# Introduction to Complexity Theory

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About *Introduction to Complexity Theory :*

Excerpts from book:

Complexity Theory is a central field of Theoretical Computer Science, with a remarkable list of celebrated achievements as well as a very vibrant present research activity. The field 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 the effect of limiting those on the class of problems that can be solved.

These lecture notes were taken by students attending my year-long introductory course on Complexity Theory, given in 1998-99 at the Weizmann Institute of Science. The course was aimed at exposing the students to the basic results and research directions in the field. The focus was on concepts and ideas, and complex technical proofs were avoided. Specific topics included:

- Revisiting NP and NPC (with emphasis on search vs decision);

- Complexity classes defined by one resource-bound - hierarchies, gaps, etc;

- 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 Hierarchy;

- The counting class #P, approximate-#P and uniqueSAT;

- 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);

- Average-case complexity;

It was assumed that students have taken a course in computability, and hence are familiar with Turing Machines.

Most of the presented material is quite independent of the specific (reasonable) model of computation, but some (e.g., Lectures 5, 16, and 19-20) depends heavily on the locality of computation of Turing machines.

Complexity Theory is a central field of Theoretical Computer Science, with a remarkable list of celebrated achievements as well as a very vibrant present research activity. The field 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 the effect of limiting those on the class of problems that can be solved.

These lecture notes were taken by students attending my year-long introductory course on Complexity Theory, given in 1998-99 at the Weizmann Institute of Science. The course was aimed at exposing the students to the basic results and research directions in the field. The focus was on concepts and ideas, and complex technical proofs were avoided. Specific topics included:

- Revisiting NP and NPC (with emphasis on search vs decision);

- Complexity classes defined by one resource-bound - hierarchies, gaps, etc;

- 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 Hierarchy;

- The counting class #P, approximate-#P and uniqueSAT;

- 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);

- Average-case complexity;

It was assumed that students have taken a course in computability, and hence are familiar with Turing Machines.

Most of the presented material is quite independent of the specific (reasonable) model of computation, but some (e.g., Lectures 5, 16, and 19-20) depends heavily on the locality of computation of Turing machines.