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.

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