Optimal simulations between unary automata

  • Carlo Mereghetti
  • Giovanni Pighizzini
Automata and Formal Languages I
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1373)

Abstract

We consider the problem of computing the costs — in terms of states — of optimal simulations between different kinds of finite automata recognizing unary languages. Our main result is a tight simulation of unary n-state two-way nondeterministic automata by \(O(e^{\sqrt {n \ln n} } )\)-state one-way deterministic automata. In addition, we show that, given a unary n-state two-way nondeterministic automaton, one can construct an equivalent O(n2)-state two-way nondeterministic automaton performing both input head reversals and nondeterministic choices at the endmarkers only. Further results on simulating unary alternating finite automata are pointed out. Our results give answers to some questions left open in the literature.

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References

  1. [Ber85]
    C. Berge. Graphs and Hypergraphs. North-Holland, 1985.Google Scholar
  2. [Bir93]
    J.-C. Birget. State-complexity of finite-state devices, state compressibility and incompressibility. Mathematical Systems Theory, 26:237–269, 1993.MATHCrossRefMathSciNetGoogle Scholar
  3. [BL77]
    P. Berman and A. Lingas. On the complexity of regular languages in terms of finite automata. Technical Report 304, Polish Academy of Sciences, 1977.Google Scholar
  4. [Chr86]
    M. Chrobak. Finite automata and unary languages. Theoretical Computer Science, 47:149–158, 1986.MATHCrossRefMathSciNetGoogle Scholar
  5. [CKS81]
    A. Chandra, D. Kozen, and L. Stockmeyer. Alternation. Journal of the ACM, 28:114–133, 1981.MATHCrossRefMathSciNetGoogle Scholar
  6. [FJY90]
    A. Fellah, H. Rirgensen, and S. Yu. Constructions for alternating finite automata. International J. Computer Math., 35:117–132, 1990.MATHCrossRefGoogle Scholar
  7. [Gef91]
    V. Geffert. Nondeterministic computations in sublogarithmic space and space constructibility. SIAM J. Computing, 20:484–498, 1991.MATHCrossRefMathSciNetGoogle Scholar
  8. [Gef97]
    V. Geffert, 1997. Private communication.Google Scholar
  9. [GI79]
    E. Gurari and O. Ibarra. Simple counter machines and number-theoretic problems. Journal of Computer and System Sciences, 19:145–162, 1979.MATHCrossRefMathSciNetGoogle Scholar
  10. [GI82]
    E. Gurari and O. Ibarra. Two-way counter machines and Diophantine equations. Journal of the ACM, 29:863–873, 1982.MATHCrossRefMathSciNetGoogle Scholar
  11. [HU79]
    J. Hopcroft and J. Ullman. Introduction to automata theory, languages, and computation. Addison-Wesley, Reading, MA, 1979.MATHGoogle Scholar
  12. [RS59]
    M.O. Rabin and D. Scott. Finite automata and their decision problems. IBM J. Res. Develop, 3:114–125, 1959.CrossRefMathSciNetGoogle Scholar
  13. [SS78]
    W. Sakoda and M. Sipser. Nondeterminism and the size of two-way finite automata. In Proc. 10th ACM Symposium on Theory of Computing, pages 275–286, 1978.Google Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Carlo Mereghetti
    • 1
  • Giovanni Pighizzini
    • 1
  1. 1.Dipartimento di Scienze dell'InformazioneUniversità degli Studi di MilanoMilanoItaly

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