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	Encyclopedia: NP (complexity)
		In computational complexity theory, the graph isomorphism problem or GI problem is the graph theory problem of determining whether, given two graphs G1 and G2, it is possible to permute (or relabel) the vertices of one graph so that it is equal to the other.
		NP can be seen as a very simple type of interactive proof system, where the prover comes up with the proof certificate and the verifier is a deterministic polynomial-time machine that checks it.
		In computational complexity theory, the complexity class PH is the union of all complexity classes in the polynomial hierarchy: PH is contained in the complexity classes PPP (the class of problems that are decidable by a polynomial time Turing machine with an access to PP oracle) and PSPACE.
	www.nationmaster.com /encyclopedia/NP-(complexity) (2388 words)

	Computational complexity theory - Internet-Encyclopedia.com (Site not responding. Last check: 2007-10-14)
		The time complexity of a problem is the number of steps that it takes to solve an instance of the problem, as a function of the size of the input, (usually measured in bits) using the most efficient algorithm.
		The complexity class P is the set of decision problems that can be solved by a deterministic machine in polynomial time.
		The complexity class NP is the set of decision problems that can be solved by a non-deterministic machine in polynomial time.
	www.internet-encyclopedia.com /ie/c/co/computational_complexity_theory.html (1045 words)

	NP-hard: Definition and links.
		Computational complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem.
		In this theory, NP-hard (Non-deterministic Polynomial-time hard) refers to the class of decision problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H.
		It is also often used to define the complexity class NP-complete which is the intersection of NP and NP-hard.
	www.encyclopedian.com /np/NP-hard.html (616 words)

	NP-complete: Definition and links.
		Assuming that P and NP are not equal, there are guaranteed to be an infinite number of problems that are in NP, but are neither NP-complete nor in P. Some of these problems may actually have higher complexity than some of the NP-complete problems.
		This holds because by their definition the classes of NP-complete and co-NP-complete problems under Turing reductions are the same and because these classes are both supersets of the same classes defined with many-one reductions.
		This implies that NP = co-NP as is shown in the proof in the article on co-NP.
	www.encyclopedian.com /np/NP-complete.html (968 words)

	NP (complexity) (Site not responding. Last check: 2007-10-14)
		The importance of this class of decision problems is that it contains many interesting searching and optimization problems where we want to know if there exists a certain solution for a certain problem or whether there exists a better solution.
		Examples are the traveling salesman problem where we want to know if there is a shorter route that goes through all the nodes in a certain network and the satisfiability problem where we want to know if a certain formula in propositional logic with propositional variables is satisfiable or not.
		Whether this is really true or not, is still one of the big open questions in computer science (see Complexity classes P and NP for an in-depth discussion).
	www.encyclopedia-1.com /n/np/np__complexity_.html (378 words)

	Encyclopedia: Complexity class
		In computational complexity theory, a complexity class is a set of problems of related complexity.
		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.
	www.nationmaster.com /encyclopedia/Complexity-class (178 words)

	Rice Algorithms & Complexity Group (Site not responding. Last check: 2007-10-14)
		The computational complexity of a problem is the amount of resources, such as time or space, required by a machine that solves the problem.
		This intimate connection was first discovered by Fagin, who showed that the complexity class NP coincides with the class of properties of finite structures expressible in existential second-order logic.
		The next discovery was by Immerman and Vardi, who proved that the complexity class P coincides with the class of properties of finite ordered structures expressible in fixpoint logic.
	www.cs.rice.edu /CS/Algorithms/complexity_theory.html (235 words)

	Dictionary of Meaning www.mauspfeil.net
		Many of these classes have a 'Co' partner which consists of the complements of all languages in the original class.
		PH (complexity) PHThe union of the classes in the polynomial hierarchy - P
		In October 2004 it was discovered that this class is in fact equal to L (complexity) L.
	www.mauspfeil.net /List_of_complexity_classes.html (731 words)

	PlanetMath: complexity class
		The most common classes are all restricted to one read-only input tape and one output/work tape (and in some cases a one-way, read-only guess tape) and are defined as follows:
		Cross-references: complexity classes, cells, polynomial time, polynomial, union, time complexity, minimal error, two-sided error, negative, one-sided error, positive, non-deterministic Turing machines, deterministic Turing machines, one-way, restricted, search problem, decides, class, decision problem, length, bounded, Turing machine, function
		This is version 1 of complexity class, born on 2002-09-06.
	planetmath.org /encyclopedia/ComplexityClass.html (210 words)

