On probabilistic ACC circuits with an exact-threshold output gate

Abstract

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 log^O(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,⋯, x_n) =h _n(p_n(x₁,⋯, x _n )) for some polynomial p _n over Z of degree log^O(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 MOD_m gates for some fixed m.

In this paper, we consider augmenting the power of ACC circuits by allowing randomness and allowing an exact-threshold gate as the output gate (an exact-threshold gate outputs 1 if exactly k of its inputs are 1, where k is a parameter; it outputs 0 otherwise), and show that every Boolean function computed by this kind of augmented ACC circuits is still in SYM⁺.

Showing that some “natural” function h does not belong to the class ACC remains an open problem in circuit complexity, and the result that ACC $\subseteq$ SYM⁺ has raised the hope that we may be able to solve this problem by exploiting the characterization of SYM⁺ in terms of polynomials, which are perhaps easier to analyze than circuits, and showing that h ∋ SYM⁺. Our new result and proof techniques suggest that the possibility that SYM⁺ contains even more Boolean functions than we currently know should also be kept in mind and explored.

By a well-known connection [FSS84], we also obtain new results about some classes related to the polynomial-time hierarchy.

Supported in part by NSP grant CCR-8958528.

Supported in part by the ESPRIT II BRA Programme of the EC under contract # 7141 (ALCOM II). Part of the work was doue while the author was a student at University of Rochester, and was supported in part by NSF grant CDA-8822724.