Adding Pebbles to Weighted Automata

  • Paul Gastin
  • Benjamin Monmege
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7381)

Abstract

We extend weighted automata and weighted rational expressions with 2-way moves and (reusable) pebbles. We show with examples from natural language modeling and quantitative model-checking that weighted expressions and automata with pebbles are more expressive and allow much more natural and intuitive specifications than classical ones. We extend Kleene-Schützenberger theorem showing that weighted expressions and automata with pebbles have the same expressive power. We focus on an efficient translation from expressions to automata. We also prove that the evaluation problem for weighted automata can be done very efficiently if the number of (reusable) pebbles is low.

Keywords

Regular Expression Expressive Power Linear Temporal Logic English Sentence Partial Unit 
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.
    Allauzen, C., Mohri, M.: A Unified Construction of the Glushkov, Follow, and Antimirov Automata. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 110–121. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Berry, G., Sethi, R.: From regular expressions to deterministic automata. Theoretical Computer Science 48, 117–126 (1986)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Berstel, J., Reutenauer, C.: Noncommutative rational series with applications, Cambridge. Encyclopedia of Mathematics & Its Applications, vol. 137 (2011)Google Scholar
  4. 4.
    Birget, J.-C.: State-complexity of finite-state devices, state compressibility and incompressibility. Theory of Computing Systems 26, 237–269 (1993)MathSciNetMATHGoogle 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.
    Bojańczyk, M., Samuelides, M., Schwentick, T., Segoufin, L.: Expressive Power of Pebble Automata. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 157–168. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Bollig, B., Gastin, P.: Weighted versus Probabilistic Logics. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 18–38. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Brüggeman-Klein, A.: Regular expressions into finite automata. Theoretical Computer Science 120, 197–213 (1993)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Brüggeman-Klein, A., Wood, D.: Caterpillars: A context specification technique. Markup Languages 2(1), 81–106 (2000)CrossRefGoogle Scholar
  11. 11.
    Brzozowski, J.A., McCluskey, E.J.: Signal flow graph techniques for sequential circuit state diagrams. IEEE Trans. on Electronic Computers 12(9), 67–76 (1963)MATHCrossRefGoogle Scholar
  12. 12.
    Buchholz, P., Kemper, P.: Model checking for a class of weighted automata. Discrete Event Dynamic Systems 20(1), 103–137 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Ciesinski, F., Größer, M.: On Probabilistic Computation Tree Logic. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 147–188. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Conway, J.: Regular Algebra and Finite Machines. Chapman & Hall (1971)Google Scholar
  15. 15.
    Droste, M., Kuich, W.: Semirings and formal power series. In: Handbook of Weighted Automata [16], ch. 1, pp. 3–27Google Scholar
  16. 16.
    Droste, M., Kuich, W., Vogler, H.: Handbook of Weighted Automata. EATCS Monographs in Theoretical Computer Science. Springer (2009)Google Scholar
  17. 17.
    Engelfriet, J., Hoogeboom, H.J.: Tree-walking pebble automata. In: Jewels are Forever, pp. 72–83. Springer (1999)Google Scholar
  18. 18.
    Ésik, Z., Kuich, W.: Modern Automata Theory. Electronic book (2007), http://dmg.tuwien.ac.at/kuich
  19. 19.
    Globerman, N., Harel, D.: Complexity results for two-way and multi-pebble automata and their logics. Theoretical Computer Science 169, 161–184 (1996)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Glushkov, V.M.: The abstract theory of automata. Russian Math. Surveys 16, 1–53 (1961)CrossRefGoogle Scholar
  21. 21.
    Knight, K., May, J.: Applications of weighted automata in natural language processing. In: Handbook of Weighted Automata [16], ch. 14, pp. 555–579Google Scholar
  22. 22.
    Kuske, D.: Schützenberger’s theorem on formal power series follows from kleene’s theorem. Theoretical Computer Science 401(1-3), 243–248 (2008)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Mandrali, E.: Weighted LTL with Discounting. In: Moreira, N., Reis, R. (eds.) CIAA 2012. LNCS, vol. 7381, pp. 353–360. Springer, Heidelberg (2012)Google Scholar
  24. 24.
    McNaughton, R., Yamada, H.: Regular expressions and state graphs for automata. IRE Trans. on Electronic Computers 9(1), 39–47 (1960)CrossRefGoogle Scholar
  25. 25.
    Meinecke, I.: A Weighted μ-Calculus on Words. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 384–395. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  26. 26.
    Ravikumar, B.: On some variations of two-way probabilistic finite automata models. Theoretical Computer Science 376(1-2), 127–136 (2007)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press (2009)Google Scholar
  28. 28.
    Sakarovitch, J.: Rational and recognisable power series. In: Handbook of Weighted Automata [16], ch. 4, pp. 103–172Google Scholar
  29. 29.
    Sakarovitch, J.: Automata and expressions. In: AutoMathA Handbook (to appear, 2012)Google Scholar
  30. 30.
    Samuelides, M., Segoufin, L.: Complexity of Pebble Tree-Walking Automata. In: Csuhaj-Varjú, E., Ésik, Z. (eds.) FCT 2007. LNCS, vol. 4639, pp. 458–469. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  31. 31.
    Schützenberger, M.-P.: On the definition of a family of automata. Information and Control 4, 245–270 (1961)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Vardi, M.: The complexity of relational query languages. In: Proceedings of STOC 1982, pp. 137–146. ACM Press (1982)Google Scholar
  33. 33.
    Vardi, M.: On the complexity of bounded-variable queries. In: Proceedings of PODS 1995, pp. 266–276. ACM Press (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Paul Gastin
    • 1
  • Benjamin Monmege
    • 1
  1. 1.CNRS, InriaLSV, ENS CachanFrance

Personalised recommendations