Advertisement

Limited Automata: Properties, Complexity and Variants

  • Giovanni PighizziniEmail author
Conference paper
  • 191 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11612)

Abstract

Limited automata are single-tape Turing machines with severe rewriting restrictions. They have been introduced in 1967 by Thomas Hibbard, who proved that they have the same computational power as pushdown automata. Hence, they provide an alternative characterization of the class of context-free languages in terms of recognizing devices. After that paper, these models have been almost forgotten for many years. Only recently limited automata were reconsidered in a series of papers, where several properties of them and of their variants have been investigated. In this work we present an overview of the most important results obtained in these researches. We also discuss some related models and possible lines for future investigations.

Notes

Acknowledgment

I am very grateful to Luca Prigioniero for his valuable and helpful comments.

References

  1. 1.
    Alur, R., Madhusudan, P.: Adding nesting structure to words. J. ACM 56(3), 16:1–16:43 (2009).  https://doi.org/10.1145/1516512.1516518MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chomsky, N., Schützenberger, M.: The algebraic theory of context-free languages. In: Braffort, P., Hirschberg, D. (eds.) Computer Programming and Formal Systems, Studies in Logic and the Foundations of Mathematics, vol. 35, pp. 118–161. Elsevier (1963).  https://doi.org/10.1016/S0049-237X(08)72023-8
  3. 3.
    Chrobak, M.: Finite automata and unary languages. Theoret. Comput. Sci. 47(3), 149–158 (1986).  https://doi.org/10.1016/0304-3975(86)90142-8. Errata: 302(1–3), 497–498 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9(3), 350–371 (1962).  https://doi.org/10.1145/321127.321132MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Guillon, B., Pighizzini, G., Prigioniero, L., Průša, D.: Two-way automata and one-tape machines. In: Hoshi, M., Seki, S. (eds.) DLT 2018. LNCS, vol. 11088, pp. 366–378. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-98654-8_30CrossRefGoogle Scholar
  6. 6.
    Guillon, B., Prigioniero, L.: Linear-time limited automata. Theoret. Comput. Sci. (2019, in press).  https://doi.org/10.1016/j.tcs.2019.03.037
  7. 7.
    Hemaspaandra, L.A., Mukherji, P., Tantau, T.: Context-free languages can be accepted with absolutely no space overhead. Inform. Comput. 203(2), 163–180 (2005).  https://doi.org/10.1016/j.ic.2005.05.005MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hennie, F.C.: One-tape, off-line Turing machine computations. Inf. Control 8(6), 553–578 (1965)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hibbard, T.N.: A generalization of context-free determinism. Inf. Control 11(1/2), 196–238 (1967)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jančar, P., Mráz, F., Plátek, M.: Characterization of context-free languages by erasing automata. In: Havel, I.M., Koubek, V. (eds.) MFCS 1992. LNCS, vol. 629, pp. 307–314. Springer, Heidelberg (1992).  https://doi.org/10.1007/3-540-55808-X_29CrossRefGoogle Scholar
  11. 11.
    Jancar, P., Mráz, F., Plátek, M.: A taxonomy of forgetting automata. In: Borzyszkowski, A.M., Sokołowski, S. (eds.) MFCS 1993. LNCS, vol. 711, pp. 527–536. Springer, Heidelberg (1993).  https://doi.org/10.1007/3-540-57182-5_44CrossRefzbMATHGoogle Scholar
  12. 12.
    Jančar, P., Mráz, F., Plátek, M.: Forgetting automata and context-free languages. Acta Inform. 33(5), 409–420 (1996).  https://doi.org/10.1007/s002360050050MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jančar, P., Mráz, F., Plátek, M., Vogel, J.: Restarting automata. In: Reichel, H. (ed.) FCT 1995. LNCS, vol. 965, pp. 283–292. Springer, Heidelberg (1995).  https://doi.org/10.1007/3-540-60249-6_60CrossRefGoogle Scholar
  14. 14.
    Kutrib, M., Pighizzini, G., Wendlandt, M.: Descriptional complexity of limited automata. Inform. Comput. 259(2), 259–276 (2018).  https://doi.org/10.1016/j.ic.2017.09.005MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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).  https://doi.org/10.1007/978-3-319-19225-3_13CrossRefzbMATHGoogle Scholar
  16. 16.
