THE P V NP PROBLEM IN THE ERA OF BIG DATA AND FAST COMPUTING Lance - PowerPoint PPT Presentation

THE P V NP PROBLEM IN THE ERA OF BIG DATA AND FAST COMPUTING Lance Fortnow Georgia Institute of Technology

Gödel Letter to von Neumann (1956) One can obviously easily construct a Turing machine, which for every formula F in first order predicate logic and every natural number n, allows one to decide if there is a proof of F of length n (length = number of symbols). Let ψ(F,n) be the number of steps the machine requires for this and let φ(n) = max F ψ(F,n). The question is how fast φ(n) grows for an optimal machine. One can show that φ(n) ≥ k n. If there really were a machine with φ(n) k n (or even k n 2 ), this would have consequences of the greatest importance. Namely, it would obviously mean that in spite of the undecidability of the Entscheidungsproblem, the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine.

■ Finite Alphabet ■ String is a sequence of characters – Can encode objects like logical formula. ■ Language is a set of strings. – Example: Set of tautologies

Turing machine M computes a language L if for all strings x – If x is in L then M(x) ends in an accept state – If x is not in L then M(X) ends in a reject state

P and NP

NP-completeness

Clique

Traveling Salesman 13,509 cities with population at least 500

Map Coloring

DNA Sequencing

Sudoku

Sudoku

Rubik’s Cube

NP-Complete

Proving P  NP

THE P V NP PROBLEM IN THE ERA OF BIG DATA AND FAST COMPUTING Lance Fortnow Georgia Institute of Technology

If P  NP: Need to Solve Hard Problems Brute Force Heuristics Approximation Solve a Different Problem Give Up

1971 2005 3000 Transistors 230 Million Transistors

SAT Solvers Can solve satisfiability problems of hundreds of variables. Does really well on problems with tens of thousands to millions of variables.

Linear Programming

Integer Programming

Mixed Integer Programming

Is P v NP Relevant Today?

Cryptography

DOG

MUFFIN

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

Occam’s Razor William of Ockham English Franciscan Friar Occam’s Razor (14 th Century) Entities must not be multiplied beyond necessity The simplest explanation is usually the best.

Data consists of a random example of some structure.

00010101000100000101 Structure: Every odd bit is a zero Random: Even bits

Kolmogorov Complexity K(x) is the smallest program that generates x x is random if K(x) is at least |x|

Kolmogorov Structure Function Minimum Description Length

Kolmogorov Structure Function Minimum Description Length

If P = NP

P versus NP is not about what is impossible but what is possible