The P=NP Question and Gödel's Lost Letter

? DoesP=NP. In just ?ve symbols Dick Karp -in 1972-captured one of the deepest and most important questions of all time. When he ?rst wrote his famous paper, I think it's fair to say he did not know the depth and importance of his question. Now over three decades later, we know P=NP is central to our understanding of compu- tion, it is a very hard problem, and its resolution will have potentially tremendous consequences. This book is a collection of some of the most popular posts from my blog- Godel ¨ Lost Letter andP=NP-which I started in early 2009. The main thrust of the blog, especially when I started, was to explore various aspects of computational complexity around the famousP=NP question. As I published posts I branched out and covered additional material, sometimes a timely event, sometimes a fun idea, sometimes a new result, and sometimes an old result. I have always tried to make the posts readable by a wide audience, and I believe I have succeeded in doing this.

List of contents

A Prologue.- A Walk In the Snow.- On the P=NP Question.- Algorithms: Tiny Yet Powerful.- Is P=NP Well Posed?.- What Would You Bet?.- What Happens When P=NP Is Resolved?.- NP Too Big or P Too Small?.- How To Solve P=NP?.- Why Believe P Not Equal To NP?.- A Nightmare About SAT.- Bait and Switch.- Who's Afraid of Natural Proofs?.- An Approach To P=NP.- Is SAT Easy?.- SAT is Not Too Easy.- Ramsey's Theorem and NP.- Can They Do That?.- Rabin Flips a Coin.- A Proof We All Missed.- Barrington Gets Simple.- Exponential Algorithms.- An EXPSPACE Lower Bound.- Randomness has Unbounded Power.- Counting Cycles and Logspace.- Ron Graham Gives a Talk.- An Approximate Counting Method.- Easy and Hard Sums.- How To Avoid O-Abuse.- How Good is The Worst Case Model?.- Savitch's Theorem.- Adaptive Sampling and Timed Adversaries.- On The Intersection of Finite Automata.- Where are the Movies?.- On Integer Factoring.- Factoring and Factorials.- BDD's.- Factoring and Fermat.- On Mathematics.- A Curious Algorithm.- Edit Distance.- Protocols.- Erd?s and the Quantum Method.- Amplifiers.- Amplifying on the PCR Amplifier.- Mathematical Embarrassments.- Mathematical Diseases.- Mathematical Surprises.- Erratum.

About the author

Richard Lipton is the Storey Professor of Computer Science at Georgia Institute of Technology. Previously he held faculty positions at Yale University, the University of California at Berkeley, and Princeton University. His research is focused primarily, but not exclusively, on theory of computation. He has made seminal contributions to many areas of computing from software engineering and program testing, to computer security and cryptography, to DNA and molecular computation, and to other areas of computer science. He is a member of The National Academy of Engineering, an ACM Fellow, and a Guggenheim fellow.

Summary

? DoesP=NP. In just ?ve symbols Dick Karp –in 1972–captured one of the deepest and most important questions of all time. When he ?rst wrote his famous paper, I think it’s fair to say he did not know the depth and importance of his question. Now over three decades later, we know P=NP is central to our understanding of compu- tion, it is a very hard problem, and its resolution will have potentially tremendous consequences. This book is a collection of some of the most popular posts from my blog— Godel ¨ Lost Letter andP=NP—which I started in early 2009. The main thrust of the blog, especially when I started, was to explore various aspects of computational complexity around the famousP=NP question. As I published posts I branched out and covered additional material, sometimes a timely event, sometimes a fun idea, sometimes a new result, and sometimes an old result. I have always tried to make the posts readable by a wide audience, and I believe I have succeeded in doing this.

Additional text

“This book is a thoroughly enjoyable read because of
the great balance between anecdotes, presentations of ‘nice’ problems and
algorithms and their solutions and proofs, ‘hard mathematics,’ and musings on
how to approach mathematical problems. After having read the book, most readers
with a background in complexity theory will most likely be unable to resist
immediately working on at least one of the many open problems presented in the
book.” (Till Tantau, Mathematical Reviews, October, 2015)
“This book … collects and edits the highlights from Lipton’s ongoing blog, rounded out by cross-references and a useful index and bibliography. … the book offers a different experience and a framed portrait of the state of the art. … Summing Up: Recommended. All levels/libraries.” (D. V. Feldman, Choice, Vol. 48 (9), May, 2011)
“The P=NP question is certainly one of the most important problems in mathematics and computer science (CS). What makes this book unique and delightful is that it gives proper weight to the question rather than the technicalities. Each chapter is based on one of Lipton’s blog posts, and readers can jump from chapter to chapter to find his beautifully written thoughts and insights. … In fact, anyone who is highly motivated by this interesting subject that relates science with reality should read it.” (Hector Zenil, ACM Computing Reviews, March, 2011)
“This book collects some entries of the author’s blog on Gödel’s lost letter and P = NP … . It is an enjoyable and lively introduction to some impressive achievements in the field of complexity theory.” (Thierry Coquand, Zentralblatt MATH, Vol. 1215, 2011)

Report

"This book is a thoroughly enjoyable read because of the great balance between anecdotes, presentations of 'nice' problems and algorithms and their solutions and proofs, 'hard mathematics,' and musings on how to approach mathematical problems. After having read the book, most readers with a background in complexity theory will most likely be unable to resist immediately working on at least one of the many open problems presented in the book." (Till Tantau, Mathematical Reviews, October, 2015)
"This book ... collects and edits the highlights from Lipton's ongoing blog, rounded out by cross-references and a useful index and bibliography. ... the book offers a different experience and a framed portrait of the state of the art. ... Summing Up: Recommended. All levels/libraries." (D. V. Feldman, Choice, Vol. 48 (9), May, 2011)
"The P=NP question is certainly one of the most important problems in mathematics and computer science (CS). What makes this book unique and delightful is that it gives proper weight to the question rather than the technicalities. Each chapter is based on one of Lipton's blog posts, and readers can jump from chapter to chapter to find his beautifully written thoughts and insights. ... In fact, anyone who is highly motivated by this interesting subject that relates science with reality should read it." (Hector Zenil, ACM Computing Reviews, March, 2011)
"This book collects some entries of the author's blog on Gödel's lost letter and P = NP ... . It is an enjoyable and lively introduction to some impressive achievements in the field of complexity theory." (Thierry Coquand, Zentralblatt MATH, Vol. 1215, 2011)

Product details

• Authors: Richard J Lipton, Richard J. Lipton
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.2014
• EAN: 9781489992727
• ISBN: 978-1-4899-9272-7
• No. of pages: 239
• Dimensions: 155 mm x 13 mm x 235 mm
• Weight: 397 g
• Illustrations: XIII, 239 p.
• Subjects: Natural sciences, medicine, IT, technology > IT, data processing > IT; A, Algorithms, computer science, Theory of Computation, History of engineering & technology, Mathematics of Computing, History of Computing, Computers, Algorithms & data structures, Numerical analysis, Mathematical theory of computation, Computer science—Mathematics, Maths for computer scientists, Algorithm Analysis and Problem Complexity, Mathematical logic, Mathematical Logic and Foundations, Mathematical foundations, Computing & Information Technology