Lower bounds, meta-algorithms, and pseudorandomness

下界、元算法和伪随机性

• 批准号: RGPIN-2019-05543
• 负责人: Kabanets, Valentine
• 金额: $ 2.99万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738361
• 关键词: Lower; bounds; meta; algorithms; pseudorandomness

Which computational problems are ``easy''  and which ones are ``hard''? Why is it so difficult to prove that our candidate hard computational problems are actually hard? Can we extract efficient algorithms from hard computational problems (from the proofs establishing their hardness)? Conversely, can we get new lower bounds by designing more efficient algorithms? What is the role of (pseudo-) randomness in all of this? Lower bounds (proving limitations of computers) predate the first computers! In 1930s, Alan Turing invented the notion of a universal computer, while proving that no such computer can solve a certain problem in logic! Despite many efficient algorithms that have been discovered over the years, there are still many natural problems (e.g., NP-complete problems) whose complexity is not known.  For the non-uniform computation model of boolean circuits, there are also many candidate hard problems, but no lower bound proofs for general enough circuit classes. The "natural proofs barrier" of Razborov and Rudich argues that certain constructive proof arguments cannot prove strong circuit lower bounds, under plausible cryptographic assumptions. At the heart of their argument is a certain average-case version of the Minimum Circuit Size Problem (MCSP) that asks to decide for a given truth table of a boolean function f, if f has circuits of size at most a given parameter s. MCSP is an important example of a meta-computational problem: the problem whose inputs are instances of other computational problems. Other examples of meta-problems include SATISFIABILITY (SAT), computational learning, and derandomization of randomized algorithms. There are many examples where circuit lower bounds lead to new algorithms, and vice versa. In particular, there are known connections between circuit lower bounds and derandomization, between constructive (in the sense of Razborov and Rudich) proofs of circuit lower bounds and learning algorithms, and between circuit lower bounds and non-trivial SAT algorithms. An important role in all these connections is played by pseudorandomness: the study of "random-like" objects that are not truly random, but "appear random" to certain computationally bounded observers. The main objective of the proposed research is to gain better understanding of the power and limitations of efficient computation by proving new lower bounds and designing new algorithms, exploring the apparent intimate connections between the two. The concepts and techniques from the pseudorandomness theory will be the main tools in this study. In particular, we will study the complexity of MCSP, look for more constructive lower bound proofs for the circuit class ACC0 and try to show that randomized polynomial-time algorithms are strictly weaker than deterministic exponential-time ones (or at least try to understand why such a separation is difficult to prove).

哪些计算问题是“简单”的，哪些是“困难”的？为什么证明我们的候选困难计算问题实际上是困难的如此困难？我们可以从困难计算问题中提取效率吗？离线，我们可以通过设计更有效的算法来获得新的下限吗？（伪）随机性在这一切中的作用是什么？下限（证明计算机的局限性）早于 20 世纪 30 年代，艾伦图灵！发明了通用计算机的概念，同时证明没有这样的计算机可以解决逻辑上的某个问题！ 尽管多年来已经发现了许多有效的算法，但仍然有许多自然问题（例如 NP 完全问题）的复杂性并不已知对于布尔电路的非均匀计算模型，也存在许多候选难题，但没有足够通用的电路类的下界证明 Razborov 和 Rudich 认为某些构造性证明论证无法证明。强电路降低他们的论点的核心是最小电路尺寸问题（MCSP）的某种平均情况版本，该问题要求确定布尔函数 f 的给定真值表（如果 f 具有大小的电路）。最多给定参数 s 是元计算问题的一个重要示例：其输入是其他计算问题的实例的问题。元问题的其他示例包括可满足性 (SAT)、计算学习和。随机算法的去随机化有许多例子，电路下界导致新算法，反之亦然，特别是，电路下界和去随机化之间、电路的构造性（在 Razborov 和 Rudich 的意义上）证明之间存在已知的联系。下界和学习算法，以及电路下界和重要的 SAT 算法之间的伪随机性在所有这些联系中发挥着重要作用：对“类随机”对象的研究，这些对象不是真正随机的，而是真正随机的。所提出的研究的主要目的是通过证明新的下限和设计新的算法来更好地理解高效计算的能力和局限性，探索两者之间明显的密切联系。伪随机理论的技术将成为本研究的主要工具，特别是，我们将研究 MCSP 的复杂性，为电路类 ACC0 寻找更具建设性的下界证明，并尝试证明随机多项式时间算法是严格的。较弱比确定性的指数时间的（或者至少尝试理解为什么这种分离很难证明）。