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We prove a lower bound of 5n − o(n) for the circuit complexity of an explicit (constructible in deterministic polynomial time) Boolean function , over the basis U2. That is, we obtain a lower bound of 5n − o(n) for the number of {and, or} gates needed to compute a certain Boolean function, over the basis {and, or, not} (where the not gates are not counted).… (More)

The minimum number of NOT gates in a Boolean circuit computing a Boolean function f is called the inversion complexity of f. In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that ⌈log 2 (n + 1)⌉ NOT gates are sufficient to compute any Boolean function on n variables. In this paper, we consider circuits… (More)

We give improved lower bounds for the size of negation-limited circuits computing Parity and for the size of negation-limited inverters. An inverter is a circuit with inputs x1,. .. , xn and outputs ¬x1,. .. , ¬xn. We show that (1) For n = 2 r − 1, circuits computing Parity with r − 1 NOT gates have size at least 6n − log 2 (n + 1) − O(1) and (2) For n = 2… (More)

The fundamental symmetric functions are EX n k (equal to 1 if the sum of n input bits is exactly k), TH n k (the sum is at least k), and MOD n m,r (the sum is congruent to r modulo m). It is well known that all these functions and in fact any symmetric Boolean function have linear circuit size. Simple counting shows that the circuit complexity of computing… (More)

Arpe and Manthey [Algorithmica'09] recently studied the minimum AND-circuit problem, which is a circuit minimization problem, and showed some results including approximation algorithms, APX-hardness and fixed parameter tractability of the problem. In this note, we show that algorithms via the k-set cover problem yield improved approximation ratios for the… (More)

In this note, we consider the minimum number of NOT operators in a Boolean formula representing a Boolean function. In circuit complexity theory, the minimum number of NOT gates in a Boolean circuit computing a Boolean function f is called the inversion complexity of f. In 1958, Markov determined the inversion complexity of every Boolean function and… (More)

Let C be a logic circuit consisting of s gates g 1 , g 2 ,. .. , g s , then the output pattern of C for an input x ∈ {0, 1} n is defined to be a vector 2 → {0, 1}, we define an f-circuit as a logic circuit where every gate computes f , and investigate computational complexity of the following counting problem: Given an f-circuit C, how many output patterns… (More)