Abstract
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.
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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)
Chrobak, M.: Finite automata and unary languages. Theor. Comput. Sci. 47, 149–158 (1986)
Culik, K. II, Gruska, J., Salomaa, A.: Systolic trellis automata, I. Int. J. Comput. Math. 15, 195–212 (1984)
Culik, K. II, Gruska, J., Salomaa, A.: Systolic trellis automata, II. Int. J. Comput. Math. 16, 3–22 (1984)
Culik, K. II, Gruska, J., Salomaa, A.: Systolic trellis automata: stability, decidability and complexity. Inf. Control 71, 218–230 (1984)
Domaratzki, M., Pighizzini, G., Shallit, J.: Simulating finite automata with context-free grammars. Inf. Process. Lett. 84, 339–344 (2002)
Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9, 350–371 (1962)
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)
Ibarra, O.H., Kim, S.M.: Characterizations and computational complexity of systolic trellis automata. Theor. Comput. Sci. 29, 123–153 (1984)
Jeż, A.: Conjunctive grammars can generate non-regular unary languages. Int. J. Found. Comput. Sci. 19(3), 597–615 (2008)
Leiss, E.L.: Unrestricted complementation in language equations over a one-letter alphabet. Theor. Comput. Sci. 132, 71–93 (1994)
Okhotin, A.: Conjunctive grammars. J. Autom. Lang. Comb. 6(4), 519–535 (2001)
Okhotin, A.: Conjunctive grammars and systems of language equations. Program. Comput. Softw. 28, 243–249 (2002)
Okhotin, A.: On the equivalence of linear conjunctive grammars to trellis automata. Inform. Théor. Appl. 38(1), 69–88 (2004)
Okhotin, A.: Nine open problems for conjunctive and Boolean grammars. Bull. EATCS 91, 96–119 (2007)
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)
Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time. In: STOC 1973, pp. 1–9 (1973)
Wotschke, D.: Personal communication to A. Okhotin, August 2000
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A. Jeż supported by MNiSW grant number N206 024 31/3826, 2006–2008.
A. Okhotin supported by the Academy of Finland under grant 118540.
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Jeż, A., Okhotin, A. Conjunctive Grammars over a Unary Alphabet: Undecidability and Unbounded Growth. Theory Comput Syst 46, 27–58 (2010). https://doi.org/10.1007/s00224-008-9139-5
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DOI: https://doi.org/10.1007/s00224-008-9139-5