Studies in Computational Complexity Theory

计算复杂性理论的研究：项目概要计算复杂性理论的重点是理解资源有限的计算能力。计算被广泛地分类为均匀计算（基于图灵机）和非均匀计算（基于电路）。研究的典型资源是均匀计算的时间步长和内存空间，以及非均匀计算的电路大小和深度。通过将计算的各种资源需求限制在一定的鲁棒界限内，我们得到复杂性类，这是复杂性理论的基本研究对象。复杂性理论的研究主要围绕两个主题：（1）以包含和分离的形式证明各种复杂性类别之间的关系;（2）量化使用计算机解决现实计算问题的难度/容易程度。这两个主题是相互联系的概念的完整性和减少。这个提议的总体目标是沿着这两个主题推进计算复杂性理论。总体目标将通过三个相关的目标来实现：目标1：调查均匀性，非均匀性和去随机化之间的相互关系。PI将调查均匀与非均匀计算中的一些问题以及它们与去随机化的关系。具体地说，PI将研究Arthur-Merlin游戏的一致去随机化，一些相关的非一致性问题，包括高电路复杂性语言的一致上界，以及某些基于覆盖的方法，以资源有界测度，并应用于低复杂性类和去随机化。PI将继续研究介于P和NP完全之间的问题，如图同构问题和一些计算群论问题。我们将研究与这些问题的低性有关的一些问题。目标3：探索复杂性理论和计算学习理论之间的相互联系。PI将探索复杂性理论和学习理论之间的相互联系。特别是，PI将调查学习问题，如学习DNF和布尔电路在教学助理模型，以及学习算法的复杂性理论的应用。更广泛的影响：拟议的研究活动将有几个更广泛的影响。复杂性理论间接影响了计算机科学的许多领域。因此，拟议的研究有可能对这些领域产生科学影响。这项赠款的研究成果将在同行评审的期刊上发表，并在国际会议上发表，从而使研究成果得到广泛传播，以提高科学认识。新课程将沿着本项目的主题进行创作和教学，从而实现教学与科研的一体化。智力上的优点：在复杂性理论方面取得的进展对于进一步认识什么可以和不可以通过计算机使用合理数量的资源来解决是必不可少的。这个提议的结果将在几个方向上扩展我们对复杂性理论的认识。对去随机化和非均匀性的研究将解决这一领域中一些重要的开放性问题，有助于更好地理解随机性在计算中的作用。部分研究内容直接涉及到现实生活中的重要问题，如图同构问题和DNF学习问题，具有实际应用的潜力.对程序检查器的研究具有一定的实际应用价值。

Studies in Computational Complexity Theory: Project SummaryComputational complexity theory focuses on understanding capabilities of resource bounded computations.Computations are broadly classi.ed as uniform computations (Turing machine based)and non-uniform computations (circuit based). Typical resources studied are time steps and memoryspace for uniform computations, and circuit size and depth for non-uniform computations. Bylimiting various resource requirements of computations to certain robust bounds we get complexityclasses, which are the fundamental objects of study in complexity theory. Research in complexitytheory broadly centers around two themes (1) proving relations, in the form of inclusions and separations,among various complexity classes (2) quantifying the di.culty/easiness of solving real-lifecomputational problems using a computer. These two themes are interconnected by the notionsof completeness and reductions. The overall goal of this proposal is to advance computationalcomplexity theory along these two themes. The overall goal will be accomplished through threerelated objectives:Objective 1: Investigate interrelations among uniformity, nonuniformity, and derandomization.The PI will investigate a number of issues in uniform vs non-uniform computations andtheir relation to derandomization. Speci.cally, the PI will investigate uniform derandomizationof Arthur-Merlin games, some related non-uniformity questions including uniform upperbounds for languages with high circuit complexity, and certain cover-based approach to resourcebounded measure with applications to lower complexity classes and derandomization.Objective 2: Investigate computational problems with intermediate complexity. The PI willcontinue his investigation of problems that are intermediate between P and NP-complete suchas Graph Isomorphism problem and some computational group-theoretic problems. A numberof questions related to lowness properties of these problems will be investigated. E.cientprogram checkers for a host of computational group-theoretic problems will be designed.Objective 3: Explore the interconnections between complexity theory and computationallearning theory. The PI will explore interconnections between complexity theory and learningtheory. In particular, the PI will investigate learning problems such as learning DNFs andBoolean Circuits in the Teaching Assistant model, and the applications of learning algorithmsto complexity theory.Broader Impacts: The proposed research activity will have several broader impacts. Complexitytheory indirectly impacts many areas of computer science. Thus proposed research has potential toscienti.cally impact these areas. Research results from this grant will be published in peer-reviewedjournals and will be presented in international conferences, thus enabling broad dissemination of thethe results to enhance scienti.c understanding. New courses will be created and taught along thetheme of this project, thus integrating teaching and research. The grant will also be used for varioushuman resource development activities such as supporting and mentoring graduate students, andinviting visitors.Intellectual Merit: Progress made in complexity theory is essential for furthering the knowledgeof what can and cannot be solved by a computer using reasonable amount of resources. The resultsfrom this proposal will extend our knowledge of complexity theory in several directions. Researchin derandomization and non-uniformity will solve some signi.cant open problems in the area andwill contribute to better understand the role of randomness in computation. Part of the proposedresearch directly relates to important real-life problems such as Graph Isomorphism problem andDNF learning problem and has potential to become practically applicable. The research on programcheckers is expected to have some practical applications.