On the computational power of PP and (+)P

Two complexity classes, PP and (+)P, are compared with PH (the polynomial-time hierarchy). The main results are as follows: (1) every set in PH is reducible in a certain sense to a set in PP, an (2) every set in PH is reducible to a set in (+)P under randomized polynomial-time reducibility with two-sided bounded error probability. It follows from these results that neither PP nor (+)P is a subset of or equivalent to PH unless PH collapses to a finite level. This is strong evidence that both classes are strictly harder than PH.<<ETX>>

[1]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[2]  Eric Allender,et al.  A note on the power of threshold circuits , 1989, 30th Annual Symposium on Foundations of Computer Science.

[3]  José L. Balcázar,et al.  The polynomial-time hierarchy and sparse oracles , 1986, JACM.

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

[5]  Timothy J. Long,et al.  Relativizing complexity classes with sparse oracles , 1986, JACM.

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

[7]  Uwe Schöning.  The power of counting , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[8]  Lane A. Hemaspaandra,et al.  On the power of probabilistic polynomial time: P/sup NP(log)/ contained in PP , 1989, [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference.

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

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

[11]  John Gill,et al.  Computational Complexity of Probabilistic Turing Machines , 1977, SIAM J. Comput..

[12]  Celia Wrathall,et al.  Complete Sets and the Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[13]  Larry J. Stockmeyer,et al.  On Approximation Algorithms for #P , 1985, SIAM J. Comput..

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

[15]  Jin-Yi Cai,et al.  On the Power of Parity Polynomial Time , 1989, STACS.

[16]  Dana Angluin,et al.  On Counting Problems and the Polynomial-Time Hierarchy , 1980, Theor. Comput. Sci..

[17]  C. Papadimitriou,et al.  Two remarks on the power of counting , 1983 .