Studies in Computational Complexity Theory

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

Studies in Computational Complexity Theory: Project Summary Computational 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 memory space for uniform computations, and circuit size and depth for non-uniform computations. By limiting various resource requirements of computations to certain robust bounds we get complexity classes, which are the fundamental objects of study in complexity theory. Research in complexity theory 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-life computational problems using a computer. These two themes are interconnected by the notions of completeness and reductions. The overall goal of this proposal is to advance computational complexity theory along these two themes. The overall goal will be accomplished through three related objectives: Objective 1: Investigate interrelations among uniformity, nonuniformity, and derandomization. The PI will investigate a number of issues in uniform vs non-uniform computations and their relation to derandomization. Speci.cally, the PI will investigate uniform derandomization of Arthur-Merlin games, some related non-uniformity questions including uniform upper bounds for languages with high circuit complexity, and certain cover-based approach to resource bounded measure with applications to lower complexity classes and derandomization. Objective 2: Investigate computational problems with intermediate complexity. The PI will continue his investigation of problems that are intermediate between P and NP-complete such as Graph Isomorphism problem and some computational group-theoretic problems. A number of questions related to lowness properties of these problems will be investigated. E.cient program checkers for a host of computational group-theoretic problems will be designed. Objective 3: Explore the interconnections between complexity theory and computational learning theory. The PI will explore interconnections between complexity theory and learning theory. In particular, the PI will investigate learning problems such as learning DNFs and Boolean Circuits in the Teaching Assistant model, and the applications of learning algorithms to complexity theory. Broader Impacts: The proposed research activity will have several broader impacts. Complexity theory indirectly impacts many areas of computer science. Thus proposed research has potential to scienti.cally impact these areas. Research results from this grant will be published in peer-reviewed journals and will be presented in international conferences, thus enabling broad dissemination of the the results to enhance scienti.c understanding. New courses will be created and taught along the theme of this project, thus integrating teaching and research. The grant will also be used for various human resource development activities such as supporting and mentoring graduate students, and inviting visitors. Intellectual Merit: Progress made in complexity theory is essential for furthering the knowledge of what can and cannot be solved by a computer using reasonable amount of resources. The results from this proposal will extend our knowledge of complexity theory in several directions. Research in derandomization and non-uniformity will solve some signi.cant open problems in the area and will contribute to better understand the role of randomness in computation. Part of the proposed research directly relates to important real-life problems such as Graph Isomorphism problem and DNF learning problem and has potential to become practically applicable. The research on program checkers is expected to have some practical applications.