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2011 26th Annual IEEE Conference on Computational Complexity
Linear Systems over Finite Abelian Groups
San Jose, California USA
June 08-June 11
ISBN: 978-0-7695-4411-3
We consider a system of linear constraints over any finite Abelian group G of the following form: l_i(x_1,...,x_n) = l_{i,1}x_1 + ... + l_{i,n}x_n in A_i for i=1,...,N and each A_i is subset of G and l_{i,j} is an element of G and x_i's are Boolean variables. Our main result shows that the subset of the Boolean cube that satisfies these constraints has exponentially small correlation with the MOD-q boolean function, when the order of G and q are co-prime numbers. Our work extends the recent result of Chattopadhyay and Wigderson (FOCS'09) who obtain such a correlation bound for linear systems over cyclic groups whose order is a product of two distinct primes or has at most one prime factor. Our result also immediately yields the first exponential bounds on the size of boolean depth-four circuits of the form MAJ of AND of ANY_{O(1)} of MOD-m for computing the MOD-q function, when m,q are co-prime. No super-polynomial lower bounds were known for such circuits for computing any explicit function. This completely solves an open problem posed by Beigel and Maciel (Complexity'97).
Index Terms:
lower bounds, boolean circuit complexity, modular gates, composite moduli, exponential sums
Arkadev Chattopadhyay, Shachar Lovett, "Linear Systems over Finite Abelian Groups," ccc, pp.300-308, 2011 26th Annual IEEE Conference on Computational Complexity, 2011
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