Symmetric polynomials over Zm and simultaneous communication protocols

摘要

We study the problem of representing symmetric Boolean functions as symmetric polynomials over Zm. We prove an equivalence between representations of Boolean functions by symmetric polynomials and simultaneous communication protocols. We show that computing a function f on 0–1 inputs with a polynomial of degree d modulo pq is equivalent to a two player simultaneous protocol for computing f where one player is given the first ⌈logpd⌉ digits of the weight in base p and the other is given the first ⌈logqd⌉ digits of the weight in base q.This equivalence allows us to show degree lower bounds by using techniques from communication complexity. For example, we show lower bounds of Ω(n) on symmetric polynomials weakly representing classes of Modr and Threshold functions. Previously the best known lower bound for such representations of any function modulo pq was Ω(n12) [D.A. Barrington, R. Beigel, S. Rudich, Representing Boolean functions as polynomials modulo composite numbers, Comput. Complexity 4 (1994) 367–382]. The equivalence also allows us to use results from number theory to prove upper bounds for Threshold-k functions. We show that proving bounds on the degree of symmetric polynomials strongly representing the Threshold-k function is equivalent to counting the number of solutions to certain Diophantine equations. We use this to show an upper bound of ɛO(nk)12+ɛ for Threshold-k assuming the abc-conjecture. We show the same bound unconditionally for k constant. Prior to this, non-trivial upper bounds were known only for the OR function [D.A. Barrington, R. Beigel, S. Rudich, Representing Boolean functions as polynomials modulo composite numbers, Comput. Complexity 4 (1994) 367–382]. We show an almost tight lower bound of Ω(nk)12, improving the previously known bound of Ω(max(k,n)) [S.-C. Tsai, Lower bounds on representing Boolean functions as polynomials in Zm, SIAM J. Discrete Math. 9 (1996) 55–62].