Describing Periodicity in Two-Way Deterministic Finite Automata Using Transformation Semigroups

  • Michal Kunc
  • Alexander Okhotin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6795)


A framework for the study of periodic behaviour of two-way deterministic finite automata (2DFA) is developed. Computations of 2DFAs are represented by a two-way analogue of transformation semigroups, every element of which describes the behaviour of a 2DFA on a certain string x. A subsemigroup generated by this element represents the behaviour on strings in x  + . The main contribution of this paper is a description of all such monogenic subsemigroups up to isomorphism. This characterization is then used to show that transforming an n-state 2DFA over a one-letter alphabet to an equivalent sweeping 2DFA requires exactly n + 1 states, and transforming it to a one-way automaton requires exactly \(\max_{0 \leqslant \ell \leqslant n} G(n-\ell)+\ell+1\) states, where G(k) is the maximum order of a permutation of k elements.


Regular Language Partial Transformation Transformation Semigroup Information Processing Letter Nondeterministic Automaton 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Michal Kunc
    • 1
  • Alexander Okhotin
    • 2
    • 3
  1. 1.Masaryk UniversityCzech Republic
  2. 2.Department of MathematicsUniversity of TurkuFinland
  3. 3.Academy of FinlandFinland

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