On Threshold Circuits and Polynomial Computation

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)

Stephen R. Tate, Professor and Department Head (Creator)

Institution

The University of North Carolina at Greensboro (UNCG )

Web Site: http://library.uncg.edu/

Abstract: A Threshold Circuit consists of an acyclic digraph of unbounded fanin, where each node computes a threshold function or its negation. This paper investigates the computational power of Threshold Circuits. A surprising relationship is uncovered between Threshold Circuits and another class of unbounded fanin circuits which are denoted Finite Field Z_P(n) Circuits, where each node computes either multiple sums or products of integers modulo a prime P(n). In particular, it is proved that all functions computed by Threshold Circuits of size S(n) ≥ n and depth D(n) can also be computed by Z_P(n) Circuits of size O(S(n) log S(n) + nP(n) log P(n)) and depth O(D(n)). Furthermore, it is shown that all functions computed by Z_P(n) Circuits of size S(n) and depth D(n) can be computed by Threshold Circuits of size O((1/∈²)(S(n) log P(n))^1+∈) and depth O((1/∈⁵)D(n)). These are the main results of this paper.

There are many useful and quite surprising consequences of this result. For example, an integer reciprocal can be computed in size n^O(1)M and depth O(1). More generally, any analytic function with a convergent rational polynomial power series (such as sine, cosine, exponentiation, square root, and logarithm) can be computed within accuracy 2^-nc , for any constant c, by Threshold Circuits of polynomial size and constant depth. In addition, integer and polynomial division, FFf, polynomial interpolation, Chinese Remaindering, all the elementary symmetric functions, banded matrix inverse, and triangular Toeplitz matrix inverse can be exactly computed by Threshold Circuits of polynomial size and constant depth. All these results and simulations hold for polytime uniform circuits. This paper also gives a corresponding simulation oflogspace uniform Z_P(n) Circuits by logspace uniform Threshold Circuits requiring an additional multiplying factor of O(log log log P(n) depth.

Finally, purely algebraic methods forlowerbounds for ZP(n) Circuits are developed. Using degree arguments, a Depth Hierarchy Theorem for Z_P(n) Circuits is proved: for any S(n) ≥ n, D(n) = O(S(n)^c') for some constant c' < 1, and prime P(n) where 6(S(n)/D(n))^D(n) < P(n) c' 2ⁿ, there exist explicitly constructible functions computable by Z_P(n) Circuits of size S(n) and depth D(n), but provably not computable by Z_P(n) Circuits of size S(n)^c and depth oD(n)) for any constant c ≥ 1.

On Threshold Circuits and Polynomial Computation
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Additional Information

Publication

SIAM Journal on Computing Vol. 21, No.5, pp. 896-908, October 1992

Language: English

Date: 1992

Keywords

Circuit complexity, Threshold circuits, Finite field circuits

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