# Complexity Theory - A Survey

### Oded Goldreich and Avi Wigderson

**Material available on-line**:
a PSfile
(768KB, written in 2004).
[in PDF].

#### Preface

The strive for efficiency is ancient and universal,
as time is always short for humans.
Computational Complexity is a mathematical study
of the what can be achieved
when time (and other resources) are scarce.
In this brief article we will introduce quite a few notions:
Formal models of computation, and measures of efficiency;
the P vs. NP problem and NP-completeness;
circuit complexity and proof complexity;
randomized computation and pseudorandomness;
probabilistic proof systems, cryptography and more.
A glossary of complexity classes is included in an appendix.
We highly recommend the given bibliography
and the references therein for more information.

#### Organization

- Introduction
- Preliminaries
2.1 Computability and Algorithms;
2.2 Efficient Computability and the class P

- The P versus NP Question
3.1 Efficient Verification and the class NP;
3.2 The Big Conjecture;
3.3 NP versus coNP

- Reducibility and NP-Completeness
- Lower Bounds
5.1 Boolean Circuit Complexity;
5.2 Arithmetic Circuits;
5.3 Proof Complexity

- Randomized Computation
6.1 Counting at Random;
6.2 Probabilistic Proof Systems;
6.3 Weak Random Sources

- The Bright Side of Hardness
7.1 Pseudorandomness;
7.2 Cryptography

- The Tip of an Iceberg
8.1 Relaxing the Requirements;
8.2 Other Complexity Measures;
8.3 Other Notions of Computation

- Concluding Remarks

Bibliography

Appendix: Glossary of Complexity Classes
A.1 Algorithm-based classes;
A.2 Circuit-based classes

#### Related Material (by Oded Goldreich)

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Copyright (C symbol) 2004 by Oded Goldreich and Avi Wigderson.
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