Say that f : {0, 1}n → {0, 1} 2-approximates g : {0, 1}n → {0, 1} if the functions disagree on at most an 2 fraction of points. This paper contains two results about approximation by DNF and other small-depth circuits: (1) For every constant 0 < 2 < 1/2 there is a DNF of size 2 √ n) that 2-approximates the Majority function on n bits, and this is optimal up to the constant in the exponent. (2) ...

In this work, we consider a public facility allocation problem decided through a voting process under the majority rule. A location of the public facility is a majority rule winner if there is no other location in the network where more than half of the voters would have been closer to than the majority rule winner. We develop fast algorithms for interesting cases with nice combinatorial struct...

We study the circuit complexity of boolean functions in a certain infinite basis. The basis consists of all functions that take value 1 on antichains over the boolean cube. We prove that the circuit complexity of the parity function and the majority function of n variables in this basis is b 2 c and ⌊ n 2 ⌋ +1 respectively. We show that the asymptotic of the maximum complexity of n-variable boo...

in this article the historical expression by martyr zaidebn e-ali on negation of forbidden majority has been described. also the majority and minority issue in the quran has been compared with current majority and minority in the political sciences. it seems that in the quran the majority which does not apply reason and does not know is the majority of opponents of prophets. thus, the majority ...

In a previous paper, the authors defined a new class ofmultiple-valued logic functions, called multiple-valued majority func-tions. This correspondence clarifies the distinction of multiple-valuedmajority functions from multiple-valued threshold functions through thedifference between a number function and an inner product of an inputvector and a weight vector. Zndcx Terns-I...

We show that Nechiporuk’s method [26] for proving lower bounds for Boolean formulas can be extended to the quantum case. This leads to an Ω(n/ log n) lower bound for quantum formulas computing an explicit function. The only known previous explicit lower bound for quantum formulas [27] states that the majority function does not have a linear–size quantum formula. We also show that quantum formul...

We study binary linear codes constructed from fifty-four Hadamard 2-(71, 35, 17) designs. The constructed codes are self-dual, doubly-even and self-complementary. Since most of these codes have large automorphism groups, they are suitable for permutation decoding. Therefore we study PD-sets of the obtained codes. We also discuss the errorcorrecting capability of the obtained codes by majority l...

This paper presents the comparative results of applying the same genetic algorithm (GA) for the evolution of both synchronous and randomly updated asynchronous cellular automata (CA) for the computationally emergent task of density classification. The present results indicate not only that these asynchronous CA evolve more quickly and consistently than their synchronous counterparts, but also t...

Dendrons and their products admit a natural, continuous median operator. We prove that there exists a two-dimensional metric continuum with a continuous median operator, for which there is no median-preserving embedding in a product of finitely many dendrons. Our method involves ideas and results concerning graph colouring and abstract convexity. The main result answers a question in [16] negat...

In this paper, we characterize self-dual elements of a free distributive lattice with n generators by means of a median operation on this lattice. Hence, we obtain a constructive characterization of Sperner's families, for which H is equal to Tr H. We construct some infinite sequences of these families. These results generalize those of N. M. Rivi6re (J. Combinatorial Theory 5 (1968), 229-234) ...