=100

2=20

8=008

------

128 bits

^

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0===P = NP===

{{Main|P versus NP problem}}

{{See also|Oracle machine}}

The complexity class '''P''' is often seen as a mathematical

abstraction modeling those computational tasks that admit an efficient

algorithm. The complexity class [[NP (complexity)|'''NP''']] is the

set of decision problems that can be solved by a [[non-deterministic

Turing machine]] in polynomial time. This class contains many

problems that people would like to solve efficiently, including the

[[Boolean satisfiability problem]], the [[Hamiltonian path problem]]

and the [[vertex cover problem]]. All the problems in this class have

the property that their solutions can be checked efficiently< name="Sipser2006" >>

Since deterministic Turing machines are special nondeterministic

Turing machines, it is easily observed that each problem in '''P''' is

also member of the class '''NP'''. The question of whether

P = NP (can problems that can be solved in non-deterministic

polynomial time also always be solved in deterministic polynomial

time?) is one of the most important open questions in theoretical

computer science because of the wide implications of a solution< name="Sipser2006>>{{cite book|last=Sipser|first=Michael|

authorlink=Michael Sipser|title=Introduction to the Theory of

Computation|edition=2nd|chapter=Time Complexity|year=2006|publisher=

[[Thomson Learning|Thomson Course Technology]]|location=USA|

isbn=0534950973}}< If the answer is yes, many important problems

can be shown to have more efficient solutions that are now used with

reluctance because of unknown [[edge case]]s. These include various

types of [[integer programming]] in [[operations research]], many

problems in [[logistics]], [[protein structure prediction]] in

[[biology]], and the ability to find formal proofs of [[pure

mathematics]] theorems<