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Quantifying the amount of verboseness (extended abstract)

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 620))

Abstract

We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the n-fold characteristic function of A. A complete characterization is obtained relating the question to finite combinatorics. In order to obtain an explicit description we encounter several interesting combinatorial problems.

Supported in part by National Science Foundation grant CCR-8958528.

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References

  1. N. Alon. On the density of sets of vectors. Discrete Mathematics, 46:199–202, 1983.

    Article  Google Scholar 

  2. A. Amir, W. I. Gasarch. Polynomial terse sets. Information and Computation, 77:37–56, 1988.

    Article  Google Scholar 

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

    Google Scholar 

  4. R. Beigel. Bi-immunity results for cheatable sets. Theoretical Computer Science, 73:249–263, 1990.

    Article  Google Scholar 

  5. R. Beigel. Bounded queries to SAT and the Boolean hierarchy. Theoretical Computer Science, 83:199–223, 1991.

    Article  Google Scholar 

  6. R. Beigel, W. I. Gasarch, J. Gill, J. C. Owings, Jr. Terse, superterse, and verbose sets. To appear in: Information and Computation.

    Google Scholar 

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

    Article  Google Scholar 

  8. B. Bollobás. Extremal Graph Theory. Academic Press, London, 1978.

    Google Scholar 

  9. 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 

  10. Yu. L. Ershov. On a hierarchy of sets I. Algebra and Logic, 7:25–43, 1968.

    Article  Google Scholar 

  11. P. Frankl. On the trace of finite sets. J. of Combinatorial Theory, Ser. A, 34, 41–45, 1983.

    Google Scholar 

  12. W. I. Gasarch. Bounded queries in recursion theory: a survey. In Proceedings of the Fifth Annual Structure in Complexity Theory Conference. IEEE Computer Society Press, 1991.

    Google Scholar 

  13. V. Harizanov, M. Kummer, J. C. Owings, Jr. Frequency computation and the cardinality theorem. To appear in: J. Symb. Log.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  17. E. B. Kinber. On frequency real-time computations. In: Teoriya Algorithmov i Programm, Vol. 2 (Edited by Ya. M. Barzdin). Latv. Valst. (Gos.) Univ. 174–182, 1975. (Russian) (MR 58:3624, Zbl 335:02023)

    Google Scholar 

  18. M. Kummer. A proof of Beigel's cardinality conjecture. To appear in: J. Symb. Log.

    Google Scholar 

  19. M. Kummer, F. Stephan. Weakly semirecursive sets and r.e. orderings. Technical Report Nr. 34/90, Fakultät für Informatik, Universität Karlsruhe, Postfach 6980, D-7500 Karlsruhe 1, 1990.

    Google Scholar 

  20. M. Kummer, F. Stephan. Some aspects of frequency computation. Technical Report Nr. 21/91, Fakultät für Informatik, Universität Karlsruhe, Postfach 6980, D-7500 Karlsruhe 1, 1991.

    Google Scholar 

  21. R. McNaughton. The theory of automata, a survey. Advances of Computers, 2:379–421, 1961.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  24. N. Sauer. On the density of families of sets. J. of Combinatorial Theory, Ser. A, 13:145–147, 1972.

    Google Scholar 

  25. S. Shelah. A combinatorial problem: Stability and order for models and theories in infmitary languages. Pacific J. of Mathematics, 41:247–261.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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Anil Nerode Mikhail Taitslin

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© 1992 Springer-Verlag Berlin Heidelberg

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Beigel, R., Kummer, M., Stephan, F. (1992). Quantifying the amount of verboseness (extended abstract). In: Nerode, A., Taitslin, M. (eds) Logical Foundations of Computer Science — Tver '92. LFCS 1992. Lecture Notes in Computer Science, vol 620. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023860

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  • DOI: https://doi.org/10.1007/BFb0023860

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55707-4

  • Online ISBN: 978-3-540-47276-6

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