Abstract
The notion of frequency computation concerns approximative computations of n distinct parallel queries to a set A. A is called (m, n)-recursive if there is an algorithm which answers any n distinct parallel queries to A such that at least m answers are correct. This paper gives natural combinatorial characterizations of the fundamental inclusion problem, namely the question for which choices of the parameters m, n, m′, n′, every (m, n)-recursive set is (m′, n′)-recursive. We also characterize the inclusion problem restricted to recursively enumerable sets and the inclusion problem for the polynomial-time bounded version of frequency computation. Furthermore, using these characterizations we obtain many explicit inclusions and noninclusions.
Supported by the Deutsche Forschungsgemeinschaft (DFG) grant Me 672/4-2.
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References
A. Amir, W. I. Gasarch. Polynomial terse sets. Information and Computation, 77:37–56, 1988.
R. Beigel. Bi-immunity results for cheatable sets. Theoretical Computer Science, 73:249–263, 1990.
R. Beigel. Bounded queries to SAT and the boolean hierarchy. Theoretical Computer Science, 83:199–223, 1991.
R. Beigel, W. I. Gasarch, J. Gill, J. C. Owings, Jr. Terse, superterse, and verbose sets. Information and Computation, 103:68–85, 1993.
R. Beigel, M. Kummer, F. Stephan. Quantifying the amount of verboseness. To appear in: Information and Computation. (A preliminary version appeared in Logical Foundations of Computer Science — Tver'92, Lecture Notes in Computer Science 620, pp. 21–32, 1992).
R. Beigel, M. Kummer, F. Stephan. Approximable sets. To appear in: Information and Computation. (A preliminary version appeared in Proceedings Structure in Complexity Theory, Ninth Annual Conference, pp. 12–23, IEEE Press, 1994.)
J. Case, S. Kaufmann, E. Kinber, M. Kummer. Learning recursive functions from approximations. In Proceedings EuroCOLT'95, Lecture Notes in Computer Science 904, pp. 140–153, 1995. Springer-Verlag
A. N. Dëgtev. On (m, n)-computable sets. In Algebraic Systems (Edited by D.I Moldavanskij). Ivanova Gos. Univ. 88–99, 1981. (Russian) (MR 86b:03049)
W. I. Gasarch. Bounded queries in recursion theory: a survey. In Proceedings of the Sixth Annual Structure in Complexity Theory Conference, pp. 62–78, IEEE Press, 1991.
V. Harizanov, M. Kummer, J. C. Owings, Jr. Frequency computation and the cardinality theorem. J. Symb. Log., 57:677–681, 1992.
E. B. Kinber. Frequency-computable functions and frequency-enumerable sets. Candidate Dissertation, Riga, 1975. (Russian)
E. B. Kinber. On frequency real-time computations. In Teoriya Algorithmov i Programm, Vol. 2 (Edited by Ya. M. Barzdin). Latv. Valst. Gos. Univ. pp. 174–182, 1975. (Russian) (MR 58:3624, Zbl 335:02023)
E. Kinber, C. Smith, M. Velauthapillai, R. Wiehagen. On learning multiple concepts in parallel. In Proceedings COLT'93, pp. 175–181, ACM Press, 1993.
M. Kummer. A proof of Beigel's cardinality conjecture. J. Symb. Log., 57:682–687, 1992.
M. Kummer, F. Stephan. Some aspects of frequency computation. Technical Report No. 21/91, Fakultät für Informatik, Universität Karlsruhe, 1991.
M. Kummer, F. Stephan. Recursion theoretic properties of frequency computation and bounded queries. To appear in: Information and Computation (A preliminary version appeared in Third Kurt Gödel Colloquium, Lecture Notes in Computer Science 713, pp. 243–254, 1993).
M. Kummer, F. Stephan. Inclusion problems in parallel learning and games. In Proceedings COLT'94, pp. 287–298, ACM Press, 1994.
T. McNicholl. A solution to the inclusion problem for frequency classes. Manuscript, 66 pp., Dept. of Mathematics, George Washington University, Washington DC, June 1994.
G. F. Rose. An extended notion of computability. In Abstr. Intern. Congr. for Logic, Meth., and Phil. of Science, Stanford, California, 1960.
B. A. Trakhtenbrot. On frequency computation of functions. Algebra i Logika, 2:25–32, 1963. (Russian)
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Kummer, M., Stephan, F. (1995). The power of frequency computation. In: Reichel, H. (eds) Fundamentals of Computation Theory. FCT 1995. Lecture Notes in Computer Science, vol 965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60249-6_64
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DOI: https://doi.org/10.1007/3-540-60249-6_64
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