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Restricted Turing Machines and Language Recognition

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Language and Automata Theory and Applications (LATA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9618))

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Abstract

In 1965 Hennie proved that one-tape deterministic Turing machines working in linear time are equivalent to finite automata, namely they characterize regular languages. This result has been improved in different directions, by obtaining optimal lower bounds for the time that one-tape deterministic and nondeterministic Turing machines need to recognize nonregular languages. On the other hand, in 1964 Kuroda showed that one-tape Turing machines that are not allowed to use any extra space, besides the part of the tape which initially contains the input, namely linear bounded automata, recognize exactly context-sensitive languages. In 1967 Hibbard proved that for each integer \(d\ge 2\), one-tape Turing machines that are allowed to rewrite each tape cell only in the first d visits are equivalent to pushdown automata. This gives a characterization of the class of context-free languages in terms of restricted Turing machines. We discuss these and other related models, by presenting an overview of some fundamental results related to them. Descriptional complexity aspects are also considered.

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Notes

  1. 1.

    For technical reasons actually we count the scans from left to right and from right to left on each cell. Hence, each transition reversing the head direction is counted as a double visit.

References

  1. Alberts, M.: Space complexity of alternating Turing machines. In: Budach, L. (ed.) FCT. LNCS, vol. 199, pp. 1–7. Springer, Heidelberg (1985)

    Chapter  Google Scholar 

  2. Bertoni, A., Mereghetti, C., Pighizzini, G.: An optimal lower bound for nonregular languages. Inf. Process. Lett. 50(6), 289–292 (1994). Corrigendum. ibid. 52(6), 339

    Article  MathSciNet  MATH  Google Scholar 

  3. 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, Amsterdam (1963)

    Chapter  Google Scholar 

  4. Gajser, D.: Verifying time complexity of Turing machines. Theor. Comput. Sci. 600, 86–97 (2015)

    Article  MathSciNet  Google Scholar 

  5. Geffert, V.: Bridging across the log(n) space frontier. Inf. Comput. 142(2), 127–158 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9(3), 350–371 (1962)

    Article  MathSciNet  Google Scholar 

  7. Hartmanis, J.: Computational complexity of one-tape Turing machine computations. J. ACM 15(2), 325–339 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hennie, F.C.: One-tape, off-line Turing machine computations. Inf. Control 8(6), 553–578 (1965)

    Article  MathSciNet  Google Scholar 

  9. Hibbard, T.N.: A generalization of context-free determinism. Inf. Control 11(1/2), 196–238 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hopcroft, J.E., Ullman, J.D.: Some results on tape-bounded Turing machines. J. ACM 16(1), 168–177 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jancar, P., Mráz, F., Plátek, M.: Forgetting automata and context-free languages. Acta Inf. 33(5), 409–420 (1996)

    Article  Google Scholar 

  12. Kobayashi, K.: On the structure of one-tape nondeterministic Turing machine time hierarchy. Theor. Comput. Sci. 40, 175–193 (1985)

    Article  MATH  Google Scholar 

  13. Kuroda, S.: Classes of languages and linear-bounded automata. Inf. Control 7(2), 207–223 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kutrib, M., Pighizzini, G., Wendlandt, M.: Descriptional complexity of limited automata (submitted)

    Google Scholar 

  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, Heidelberg (2015)

    Chapter  Google Scholar 

  16. Lewis II., P.M., Stearns, R.E., Hartmanis, J.: Memory bounds for recognition of context-free and context-sensitive languages. In: FOCS, pp. 191–202. IEEE (1965)

    Google Scholar 

  17. Mereghetti, C.: Testing the descriptional power of small Turing machines on nonregular language acceptance. Int. J. Found. Comput. Sci. 19(4), 827–843 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Michel, P.: An NP-complete language accepted in linear time by a one-tape Turing machine. Theor. Comput. Sci. 85(1), 205–212 (1991)

