  12. Fagin, R., Halpern, J., Moses, Y., Vardi, M.: Reasoning about knowledge. The MIT Press, Boston (2004)

    Google Scholar 

  13. Grädel, E.: Model-checking games for logics of imperfect information. Theor. Comput. Sci. 493, 2–14 (2013)

    Article  MATH  Google Scholar 

  14. Farwer, B.: 1 omega-Automata. In: Grädel, E., Thomas, W., Wilke, T. (eds.) Automata, Logics, and Infinite Games. LNCS, vol. 2500, pp. 3–21. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  15. Halpern, J.Y., van der Meyden, R., Vardi, M.Y.: Complete axiomatizations for reasoning about knowledge and time. SIAM J. Comput. 33(3), 674–703 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Halpern, J.Y., Vardi, M.Y.: The complexity of reasoning about knowledge and time. 1. Lower bounds. J. Comp. Sys. Sci. 38(1), 195–237 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Janin, D., Walukiewicz, I.: On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119. Springer, Heidelberg (1996)

    Google Scholar 

  18. Matz, O., Schweikardt, N., Thomas, W.: The monadic quantifier alternation hierarchy over grids and graphs. Inf. Comput. 179(2), 356–383 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Maubert, B., Pinchinat, S.: Jumping automata for uniform strategies. In: Proceedings of FSTTCS 2013, pp. 287–298 (2013)

    Google Scholar 

  20. Park, D.: Concurrency and automata on infinite sequences. In: Deussen, P. (ed.) Proceedings of 5th GI Conference on Theoretical Computer Science. Lecture Notes in Computer Science, vol. 104, pp. 167–183. Springer, Heidelberg (1981)

    Chapter  Google Scholar 

  21. Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: POPL 1989, pp. 179–190. ACM Press (1989)

    Google Scholar 

  22. Puchala, B.: Asynchronous omega-regular games with partial information. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 592–603. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  23. Reiter, F.: Distributed graph automata. CoRR: abs/1404.6503 (2014)

  24. Shilov, N.V., Garanina, N.O.: Combining knowledge and fixpoints. Technical report Preprint n.98, A.P. Ershov Institute of Informatics Systems, Novosibirsk 2002. http://www.iis.nsk.su/files/preprints/098.pdf

  25. Thomas, W.: Infinite trees and automaton-definable relations over omega-words. Theor. Comput. Sci. 103(1), 143–159 (1992)

    Article  MATH  Google Scholar 

  26. Thomas, W.: Languages, automata, and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages: Volume 3 Beyond Words, pp. 389–455. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  27. van der Meyden, R., Shilov, N.V.: Model checking knowledge and time in systems with perfect recall. In: Pandu Rangan, C., Raman, V., Sarukkai, S. (eds.) FST TCS 1999. LNCS, vol. 1738, pp. 432–445. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  28. Vardi, M.Y.: Reasoning about the past with two-way automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 628–641. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  29. Walukiewicz, I.: Monadic second-order logic on tree-like structures. Theor. Comput. Sci. 275(1–2), 311–346 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Université Paris Est, LACL, UPEC, Créteil, France

    Cătălin Dima

  2. LORIA - CNRS / Université de Lorraine, Nancy, France

    Bastien Maubert

  3. IRISA, Université de Rennes 1, Rennes, France

    Sophie Pinchinat

Authors
  1. Cătălin Dima
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  2. Bastien Maubert
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  3. Sophie Pinchinat
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Corresponding author

Correspondence to Sophie Pinchinat .

Editor information

Editors and Affiliations

  1. Università di Roma "Tor Vergata", Rome, Italy

    Giuseppe F Italiano

  2. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

  3. University of Edinburgh, Edinburgh, United Kingdom

    Donald T. Sannella

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Dima, C., Maubert, B., Pinchinat, S. (2015). Relating Paths in Transition Systems: The Fall of the Modal Mu-Calculus. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9234. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48057-1_14

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  • DOI: https://doi.org/10.1007/978-3-662-48057-1_14

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  • Print ISBN: 978-3-662-48056-4

  • Online ISBN: 978-3-662-48057-1

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