Limited Automata and Unary Languages

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10396)

Abstract

Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. When \(d=1\) these models characterize regular languages. An exponential gap between the size of limited automata accepting unary languages and the size of equivalent finite automata is proved. Since a similar gap was already known from unary context-free grammars to finite automata, also the conversion of such grammars into limited automata is investigated. It is proved that from each unary context-free grammar it is possible to obtain an equivalent 1-limited automaton whose description has a size which is polynomial in the size of the grammar. Furthermore, despite the exponential gap between the sizes of limited automata and of equivalent unary finite automata, there are unary regular languages for which d-limited automata cannot be significantly smaller than equivalent finite automata, for any arbitrarily large d.

References

  1. 1.
    Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9(3), 350–371 (1962). http://doi.acm.org/10.1145/321127.321132 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Hibbard, T.N.: A generalization of context-free determinism. Inf. Control 11(1/2), 196–238 (1967)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)MATHGoogle Scholar
  4. 4.
    Kutrib, M., Pighizzini, G., Wendlandt, M.: Descriptional complexity of limited automata. Inf. Comput. (to appear)Google Scholar
  5. 5.
    Kutrib, M., Wendlandt, M.: On simulation cost of unary limited automata. In: Shallit, J., Okhotin, A. (eds.) DCFS 2015. LNCS, vol. 9118, pp. 153–164. Springer, Cham (2015). doi:10.1007/978-3-319-19225-3_13 CrossRefGoogle Scholar
  6. 6.
    McNaughton, R., Papert, S.A.: Counter-Free Automata. M.I.T. Research Monograph, vol. 65. The MIT Press, Cambridge (1971)MATHGoogle Scholar
  7. 7.
    Mereghetti, C., Pighizzini, G.: Two-way automata simulations and unary languages. J. Autom. Lang. Comb. 5(3), 287–300 (2000)MathSciNetMATHGoogle Scholar
  8. 8.
    Okhotin, A.: Non-erasing variants of the Chomsky–Schützenberger theorem. In: Yen, H.-C., Ibarra, O.H. (eds.) DLT 2012. LNCS, vol. 7410, pp. 121–129. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31653-1_12 CrossRefGoogle Scholar
  9. 9.
    Pighizzini, G., Shallit, J., Wang, M.: Unary context-free grammars and pushdown automata, descriptional complexity and auxiliary space lower bounds. J. Comput. Syst. Sci. 65(2), 393–414 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Pighizzini, G.: Strongly limited automata. Fundam. Inform. 148(3–4), 369–392 (2016). http://dx.doi.org/10.3233/FI-2016-1439 MathSciNetGoogle Scholar
  11. 11.
    Pighizzini, G., Pisoni, A.: Limited automata and regular languages. Int. J. Found. Comput. Sci. 25(7), 897–916 (2014). http://dx.doi.org/10.1142/S0129054114400140 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Pighizzini, G., Pisoni, A.: Limited automata and context-free languages. Fundam. Inf. 136(1–2), 157–176 (2015). http://dx.doi.org/10.3233/FI-2015-1148 MathSciNetMATHGoogle Scholar
  13. 13.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences. http://oeis.org/A007814
  14. 14.
    Wagner, K.W., Wechsung, G.: Computational Complexity. D. Reidel Publishing Company, Dordrecht (1986)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità degli Studi di MilanoMilanItaly

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