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Diamonds Are Forever: The Variety DA
 Semigroups, Algorithms, Automata and Languages, Coimbra (Portugal) 2001
, 2002
"... We survey different characterizations (algebraic, combinatorial,... ..."
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Cited by 34 (5 self)
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We survey different characterizations (algebraic, combinatorial,...
Separating the Communication Complexities of MOD m and MOD p Circuits
 IN PROC. 33RD IEEE FOCS
, 1995
"... We prove in this paper that it is much harder to evaluate depth2, sizeN circuits with MOD m gates than with MOD p gates by kparty communication protocols: we show a kparty protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates ..."
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Cited by 21 (5 self)
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We prove in this paper that it is much harder to evaluate depth2, sizeN circuits with MOD m gates than with MOD p gates by kparty communication protocols: we show a kparty protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates needs\Omega\Gamma N) bits, where p denotes a prime, and m a composite, nonprime power number. As a corollary, for all m, we show a function, computable with a depth2 circuit with MODm gates, but not with any depth2 circuit with MOD p gates. Obviously, the kparty protocols are not weaker than the k 0 party protocols, for k 0 ? k. Our results imply that if there is a prime p between k and k 0 : k ! p k 0 , then there exists a function which can be computed by a k 0 party protocol with a constant number of communicated bits, while any kparty protocol needs linearly many bits of communication. This result gives a hierarchy theorem for multiparty protocols.
Harmonic Analysis, Real Approximation, and the Communication Complexity of Boolean Functions
 Algorithmica
, 1996
"... The 2party communication complexity of Boolean function f is known to be at least log rank(M f ), i.e. the logarithm of the rank of the communication matrix of f [19]. Lov'asz and Saks [17] asked whether the communication complexity of f can be bounded from above by ( log rank(M f )) c , for ..."
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Cited by 5 (0 self)
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The 2party communication complexity of Boolean function f is known to be at least log rank(M f ), i.e. the logarithm of the rank of the communication matrix of f [19]. Lov'asz and Saks [17] asked whether the communication complexity of f can be bounded from above by ( log rank(M f )) c , for some constant c. The question was answered affirmatively for a special class of functions f in [17], and Nisan and Wigderson proved nice results related to this problem [20], but for arbitrary f , it remained a difficult open problem. We prove here an analogous polylogarithmic upper bound in the stronger multiparty communication model of Chandra, Furst and Lipton [6], which, instead of the rank of the communication matrix, depends on the L 1 norm of function f , for arbitrary Boolean function f .
The NOF Multiparty Communication Complexity of Composed Functions
"... We study the kparty ‘number on the forehead ’ communication complexity of composed functions f ◦ g, where f: {0,1} n → {±1}, g: {0,1} k → {0,1} and for (x1,...,xk) ∈ ({0,1} n) k, f ◦g(x1,...,xk) = f (...,g(x1,i,...,xk,i),...). We show that there is an O(log 3 n) cost simultaneous protocol for SYM ..."
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Cited by 3 (0 self)
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We study the kparty ‘number on the forehead ’ communication complexity of composed functions f ◦ g, where f: {0,1} n → {±1}, g: {0,1} k → {0,1} and for (x1,...,xk) ∈ ({0,1} n) k, f ◦g(x1,...,xk) = f (...,g(x1,i,...,xk,i),...). We show that there is an O(log 3 n) cost simultaneous protocol for SYM ◦ g when k> 1 + logn, SYM is any symmetric function and g is any function. Previously, an efficient protocol was only known for SYM ◦ g when g is symmetric and “compressible”. We also get a nonsimultaneous protocol for SYM ◦ g of cost O(n/2 k · logn + k logn) for any k ≥ 2. In the setting of k ≤ 1 + logn, we study more closely functions of the form MAJORITY ◦g, MODm ◦g, and NOR ◦g, where the latter two are generalizations of the wellknown and studied functions Generalized Inner Product and Disjointness respectively. We characterize the communication complexity of these functions with respect to the choice of g. In doing so, we answer a question posed by Babai et al. (SIAM Journal on Computing, 33:137–166, 2004) and determine the communication complexity of MAJORITY ◦ QCSBk, where QCSBk is the “quadratic character of the sum of the bits” function.
Multiparty Communication Complexity of Finite Monoids
 In Birget et al
"... We study the relationship between the complexity of languages, in Yao's 2party communication game and its extensions, and the algebraic properties of finite monoids that can recognize them. ..."
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We study the relationship between the complexity of languages, in Yao's 2party communication game and its extensions, and the algebraic properties of finite monoids that can recognize them.
ABSTRACT: Separating the Communication Complexities of MOD m and MOD p Circuits
"... We prove in this paper that it is much harder to evaluate depth–2, size–N circuits with MOD m gates than with MOD p gates by k–party communication protocols: we show a k–party protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates need ..."
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We prove in this paper that it is much harder to evaluate depth–2, size–N circuits with MOD m gates than with MOD p gates by k–party communication protocols: we show a k–party protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates needs Ω(N) bits, where p denotes a prime, and m a composite, nonprime power number. As a corollary, for all m, we show a function, computable with a depth–2 circuit with MOD m gates, but not with any depth–2 circuit with MOD p gates. Obviously, the k–party protocols are not weaker than the k ′ –party protocols, for k ′> k. Our results imply that if there is a prime p between k and k ′ : k < p ≤ k ′ , then there exists a function which can be computed by a k ′ –party protocol with a constant number of communicated bits, while any k–party protocol needs linearly many bits of communication. This result gives a hierarchy theorem for multi–party protocols. 1 1.
MOD m Gates do not Help on the Ground Floor
, 1993
"... We prove that any depth3 circuit with MOD m gates of unbounded fanin on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we prove ..."
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We prove that any depth3 circuit with MOD m gates of unbounded fanin on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we prove that n(nlog n):::; logw log M, where w is the sum of the absolute values of the weights, and M is the maximum fanin of the AND gates on level 2. Setting all weights to 1, we got a tradeoff between the logarithms of the topfanin and the maximum fanin on level 2. In contrast, with n AND gates at the bottom and a Jingle MOD 2 gate at the top one can compute the inner product function. The lowerbound proof does not use any monotonicity or uniformity assumptions, and all of our gates have unbounded fanin. The key step in the proof is a random evaluation protocol of a circuit with MOD m gates.