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Recursion theoretic properties of frequency computation and bounded queries (extended abstract)

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Computational Logic and Proof Theory (KGC 1993)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 713))

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Abstract

The notion of frequency computation captures the class Ω of all sets A such that for some n the n-fold characteristic function of A can be computed with less than n errors. Alternatively, it can be computed by a total oracle machine with less than n queries to the oracle. We consider the recursion theoretic properties of Ω with special emphasis on r.e. sets.

Supported by the Deutsche Forschungsgemeinschaft (DFG) grant Me 672/4-1

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References

  1. K. Ambos-Spies, A. Nies, R. A. Shore. The theory of the recursively enumerable weak truth-table degrees is undecidable. J. Symb. Logic, 57:864–874, 1992.

    Google Scholar 

  2. P. Assouad. Densité et dimension. Ann. Inst. Fourier, 33(3):233–282, 1983.

    Google Scholar 

  3. R. Beigel. Query-limited reducibilities. Ph.D. thesis, Stanford University, Stanford, USA, 1987.

    Google Scholar 

  4. R. Beigel, W. I. Gasarch, J. Gill, J. C. Owings, Jr. Terse, superterse, and verbose sets. Information and Computation, 103:68–85, 1993.

    Google Scholar 

  5. R. Beigel, M. Kummer, F. Stephan. Quantifying the amount of verboseness. To appear in: Information and Computation (a preliminary version appeared in: Lecture Notes for Computer Science (“Logic at Tver”), Vol.620, pp. 21–32, 1992).

    Google Scholar 

  6. R. Beigel, W. I. Gasarch, L. Hay. Bounded query classes and the difference hierarchy. Arch. Math. Logic, 29:69–84, 1989.

    Google Scholar 

  7. R. Beigel, W. I. Gasarch, J. C. Owings, Jr. Nondeterministic bounded query reducibilities. Annals of Pure and Applied Logic, 41:107–118, 1989.

    Google Scholar 

  8. A. Blumer, A. Ehrenfeucht, D. Haussler, M. K. Warmuth. Learnability and the Vapnik-Chervonenkis dimension. J. of the ACM, 36:929–965, 1989.

    Google Scholar 

  9. V. K. Bulitko. Reducibility by Zhegalkin linear tables. Sib. Math. J., 21:332–339.

    Google Scholar 

  10. A. N. Degtev. Hypersimple sets with retraceable complements. Algebra and Logic, 10:235–246, 1971.

    Google Scholar 

  11. A. N. Degtev. tt- and m-degrees. Algebra and Logic, 12:78–89, 1973.

    Google Scholar 

  12. A. N. Degtev. Reducibilities of tabular type in the theory of algorithms. Russian Math. Surveys, 34(3):155–192, 1979.

    Google Scholar 

  13. A. N. Degtev. On (m,n)-computable sets. In Algebraic Systems (Edited by D.I. Moldavanskij). Ivanova Gos. Univ. 88–99, 1981. (Russian) (MR 86b:03049)

    Google Scholar 

  14. W. I. Gasarch. Bounded queries in recursion theory: a survey. In Proceedings of the Sixth Annual Structure in Complexity Theory Conference. IEEE Computer Society Press, 62–78, 1991.

    Google Scholar 

  15. V. Harizanov, M. Kummer, J. C. Owings, Jr. Frequency computation and the cardinality theorem. J. Symb. Log., 57:677–681, 1992.

    Google Scholar 

  16. D. Haussler, E. Welzl. ε-nets and simplex range queries. Discrete Comput. Geom., 2:127–151, 1987.

    Google Scholar 

  17. C. G. Jockusch, Jr. Semirecursive sets and positive reducibility. Trans. Amer. Math. Soc., 131:420–436, 1968.

    Google Scholar 

  18. C. G. Jockusch, Jr. Three easy constructions of recursively enumerable degrees. In Lecture Notes in Mathematics, Vol. 859, 83–91, Springer-Verlag, Berlin, 1981.

    Google Scholar 

  19. C. G. Jockusch, Jr., J. C. Owings, Jr. Weakly semirecursive sets. J. Symb. Log., 55:637–644, 1990.

    Google Scholar 

  20. E. B. Kinber. On frequency calculations of general recursive predicates. Sov. Math. Dokl., 13:873–876, 1972.

    Google Scholar 

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

    Google Scholar 

  22. M. Kummer. A proof of Beigel's cardinality conjecture. J. Symb. Log., 57:682–687, 1992.

    Google Scholar 

  23. M. Kummer, F. Stephan. Some aspects of frequency computation. Technical Report Nr. 21/91, Fakultät für Informatik, Universität Karlsruhe, 1991.

    Google Scholar 

  24. P. H. Morris. A reducibility condition for recursiveness. Proceedings of the American Math. Soc., 60:270–272, 1976.

    Google Scholar 

  25. P. Odifreddi. Classical Recursion Theory. North-Holland, Amsterdam, 1989.

    Google Scholar 

  26. G. F. Rose. An extended notion of computability. In Abstr. Intern. Congr. for Logic, Meth., and Phil. of Science, Stanford, California, 1960.

    Google Scholar 

  27. R. I. Soare. Recursively Enumerable Sets and Degrees. Springer-Verlag, Berlin, 1987.

    Google Scholar 

  28. B. A. Trakhtenbrot. On frequency computation of functions. Algebra i Logika, 2:25–32, 1963. (Russian)

    Google Scholar 

  29. B. A. Trakhtenbrot. On autoreducibility. Sov. Math. Dokl., 11:814–817, 1970.

    Google Scholar 

  30. V. N. Vapnik, A. Ya. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl., 16:264–280, 1971.

    Google Scholar 

  31. C. E. M. Yates. Recursively enumerable sets and retracing functions. Zeit. Math. Log. Grund. Math., 8:331–345, 1962.

    Google Scholar 

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Authors and Affiliations

  1. Institut für Logik, Komplexität und Deduktionssysteme, Universität Karlsruhe, D-76128, Karlsruhe, Germany

    Martin Kummer & Frank Stephan

Authors
  1. Martin Kummer
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  2. Frank Stephan
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Georg Gottlob Alexander Leitsch Daniele Mundici

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© 1993 Springer-Verlag Berlin Heidelberg

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Kummer, M., Stephan, F. (1993). Recursion theoretic properties of frequency computation and bounded queries (extended abstract). In: Gottlob, G., Leitsch, A., Mundici, D. (eds) Computational Logic and Proof Theory. KGC 1993. Lecture Notes in Computer Science, vol 713. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0022573

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  • DOI: https://doi.org/10.1007/BFb0022573

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57184-1

  • Online ISBN: 978-3-540-47943-7

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