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

LanguagesLn={1x2ix: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 languageLn 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 recognizingLf(k). This is a solution of the problem stated by Chrobak.

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