Closure Properties and Witness Reduction

Witness reduction has played a crucial role in several recent results in complexity theory. These include Toda?s result that PH ? BP · ?P, the "collapsing" of PH into ?P with a high probability; Toda and Ogiwara?s results which "collapse" PH into various counting classes with a high probability; and Ogiwara and Hemachandra?s results describing hard functions for various function classes. Ogiwara and Hemachandra?s results establish a connection between functions being hard for #P and functions interacting with the class to effect witness reduction. In fact, we believe that the ability to achieve some form of witness reduction is what makes a function hard for a class of functions. To support our thesis we define new function classes and obtain results analogous to those of Ogiwara and Hemachandra. We also introduce the notion of randomly hard functions and obtain similar results.

[1]  Janos Simon.  On some central problems in computational complexity , 1975 .

[2]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[3]  John Gill,et al.  Relative to a Random Oracle A, PA != NPA != co-NPA with Probability 1 , 1981, SIAM J. Comput..

[4]  Christos H. Papadimitriou,et al.  Two remarks on the power of counting , 1983, Theoretical Computer Science.

[5]  Leslie G. Valiant,et al.  NP is as easy as detecting unique solutions , 1985, STOC '85.

[6]  Mark W. Krentel.  The complexity of optimization problems , 1986, STOC '86.

[7]  Jacobo Torán,et al.  On Counting and Approximation , 1988, CAAP.

[8]  Jacobo Torán.  An oracle characterization of the counting hierarchy , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[9]  Uwe Schöning.  Probabilistic Complexity Classes and Lowness , 1989, J. Comput. Syst. Sci..

[10]  Juris Hartmanis,et al.  The Boolean Hierarchy II: Applications , 1989, SIAM J. Comput..

[11]  Seinosuke Toda.  On the computational power of PP and (+)P , 1989, 30th Annual Symposium on Foundations of Computer Science.

[12]  U. Schoning.  Probalisitic complexity classes and lowness , 1989 .

[13]  Kenneth W. Regan,et al.  A Simpler Proof of PH C BP[ӨP] , 1990 .

[14]  Seinosuke Toda.  The complexity of finding medians , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[15]  Daniel A. Spielman,et al.  PP is closed under intersection , 1991, STOC '91.

[16]  Stuart A. Kurtz,et al.  Gap-definable counting classes , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[17]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[18]  Stephen A. Fenner,et al.  Gap-deenable Counting Classes , 1991 .

[19]  John Gill,et al.  Counting Classes: Thresholds, Parity, Mods, and Fewness , 1990, Theor. Comput. Sci..

[20]  Mitsunori Ogihara,et al.  Counting Classes are at Least as Hard as the Polynomial-Time Hierarchy , 1992, SIAM J. Comput..

[21]  Lane A. Hemaspaandra,et al.  A Complexity Theory for Feasible Closure Properties , 1991, J. Comput. Syst. Sci..

[22]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..

[23]  Lance Fortnow,et al.  PP is Closed Under Truth-Table Reductions , 1996, Inf. Comput..