A Fixed-Depth Size-Hierarchy Theorem for AC$^0[\\oplus]$ via the Coin\n Problem Article Swipe
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· 2018
· Open Access
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· DOI: https://doi.org/10.48550/arxiv.1809.04092
· OA: W2966418478
We prove the first Fixed-depth Size-hierarchy Theorem for uniform\nAC$^0[\\oplus]$ circuits; in particular, for fixed $d$, the class\n$\\mathcal{C}_{d,k}$ of uniform AC$^0[\\oplus]$ formulas of depth $d$ and size\n$n^k$ form an infinite hierarchy. For this, we find the first class of explicit\nfunctions giving (up to polynomial factor) matching upper and lower bounds for\nAC$^0[\\oplus]$ formulas, derived from the $\\delta$-Coin Problem, the\ncomputational problem of distinguishing between coins that are heads with\nprobability $(1+\\delta)/2$ or $(1-\\delta)/2,$ where $\\delta$ is a parameter\ngoing to $0$. We study this problem's complexity and make progress on both\nupper bounds and lower bounds.\n Upper bounds. We find explicit monotone AC$^0$ formulas solving the\n$\\delta$-coin problem, having depth $d$, size $\\exp(O(d(1/\\delta)^{1/(d-1)}))$,\nand sample complexity poly$(1/\\delta)$, for constant $d\\ge2$. This matches\nprevious upper bounds of O'Donnell and Wimmer (ICALP 2007) and Amano (ICALP\n2009) in terms of size and improves the sample complexity.\n Lower bounds. The upper bounds are nearly tight even for the stronger model\nof AC$^0[\\oplus]$ formulas (which allow NOT and Parity gates): any\nAC$^0[\\oplus]$ formula solving the $\\delta$-coin problem must have size\n$\\exp(\\Omega(d(1/\\delta)^{1/(d-1)})).$ This strengthens a result of Cohen,\nGanor and Raz (APPROX-RANDOM 2014), who prove a similar result for AC$^0$, and\na result of Shaltiel and Viola (SICOMP 2010), who give a superpolynomially\nweaker (still exponential) lower bound.\n The upper bound is a derandomization involving a use of Janson's inequality\n(as far as we know, the first such use of the inequality) and classical\ncombinatorial designs. For the lower bound, we prove an optimal (up to constant\nfactor) degree lower bound for multivariate polynomials over $\\mathbb{F}_2$\nsolving the $\\delta$-coin problem, which may be of independent interest.\n
