Introduction to Complexity Theory Notes for a One Semester Course

Permission to make copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for proot or commercial advantage and that new copies bear this notice and the full citation on the rst page. Abstracting with credit is permitted. I Preface Complexity Theory is a central eld of Theoretical Computer Science, with a remarkable list of celebrated achievements as well as a very vibrant present research activity. T h e e e l d i s 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 the eeect of limiting those on the c l a s s o f p r oblems that can be solved. P u t i n o t h e r w ords, Complexity Theory aims at understanding the nature of eecient computation. Topics: In my opinion, a introductory course in complexity theory should aim at exposing the students to the basic results and research directions in the eld. The focus should be on concepts and ideas, and complex technical proofs should be avoided. Speciic topics may include Revisiting NP and NPC (with emphasis on search vs decision)) Complexity classes deened by one resource-bound { hierarchies, gaps, etcc Non-deterministic Space complexity (with emphasis on NL)) Randomized Computations (e.g., ZPP, RP and BPP)) Non-uniform complexity (e.g., P/poly, and lower bounds on restricted circuit classes)) The Polynomial-time Hierarchyy The counting class #P, approximate-#P and uniqueSATT Probabilistic proof systems (i.e., IP, PCP and ZK)) Pseudorandomness (generators and derandomization)) Time versus Space (in Turing Machines)) Circuit-depth versus TM-space (e.g., AC, NC, SC)) Communication complexityy Average-case complexityy Of course, it would be hard (if not impossible) to cover all the above topics (even brieey) in a single-semester course (of two hours a week). Thus, a choice of topics has to bemade, and the rest may be merely mentioned in a relevant lecture or in the concluding lecture. The choice may depend on other courses given in the institutee in fact, my o wn choice was strongly eeected by this aspect. II Prerequisites: It is assumed that students have t a k en a course in computability, and hence are familiar with Turing Machines. Model of Computation: Most of the presented material is quite independent of the speciic (reasonable) model …