Lower bounds, meta-algorithms, and pseudorandomness

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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).