Investigations on Automata and Languages over a Unary Alphabet

  • Giovanni Pighizzini
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8587)


The investigation of automata and languages defined over a one letter alphabet shows interesting differences with respect to the case of alphabets with at least two letters. Probably, the oldest example emphasizing one of these differences is the collapse of the classes of regular and context-free languages in the unary case (Ginsburg and Rice, 1962). Many differences have been proved concerning the state costs of the simulations between different variants of unary finite state automata (Chrobak, 1986, Mereghetti and Pighizzini, 2001). We present an overview of those results. Because important connections with fundamental questions in space complexity, we give emphasis to unary two-way automata. Furthermore, we discuss unary versions of other computational models, as one-way and two-way pushdown automata, even extended with auxiliary workspace, and multi-head automata.


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  1. 1.
    Brandenburg, F.J.: On one-way auxiliary pushdown automata. In: Tzschach, H., Walter, H.K.-G., Waldschmidt, H. (eds.) GI-TCS 1977. LNCS, vol. 48, pp. 132–144. Springer, Heidelberg (1977)CrossRefGoogle Scholar
  2. 2.
    Chistikov, D., Majumdar, R.: Unary pushdown automata and straight-line programs. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014, Part II. LNCS, vol. 8573, pp. 146–157. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  3. 3.
    Chrobak, M.: A note on bounded-reversal multipushdown machines. Inf. Process. Lett. 19(4), 179–180 (1984)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Chrobak, M.: Finite automata and unary languages. Theor. Comput. Sci. 47(3), 149–158 (1986); errata: [5] Google Scholar
  5. 5.
    Chrobak, M.: Errata to: Finite automata and unary languages. Theor. Comput. Sci. 302(1-3), 497–498 (2003)Google Scholar
  6. 6.
    Chytil, M.: Almost context-free languages. Fundamenta Informaticae IX, 283–322 (1986)Google Scholar
  7. 7.
    Gawrychowski, P.: Chrobak normal form revisited, with applications. In: Bouchou-Markhoff, B., Caron, P., Champarnaud, J.-M., Maurel, D. (eds.) CIAA 2011. LNCS, vol. 6807, pp. 142–153. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Geffert, V.: Magic numbers in the state hierarchy of finite automata. Inf. Comput. 205(11), 1652–1670 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Geffert, V., Mereghetti, C., Pighizzini, G.: Converting two-way nondeterministic unary automata into simpler automata. Theor. Comput. Sci. 295, 189–203 (2003)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Geffert, V., Mereghetti, C., Pighizzini, G.: Complementing two-way finite automata. Inf. Comput. 205(8), 1173–1187 (2007)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Geffert, V., Pighizzini, G.: Two-way unary automata versus logarithmic space. Inf. Comput. 209(7), 1016–1025 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Ginsburg, S., Greibach, S.A.: Deterministic context free languages. Information and Control 9(6), 620–648 (1966)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9(3), 350–371 (1962)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Harrison, M.A.: Introduction to Formal Language Theory. Addison-Wesley Longman Publishing Co., Inc., Boston (1978)MATHGoogle Scholar
  15. 15.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979)Google Scholar
  16. 16.
    Ibarra, O.H.: A note on semilinear sets and bounded-reversal multihead pushdown automata. Inf. Process. Lett. 3(1), 25–28 (1974)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Immerman, N.: Nondeterministic space is closed under complementation. SIAM J. Comput. 17(5), 935–938 (1988)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Jones, N.D.: Space-bounded reducibility among combinatorial problems. J. Comput. Syst. Sci. 11(1), 68–85 (1975)CrossRefMATHGoogle Scholar
  19. 19.
    Kapoutsis, C.A.: Minicomplexity. In: Kutrib, M., Moreira, N., Reis, R. (eds.) DCFS 2012. LNCS, vol. 7386, pp. 20–42. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  20. 20.
    Kapoutsis, C.A., Pighizzini, G.: Two-way automata characterizations of l/poly versus nl. In: Hirsch, E.A., Karhumäki, J., Lepistö, A., Prilutskii, M. (eds.) CSR 2012. LNCS, vol. 7353, pp. 217–228. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  21. 21.
    Karp, R., Lipton, R.: Turing machines that take advice. In: Engeler, E., et al. (eds.) Logic and Algorithmic, pp. 191–209. L’Enseignement Mathématique, Genève (1982)Google Scholar
  22. 22.
    Kopczynski, E., To, A.W.: Parikh images of grammars: Complexity and applications. In: LICS, pp. 80–89. IEEE Computer Society (2010)Google Scholar
  23. 23.
    Kutrib, M., Malcher, A., Wendlandt, M.: Size of unary one-way multi-head finite automata. In: Jurgensen, H., Reis, R. (eds.) DCFS 2013. LNCS, vol. 8031, pp. 148–159. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  24. 24.
    Landau, E.: Über die maximalordnung der permutation gegebenen grades. Archiv der Mathematik und Physik 3, 92–103 (1903)Google Scholar
  25. 25.
