A quorum system is a collection of sets (quorums) every two of which intersect. Quorum systems have been used for many applications in the area of distributed systems, including mutual exclusion, data replication and dissemination of information Given a strategy to pick quorums, the load L(S) is the minimal access probability of the busiest element, minimizing over the strategies. The capacity Cap(S) is the highest quorum accesses rate that S can handle, so Cap(S) = 1=L(S).

We study the Fourier representation of Boolean functions. The goal is to look at the frequency domain of Boolean functions to get complexity properties. Preliminary results indicate that this might be fruitful. In addition to presenting new results, we review some of the most significant work on the subject. Istituto di Matematica Computazionale, Consiglio Nazionale delle Ricerche, and Dipartimento di Informatica, Pisa (Italy). y Istituto di Matematica Computazionale, Consiglio Nazionale delle Ricerche, Pisa (Italy). e-mail: codenotti@iei.pi.cnr.it z Department of Computer Science, The University of Chicago. Portions of this work were done while visiting IEI-CNR in Pisa, sponsored by a grant from CNR. 1 Introduction The Fourier transform of a Boolean function is an invertible linear mapping of the values of the function onto a set of coefficients, known as Fourier coefficients. This transformation is such that the Fourier coefficients contain information about the regularitie...

Let f be a de Morgan read-once function of n variables. Let f " be the random restriction obtained by independently assigning to each variable of f , the value 0 with probability (1 \Gamma ")=2, the value 1 with the same probability, and leaving it unassigned with probability ". We show that f " depends, on the average, on only O(" ff n + "n 1=ff ) variables, where ff = log p 5\Gamma1 2 ' 3:27. This result is asymptotically the tightest possible. It improves a similar result obtained recently by Hastad, Razborov and Yao. 1 Introduction Obtaining non-trivial lower bounds on the complexity of Boolean functions is currently a very difficult task. Only a handful of methods yielding such lower bounds are currently known and even they work only in suitably restricted models. The current state of affairs in this respect is summerized in the books of Dunne [5] and Wegener [15] and the survey paper of Boppana and Sipser [2]. Many of the currently known methods for obtaining complexity l...

In this note we consider the problem of computing threshold functions using directed monotone contact networks. We give constructions of monotone contact networks of size (k − 1)(n − k + 2) ⌈log(n − k + 2) ⌉ computing T n k, for 2 ≤ k ≤ n − 1. Our upper bound is close to the Ω(kn log(n/(k − 1))) lower bound for small thresholds and the k(n − k + 1) lower bound for large thresholds. Our networks are described explicitly; we do not use probabilistic existence arguments.

Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone readonce formulae of size O(n ff+2 ) (where ff = log p 5\Gamma1 2 ' 3:27) that amplify (/ \Gamma 1 n ; / + 1 n ) (where / = p 5\Gamma1 2 ' 0:62) to (2 \Gamman ; 1 \Gamma 2 \Gamman ) and deduced as a consequence the existence of monotone formulae of the same size that compute the majority of n bits. Boppana has shown that any monotone read-once formula that amplifies (p \Gamma 1 n ; p + 1 n ) to ( 1 4 ; 3 4 ) (where 0 ! p ! 1 is constant) has size\Omega\Gamma n ff ) and that any monotone, not necessarily read-once, contact network (and in particular any monotone formula) that amplifies ( 1 4 ; 3 4 ) to (2 \Gamman ; 1 \Gamma 2 \Gamman ) has size\Omega\Gamma n 2 ). We extend Boppana's results in two ways. We first show that his two lower bounds hold for general read-onc...

We show that every monotone formula that computes the threshold function THk,n, 2 ≤ k ≤ n 2, has size at least � � k n 2 n log ( k−1). The same lower bound is shown to hold in the stronger monotone directed contact networks model. 1

Abstract We prove that the size of any read-once de Morgan formula reduces in average by a factor of at least p ff\Gamma o(1) when all but a fraction p of the input variables are randomly assigned to f0; 1g (here ff *) 1 = log2( p 5 \Gamma 1) ss 3:27). This resolves in the

. Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone read-once formulas of size O(n #+2 ) (where # = log # 5-1 2 # 3.27) that amplify (#- 1 n ,#+ 1 n ) (where # =( # 5-1)/2 # 0.62) to (2 -n , 1 - 2 -n ) and deduced as a consequence the existence of monotone formulas of the same size that compute the majority of n bits. Boppana has shown that any monotone read-once formula that amplifies (p - 1 n ,p+ 1 n )to( 1 4 , 3 4 ) (where 0 <p<1 is constant) has size ## n # ) and that any monotone, not necessarily read-once, contact network (and in particular any monotone formula) that amplifies ( 1 4 , 3 4 )to(2 -n ,1-2 -n ) has size## n 2 ). We extend Boppana's results in two ways. We first show that his two lower bounds hold for general read-once formulas, not necessarily monotone, that may even include exclusive-or gates. We...