Towards a Unified Theory of Proof and Circuit Complexity

走向证明和电路复杂性的统一理论

• 批准号: RGPIN-2021-03036
• 负责人: Robere, Robert
• 金额: $ 3.35万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=750379
• 关键词: Towards; Unified; Theory; Proof; Circuit

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.

考虑以下问题：给你一个非常大的数字（例如 10,000 位数字），并要求你找到能整除该数字的任何数字（除了 1 或它本身）。显然，这个问题对于强大的计算机来说很难有效地解决。此外，现代世界的许多技术都依赖于这样一个事实：这个问题很难解决，因为它的硬度是保证我们所有互联网通信安全的密码学的基础！ 这些关于计算任务的内在难度的问题是计算复杂性理论的研究，这是我的研究领域。在复杂性理论中，我们研究计算机执行计算所必须消耗的资源量，例如运行时间、内存或能量。这样，计算复杂性是算法设计的另一面，算法设计涉及为给定任务找到最有效的算法。这项研究提案概述了计算复杂性理论中发展起来的一系列强大的新技术——称为提升定理。这些新技术已经有了令人眼花缭乱的应用，解决了计算复杂性理论和离散数学中的大量难题。此外，他们还揭示了算法和证明之间深刻的新联系；表明，在许多感兴趣的情况下，算法就是证明，证明也是算法，因此分析一个对象可以让我们分析另一个对象。 “提升革命”还远未结束，在本提案中，我们确定了三个具体方向，我们认为这些方向已经成熟，可以利用这些新技术进行攻击。首先是加深对优化常用算法的理解；第二个是量子计算；最后是数字电路理论（与运行计算机的电路类型相同）。最后，我们相信还可以开发一种更深入的理论（目前仅部分理解）来解释这些应用程序的惊人广度，并表明在这个方向上有一些有前途的工作。