    Kutrib, M., Wendlandt, M.: Reversible limited automata. Fund. Inform. 155(1–2), 31–58 (2017).  https://doi.org/10.3233/FI-2017-1575MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mehlhorn, K.: Pebbling mountain ranges and its application to DCFL-recognition. In: de Bakker, J., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 422–435. Springer, Heidelberg (1980).  https://doi.org/10.1007/3-540-10003-2_89CrossRefGoogle Scholar
  18. 18.
    Nasyrov, I.R.: Deterministic realization of nondeterministic computations with a low measure of nondeterminism. Cybernetics 27(2), 170–179 (1991).  https://doi.org/10.1007/BF01068368MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ogden, W.F., Ross, R.J., Winklmann, K.: An “interchange lemma” for context-free languages. SIAM J. Comput. 14(2), 410–415 (1985).  https://doi.org/10.1137/0214031MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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).  https://doi.org/10.1007/978-3-642-31653-1_12CrossRefzbMATHGoogle Scholar
  21. 21.
    Otto, F.: Restarting automata and their relations to the Chomsky hierarchy. In: Ésik, Z., Fülöp, Z. (eds.) DLT 2003. LNCS, vol. 2710, pp. 55–74. Springer, Heidelberg (2003).  https://doi.org/10.1007/3-540-45007-6_5CrossRefGoogle Scholar
  22. 22.
    Peckel, J.: On a deterministic subclass of context-free languages. In: Gruska, J. (ed.) MFCS 1977. LNCS, vol. 53, pp. 430–434. Springer, Heidelberg (1977).  https://doi.org/10.1007/3-540-08353-7_164CrossRefGoogle Scholar
  23. 23.
    Peckel, J.: A deterministic subclass of context-free languages. Časopis pro pěstování matematiky 103(1), 43–52 (1978). http://eudml.org/doc/21335MathSciNetzbMATHGoogle Scholar
  24. 24.
    Pighizzini, G.: Nondeterministic one-tape off-line Turing machines. J. Autom. Lang. Comb. 14(1), 107–124 (2009).  https://doi.org/10.25596/jalc-2009-107. http://arXiv.org/abs/0905.1271
  25. 25.
    Pighizzini, G.: Two-way finite automata: old and recent results. Fund. Inform. 126(2–3), 225–246 (2013).  https://doi.org/10.3233/FI-2013-879MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pighizzini, G.: Guest column: one-tape Turing machine variants and language recognition. SIGACT News 46(3), 37–55 (2015).  https://doi.org/10.1145/2818936.2818947MathSciNetCrossRefGoogle Scholar
  27. 27.
    Pighizzini, G.: Strongly limited automata. Fund. Inform. 148(3–4), 369–392 (2016).  https://doi.org/10.3233/FI-2016-1439MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pighizzini, G., Pisoni, A.: Limited automata and regular languages. Internat. J. Found. Comput. Sci. 25(7), 897–916 (2014).  https://doi.org/10.1142/S0129054114400140MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pighizzini, G., Pisoni, A.: Limited automata and context-free languages. Fund. Inform. 136(1–2), 157–176 (2015).  https://doi.org/10.3233/FI-2015-1148MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Pighizzini, G., Prigioniero, L.: Limited automata and unary languages. Inform. Comput. 266, 60–74 (2019).  https://doi.org/10.1016/j.ic.2019.01.002MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    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.) Proceedings 10th Annual ACM Symposium on Theory of Computing (STOC 1978), pp. 275–286. ACM (1978).  https://doi.org/10.1145/800133.804357
  32. 32.
    Shepherdson, J.C.: The reduction of two-way automata to one-way automata. IBM J. Res. Dev. 3(2), 198–200 (1959).  https://doi.org/10.1147/rd.32.0198MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences. http://oeis.org/A007814
  34. 34.
    Wagner, K.W., Wechsung, G.: Computational Complexity. D. Reidel Publishing Company, Dordrecht (1986)zbMATHGoogle Scholar
  35. 35.
    Wechsung, G.: Characterization of some classes of context-free languages in terms of complexity classes. In: Bečvář, J. (ed.) MFCS 1975. LNCS, vol. 32, pp. 457–461. Springer, Heidelberg (1975).  https://doi.org/10.1007/3-540-07389-2_233CrossRefGoogle Scholar
  36. 36.
    Wechsung, G., Brandstädt, A.: A relation between space, return and dual return complexities. Theoret. Comput. Sci. 9, 127–140 (1979).  https://doi.org/10.1016/0304-3975(79)90010-0MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Yamakami, T.: Behavioral strengths and weaknesses of various models of limited automata. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds.) SOFSEM 2019. LNCS, vol. 11376, pp. 519–530. Springer, Cham (2019).  https://doi.org/10.1007/978-3-030-10801-4_40CrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2019

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità degli Studi di MilanoMilanItaly

Personalised recommendations