State Complexity of Operations on Two-Way Deterministic Finite Automata over a Unary Alphabet

  • Michal Kunc
  • Alexander Okhotin
Conference paper

DOI: 10.1007/978-3-642-22600-7_18

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6808)
Cite this paper as:
Kunc M., Okhotin A. (2011) State Complexity of Operations on Two-Way Deterministic Finite Automata over a Unary Alphabet. In: Holzer M., Kutrib M., Pighizzini G. (eds) Descriptional Complexity of Formal Systems. DCFS 2011. Lecture Notes in Computer Science, vol 6808. Springer, Berlin, Heidelberg

Abstract

The paper determines the number of states in a two-way deterministic finite automaton (2DFA) over a one-letter alphabet sufficient and in the worst case necessary to represent the results of the following operations: (i) intersection of an m-state 2DFA and an n-state 2DFA requires between m + n and m + n + 1 states; (ii) union of an m-state 2DFA and an n-state 2DFA, between m + n and 2m + n + 4 states; (iii) Kleene star of an n-state 2DFA, (g(n) + O(n))2 states, where \(g(n)=e^{\sqrt{n \ln n}(1+o(1))}\) is the maximum value of lcm(p1, …, pk) for \(\sum p_i \leqslant n\), known as Landau’s function; (iv) k-th power of an n-state 2DFA, between (k − 1)g(n) − k and k(g(n) + n) states; (v) concatenation of an m-state and an n-state 2DFAs, \(e^{(1+o(1)) \sqrt{(m+n)\ln(m+n)}}\) states.

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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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