Some Minimality Results on Biresidual and Biseparable Automata

  • Hellis Tamm
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6031)

Abstract

Residual finite state automata (RFSA) are a subclass of nondeterministic finite automata (NFA) with the property that every state of an RFSA defines a residual language of the language accepted by the RFSA. Recently, a notion of a biresidual automaton (biRFSA) – an RFSA such that its reversal automaton is also an RFSA – was introduced by Latteux, Roos, and Terlutte, who also showed that a subclass of biRFSAs called biseparable automata consists of unique state-minimal NFAs for their languages. In this paper, we present some new minimality results concerning biRFSAs and biseparable automata. We consider two lower bound methods for the number of states of NFAs – the fooling set and the extended fooling set technique – and present two results related to these methods. First, we show that the lower bound provided by the fooling set technique is tight for and only for biseparable automata. And second, we prove that the lower bound provided by the extended fooling set technique is tight for any language accepted by a biRFSA. Also, as a third result of this paper, we show that any reversible canonical biRFSA is a transition-minimal ε-NFA. To prove this result, the theory of transition-minimal ε-NFAs by S. John is extended.

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References

  1. 1.
    Birget, J.C.: Intersection and union of regular languages and state complexity. Information Processing Letters 43, 185–190 (1992)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Denis, F., Lemay, A., Terlutte, A.: Residual finite state automata. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 144–157. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  3. 3.
    Glaister, I., Shallit, J.: A lower bound technique for the size of nondeterministic finite automata. Information Processing Letters 59, 75–77 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gruber, H., Holzer, M.: Finding lower bounds for nondeterministic state complexity is hard. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 363–374. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Han, Y.S., Salomaa, K., Wood, D.: Nondeterministic state complexity of basic operations for prefix-free regular languages. Fundam. Inform. 90, 93–106 (2009)MATHMathSciNetGoogle Scholar
  6. 6.
    John, S.: Minimal unambiguous ε-NFA. In: Domaratzki, M., Okhotin, A., Salomaa, K., Yu, S. (eds.) CIAA 2004. LNCS, vol. 3317, pp. 190–201. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Latteux, M., Lemay, A., Roos, Y., Terlutte, A.: Identification of biRFSA languages. Theoretical Computer Science 356, 212–223 (2006)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Latteux, M., Roos, Y., Terlutte, A.: BiRFSA languages and minimal NFAs. Technical Report GRAPPA-0205, GRAPPA (2005)Google Scholar
  9. 9.
    Latteux, M., Roos, Y., Terlutte, A.: Minimal NFA and biRFSA languages. RAIRO - Theoretical Informatics and Applications 43(2), 221–237 (2009)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Tamm, H.: On transition minimality of bideterministic automata. International Journal of Foundations of Computer Science 19(3), 677–690 (2008)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Tamm, H., Ukkonen, E.: Bideterministic automata and minimal representations of regular languages. Theoretical Computer Science 328, 135–149 (2004)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Hellis Tamm
    • 1
  1. 1.Institute of CyberneticsTallinn University of TechnologyTallinnEstonia

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