Name
Abstract | ||
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We study approximation of Boolean functions by low-degree polynomials over the ring Z/2^kZ. More precisely, given a Boolean function F:{0,1}^n -u003e {0,1}, define its k-lift to be F_k:{0,1}^n -u003e {0,2^(k-1)} by F_k(x) = 2^(k-F(x)) (mod 2^k). We consider the fractional agreement (which we refer to as gamma_{d,k}(F)) of F_k with degree d polynomials from Z/2^kZ[x_1,..,x_n].Our results are the following:* Increasing k can help: We observe that as k increases, gamma_{d,k}(F) cannot decrease. We give two kinds of examples where gamma_{d,k}(F) actually increases. The first is an infinite family of functions F such that gamma_{2d,2}(F) - gamma_{3d-1,1}(F) u003e= Omega(1). The second is an infinite family of functions F such that gamma_{d,1}(F) = 1/2 + Omega(1).* Increasing k doesnu0027t always help: Adapting a proof of Green [Comput. Complexity, 9(1):16--38, 2000], we show that irrespective of the value of k, the Majority function Maj_n satisfies gamma_{d,k}(Maj_n) u003c= 1/2+ O(d)/sqrt{n}. In other words, polynomials over Z/2^kZ for large k do not approximate the majority function any better than polynomials over Z/2Z.We observe that the model we study subsumes the model of non-classical polynomials, in the sense that proving bounds in our model implies bounds on the agreement of non-classical polynomials with Boolean functions. In particular, our results answer questions raised by Bhowmick and Lovett [In Proc. 30th Computational Complexity Conf., pages 72-87, 2015] that ask whether non-classical polynomials approximate Boolean functions better than classical polynomials of the same degree. |
Year | Venue | Field |
|---|---|---|
2017 | STACS | Boolean function,Discrete mathematics,Polynomial approximations,Combinatorics,Polynomial,Omega,Mathematics,Majority function,Computational complexity theory |
DocType | Citations | PageRank |
Conference | 0 | 0.34 |
References | Authors | |
0 | 3 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Abhishek Bhrushundi | 1 | 5 | 2.49 |
| Prahladh Harsha | 2 | 371 | 32.06 |
| Srikanth Srinivasan | 3 | 2 | 3.75 |