P=NP question
P=NP question One of the major open questions in theoretical computer science at present.
P is the class of formal languages that are recognizable in polynomial time. More precisely a language L is in P if there exists a Turing machine program M and a polynomial p(n) such that M recognizes L and T_{M}(n) ← p(n)
for all nonnegative integers n, where T_{M} is the time complexity of M (see complexity measure). It is generally accepted that if a language is not in P then there is no algorithm that recognizes it and is guaranteed to be always “fast”.
NP is the class of languages that are recognizable in polynomial time on a nondeterministic Turing machine.
Clearly P ⊆ NP
but the question of whether or not P = NP
has not been solved despite a great amount of research.
Contained in NP is a set NPC of languages that are called NPcomplete. A language L_{1} is in NPC if every language L_{2} in NP can be polynomially reduced to L_{1}, i.e. there is some function f such that
(a) x ∈ L_{1} iff f(x) ∈ L_{2}
(b) f(x) is computable by a Turing machine in time bounded by a polynomial in the length of x.
It can be shown that if any NPcomplete language is also in P then P = NP.
A wide variety of problems occurring in computer science, mathematics, and operations research are now known to be NPcomplete. As an example the problem of determining whether a Boolean expression in conjunctive normal form (see conjunction) can be satisfied by a truth assignment was the first problem found to be NPcomplete; this is generally referred to as the satisfiability (or CNF satisfiability) problem. Despite considerable effort none of these NPcomplete problems have been shown to be polynomially solvable. Thus it is widely conjectured that no NPcomplete problem is polynomially solvable and P ≠ NP.
A language is said to be NPhard if any language in NP can be polynomially reduced to it, even if the language itself is not in NP.
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