P versus NP

1
P versus NP

• Ben Dalziel
• Tuesday 31st January 2006

2
Agenda

• Background
• P, solved in Polynomial time
• NP, Nondeterministic Polynomial
• The million dollar question
• P versus NP
• Examples
• Discussion
• Conclusion

3
Background

• P deterministic Polynomial time
• Set of efficiently solvable problems
• Solvable in polynomial time by a deterministic
  Turing machine

12 X 34 48
12 X 34 48 360
48 360 408
n22n Steps, MAX
4
Background

• NP Nondeterministic Polynomial time
• Set of efficiently checkable problems
• Solvable in polynomial time by a nondeterministic
  Turing machine

100000000000000000000000000 000000000000000000000
000000 000000000000000000000000000 000000000000000
00000 steps
10n
Trial and Error
2n digit number
Multiply two n-digit numbers
10,200 steps
n100
Factorise 2n digit number
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The Million Dollar Question

• P versus NP, 1971
• P?NP
• Hard problems do exist
• P necessarily in NP
• PNP
• Digital encryption in jeopardy
• E-commerce meltdown
• CMI

6
Discussion

• People have only hypothesised P NP
• NP Computers
• Nondeterministic Turing Machines
• Nobody has proven P ? NP
• Many believe it is only a matter of time
• William Gasarch, The P?NP poll

7
Conclusion

• Still a fascinating time for mathematics
• Demonstrates that Mathematics is still
• Relevant
• Active
• Interesting
• Impossible to say
• Who knows what the future will hold, but as
  mathematicians, its our job to find out

8
Thank you

• Any Question?