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Lance Fortnow Georgia Institute of Technology

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Step 1: Post Elusive Proof. Step 2: Watch Fireworks. By John Markoff … Vinay Deolalikar, a mathematician and electrical engineer at Hewlett-Packard, posted a proposed proof of what is known as the P versus NP problem on a Web site, and quietly notified a number of the key researchers. Email: August 6, 2010 From: Deolalikar, Vinay To: 22 people Dear Fellow Researchers, I am pleased to announce a proof that P is not equal to NP, which is attached in 10pt and 12pt fonts…

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$1 Million Award for solving any of these problems. Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equations P vs NP Poincaré Conjecture Riemann Hypothesis Yang-Mills Theory

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We can efficiently find a matching even among millions of men and women avoiding having to search all the possibilities. P

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Given a solution to a clique problem we can check it quickly NP

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Easy to Solve Easy to Verify NPP

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P = NP

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P NP

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P = NP ?

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Transition Function Tape Alphabet Blank Symbol Input Alphabet State Space Start State Accept State

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Transition function (state, symbol) (state, symbol, direction) Nondeterministic Can map to multiple possibilities

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Instead of Turing machine Multiple tapes Random access λ – calculus C++ LaTeX Probabilistic and Quantum computers might not define the same class

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A B

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Hardest problems in NP Cook-Levin 1971 Boolean Formula Satisfiability

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1935: Turings Machine 1962: Hartmanis-Stearns: Computation time depends on size of problem 1966: Edmonds, Cobham: Models of efficient computation 1971: Steve Cook defines first NP-complete problem 1972: Richard Karp shows 22 common problems NP- complete 1971: Leonid Levin similar work in Russia 1979: Garey and Johnson publish list of 100s of NP- complete problems Now thousands of NP-complete problems over many disciplines

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WE CURE CANCER

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William of Ockham, English Franciscan Friar Occams Razor (14 th Century) Entia non sunt multiplicanda praeter necessitatem

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William of Ockham English Franciscan Friar Occams Razor (14 th Century) Entities must not be multiplied beyond necessity The simplest explanation is usually the best. If P = NP we can find that simplest explanation.

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Rosetta Stone 196 BC Decree in three languages Greek Deomotic Hieroglyphic In 1822, Jean-François Champollion found a simple grammar.

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How do you deal with NP-completeness?

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Brute Force Heuristics Small Parameters Approximation Solve a Different Problem Give Up

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123456 S1S1 InOutInOutIn S2S2 OutInOut InOut S3S3 Out S4S4 InOutInOutInOut S5S5 In S6S6 OutInOut In

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NP doesnt have enough power to simulate P Relativized world where P = NP. Can get weaker time/space results: No algorithm for satisfiability that uses logarithmic space and n 1.8 time.

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Measure complexity by size of circuit. Different circuits for each input length. Efficient computation essentially equivalent to small circuits.

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Idea: Show no single gate changes things much so needs lots of gates for NP-complete problems Works for circuits of limited depth or negations. Natural Proofs give some limitations on this technique.

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( x AND y ) OR (NOT x) OR (Not y) If P = NP (or even NP = co-NP) then every tautology has a short proof. Try to show tautologies only have long proofs. Works only for limited proof systems like resolution.

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