Error-bounded probabilistic computations between MA and AM

We introduce the probabilistic class SBP. This class emerges from BPP by keeping the promise of a probability gap but decreasing the probability limit from 1/2 to exponentially small values. We show that SBP is in the polynomial-time hierarchy, between MA and AM on the one hand and between BPP and BPPpath on the other hand. We provide evidence that SBP does not coincide with these and other known complexity classes. In particular, in a suitable relativized world SBP is not contained in Σ2P. In the same world, BPPpath is not contained in Σ2P, which solves an open question raised by Han, Hemaspaandra, and Thierauf. We study the question of whether SBP has many-one complete sets. We relate this question to the existence of uniform enumerations and construct an oracle relative to which SBP and AM do not have many-one complete sets. We introduce the operator SB. and prove that, for any class C with certain properties, BP ċ ∃ ċ C contains every class defined by applying an operator sequence over {Uċ, ∃ċ, BPċ, SBċ} to C.

[1]  Lane A. Hemachandra,et al.  A complexity theory for feasible closure properties , 1993 .

[2]  Stuart A. Kurtz,et al.  Gap-Definable Counting Classes , 1994, J. Comput. Syst. Sci..

[3]  Claude E. Shannon,et al.  Computability by Probabilistic Machines , 1970 .

[4]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[5]  Stathis Zachos,et al.  Robustness of Probabilistic Computational Complexity Classes under Definitional Perturbations , 1982, Inf. Control..

[6]  Lance Fortnow.  Relativized Worlds with an Infinite Hierarchy , 1999, Inf. Process. Lett..

[7]  Klaus Ambos-Spies.  A Note on the Complete Problems for Complexity Classes , 1986, Inf. Process. Lett..

[8]  Larry Carter,et al.  Universal Classes of Hash Functions , 1979, J. Comput. Syst. Sci..

[9]  Lide Li.  On PP-Low Classes , 1993 .

[10]  A. Paz.  Probabilistic algorithms , 2003 .

[11]  Uwe Schöning.  Graph Isomorphism is in the Low Hierarchy , 1988, J. Comput. Syst. Sci..

[12]  Stathis Zachos,et al.  Does co-NP Have Short Interactive Proofs? , 1987, Inf. Process. Lett..

[13]  Osamu Watanabe,et al.  New Collapse Consequences of NP Having Small Circuits , 1995, ICALP.

[14]  Klaus W. Wagner,et al.  The Difference and Truth-Table Hierarchies for NP , 1987, RAIRO Theor. Informatics Appl..

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

[16]  R.E. Ladner,et al.  A Comparison of Polynomial Time Reducibilities , 1975, Theor. Comput. Sci..

[17]  J. Köbler,et al.  New Collapse Consequences of NP Having Small Circuits , 1999, SIAM J. Comput..

[18]  Aaron D. Wyner,et al.  Computability by Probabilistic Machines , 1993 .

[19]  Stuart A. Kurtz,et al.  On oracle builder's toolkit , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[20]  Michael Sipser,et al.  On Relativization and the Existence of Complete Sets , 1982, ICALP.

[21]  N. K. Vereschchagin.  On the power of PP , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[22]  Lance Fortnow,et al.  Complexity limitations on quantum computation , 1999, J. Comput. Syst. Sci..

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

[24]  Klaus W. Wagner,et al.  Bounded Query Classes , 1990, SIAM J. Comput..

[25]  Nikolai K. Vereshchagin,et al.  Banishing Robust Turing Completeness , 1993, Int. J. Found. Comput. Sci..

[26]  Michael Sipser,et al.  A complexity theoretic approach to randomness , 1983, STOC.

[27]  Lance Fortnow,et al.  Are There Interactive Protocols for CO-NP Languages? , 1988, Inf. Process. Lett..

[28]  Leslie G. Valiant,et al.  Relative Complexity of Checking and Evaluating , 1976, Inf. Process. Lett..

[29]  Richard Beigel.  Perceptrons, PP, and the polynomial hierarchy , 2005, computational complexity.

[30]  Yuri Gurevich,et al.  Algebras of feasible functions , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

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

[32]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

[33]  Randall Pruim,et al.  Relativized separation of EQP from PNP , 2001, Inf. Process. Lett..

[34]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .

[35]  Silvio Micali,et al.  Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems , 1991, JACM.

[36]  Nikolai K. Vereshchagin.  On The Power of PP , 1992, Computational Complexity Conference.

[37]  Clemens Lautemann,et al.  BPP and the Polynomial Hierarchy , 1983, Inf. Process. Lett..

[38]  Theodore P. Baker,et al.  A second step toward the polynomial hierarchy , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[39]  Lane A. Hemaspaandra,et al.  Threshold Computation and Cryptographic Security , 1993, ISAAC.

[40]  Sanjay Gupta.  The power of witness reduction , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[41]  Frederick A. Johnson.  Counting Functions , 1992, Notre Dame J. Formal Log..

[42]  John T. Gill,et al.  Computational complexity of probabilistic Turing machines , 1974, STOC '74.

[43]  László Babai,et al.  Arthur-Merlin Games: A Randomized Proof System, and a Hierarchy of Complexity Classes , 1988, J. Comput. Syst. Sci..

[44]  Ker-I Ko,et al.  Some Observations on the Probabilistic Algorithms and NP-hard Problems , 1982, Inf. Process. Lett..

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

[46]  Juris Hartmanis,et al.  Complexity Classes without Machines: On Complete Languages for UP , 1986, Theor. Comput. Sci..

[47]  Lance Fortnow,et al.  Optimal Proof Systems and Sparse Sets , 2000, STACS.

[48]  Miklos Santha.  Relativized Arthur-Merlin versus Merlin-Arthur Games , 1989, Inf. Comput..

[49]  J. Hartmanis,et al.  Computation Times of NP Sets of Different Densities , 1984, Theor. Comput. Sci..

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

[51]  Sanjay Gupta,et al.  Closure Properties and Witness Reduction , 1995, J. Comput. Syst. Sci..

[52]  Ming-Yang Kao,et al.  Towards understanding the predictability of stock markets from the perspective of computational complexity , 2000, SODA '01.

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

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

[55]  Volker Strassen,et al.  A Fast Monte-Carlo Test for Primality , 1977, SIAM J. Comput..

[56]  Stephen A. Fenner.  PP-Lowness and a Simple Definition of AWPP , 2002, Theory of Computing Systems.

[57]  Christian Glaßer,et al.  Error-Bounded Probabilistic Computations between MA and AM , 2003, MFCS.