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Mining Circuit Lower Bound Proofs for Meta-algorithms

机译：Meta算法的采矿电路下限

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We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for easy Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an n-variate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2^n) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size 2^n/n. We get non-trivial compression for functions computable by AC0 circuits, (de Morgan) formulas, and (read-once) branching programs of the size for which the lower bounds for the corresponding circuit class are known. These compression algorithms rely on the structural characterizations of "easy" functions, which are useful both for proving circuit lower bounds and for designing "meta-algorithms" (such as Circuit-SAT). For (de Morgan) formulas, such structural characterization is provided by the "shrinkage under random restrictions" results cite {Sub61, Has98-shrinkage}, strengthened to the "high-probability" version by {San10, IMZ12, KR13}. We give a new, simple proof of the "high-probability" version of the shrinkage result for (de Morgan) formulas, with improved parameters. We use this shrinkage result to get both compression and #SAT algorithms for (de Morgan) formulas of size about n^2. We also use this shrinkage result to get an alternative proof of the recent result by Komargodski and Raz {KR13} of the average-case lower bound against small (de Morgan) formulas. Finally, we show that the existence of any non-trivial compression algorithm for a circuit class mathcal {C}subseteqP/poly would imply the circuit lower bound NEXPnotsubseteq mathcal {C}. This complements Williams's result {Wil10} that any non-trivial Circuit-SAT algorithm for a circuit class mathcal {C- would imply a super polynomial lower bound against mathcal {C} for a language in NEXP also proves such an implication in NEXP.
机译：我们表明，基于随机限制方法的电路下限校验产生了来自相应电路类的易于布尔函数的非普通压缩算法。压缩问题定义如下：给定由某些未知的小电路计算的N变形布尔函数F的真相表来自来自已知电路的一个未知的小电路，在确定性时间聚（2 ^ n）电路C（没有限制C）计算F的类型使得C的尺寸小于微小电路尺寸2 ^ n / n。对于AC0电路（de摩根）公式和（读一次）的尺寸的分支程序，我们获得了非琐碎的压缩，并且（读取一次）分支程序是已知的，对应电路类的下限。这些压缩算法依赖于“易于”功能的结构特性，这对于证明电路下限和设计“元算法”（例如电路SAT）。对于（de摩根）公式，这种结构表征由“随机限制下的收缩”结果引用{sub61，has98-shrinkage}，加强了{san10，imz12，kr13}的“高概率”版本。我们为（de Morgan）公式的收缩结果的“高概率”版本提供了一种新的，简单的证据，具有改进的参数。我们使用此收缩结果来获得（de摩根）大小的压缩和#sat算法约为n ^ 2。我们还使用这种收缩结果来获得Komargodski和RAZ {KR13}的最近结果的替代证据，其平均案例下限对阵小（de Morgan）公式。最后，我们表明，对于电路类Mathcal {C} subseteqp / poly的任何非智能压缩算法的存在意味着电路下方NexpnotsubseDeq Mathcal {C}。这补充了Williams的结果{Wil10}电路类Mathcal {C-的任何非平凡电路 - SAT算法{C-将暗示对Nexp中的语言的Mathcal {C}的超大多项式下限也证明了在Nexp中的含义。

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