Towards a Unified Theory of Proof and Circuit Complexity

点击翻译按钮获取中文摘要

Consider the following problem: you are given an extremely large (say, 10,000 digit) number, and are asked to find any number (other than 1 or itself) that divides it. This problem is, apparently, very hard for powerful computers to solve efficiently. Moreover, much of the technology of the modern world crucially relies on the fact that this problem is hard to solve, since its hardness is the foundation of the cryptography that keeps all of our internet communications safe! Such questions about the intrinsic hardness of computational tasks are the study of computational complexity theory, which is my area of research. In complexity theory, we study the amounts of resources --- like running time, memory, or energy --- that computers must consume to perform their computations. In this way, computational complexity is the flip side of the coin to algorithm design, which is concerned with finding the most efficient algorithms for a given task. This research proposal outlines a new and powerful family of techniques --- called lifting theorems --- that have been developed in computational complexity theory. These new techniques have had a dazzling number of applications, leading to the resolution of a large number of hard problems in both computational complexity theory and discrete mathematics. Furthermore, they have revealed deep new connections between algorithms and proofs; showing that, in many cases of interests, algorithms are proofs and proofs are algorithms, and so analyzing one object allows us to analyze the other. The "lifting revolution" is far from being over and, in this proposal, we identify three concrete directions that we believe are ripe for attack with these new techniques. The first is deepening our understanding of algorithms commonly used in optimization; the second, in quantum computation; and the last, the theory of digital circuits (the same types of circuits that run your computer). Finally, we believe that a deeper --- and currently, only partially understood --- theory that explains the surprising breadth of these applications can also be developed, and indicate some promising work in this direction.