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Towards a Cardinality Theorem for Finite Automata

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Mathematical Foundations of Computer Science 2002 (MFCS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2420))

Abstract

Kummer’s cardinality theorem states that a language is recursive if a Turing machine can exclude for any n words one of the n + 1 possibilities for the number of words in the language. This paper gathers evidence that the cardinality theorem might also hold for finite automata. Three reasons are given. First, Beigel’s nonspeedup theorem also holds for finite automata. Second, the cardinality theorem for finite automata holds for n = 2. Third, the restricted cardinality theorem for finite automata holds for all n.

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References

  1. H. Austinat, V. Diekert, and U. Hertrampf. A structural property of regular frequency computations. Theoretical Comput. Sci., to appear 2002.

    Google Scholar 

  2. H. Austinat, V. Diekert, U. Hertrampf, and H. Petersen. Regular frequency computations. In Proc. RIMS Symposium on Algebraic Syst., Formal Languages and Computation, volume 1166 of RIMS Kokyuroku, pages 35–42, Research Inst. for Math. Sci., Kyoto University, Japan, 2000.

    Google Scholar 

  3. R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, USA, 1987.

    Google Scholar 

  4. R. Beigel, W. Gasarch, M. Kummer, G. Martin, T. McNicholl, and F. Stephan. The complexity of ODD An . J. Symbolic Logic, 65(1):1–18, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Cai and L. Hemachandra. Enumerative counting is hard. Inform. Computation, 82(1):34–44, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  6. W. Gasarch. Bounded queries in recursion theory: A survey. In Proc. 6th Structure in Complexity Theory Conf., pages 62–78, 1991. IEEE Computer Society Press.

    Google Scholar 

  7. V. Harizanov, M. Kummer, and J. Owings. Frequency computations and the cardinality theorem. J. Symbolic Logic, 52(2):682–687, 1992.

    MathSciNet  Google Scholar 

  8. L. Hemachandra. The strong exponential hierarchy collapses. J. Comput. Syst. Sci., 39(3):299–322, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  9. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. A downward collapse in the polynomial hierarchy. SIAM J. Comput., 28(2):383–393, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Hoene and A. Nickelsen. Counting, selecting, and sorting by query-bounded machines. In Proc. 10th Symposium on Theoretical Aspects of Comput. Sci., volume 665 of Lecture Notes in Comput. Sci., pages 196–205. Springer-Verlag, 1993.

    Google Scholar 

  11. N. Immerman. Nondeterministic space is closed under complementation. SIAM J. Comput., 17(5):935–938, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  12. C. Jockusch, Jr. Reducibilities in Recursive Function Theory. PhD thesis, Massachusetts Inst. of Technology, USA, 1966.

    Google Scholar 

  13. J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM J. Comput., 17(6):1263–1282, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Kadin. PNP[O(logn)] and sparse Turing-complete sets for NP. J. Comput. Syst. Sci., 39(3):282–298, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  15. E. Kinber. Frequency computations in finite automata. Cybernetics, 2:179–187, 1976.

    Google Scholar 

  16. M. Kummer. A proof of Beigel’s cardinality conjecture. J. Symbolic Logic, 57(2): 677–681, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Kummer and F. Stephan. Effective search problems. Math. Logic Quarterly, 40(2):224–236, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Mahaney. Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis. J. Comput. Syst. Sci., 25(2):130–143, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Nickelsen. On polynomially \( \mathcal{D} \) -verbose sets. In Proc. 14th Symposium on Theoretical Aspects of Comput. Sci., volume 1200 of Lecture Notes in Comput. Sci., pages 307–318. Springer-Verlag, 1997.

    Google Scholar 

  20. J. Owings, Jr. A cardinality version of Beigel’s nonspeedup theorem. J. Symbolic Logic, 54(3):761–767, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. Szelepcsényi. The method of forced enumeration for nondeterministic automata. Acta Informatica, 26(3):279–284, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  22. T. Tantau. Comparing verboseness for finite automata and Turing machines. In Proc. 19th Symposium on Theoretical Aspects of Comput. Sci., volume 2285 of Lecture Notes in Comput. Sci., pages 465–476. Springer-Verlag, 2002.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Fakultät für Elektrotechnik und Informatik, Technische Universität Berlin, 10623, Berlin, Germany

    Till Tantau

Authors
  1. Till Tantau
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Editor information

Editors and Affiliations

  1. Institute of Informatics, Warsaw University, ul. Banacha 2, 02-097, Warszawa, Poland

    Krzysztof Diks  & Wojciech Rytter  & 

  2. Department of Computer Science, Liverpool University, Peach Street, Liverpool, L69 7ZF, UK

    Wojciech Rytter

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© 2002 Springer-Verlag Berlin Heidelberg

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Tantau, T. (2002). Towards a Cardinality Theorem for Finite Automata. In: Diks, K., Rytter, W. (eds) Mathematical Foundations of Computer Science 2002. MFCS 2002. Lecture Notes in Computer Science, vol 2420. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45687-2_52

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  • DOI: https://doi.org/10.1007/3-540-45687-2_52

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44040-6

  • Online ISBN: 978-3-540-45687-2

  • eBook Packages: Springer Book Archive

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