Computational Complexity

Computational Complexity a k a Complexity Theory is a central eld of Com puter Science with a remarkable list of celebrated achievements as well as a very vibrant research activity The eld is concerned with the study of the intrinsic complexity of computational tasks and this study tend to aim at generality It focuses on natural computational resources and considers the e ect of limiting these resources on the class of problems that can be solved The half century history of Complexity theory has witnessed two main research e orts or directions The rst direction is aimed towards actually establishing concrete lower bounds on the complexity of problems via an anal ysis of the evolution or e ect of the process of computation Thus in a sense the heart of this direction is a low level analysis of computation Most re search in circuit complexity cf and in proof complexity cf falls within this category In contrast a second research e ort is aimed at exploring the connections among computational problems and notions without being able to provide absolute statements on either of the problems or notions being re lated The current exposition focuses on the latter e ort or direction which may be viewed as a high level study of computation We present a few of the interesting notions results and open problems in this direction We list several reasons for our choice to focus on the high level direction The rst is the great conceptual signi cance of the know results that is as we