    Landau, E.: Handbuch der Lehre von der Verteilung der Primzahlen I. Teubner, Leipzig (1909)Google Scholar
  26. 26.
    Lavado, G.J., Pighizzini, G., Seki, S.: Converting nondeterministic automata and context-free grammars into Parikh equivalent one-way and two-way deterministic automata. Inf. Comput. 228, 1–15 (2013)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Liu, L., Weiner, P.: Finite-reversal pushdown automata and semi-linear sets. In: Proc. of Sec. Ann. Princeton Conf. on Inf. Sciences and Systems, pp. 334–338 (1968)Google Scholar
  28. 28.
    Ljubič, J.: Bounds for the optimal determinization of nondeterministic autonomous automata. Sibirskij Matematičeskij Žurnal 2, 337–355 (1964) (in Russian)Google Scholar
  29. 29.
    Lupanov, O.: A comparison of two types of finite automata. Problemy Kibernet 9, 321–326 (1963) (in Russian); German translation: Über den Vergleich zweier Typen endlicher Quellen. Probleme der Kybernetik 6, 329–335 (1966)Google Scholar
  30. 30.
    Malcher, A., Mereghetti, C., Palano, B.: Descriptional complexity of two-way pushdown automata with restricted head reversals. Theor. Comput. Sci. 449, 119–133 (2012)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Martinez, A.: Efficient computation of regular expressions from unary nfas. In: Dassow, J., Hoeberechts, M., Jürgensen, H., Wotschke, D. (eds.) DCFS, vol. Report No. 586, pp. 174–187. Department of Computer Science, The University of Western Ontario, Canada (2002)Google Scholar
  32. 32.
    Mereghetti, C.: Testing the descriptional power of small Turing machines on nonregular language acceptance. Int. J. Found. Comput. Sci. 19(4), 827–843 (2008)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Mereghetti, C., Pighizzini, G.: Two-way automata simulations and unary languages. Journal of Automata, Languages and Combinatorics 5(3), 287–300 (2000)MATHMathSciNetGoogle Scholar
  34. 34.
    Mereghetti, C., Pighizzini, G.: Optimal simulations between unary automata. SIAM J. Comput. 30(6), 1976–1992 (2001)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Meyer, A.R., Fischer, M.J.: Economy of description by automata, grammars, and formal systems. In: FOCS. pp. 188–191. IEEE (1971)Google Scholar
  36. 36.
    Monien, B.: Deterministic two-way one-head pushdown automata are very powerful. Inf. Process. Lett. 18(5), 239–242 (1984)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Moore, F.: On the bounds for state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way finite automata. IEEE Transactions on Computers C-20(10), 1211–1214 (1971)CrossRefGoogle Scholar
  38. 38.
    Pighizzini, G.: Deterministic pushdown automata and unary languages. Int. J. Found. Comput. Sci. 20(4), 629–645 (2009)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Pighizzini, G.: Two-way finite automata: Old and recent results. Fundam. Inform. 126(2-3), 225–246 (2013)MATHMathSciNetGoogle Scholar
  40. 40.
    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)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM J. Res. Dev. 3(2), 114–125 (1959)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Reinhardt, K., Allender, E.: Making nondeterminism unambiguous. SIAM Journal on Computing 29(4), 1118–1131 (2000)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Sakoda, W.J., Sipser, M.: Nondeterminism and the size of two way finite automata. In: Lipton, R.J., Burkhard, W.A., Savitch, W.J., Friedman, E.P., Aho, A.V. (eds.) STOC, pp. 275–286. ACM (1978)Google Scholar
  44. 44.
    Savitch, W.J.: Relationships between nondeterministic and deterministic tape complexities. J. Comput. Syst. Sci. 4(2), 177–192 (1970)CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Sawa, Z.: Efficient construction of semilinear representations of languages accepted by unary nondeterministic finite automata. Fundam. Inform. 123(1), 97–106 (2013)MATHMathSciNetGoogle Scholar
  46. 46.
    Shallit, J.O.: A Second Course in Formal Languages and Automata Theory. Cambridge University Press (2008)Google Scholar
  47. 47.
    Shepherdson, J.C.: The reduction of two-way automata to one-way automata. IBM J. Res. Dev. 3(2), 198–200 (1959)CrossRefMathSciNetGoogle Scholar
  48. 48.
    Sudborough, I.H.: Bounded-reversal multihead finite automata languages. Information and Control 25(4), 317–328 (1974)CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Szalay, M.: On the maximal order in S_n and S_n *. Acta Arithmetica 37, 321–331 (1980)MATHMathSciNetGoogle Scholar
  50. 50.
    Szelepcsényi, R.: The method of forced enumeration for nondeterministic automata. Acta Inf. 26(3), 279–284 (1988)CrossRefMATHGoogle Scholar
  51. 51.
    To, A.W.: Unary finite automata vs. arithmetic progressions. Inf. Process. Lett. 109(17), 1010–1014 (2009)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Valiant, L.G.: Regularity and related problems for deterministic pushdown automata. J. ACM 22(1), 1–10 (1975)CrossRefMATHMathSciNetGoogle Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Giovanni Pighizzini
    • 1
  1. 1.Dipartimento di InformaticaUniversità degli Studi di MilanoItaly

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