Questions like this motivate the concepts of ''hard'' and ''[[complete (complexity)|complete]]''.

Questions like this motivate the concepts of ''hard'' and ''[[complete (complexity)|complete]]''.

Post by http://ale » Sun, 25 Oct 2009 10:39:14


=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<{{