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ORGANIZATION AND CHAPTER SUMMARIES generate, create code 3/9 none for .net projects .NET Framework 3.5 average value of an .NET 3 of 9 arbitrary function, and the problem of randomness extraction (i.e.

, procedures for extracting almost perfect randomness from sources of weak or defected randomness). Appendix E: Explicit Constructions. Complexity Theory provides a clear perspective on the intuitive notion of an explicit construction.

This perspective is demonstrated with respect to error-correcting codes and expander graphs. Starting with codes, the appendix focuses on various computational aspects, and offers a review of several popular constructions as well as a construction of a binary code of constant rate and constant relative distance. Also included are a brief review of the notions of locally testable and locally decodable codes, and a useful upper bound on the number of codewords that are close to any single sequence.

Turning to expander graphs, the appendix contains a review of two standard de nitions of expanders, two levels of explicitness, two properties of expanders that are related to (single-step and multi-step) random walks on them, and two explicit constructions of expander graphs. Appendix F: Some Omitted Proofs. This appendix contains some proofs that were not included in the main text (for a variety of reasons) and still are bene cial as alternatives to the original and/or standard presentations.

Included are a proof that PH is reducible to #P via randomized Karp-reductions, and the presentation of two useful transformations regarding interactive proof systems. Appendix G: Some Computational Problems. This appendix includes de nitions of most of the speci c computational problems that are referred to in the main text.

In particular, it contains a brief introduction to graph algorithms, Boolean formulae, and nite elds.. Acknowledgments My perspective on Co .net framework barcode 39 mplexity Theory was most in uenced by Shimon Even and Leonid Levin. In fact, it was hard not to be in uenced by these two remarkable and highly opinionated researchers (especially for somebody like me who was fortunate to spend a lot of time with them).

1 Shimon Even viewed Complexity Theory as the study of the limitations of algorithms, a study concerned with natural computational resources and natural computational tasks. Complexity Theory was there to guide the engineer and to address the deepest questions that bother an intellectually curious computer scientist. I believe that this book shares Shimon s perspective of Complexity Theory as evolving around such questions.

Leonid Levin emphasized the general principles that underlie Complexity Theory, rejecting any model-dependent effects as well as the common coupling of Complexity Theory with the theory of automata and formal languages. In my opinion, this book is greatly in uenced by these perspectives of Leonid. I wish to acknowledge the in uence of numerous other colleagues on my professional perspectives and attitudes.

These include Sha Goldwasser, Dick Karp, Silvio Micali, and Avi Wigderson. Needless to say, this is but a partial list that re ects in uences of which I am most aware. The year I spent at Radcliffe Institute for Advanced Study (of Harvard University) was instrumental in my decision to undertake the writing of this book.

I am grateful to Radcliffe for creating such an empowering atmosphere. I also wish to thank many colleagues for their comments and advice (or help) regarding earlier versions of this text. A partial list includes Noga Alon, Noam Livne, Dieter van Melkebeek, Omer Reingold, Dana Ron, Ronen Shaltiel, Amir Shpilka, Madhu Sudan, Salil Vadhan, and Avi Wigderson.

Lastly, I am grateful to Mohammad Mahmoody Ghidary and Or Meir for their careful reading of drafts of this manuscript and for the numerous corrections and suggestions that they have provided. Relation to previous texts. Some of the text of this book has been adapted from previous texts of mine.

In particular, s 8 and 9 were written based on my surveys [90, Chap. 3] and [90, Chap. 2], respectively; but the exposition has been extensively revised.

Shimon Even was my g raduate studies adviser (at the Technion, 1980 83), whereas I had a lot of meetings with Leonid Levin during my postdoctoral period (at MIT, 1983 86)..
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