Presentation on theme: "Lance Fortnow Georgia Institute of Technology. Step 1: Post Elusive Proof. Step 2: Watch Fireworks. By John Markoff … Vinay Deolalikar, a mathematician."— Presentation transcript:
Lance Fortnow Georgia Institute of Technology
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. 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…
$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
We can efficiently find a matching even among millions of men and women avoiding having to search all the possibilities. P
Given a solution to a clique problem we can check it quickly NP
Easy to Solve Easy to Verify NPP
P = NP
P = NP ?
Transition Function Tape Alphabet Blank Symbol Input Alphabet State Space Start State Accept State
Transition function (state, symbol) (state, symbol, direction) Nondeterministic Can map to multiple possibilities
Instead of Turing machine Multiple tapes Random access λ – calculus C++ LaTeX Probabilistic and Quantum computers might not define the same class
Hardest problems in NP Cook-Levin 1971 Boolean Formula Satisfiability
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
WE CURE CANCER
William of Ockham, English Franciscan Friar Occams Razor (14 th Century) Entia non sunt multiplicanda praeter necessitatem
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.
Rosetta Stone 196 BC Decree in three languages Greek Deomotic Hieroglyphic In 1822, Jean-François Champollion found a simple grammar.
How do you deal with NP-completeness?
Brute Force Heuristics Small Parameters Approximation Solve a Different Problem Give Up
S1S1 InOutInOutIn S2S2 OutInOut InOut S3S3 Out S4S4 InOutInOutInOut S5S5 In S6S6 OutInOut In
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.
Measure complexity by size of circuit. Different circuits for each input length. Efficient computation essentially equivalent to small circuits.
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.
( 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.