We prove that a sufficiently strong parallel repetition theorem for a special case of multiplayer (multiprover) games implies super-linear lower bounds for multi-tape Turing machines with advice. To the best of our knowledge, this is the first connection between parallel repetition and lower bounds for time complexity and the first major potential implication of a parallel repetition theorem with more than two players. Along the way to proving this result, we define and initiate a study of block rigidity, extending Valiant's notion of rigidity. While rigidity was originally defined for matrices, or, equivalently, for (multi-output) linear functions, we extend and study both rigidity and block rigidity for general (multi-output) functions. Using techniques of Paul, Pippenger, Szemer\'edi and Trotter, we show that a block-rigid function cannot be computed by multi-tape Turing machines that run in linear (or slightly super-linear) time, even in the non-uniform setting, where the machine gets an arbitrary advice tape. We then describe a class of multiplayer games, such that, a sufficiently strong parallel repetition theorem for that class of games implies an explicit block-rigid function. The games in that class have the following property that may be of independent interest: for every random string for the verifier (which, in particular, determines the vector of queries to the players), there is a unique correct answer for each of the players, and the verifier accepts if and only if all answers are correct. We refer to such games as independent games. The theorem that we need is that parallel repetition reduces the value of games in this class from $v$ to $v^{\Omega(n)}$, where $n$ is the number of repetitions. As another application of block rigidity, we show conditional size-depth tradeoffs for boolean circuits, where the gates compute arbitrary functions over large sets.

中文翻译：

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块刚性：强大的多人并行重复意味着图灵机的超线性下界

我们证明，对于多人（multiprover）游戏的特殊情况，足够强的并行重复定理意味着具有建议的多磁带图灵机的超线性下界。据我们所知，这是并行重复与时间复杂度下界之间的第一个联系，也是多于两个参与者的并行重复定理的第一个主要潜在含义。在证明这一结果的过程中，我们定义并启动了对块刚性的研究，扩展了 Valiant 的刚性概念。虽然刚性最初是为矩阵定义的，或者等效地为（多输出）线性函数定义，但我们扩展和研究了一般（多输出）函数的刚性和块刚性。使用 Paul、Pippenger、Szemer\'edi 和 Trotter 的技术，我们表明，即使在机器获得任意建议磁带的非均匀设置中，也无法通过在线性（或略微超线性）时间内运行的多带图灵机计算块刚性函数。然后，我们描述了一类多人游戏，这样，该类游戏的足够强的并行重复定理意味着明确的块刚性函数。该类中的游戏具有以下可能独立感兴趣的属性：对于验证者的每个随机字符串（特别是确定对玩家的查询向量），每个玩家都有唯一的正确答案，并且验证者接受当且仅当所有答案都正确时。我们将此类游戏称为独立游戏。我们需要的定理是并行重复将此类游戏的价值从 $v$ 减少到 $v^{\Omega(n)}$，其中 $n$ 是重复次数。作为块刚性的另一个应用，我们展示了布尔电路的条件大小深度权衡，其中门在大集合上计算任意函数。