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On Separating Constant from Polynomial Ambiguity of Finite Automata

(Extended Abstract)
  • Joachim Kupke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3831)

Abstract

The degree of nondeterminism of a finite automaton can be measured by means of its ambiguity function. In many instances, whenever automata are allowed to be (substantially) less ambiguous, it is known that the number of states needed to recognize at least some languages increases exponentially. However, when comparing constantly ambiguous automata with polynomially ambiguous ones, the question whether there are languages such that the inferior class of automata requires exponentially many states more than the superior class to recognize them is still an open problem. The purpose of this paper is to suggest a family of languages that seems apt for a proof of this (conjectured) gap. As a byproduct, we derive a new variant of the proof of the existence of a superpolynomial gap between polynomial and fixed-constant ambiguity. Although our candidate languages are defined over a huge alphabet, we show how to overcome this drawback.

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References

  1. 1.
    Hopcroft, J., Motwani, R., Ullman, J.: Introduction to Automata and Language Theory. Addison-Wesley, Reading (2000)Google Scholar
  2. 2.
    Hromkovič, J.: Communication Complexity and Parallel Computing. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  3. 3.
    Hromkovi č, J., Karhumäki, J., Klauck, H., Seibert, S., Schnitger, G.: Measures on Nondeterminism in Finite Automata. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 199–210. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. 4.
    Kupke, J.: Limiting the Ambiguity of Non-Deterministic Finite Automata. Diploma Thesis, RWTH Aachen (2002), Available online at: http://www-i1.informatik.rwth-aachen.de/~joachimk/ltaondfa.ps
  5. 5.
    Kupke, J.: On Separating Constant from Polynomial Ambiguity of Finite Automata, http://www.ite.ethz.ch/people/jkupke/publications/oscfpaofa.ps
  6. 6.
    Gramlich, G., Schnitger, G.: Minimizing NFAs and Regular Expressions. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 399–411. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Leung, H.: Separating Exponentially Ambiguous Finite Automata from Polynomially Ambiguous Finite Automata. SIAM J. Comp. 27, 1073–1082 (1998)zbMATHCrossRefGoogle Scholar
  8. 8.
    Moore, F.: On the Bounds for State-Set Size in the Proofs of Equivalence between Deterministic, Nondeterministic, and Two-Way Finite Automata. IEEE Trans. Comput. 20, 1211–1214 (1971)zbMATHCrossRefGoogle Scholar
  9. 9.
    Ravikumar, B., Ibarra, O.: Relating the Type of Ambiguity of Finite Automata to the Succinctness of Their Representation. SIAM J. Comp. 18, 1263–1282 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Sipser, M.: Lower Bounds on the Size of Sweeping Automata. J. Comp. and Sys. Sci. 21(2), 195–202 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Weber, A., Seidl, H.: On the Ambiguity of Finite Automata. In: Wiedermann, J., Gruska, J., Rovan, B. (eds.) MFCS 1986. LNCS, vol. 233, pp. 620–629. Springer, Heidelberg (1986)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Joachim Kupke
    • 1
  1. 1.ETH ZurichZurichSwitzerland

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