# 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*(x_{1},⋯ *,x*_{ n }) can be expressed as *h*(x_{1},⋯, x_{n}) =h _{n}(p_{n}(x_{1},⋯, *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.

## Keywords

Boolean Function IEEE Computer Society Chinese Remainder Theorem Count Gate Output Gate## Preview

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