A lower bound on the mod 6 degree of the OR function

Abstract.

We examine the computational power of modular counting, where the modulus m is not a prime power, in the setting of polynomials in Boolean variables over Z m . In particular, we say that a polynomial P weakly represents a Boolean function f (both have n variables) if for any inputs x and y in {0,1}n, we have \(P(x)\neq P(y)\) whenever \(f(x)\neq f(y)\). Barrington et al. (1994) investigated the minimal degree of a polynomial representing the OR function in this way, proving an upper bound of O(n 1/ r) (where r is the number of distinct primes dividing m) and a lower bound of \(\omega (1)\). Here, we show a lower bound of \(\Omega (\log n)\) when m is a product of two primes and \(\Omega ((\log n)^{1/(r-1)})\) in general. While many lower bounds are known for a much stronger form of representation of a function by a polynomial (Barrington et al. 1994, Tsai 1996), very little is known using this liberal (and, we argue, more natural) definition. While the degree is known to be \(\Omega (\log n)\) for the generalized inner product because of its high communication complexity (Grolmusz 1995), our bound is the best known for any function of low communication complexity and any modulus not a prime power.

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received 29 September 1994

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Tardos, G., Mix Barrington, D. A lower bound on the mod 6 degree of the OR function. Comput. complex. 7, 99–108 (1998). https://doi.org/10.1007/PL00001597

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  • Key words. Circuit complexity; modular counting; Boolean function complexity.