Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas
Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas
复制标题
有界深度多线性公式的次指数大小命中集
DOI：
10.1007/s00037-016-0131-1
复制
发表时间：
2014
影响因子：
1.4
通讯作者：
Ben lee Volk
中科院分区：
文献类型：
--
作者：
R. Oliveira;Amir Shpilka;Ben lee Volk
在本文中，我们使用Black-Box Pit和下边界之间的限制性深度算法进行了次数尺寸。 ）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {asysym} \ usepackage {amsfonts} \ usepackage {amsymb} \ setLength {\ oddSideMargin} { - 69pt} \ begin {document} $$ {（n^\ delta）} $ teen文档class [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasysym} \ usepackage {amsfonts} \ usepackage {amssymb} dsidemargin} { - 69pt} \ begin {document} $$ {\ left（\ tilde {o} \ left（n^{2/3 +） 2 \ delta/3} \ right）\ right）} $$ \ end {document}。 \ usepackage {wasysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {amsbsy} \ usepackage {mathrsfs} \ \ tilde {\ omega}（n^{1/2}））} $$一些明确的多项式。 \ usepackage {mathrsfs} \ usepackage {upgreek} \ setLength {\ oddsIdemargin} { - 69pt} \ begin {document {document} $ $ {（n^\ delta）} Expo〜N2/3+4δ/3 \ DocumentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {asysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} } \ begin {document} $$ {\ left（\ tilde {o} \ left（n^{2/3 + 4 \ delta/3} \ right）\ right）} $$ \ END {document {document}。 ω〜（n1/4））\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {amsbsy} \ usepackage {mathrsfs} \ \ tilde {\ omega}（n^{1/4}）} $$ \ end {document} depth-4多线性公式，用于某些显式多项式。 ] {minimal} \ usepackage {amsmath} \ usepackage {wasysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {mathrsfs} ，，，， \ times} $$ \ end {document}大门，其中所有门的所有门都具有相同的扇形，我们给出了一组大小（大致）expn1-Δ\ documentClass [12pt] {minimal} \ usepackage } \ usepackage {asysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {mathrsfs} \ usepackage { } $$ \ end {document}，对于常规深度-D多线性公式，最多为n和size exp（nδ）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasysym} {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {mathrsfs} \ usepackage {upgreek} \ setLength {\ oddSidemargin} { - 69pt} \ begin {document {document} $ $ {（n^\ delta）} 5d）\文档class [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasysym} \ usepackage {amsfonts} \ usepackage {amssymb} {\ oddsidemargin } { - 69pt} \ begin {document} $$ {\ delta = o（1/{\ sqrt {5}^d}）} $$ \ end {document}。 usepackage {amsmath} \ usepackage {asysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ begin {document} $$ {（\ tilde {\ omega}（n^{1/{\ sqrt {5}^d}}）} $$这些模型的界限是众所周知的，但这些界限都没有通过命中集来实现。本质上并依靠降低基础公式，首先到深度4公式，然后转到一个读取代数分支程序（从depth-3公式中，我们直接转到一开始的代数分支程序）。
In this paper, we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds, we obtain lower bounds for these models. For depth-3 multilinear formulas, of size exp(nδ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(n^\delta)}$$\end{document}, we give a hitting set of size expO~n2/3+2δ/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left(\tilde{O}\left(n^{2/3 + 2\delta/3}\right) \right)}$$\end{document}. This implies a lower bound of exp(Ω~(n1/2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\tilde{\Omega}(n^{1/2}))}$$\end{document} for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp(nδ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(n^\delta)}$$\end{document}, we give a hitting set of size expO~n2/3+4δ/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left(\tilde{O}\left(n^{2/3 + 4\delta/3}\right) \right)}$$\end{document}. This implies a lower bound of exp(Ω~(n1/4))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\tilde{\Omega}(n^{1/4}))}$$\end{document} for depth-4 multilinear formulas, for some explicit polynomial. A regular formula consists of alternating layers of +,×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${+,\times}$$\end{document} gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) expn1-δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left(n^{1- \delta}\right)}$$\end{document}, for regular depth-d multilinear formulas with formal degree at most n and size exp(nδ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(n^\delta)}$$\end{document}, where δ=O(1/5d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta = O(1/{\sqrt{5}^d})}$$\end{document}. This result implies a lower bound of roughly exp(Ω~(n1/5d))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\tilde{\Omega}(n^{1/{\sqrt{5}^d}}))}$$\end{document} for such formulas. We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known. Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas, we go straight to read-once algebraic branching programs).
一次读取的不经意算术分支程序之和的确定性身份测试
影响因子：
1.4
作者：
Rohit Gurjar;Arpita Korwar;Nitin Saxena;Thomas Thierauf
通讯作者：
Thomas Thierauf