On Threshold Circuits and Polynomial Computation

Abstract

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 Sn n and depth Dn can also be computed by Z P n Circuits of size OS nlog Sn + nP n log P n and depth ODn. Furthermore, it is shown that all functions computed by Z P n Circuits of size S n and depth Dn can be computed by Threshold Circuits of size O 1 2 Sn log P n 1+ and depth O 1 5 Dn.These are the main results of this paper.There are many useful and quite surprising consequences of this result. For example, integer reciprocal can be computed in size n O1 and depth O1. More generally, a n y analytic function with a convergent rational polynomial power series such as sine, cosine, exponentiation, square root, and logarithm can be computed within accuracy 2 ,n c , for any constant c, b y Threshold Circuits of polynomial size and constant depth. In addition, integer and polynomial division, FFT, polynomial interpolation, Chinese Remaindering, all the elementarysymmetric 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 of logspace uniform Z P n Circuits by logspace uniform Threshold Circuits requiring an additional multiplying factor of Olog log log P n depth.Finally, purely algebraic methods for lower bounds for Z P n Circuits are developed. Using degree arguments, a Depth Hierarchy Theorem for Z P n Circuits is proved: for any S n n, Dn = O S n c 0 for some constant c 0 1, and prime P n where 6Sn=Dn Dn P n 2 n , there exists explicitly constructible functions computable by Z P n Circuits of size Sn and depth Dn, but provably not computable by Z P n Circuits of size Sn c and depth oDn for any constant c 1:

Pseudorandom Functions and Lattices

“…The subsequent rounding step simply amounts to dropping some of the least-significant digits if q and p are both powers of the same small base, or more generally, multiplying by p/q (under suitable precision) and truncating. Both operations can be performed in TC 0 , for any q = 2 poly(n) [RT92].…”

“…Generally speaking, matrix multi-product does not appear to be computable in TC 0 (if it were, then TC 0 would equal NC 1 [MP00]). However, in our case the matrices are known in advance (the variable input is the subset, so it may be possible to reduce the depth of the computation via preprocessing, using ideas from [RT92]. As described in Section 4.3, both binary matrix product and rounding can be implemented with simple depth-2 arithmetic circuits, and hence in TC 0 , so at worst F can be computed in TC 1 by computing the subset product in a tree-like fashion, followed by a final rounding step.…”

Fast arithmetic computing with neural networks

“…This would justify research in device technology to investigate the feasibility of building such elements with small cost. In fact, there has already been some research progress in this area as mentioned in [9].…”

Optimal-depth threshold circuits for multiplication and related problems

“…Threshold circuits constitute a powerful computational model for arithmetic and other computations [ 12,13,16,201. A linear threshold function is defined as a Boolean %YQYX) > = 1 if F(X) > 0; 0 if F(X) < 0, function awhere X = (~1, .…”