    Article  MATH  Google Scholar 

  19. Michel, P.: A survey of space complexity. Theor. Comput. Sci. 101(1), 99–132 (1992)

    Article  MATH  Google Scholar 

  20. Mill, J.R.: Linear bounded automata. WADD Tech. Note, pp. 60–165, Wright Patterson Air Force Base, Ohio (1960)

    Google Scholar 

  21. Peckel, J.: On a deterministic subclass of context-free languages. In: Gruska, J. (ed.) Mathematical Foundations of Computer Science 1977. LNCS, vol. 53, pp. 430–434. Springer, Heidelberg (1977)

    Chapter  Google Scholar 

  22. Pighizzini, G.: Nondeterministic one-tape off-line Turing machines. J. Automata, Lang. Combinatorics 14(1), 107–124 (2009). arxiv:0905.1271

    MathSciNet  MATH  Google Scholar 

  23. Pighizzini, G.: Guest column: one-tape Turing machine variants and language recognition. SIGACT News 46(3), 37–55 (2015). arxiv:1507.08582

    Article  MathSciNet  Google Scholar 

  24. Pighizzini, G.: Strongly limited automata. Fundam. Inform. (to appear). A preliminaryversion appeared In: Bensch, S., Freund, R., Otto, F. (eds.) Proceedings of the Sixth Workshop on Non-Classical Models for Automata and Applications - NCMA 2014, 28–29 July 2014, Kassel, Germany, vol. 304, pp. 191–206. Österreichische Computer Gesellschaft (2014). books@ocg.at

    Google Scholar 

  25. Pighizzini, G., Pisoni, A.: Limited automata and regular languages. Int. J. Found. Comput. Sci. 25(7), 897–916 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pighizzini, G., Pisoni, A.: Limited automata and context-free languages. Fundam. Inform. 136(1–2), 157–176 (2015)

    MathSciNet  Google Scholar 

  27. Průša, D.: Weight-Reducing hennie machines and their descriptional complexity. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 553–564. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  28. Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM J. Res. Dev. 3(2), 114–125 (1959)

    Article  MathSciNet  Google Scholar 

  29. Stearns, R.E., Hartmanis, J., Lewis II., P.M.: Hierarchies of memory limited computations. In: FOCS, pp. 179–190. IEEE (1965)

    Google Scholar 

  30. Szepietowski, A.: Turing Machines with Sublogarithmic Space. LNCS, vol. 843. Springer, Heidelberg (1994)

    MATH  Google Scholar 

  31. Tadaki, K., Yamakami, T., Lin, J.C.H.: Theory of one-tape linear-time Turing machines. Theor. Comput. Sci. 411(1), 22–43 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Trakhtenbrot, B.A.: Turing machine computations with logarithmic delay (in russian). Algebra I Logica 3, 33–48 (1964)

    MATH  Google Scholar 

  33. Wagner, K.W., Wechsung, G.: Computational Complexity. D. Reidel Publishing Company, Dordrecht (1986)

    Google Scholar 

  34. Wechsung, G.: Characterization of some classes of context-free languages in terms of complexity classes. In: Becvár, J. (ed.) Mathematical Foundations of Computer Science 1975. LNCS, vol. 32, pp. 457–461. Springer, Heidelberg (1975)

    Google Scholar 

  35. Wechsung, G., Brandstädt, A.: A relation between space, return and dual return complexities. Theor. Comput. Sci. 9, 127–140 (1979)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Milano, Milano, Italy

    Giovanni Pighizzini

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  1. Giovanni Pighizzini
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Correspondence to Giovanni Pighizzini .

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Editors and Affiliations

  1. Rovira i Virgili University, Tarragona, Spain

    Adrian-Horia Dediu

  2. Czech Technical University, Prague, Czech Republic

    Jan Janoušek

  3. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  4. Justus Liebig University Giessen, Gießen, Germany

    Bianca Truthe

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Pighizzini, G. (2016). Restricted Turing Machines and Language Recognition. In: Dediu, AH., Janoušek, J., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2016. Lecture Notes in Computer Science(), vol 9618. Springer, Cham. https://doi.org/10.1007/978-3-319-30000-9_3

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  • DOI: https://doi.org/10.1007/978-3-319-30000-9_3

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