# One-way multihead finite automata and 2-bounded languages

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## Abstract

Languages that is,

*L*_{ n }={1^{ x }2^{ ix }:*i, x*∈ ℕ, 1≤*i*≤*n*} were used to show that, for each*k*, one-way non-sensing deterministic finite automata (1-MFA) with*k*+1 heads are more powerful than such automata with*k*heads, even if we consider only 2-bounded languages (Chrobak). For*k*∈ ℕ let*f*(*k*) be the maximal number*n*such that language*L*_{ n }can be recognized by a 1-MFA with*k*heads. We present a precise inductive formula for*f*(*k*). It may be shown that, for*k*≥3,$$\frac{{(2k - 5)! \cdot (k - 2) \cdot (k - 1)}}{{2^{k - 3} }} \leqslant f(k) \leqslant \frac{{(2k - 5)! \cdot (k - 2) \cdot (k - 1) \cdot 3k^2 }}{{2^{k - 3} }}$$

*f*(*k*)≈*k*^{2k}. The proof is constructive in the sense that it shows how to construct a*k*-head automaton recognizing*L*_{ 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.

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## References

- [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]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]Chrobak, M., and Li, M.,
*k*+1 heads are better than*k*for PDAs,*Journal of Computer and System Sciences*,**37**(1988), pp. 144–155.Google Scholar - [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]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]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]Kutyłowski, M., One-way multihead finite automata,
*Theoretical Computer Science*, to appear.Google Scholar - [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]Piatkowski, T. F.,
*N-head Finite-State Machines*, Ph.D. Dissertation, University of Michigan, 1963.Google Scholar - [10]Rosenberg, A. L.,
*Nonwriting Extensions of Finite Automata*, Ph.D. Dissertation, Harvard University, 1965.Google Scholar - [11]Rosenberg, A. L., On multihead finite automata,
*IBM Journal of Research and Development*,**10**(1966), pp. 388–394.Google Scholar - [12]Yao, A. C., and Rivest, R. L.,
*K*+1 heads are better than*K, Journal of Association for Computing Machinery*,**25**(1978), pp. 337–340.Google Scholar

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© Springer-Verlag New York Inc. 1990