On Probabilistic ACC Circuits with an Exact-Threshold Output Gate

Let SYM+ denote the class of Boolean functions computable by depth-two size-\(n^{log^{O(1)} n}\) circuits with a symmetric-function gate at the root and AND gates of fan-in logO(1)n at the next level, or equivalently, the class of Boolean functions h such that h(x1,⋯ ,x n ) can be expressed as h(x1,⋯, xn) =h n(pn(x1,⋯, x n )) for some polynomial p n over Z of degree logO(1)n and norm (the sum of the absolute values of its coefficients) \(n^{log^{O(1)} n}\) and some function h n : Z → {0,1}. Building on work of Yao [Yao90], Beigel and Tarui [BT91] showed that ACC \(\subseteq\) SYM+, where ACC is the class of Boolean functions computable by constant-depth polynomial-size circuits with NOT, AND, OR, and MODm gates for some fixed m.

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

[2]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[3]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

[4]  Eric Allender,et al.  On the Power of Uniform Families of Constant Depth Treshold Circuits , 1990, MFCS.

[5]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

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

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

[8]  J. Håstad.  Computational limitations of small-depth circuits , 1987 .

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

[10]  A. Razborov.  Lower bounds on the size of bounded depth circuits over a complete basis with logical addition , 1987 .

[11]  Michael Ben-Or,et al.  A theorem on probabilistic constant depth Computations , 1984, STOC '84.

[12]  David A. Mix Barrington.  Quasipolynomial size circuit classes , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[13]  David A. Mix Barrington,et al.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1986, STOC '86.

[14]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[15]  David S. Johnson,et al.  A Catalog of Complexity Classes , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[16]  Andrew Chi-Chih Yao,et al.  ON ACC and threshold circuits , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[17]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

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

[19]  Ravi B. Boppana,et al.  The Complexity of Finite Functions , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[20]  Michael Sipser,et al.  The history and status of the P versus NP question , 1992, STOC '92.

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

[22]  M. A. Epstein.  Lower Bounds for the Size of Circuits of Bounded Depth in Basis F^; G , 1986 .