Lower bounds, meta-algorithms, and pseudorandomness

哪些计算问题是“容易”的，哪些是“难”的？为什么要证明我们的候选难计算问题实际上很难呢？我们能否从困难的计算问题中提取有效的算法（从建立其硬度的证明中）？反过来说，我们能否通过设计更有效的算法得到新的下界？（伪）随机性在这一切中的作用是什么？下界（证明计算机的局限性）早于第一台计算机！在20世纪30年代，艾伦·图灵发明了通用计算机的概念，同时证明了没有这样的计算机可以解决某个逻辑问题！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）的复杂性未知。对于布尔电路的非均匀计算模型，也有许多候选难题，但对于足够一般的电路类，没有下界证明。Razborov和Rudich的“自然证明屏障”认为，在可信的密码学假设下，某些建设性证明论证不能证明强电路下界。他们争论的核心是最小电路尺寸问题（MCSP）的一个特定的平均情况版本，该问题要求决定布尔函数f的给定真值表，如果f的电路大小最多为给定参数s。MCSP是元计算问题的一个重要例子：输入是其他计算问题实例的问题。元问题的其他例子包括SATISFIABILITY （SAT）、计算学习和随机算法的非随机化。有很多电路下界导致新算法的例子，反之亦然。特别是，在电路下界和非随机化之间，在电路下界的建设性（在Razborov和Rudich的意义上）证明和学习算法之间，以及在电路下界和非平凡SAT算法之间，都有已知的联系。在所有这些联系中，伪随机性扮演了一个重要的角色：研究“类随机”的对象，这些对象不是真正随机的，但对某些计算有限的观察者来说“显得随机”。提出的研究的主要目的是通过证明新的下界和设计新的算法来更好地理解有效计算的能力和局限性，探索两者之间明显的密切联系。伪随机理论的概念和技术将是本研究的主要工具。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，并试图表明随机多项式时间算法严格弱于确定性指数时间算法（或至少试图理解为什么这种分离难以证明）。

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).