TY - JOUR

T1 - An arithmetic model of computation equivalent to threshold circuits

AU - Boyar, Joan

AU - Frandsen, Gudmund

AU - Sturtivant, Carl

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1992/2/17

Y1 - 1992/2/17

N2 - We define a new structured and general model of computation: circuits using arbitrary fan-in arithmetic gates over the characteristic-two finite fields (F2n). These circuits have only one input and one output. We show how they correspond naturally to boolean computations with n inputs and n outputs. We show that if circuit sizes are polynomially related, then the arithmetic circuit depth and the threshold circuit depth to compute a given function differ by at most a constant factor. We use threshold functions with arbitrary weights; however, we show that when compared to the usual threshold model, the depth measure of this generalised model differs only by at most a constant factor (at polynomial size). The fan-in of our arithmetic model is also unbounded in the most generous sense: circuit size is measured as the number of Σ and Π-gates; there is no bound on the number of "wires". We show that these results are provable for any reasonable correspondence between strings of n-bits and elements of F2n. And we find two such distinct characterizations. Thus, we show that arbitrary fan-in arithmetic computations over F2n constitute a precise abstraction of Boolean threshold computations with the pleasant property that various algebraic laws have been recovered.

AB - We define a new structured and general model of computation: circuits using arbitrary fan-in arithmetic gates over the characteristic-two finite fields (F2n). These circuits have only one input and one output. We show how they correspond naturally to boolean computations with n inputs and n outputs. We show that if circuit sizes are polynomially related, then the arithmetic circuit depth and the threshold circuit depth to compute a given function differ by at most a constant factor. We use threshold functions with arbitrary weights; however, we show that when compared to the usual threshold model, the depth measure of this generalised model differs only by at most a constant factor (at polynomial size). The fan-in of our arithmetic model is also unbounded in the most generous sense: circuit size is measured as the number of Σ and Π-gates; there is no bound on the number of "wires". We show that these results are provable for any reasonable correspondence between strings of n-bits and elements of F2n. And we find two such distinct characterizations. Thus, we show that arbitrary fan-in arithmetic computations over F2n constitute a precise abstraction of Boolean threshold computations with the pleasant property that various algebraic laws have been recovered.

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U2 - 10.1016/0304-3975(92)90335-D

DO - 10.1016/0304-3975(92)90335-D

M3 - Article

AN - SCOPUS:0026813292

VL - 93

SP - 303

EP - 319

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 2

ER -