Closure of Polynomial Time Partial Information Classes under Polynomial Time Reductions

Polynomial time partial information classes are extensions of the class P of languages decidable in polynomial time. A partial information algorithm for a language A computes, for fixed n ∈ N, on input of words x1, . . . , xn a set P of bitstrings, called a pool, such that XA(x1, . . . , xn) ∈ P, where P is chosen from a family D of pools. A language A is in P[D], if there is a polynomial time partial information algorithm which for all inputs (x1, . . . , xn) outputs a pool P ∈ D with XA(x1, . . . , xn) ∈ P. Manyextensions of P studied in the literature, including approximable languages, cheatability, p-selectivity and frequency computations, form a class P[D] for an appropriate family D. We characterise those families D for which P[D] is closed under certain polynomial time reductions, namely bounded truth-table, truth-table, and Turing reductions. We also treat positive reductions. A class P[D] is presented which strictlycon tains the class P-sel of p-selective languages and is closed under positive truth-table reductions.

[1]  Lane A. Hemaspaandra,et al.  Reducibility Classes of P-Selective Sets , 1996, Theor. Comput. Sci..

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

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

[4]  Juris Hartmanis,et al.  On isomorphisms and density of NP and other complete sets , 1976, STOC '76.

[5]  Gerd Wechsung,et al.  Time bounded frequency computations , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[6]  Efim B. Kinber,et al.  Frequency computation and bounded queries , 1995, Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference.

[7]  Arfst Nickelsen.  On Polynomially D-Verbose Sets , 1997, STACS.

[8]  Jörg Rothe,et al.  Polynomial-Time Multi-Selectivity , 1997, J. Univers. Comput. Sci..

[9]  Arfst Nickelsen,et al.  Counting, Selecting, adn Sorting by Query-Bounded Machines , 1993, STACS.

[10]  Johannes Köbler,et al.  On the Structure of Low Sets , 1995, SCT.

[11]  Alan L. Selman,et al.  P-Selective Sets, Tally Languages, and the Behavior of Polynomial Time Reducibilities on NP , 1979, ICALP.

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

[13]  R. Beigel.  Query-limited reducibilities , 1988 .

[14]  Efim B. Kinber,et al.  Frequency Computation and Bounded Queries , 1996, Theor. Comput. Sci..

[15]  Juris Hartmanis,et al.  On Isomorphisms and Density of NP and Other Complete Sets , 1977, SIAM J. Comput..

[16]  Amihood Amir,et al.  Some connections between bounded query classes and nonuniform complexity , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[17]  Ker-I Ko.  On Self-Reducibility and Weak P-Selectivity , 1983, J. Comput. Syst. Sci..

[18]  Amihood Amir,et al.  Some connections between bounded query classes and non-uniform complexity , 2000, Inf. Comput..

[19]  Paul Young,et al.  Using Self-Reducibilities to Characterize Polynomial Time , 1993, Inf. Comput..

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

[21]  Nancy A. Lynch,et al.  Comparison of polynomial-time reducibilities , 1974, STOC '74.

[22]  Paul R. Young,et al.  On semi-cylinders, splinters, and bounded-truth-table reducibility , 1965 .

[23]  Alan L. Selman,et al.  Reductions on NP and P-Selective Sets , 1982, Theor. Comput. Sci..