The Power of Frequency Computation (Extended 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.

[1]  Frank Stephan,et al.  Recursion Theoretic Properties of Frequency Computation and Bounded Queries (Extended Abstract) , 1993, Kurt Gödel Colloquium.

[2]  Frank Stephan,et al.  Quantifying the Amount of Verboseness , 1992, Inf. Comput..

[3]  Frank Stephan,et al.  Approximable Sets , 1995, Inf. Comput..

[4]  R. Beigel,et al.  Bounded Queries to SAT and the Boolean Hierarchy , 1991, Theor. Comput. Sci..

[5]  William I. Gasarch,et al.  Bounded queries in recursion theory: a survey , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[6]  Carl H. Smith,et al.  On learning multiple concepts in parallel , 1993, COLT '93.

[7]  Frank Stephan,et al.  Inclusion problems in parallel learning and games (extended abstract) , 1994, Annual Conference Computational Learning Theory.

[8]  Martin Kummer.  A Proof of Beigel's Cardinality Conjecture , 1992, J. Symb. Log..

[9]  John Case,et al.  Learning Recursive Functions from Approximations , 1997, J. Comput. Syst. Sci..

[10]  Valentina S. Harizanov,et al.  Frequency Computations and the Cardinality Theorem , 1992, J. Symb. Log..

[11]  Richard Beigel,et al.  Bi-Immunity Results for Cheatable Sets , 1990, Theor. Comput. Sci..

[12]  Frank Stephan,et al.  Recursion Theoretic Properties of Frequency Computation and Bounded Queries , 1995, Inf. Comput..

[13]  Frank Stephan,et al.  Quantifying the Amount of Verboseness , 1995, Inf. Comput..

[14]  John Gill,et al.  Terse, Superterse, and Verbose Sets , 1993, Inf. Comput..

[15]  Frank Stephan,et al.  Inclusion Problems in Parallel Learning and Games , 1996, J. Comput. Syst. Sci..

[16]  Amihood Amir,et al.  Polynomial Terse Sets , 1988, Inf. Comput..