Advertisement

A Probabilistic Kleene Theorem

  • Benedikt Bollig
  • Paul Gastin
  • Benjamin Monmege
  • Marc Zeitoun
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7561)

Abstract

We provide a Kleene Theorem for (Rabin) probabilistic automata over finite words. Probabilistic automata generalize deterministic finite automata and assign to a word an acceptance probability. We provide probabilistic expressions with probabilistic choice, guarded choice, concatenation, and a star operator. We prove that probabilistic expressions and probabilistic automata are expressively equivalent. Our result actually extends to two-way probabilistic automata with pebbles and corresponding expressions.

Keywords

Probabilistic Choice Star Operator Probabilistic Expression Deterministic Finite Automaton XPath Query 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baier, C., Bertrand, N., Größer, M.: On Decision Problems for Probabilistic Büchi Automata. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 287–301. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Baier, C., Größer, M.: Recognizing ω-regular languages with probabilistic automata. In: Proc. of LICS 2005, pp. 137–146. IEEE Computer Society (2005)Google Scholar
  3. 3.
    Baier, C., Katoen, J.-P.: Principles of model checking. MIT Press (2008)Google Scholar
  4. 4.
    Berstel, J., Reutenauer, C.: Noncommutative rational series with applications, Cambridge. Encyclopedia of Mathematics & Its Applications, vol. 137, Cambridge (2011)Google Scholar
  5. 5.
    Bojańczyk, M.: Tree-Walking Automata. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 1–2. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Bollig, B., Gastin, P., Monmege, B., Zeitoun, M.: Pebble Weighted Automata and Transitive Closure Logics. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 587–598. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Buchholz, P., Kemper, P.: Quantifying the Dynamic Behavior of Process Algebras. In: de Luca, L., Gilmore, S. (eds.) PAPM-PROBMIV 2001. LNCS, vol. 2165, pp. 184–199. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Chadha, R., Sistla, A.P., Viswanathan, M.: Power of randomization in automata on infinite strings. Logical Methods in Computer Science 7(3:22) (2011)Google Scholar
  9. 9.
    Cortes, C., Mohri, M., Rastogi, A.: Lp distance and equivalence of probabilistic automata. Int. J. Found. Comput. Sci. 18(4), 761–779 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Cortes, C., Mohri, M., Rastogi, A., Riley, M.: On the computation of the relative entropy of probabilistic automata. Int. J. Found. Comput. Sci. 19(1), 219–242 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Deng, Y., Palamidessi, C.: Axiomatizations for probabilistic finite-state behaviors. Theor. Comput. Sci. 373(1-2), 92–114 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Deng, Y., Palamidessi, C., Pang, J.: Compositional Reasoning for Probabilistic Finite-State Behaviors. In: Middeldorp, A., van Oostrom, V., van Raamsdonk, F., de Vrijer, R. (eds.) Processes, Terms and Cycles: Steps on the Road to Infinity. LNCS, vol. 3838, pp. 309–337. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Droste, M., Gastin, P.: Weighted automata and weighted logics. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata. EATCS Monographs in Theoretical Computer Science, ch. 5, pp. 175–211. Springer (2009)Google Scholar
  14. 14.
    Dwork, C., Stockmeyer, L.: On the power of 2-way probabilistic finite state automata. In: Proc. of FoCS 1989, pp. 480–485. IEEE Computer Society (1989)Google Scholar
  15. 15.
    Flesca, S., Furfaro, F., Greco, S.: Weighted path queries on semistructured databases. Inform. and Comput. 204(5), 679–696 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Gastin, P., Monmege, B.: Adding Pebbles to Weighted Automata. In: Moreira, N., Reis, R. (eds.) CIAA 2012. LNCS, vol. 7381, pp. 28–51. Springer, Heidelberg (2012)Google Scholar
  17. 17.
    Gimbert, H., Oualhadj, Y.: Probabilistic Automata on Finite Words: Decidable and Undecidable Problems. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 527–538. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Kiefer, S., Murawski, A.S., Ouaknine, J., Wachter, B., Worrell, J.: On the Complexity of the Equivalence Problem for Probabilistic Automata. In: Birkedal, L. (ed.) FOSSACS 2012. LNCS, vol. 7213, pp. 467–481. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  19. 19.
    Marx, M.: Conditional XPath. ACM Transactions on Database Systems 30(4), 929–959 (2005)CrossRefGoogle Scholar
  20. 20.
    Paz, A.: Introduction to probabilistic automata (Computer science and applied mathematics). Academic Press (1971)Google Scholar
  21. 21.
    Rabin, M.O.: Probabilistic automata. Inform. and Control 6, 230–245 (1963)CrossRefzbMATHGoogle Scholar
  22. 22.
    Ravikumar, B.: On some variations of two-way probabilistic finite automata models. Theor. Comput. Sci. 376, 127–136 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Sakarovitch, J.: Rational and recognizable power series. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata. EATCS Monographs in Theoretical Computer Science, ch. 4, pp. 103–172. Springer (2009)Google Scholar
  24. 24.
    Schützenberger, M.P.: On the definition of a family of automata. Inform. and Control 4, 245–270 (1961)CrossRefzbMATHGoogle Scholar
  25. 25.
    Segala, R.: Probability and Nondeterminism in Operational Models of Concurrency. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 64–78. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  26. 26.
    Silva, A., Bonchi, F., Bonsangue, M.M., Rutten, J.: Quantitative Kleene coalgebras. Inf. Comput. 209(5), 822–849 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    ten Cate, B., Segoufin, L.: XPath, transitive closure logic, and nested tree walking automata. In: Proc. of PODS 2008, pp. 251–260. ACM (2008)Google Scholar
  28. 28.
    Tzeng, W.-G.: A polynomial-time algorithm for the equivalence of probabilistic automata. SIAM J. Comput. 21(2), 216–227 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    van Glabbeek, R.J., Smolka, S.A., Steffen, B.: Reactive, generative and stratified models of probabilistic processes. Inform. and Comput. 121(1), 59–80 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Weidner, T.: Probabilistic Automata and Probabilistic Logic. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 813–824. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Benedikt Bollig
    • 1
  • Paul Gastin
    • 1
  • Benjamin Monmege
    • 1
  • Marc Zeitoun
    • 2
  1. 1.LSV, ENS Cachan, CNRS & InriaFrance
  2. 2.LaBRIUniv. Bordeaux & CNRSFrance

Personalised recommendations