Advertisement

Mathematical systems theory

, Volume 23, Issue 1, pp 107–139 | Cite as

One-way multihead finite automata and 2-bounded languages

  • Mirosław Kutyłowski
Article

Abstract

LanguagesL n ={1 x 2 ix :i, x ∈ ℕ, 1≤in} were used to show that, for eachk, one-way non-sensing deterministic finite automata (1-MFA) withk+1 heads are more powerful than such automata withk heads, even if we consider only 2-bounded languages (Chrobak). Fork ∈ ℕ letf(k) be the maximal numbern such that languageL n can be recognized by a 1-MFA withk heads. We present a precise inductive formula forf(k). It may be shown that, fork≥3,
$$\frac{{(2k - 5)! \cdot (k - 2) \cdot (k - 1)}}{{2^{k - 3} }} \leqslant f(k) \leqslant \frac{{(2k - 5)! \cdot (k - 2) \cdot (k - 1) \cdot 3k^2 }}{{2^{k - 3} }}$$
that is,f(k)≈k2k. The proof is constructive in the sense that it shows how to construct ak-head automaton recognizingL f(k) . This is a solution of the problem stated by Chrobak.

Keywords

Computational Mathematic Finite Automaton Deterministic Finite Automaton Inductive Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Chrobak, M., Hierarchies of one-way multihead finite languages, in:Proc. ICALP'85, Lecture Notes in Computer Science, vol. 194, Springer-Verlag, Berlin, 1985, pp. 101–110.Google Scholar
  2. [2]
    Chrobak, M., Hierarchies of one-way multihead automata languages,Theoretical Computer Science,48 (1986), pp. 153–181 (a full version of [1]).Google Scholar
  3. [3]
    Chrobak, M., and Li, M.,k+1 heads are better thank for PDAs,Journal of Computer and System Sciences,37 (1988), pp. 144–155.Google Scholar
  4. [4]
    Chrobak, M., and Rytter, W., Remarks on string-matching and one-way multihead automata,Information Processing Letters,24 (1987), pp. 325–329.Google Scholar
  5. [5]
    Galil, Z., Open problems in stringology, in:Combinatorial Algorithms on Words (A. Apostolico and Z. Galil, eds.), Springer-Verlag, Berlin, 1974, pp. 350–359.Google Scholar
  6. [6]
    Ibarra, O. H., and Kim, C. E., On 3-head versus 2-head finite automata,Acta Informatica,4 (1975), pp. 173–200.Google Scholar
  7. [7]
    Kutyłowski, M., One-way multihead finite automata,Theoretical Computer Science, to appear.Google Scholar
  8. [8]
    Kutyłowski, M., One-Way Multihead Finite Automata and 2-Bounded Languages, Technical Report, Institut für Theoretische Informatik, Technische Hochschule Darmstadt, March 1989 (a revised version) (an extended version of this paper).Google Scholar
  9. [9]
    Piatkowski, T. F.,N-head Finite-State Machines, Ph.D. Dissertation, University of Michigan, 1963.Google Scholar
  10. [10]
    Rosenberg, A. L.,Nonwriting Extensions of Finite Automata, Ph.D. Dissertation, Harvard University, 1965.Google Scholar
  11. [11]
    Rosenberg, A. L., On multihead finite automata,IBM Journal of Research and Development,10 (1966), pp. 388–394.Google Scholar
  12. [12]
    Yao, A. C., and Rivest, R. L.,K+1 heads are better thanK, Journal of Association for Computing Machinery,25 (1978), pp. 337–340.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Mirosław Kutyłowski
    • 1
  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland

Personalised recommendations