Conjunctive Grammars over a Unary Alphabet: Undecidability and Unbounded Growth

  • Artur Jeż
  • Alexander Okhotin


It has recently been proved (Jeż, DLT 2007) that conjunctive grammars (that is, context-free grammars augmented by conjunction) generate some non-regular languages over a one-letter alphabet. The present paper improves this result by constructing conjunctive grammars for a larger class of unary languages. The results imply undecidability of a number of decision problems of unary conjunctive grammars, as well as non-existence of a recursive function bounding the growth rate of the generated languages. An essential step of the argument is a simulation of a cellular automaton recognizing positional notation of numbers using language equations.


Conjunctive grammars Unary languages Language equations Trellis automata Cellular automata 


  1. 1.
    Autebert, J., Berstel, J., Boasson, L.: Context-free languages and pushdown automata. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 111–174. Springer, New York (1997) Google Scholar
  2. 2.
    Chrobak, M.: Finite automata and unary languages. Theor. Comput. Sci. 47, 149–158 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Culik, K. II, Gruska, J., Salomaa, A.: Systolic trellis automata, I. Int. J. Comput. Math. 15, 195–212 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Culik, K. II, Gruska, J., Salomaa, A.: Systolic trellis automata, II. Int. J. Comput. Math. 16, 3–22 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Culik, K. II, Gruska, J., Salomaa, A.: Systolic trellis automata: stability, decidability and complexity. Inf. Control 71, 218–230 (1984) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Domaratzki, M., Pighizzini, G., Shallit, J.: Simulating finite automata with context-free grammars. Inf. Process. Lett. 84, 339–344 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9, 350–371 (1962) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Hartmanis, J.: Context-free languages and Turing machine computations. In: Proceedings of Symposia in Applied Mathematics, vol. 19, pp. 42–51. Am. Math. Soc., Providence (1967) Google Scholar
  9. 9.
    Ibarra, O.H., Kim, S.M.: Characterizations and computational complexity of systolic trellis automata. Theor. Comput. Sci. 29, 123–153 (1984) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Jeż, A.: Conjunctive grammars can generate non-regular unary languages. Int. J. Found. Comput. Sci. 19(3), 597–615 (2008) zbMATHCrossRefGoogle Scholar
  11. 11.
    Leiss, E.L.: Unrestricted complementation in language equations over a one-letter alphabet. Theor. Comput. Sci. 132, 71–93 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Okhotin, A.: Conjunctive grammars. J. Autom. Lang. Comb. 6(4), 519–535 (2001) zbMATHMathSciNetGoogle Scholar
  13. 13.
    Okhotin, A.: Conjunctive grammars and systems of language equations. Program. Comput. Softw. 28, 243–249 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Okhotin, A.: On the equivalence of linear conjunctive grammars to trellis automata. Inform. Théor. Appl. 38(1), 69–88 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Okhotin, A.: Nine open problems for conjunctive and Boolean grammars. Bull. EATCS 91, 96–119 (2007) zbMATHMathSciNetGoogle Scholar
  16. 16.
    Okhotin, A., Yakimova, O.: On language equations with complementation. In: Developments in Language Theory, DLT 2006, Santa Barbara, USA, June 26–29, 2006. LNCS, vol. 4036, pp. 420–432. Springer, Berlin (2006) CrossRefGoogle Scholar
  17. 17.
    Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time. In: STOC 1973, pp. 1–9 (1973) Google Scholar
  18. 18.
    Wotschke, D.: Personal communication to A. Okhotin, August 2000 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland
  2. 2.Department of MathematicsUniversity of TurkuTurkuFinland
  3. 3.Academy of FinlandHelsinkiFinland

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