UNSPECIFIED (1992) ON PROBABILISTIC ACC CIRCUITS WITH AN EXACT-THRESHOLD OUTPUT GATE. LECTURE NOTES IN COMPUTER SCIENCE, 650 . pp. 420-429. ISSN 0302-9743

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Abstract

Let SYM+ denote the class of Boolean functions computable by depth-two size-n(logO(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 f such that f(x1,...,x(n)) can be expressed as f(x1,...,x(n)) = h(n)(p(n)(x1,..., x(n))) for some polynomial p. over Z of degree log(O(1)n) and norm (the sum of the absolute values of its coefficients) n(logO(1)n) and some function h(n) : Z --> {0, 1}. Building on work of Yao [Yao90], Beigel and Tarui [BT91] showed that ACC subset-or-equal-to 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 f docs not belong to the class ACC remains an open problem in circuit complexity, and the result that ACC subset-or-equal-to SYM+ has raised the hope that we may be able to solve this problem by exploiting the characterization of SYM+ iii terms of polynomials, which are perhaps easier to analyze than circuits, and showing that f is-not-an-element-of 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 iu mind and explored. By a well-known connection [FSS84], we also obtain new results about some classes related to the polynomial-time hierarchy.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software
Journal or Publication Title:	LECTURE NOTES IN COMPUTER SCIENCE
Publisher:	SPRINGER VERLAG
ISSN:	0302-9743
Date:	1992
Volume:	650
Number of Pages:	10
Page Range:	pp. 420-429
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/21279

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