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In computational complexity theory, the complexity class NP-complete, also known as NP-C or NPC, is a subset of NP ("non-deterministic polynomial time"). Problems are designated "NP-complete" if their solutions can be quickly checked for correctness, and if the same solving algorithm used can solve all other NP problems. They are the most difficult problems in NP in the sense that a deterministic, polynomial-time solution to any NP-complete problem would provide a solution to every other problem in NP (and conversely, if any one of them provably lacks a deterministic polynomial-time solution, none of them has one). Problems in NP-complete are known as NP-complete problems. A more formal definition is given below.

One example of an NP-complete problem is the subset sum problem which is: given a finite set of integers, determine whether any non-empty subset of them sums to zero. A supposed answer is very easy to verify for correctness, but there is no known efficient algorithm to find an answer; that is, all known algorithms are impractically slow for large sets of integers.

Reference:
http://en.wikipedia.org/wiki/NP-complete

In computational complexity theory, NP is one of the most fundamental complexity classes. The abbreviation NP refers to "Non-deterministic Polynomial time".

Intuitively, NP contains all decision problems for which the 'yes'-answers have simple proofs of the fact that the answer is indeed 'yes'. More precisely, these proofs have to be verifiable in polynomial time by a deterministic Turing machine. In an equivalent formal definition, NP is the set of decision problems solvable in polynomial time by a non-deterministic Turing machine.

The complexity class P is contained in NP, but NP contains many important problems, called NP-complete problems, for which no polynomial-time algorithms are known. The most important open question in complexity theory, the P = NP problem, asks whether such algorithms actually exist for NP-complete problems. It is widely believed that this is not the case.

Reference:
http://en.wikipedia.org/wiki/NP_%28complexity%29

In computational complexity theory, P is one of the most fundamental complexity classes. It contains all decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time.

P is often taken to be the class of computational problems which are "efficiently solvable" or "tractable", although there are potentially larger classes that are also considered tractable such as RP and BPP. Also, there exist problems in P which are intractable in practical terms; for example, some require at least n^1000000 operations. See even harder problems of complexity classes for further discussion.

Reference:
http://en.wikipedia.org/wiki/P_%28complexity%29

In computational complexity theory, a complexity class is a set of problems of related complexity. A typical complexity class has a definition of the form:

the set of problems that can be solved by abstract machine M using O(f(n)) of resource R (n is the size of the input)
For example, the class NP is the set of decision problems that can be solved by a non-deterministic Turing machine in polynomial time, while the class PSPACE is the set of decision problems that can be solved by a deterministic Turing machine in polynomial space. Some complexity classes are sets of function problems, such as FP.

Many complexity classes can be characterized in terms of the mathematical logic needed to express them; see descriptive complexity.

The Blum axioms can be used to define complexity classes without referring to a concrete computational model.

Reference:
http://en.wikipedia.org/wiki/Complexity_class

In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by some sort of definability property; for example, the collection of all open sets in some fixed collection of Polish spaces is a pointclass. (An open set may be seen as in some sense definable because it cannot be a purely arbitrary collection of points; for any point in the set, all points sufficiently close to that point must also be in the set.)

Pointclasses find application in formulating many important principles and theorems from set theory and real analysis. Strong set-theoretic principles may be stated in terms of the determinacy of various pointclasses, which in turn implies that sets in those pointclasses (or sometimes larger ones) have regularity properties such as Lebesgue measurability (and indeed universal measurability), the property of Baire, and the perfect set property.

Reference:
http://en.wikipedia.org/wiki/Lightface

In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof.

The notion of analytic proof was introduced by Gerhard Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as was shown by Dag Prawitz; the definition is slightly more complex — we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting.

Reference:
http://en.wikipedia.org/wiki/Structural_proof_theory