Unambiguous Conjunctive Grammars over a One-Letter Alphabet

  • Artur Jeż
  • Alexander Okhotin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7907)

Abstract

It is demonstrated that unambiguous conjunctive grammars over a unary alphabet Σ = {a} have non-trivial expressive power, and that their basic properties are undecidable. The key result is that for every base \(k \geqslant 11\) and for every one-way real-time cellular automaton operating over the alphabet of base-k digits \(\big\{\lfloor\frac{k+9}{4}\rfloor, \ldots, \lfloor\frac{k+1}{2}\rfloor\big\}\), the language of all strings an with the base-k notation of the form 1w1, where w is accepted by the automaton, is generated by an unambiguous conjunctive grammar. Another encoding is used to simulate a cellular automaton in a unary language containing almost all strings. These constructions are used to show that for every fixed unambiguous conjunctive language L0, testing whether a given unambiguous conjunctive grammar generates L0 is undecidable.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aizikowitz, T., Kaminski, M.: LR(0) conjunctive grammars and deterministic synchronized alternating pushdown automata. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 345–358. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press (2003)Google Scholar
  3. 3.
    Autebert, J., Berstel, J., Boasson, L.: Context-free languages and pushdown automata. In: Rozenberg, Salomaa (eds.) Handbook of Formal Languages, vol. 1, pp. 111–174. Springer (1997)Google Scholar
  4. 4.
    Culik II, K., Gruska, J., Salomaa, A.: Systolic trellis automata, I and II. International Journal of Computer Mathematics 15, 16, 195–212, 3–22 (1984)Google Scholar
  5. 5.
    Enflo, P., Granville, A., Shallit, J., Yu, S.: On sparse languages L such that LL = Σ*. Discrete Applied Mathematics 52, 275–285 (1994)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Ibarra, O.H., Kim, S.M.: Characterizations and computational complexity of systolic trellis automata. Theoretical Computer Science 29, 123–153 (1984)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jeż, A.: Conjunctive grammars can generate non-regular unary languages. International Journal of Foundations of Computer Science 19(3), 597–615 (2008)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Jeż, A., Okhotin, A.: Conjunctive grammars over a unary alphabet: undecidability and unbounded growth. Theory of Computing Systems 46(1), 27–58 (2010)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Jeż, A., Okhotin, A.: Complexity of equations over sets of natural numbers. Theory of Computing Systems 48(2), 319–342 (2011)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Jeż, A., Okhotin, A.: On the computational completeness of equations over sets of natural numbers. In: Aceto, L., et al. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 63–74. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Jeż, A., Okhotin, A.: On the number of nonterminal symbols in unambiguous conjunctive grammars. In: Kutrib, M., Moreira, N., Reis, R. (eds.) DCFS 2012. LNCS, vol. 7386, pp. 183–195. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Kountouriotis, V., Nomikos, C., Rondogiannis, P.: Well-founded semantics for Boolean grammars. Information and Computation 207(9), 945–967 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kunc, M.: What do we know about language equations? In: Harju, T., Karhumäki, J., Lepistö, A. (eds.) DLT 2007. LNCS, vol. 4588, pp. 23–27. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    McKenzie, P., Wagner, K.W.: The complexity of membership problems for circuits over sets of natural numbers. Computational Complexity 16, 211–244 (2007)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Okhotin, A.: Conjunctive grammars. Journal of Automata, Languages and Combinatorics 4, 519–535 (2001)MathSciNetGoogle Scholar
  16. 16.
    Okhotin, A.: On the equivalence of linear conjunctive grammars to trellis automata. Informatique Théorique et Applications 38(1), 69–88 (2004)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Okhotin, A.: Unambiguous Boolean grammars. Information and Computation 206, 1234–1247 (2008)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Okhotin, A.: Decision problems for language equations. Journal of Computer and System Sciences 76(3-4), 251–266 (2010)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Okhotin, A.: Fast parsing for Boolean grammars: a generalization of Valiant’s algorithm. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 340–351. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  20. 20.
    Okhotin, A., Reitwießner, C.: Parsing Boolean grammars over a one-letter alphabet using online convolution. Theoretical Computer Science 457, 149–157 (2012)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Okhotin, A., Rondogiannis, P.: On the expressive power of univariate equations over sets of natural numbers. Information and Computation 212, 1–14 (2012)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Semenov, A.L.: Algorithmic problems for power series and for context-free grammars. Doklady Akademii Nauk SSSR 212, 50–52 (1973)MathSciNetGoogle Scholar
  23. 23.
    Terrier, V.: On real-time one-way cellular array. Theoretical Computer Science 141, 331–335 (1995)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Artur Jeż
    • 1
    • 2
  • Alexander Okhotin
    • 3
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.Institute of Computer ScienceUniversity of WrocławPoland
  3. 3.Department of Mathematics and StatisticsUniversity of TurkuFinland

Personalised recommendations