Advertisement

One-Way Bounded-Error Probabilistic Pushdown Automata and Kolmogorov Complexity

(Preliminary Report)
  • Tomoyuki YamakamiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10396)

Abstract

One-way probabilistic pushdown automata (or ppda’s) are a simple model of randomized computation with last-in first-out memory device known as stacks and, when error probabilities are bounded away from 1 / 2, ppda’s can characterize a family of bounded-error probabilistic context-free languages (BPCFL). We resolve a fundamental question raised by Hromkovič and Schnitger [Inf. Comput. 208 (2010) 982–995] concerning the limitation of the language recognition power of bounded-error ppda’s. More specifically, we prove that a well-known language—the set of palindromes—cannot be recognized by any bounded-error ppda; in other words, this language stays outside of BPCFL. Furthermore, we show that, with bounded-error probability, no ppda can determine whether the center bit of input string is 1 (one). For those impossibility results, we utilize a complexity measure of algorithmic information known as Kolmogorov complexity. In our proofs, we first transform ppda’s into an ideal shape and then lead to a key lemma by employing a Kolmogorov complexity argument.

Keywords

Probabilistic pushdown automata Bounded error probability BPCFL Palindromes Kolmogorov complexity 

References

  1. 1.
    Bar-Hillel, Y., Perles, M., Shamir, E.: On formal properties of simple phrase-structure grammars. Z. Phonetik Sprachwiss. Kommunikationsforsch 14, 143–172 (1961)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Glier, O.: Kolmogorov complexity and deterministic context-free languages. SIAM J. Comput. 32, 1389–1394 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  4. 4.
    Hromkovič, J., Schnitger, G.: On probabilistic pushdown automata. Inf. Comput. 208, 982–995 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kaņeps, J., Geidmanis, D., Freivalds, R.: Tally languages accepted by Monte Carlo pushdown automata. In: Rolim, J. (ed.) RANDOM 1997. LNCS, vol. 1269, pp. 187–195. Springer, Heidelberg (1997). doi: 10.1007/3-540-63248-4_16 CrossRefGoogle Scholar
  6. 6.
    Lewis, H.R., Papadimitriou, C.H.: Elements of the Theory of Computation, 2nd edn. Prentice-Hall, Englewood Cliffs (1998)Google Scholar
  7. 7.
    Li, M., Vitányi, P.: A new approach to formal language theory by Kolmogorov complexity. SIAM J. Comput. 24, 398–410 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Li, M., Vitányi, P.: Kolmogorov Complexity and Its Applications, 2nd edn. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    Macarie, I.I., Ogihara, M.: Properties of probabilistic pushdown automata. Theor. Comput. Sci. 207, 117–130 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pighizzini, G., Pisoni, A.: Limited automata and context-free languages. Fundam. Inf. 136, 157–176 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Yamakami, T.: Swapping lemmas for regular and context-free languages. arXiv:0808.4122 (2008)
  12. 12.
    Yamakami, T.: The roles of advice to one-tape linear-time turing machines and finite automata. Int. J. Found. Comput. Sci. 21, 941–962 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yamakami, T.: Immunity and pseudorandomness of context-free languages. Theor. Comput. Sci. 412, 6432–6450 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Yamakami, T.: Oracle pushdown automata, nondeterministic reducibilities, and the hierarchy over the family of context-free languages. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds.) SOFSEM 2014. LNCS, vol. 8327, pp. 514–525. Springer, Cham (2014). doi: 10.1007/978-3-319-04298-5_45. A complete version arXiv:1303.1717v2 under a slightly different titleCrossRefGoogle Scholar
  15. 15.
    Yamakami, T.: Pseudorandom generators against advised context-free languages. Theor. Comput. Sci. 613, 1–27 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of EngineeringUniversity of FukuiFukuiJapan

Personalised recommendations