TY - JOUR

T1 - Sizeenergy tradeoffs for unate circuits computing symmetric Boolean functions

AU - Uchizawa, Kei

AU - Takimoto, Eiji

AU - Nishizeki, Takao

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/3/4

Y1 - 2011/3/4

N2 - A unate gate is a logical gate computing a unate Boolean function, which is monotone in each variable. Examples of unate gates are AND gates, OR gates, NOT gates, threshold gates, etc. A unate circuit C is a combinatorial logic circuit consisting of unate gates. Let f be a symmetric Boolean function of n variables, such as the Parity function, MOD function, and Majority function. Let m0 and m1 be the maximum numbers of consecutive 0's and consecutive 1's in the value vector of f, respectively, and let l=minm0,m1 and m=maxm0,m1. Let C be a unate circuit computing f. Let s be the size of the circuit C, that is, C consists of s unate gates. Let e be the energy of C, that is, e is the maximum number of gates outputting "1" over all inputs to C. In this paper, we show that there is a tradeoff between the size s and the energy e of C. More precisely, we show that (n+1-l)m≤se. We also present lower bounds on the size s of C represented in terms of n, l and m. Our tradeoff immediately implies that logn≤elogs for every unate circuit C computing the Parity function of n variables.

AB - A unate gate is a logical gate computing a unate Boolean function, which is monotone in each variable. Examples of unate gates are AND gates, OR gates, NOT gates, threshold gates, etc. A unate circuit C is a combinatorial logic circuit consisting of unate gates. Let f be a symmetric Boolean function of n variables, such as the Parity function, MOD function, and Majority function. Let m0 and m1 be the maximum numbers of consecutive 0's and consecutive 1's in the value vector of f, respectively, and let l=minm0,m1 and m=maxm0,m1. Let C be a unate circuit computing f. Let s be the size of the circuit C, that is, C consists of s unate gates. Let e be the energy of C, that is, e is the maximum number of gates outputting "1" over all inputs to C. In this paper, we show that there is a tradeoff between the size s and the energy e of C. More precisely, we show that (n+1-l)m≤se. We also present lower bounds on the size s of C represented in terms of n, l and m. Our tradeoff immediately implies that logn≤elogs for every unate circuit C computing the Parity function of n variables.

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U2 - 10.1016/j.tcs.2010.11.022

DO - 10.1016/j.tcs.2010.11.022

M3 - Article

AN - SCOPUS:79151486492

VL - 412

SP - 773

EP - 782

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 8-10

ER -