	NOTES ON COMPLEXITY AND NP COMPLETENESS
		BACKGROUND P is the class of decision problems that are solvable by algorithms that run in polynomial time (as a function of input size).
		THE CLASS NP The above examples indicate the potential interest of problems that, although they may be hard to solve by algorithms that run fast, they can be verified quickly.
		And of course, P is a subset of NP.
	www.cs.umd.edu /class/spring2003/cmsc351/notes/complexity.html (1866 words)

	Math Thesis: P versus NP
		The problem of P versus NP is essentially an in depth look at many hard problems and their relations to easier problems.
		Complexity classes are used to show a hierarchy of the complexities of problems.
		Whether or not the grand mystery of the complexity classes P and NP will ever be solved, the discoveries that have arisen and the ones yet to be found in the exploration of computation theory will continue to be significant contributions to both computer science and mathematics.
	www.sccs.swarthmore.edu /users/01/jimmy/writing/thesis.htm (3596 words)

	comp7713 NP-Completeness (Site not responding. Last check: 2007-10-14)
		Complexity Class P is the set of all problems that are solvable by polynomial time algorithms.
		Only if the time complexity of the reduction algorithm is polynomial in the length of the instance, the polynomial complexity of the problem is preserved by the reduction.
		If the complexity of reduction algorithm itself is larger than that of the reduced problem, then it will dominate the complexity of the whole algorithm, and becomes less useful.
	www.msci.memphis.edu /~giri/7713/f99/jizhou2.htm (927 words)

	EXPSPACE - Wikipedia, the free encyclopedia
		The complexity class EXPSPACE-complete is also a set of decision problems.
		EXPSPACE is a strict superset of PSPACE, NP-complete, NP, and P and is believed to be a strict superset of EXPTIME.
		An example of an EXPSPACE-complete problem is the problem of recognizing whether two regular expressions represent different languages, where the expressions are limited to four operators: union, concatenation, the Kleene star (zero or more copies of an expression), and squaring (two copies of an expression).
	en.wikipedia.org /wiki/EXPSPACE (302 words)

	complexity class (Site not responding. Last check: 2007-10-14)
		complexity class of decision problems for which answers can be checked by an algorithm whose run time is polynomial in the size of the input.
		NP) and no other NP problem is more than a polynomial factor harder.
		complexity class of decision problems that are intrinsically harder than those that can be solved by a nondeterministic Turing machine in
	www.cs.binghamton.edu /~hzeng/CS552/complexity%20class.htm (796 words)

	The Complexity Zoo (Site not responding. Last check: 2007-10-14)
		The class of function problems of the form "compute f(X)," where f is the number of accepting paths of an NP machine.
		The class of problems solvable by a BPP machine that is given O(log n) advice bits, which can depend on both the machine's random coin flips and the input length n, but not on the input itself.
		The class of NPO problems which have the property that for all locally optimal solutions, the ratio between the values of the local and global optima is upper-bounded by a constant.
	www.complexityzoo.com (11024 words)

	NP - Wikipedia, the free encyclopedia
		The complexity class NP in computational complexity theory
		NP is also a book by Japanese author Yoshimoto Banana.
		This is a disambiguation page — a navigational aid which lists pages that might otherwise share the same title.
	www.wikipedia.org /wiki/NP (107 words)

	NP-Complete - Wikipedia
		In complexity theory, the complexity class NP-complete is the set of problems that are the hardest problems in NP, in the sense that they are the ones most likely not to be in P.
		One example of an NP-complete problem is Subset Sum which is: given a finite set of integers, determine whether any subset of them adds up to zero.
		Assuming that P and NP are not equal, there are guaranteed to be an infinite number of problems that are in NP, but are neither NP-complete nor in P.
	nostalgia.wikipedia.org /wiki/NP-Complete (530 words)

	NP (complexity) (Site not responding. Last check: 2007-10-14)
		In computational complexitytheory, NP ("non-deterministic polynomial-time") is the set of decision problems solvable in polynomial time on a non-deterministic Turing machine.
		In all these cases there is a certificate (apath that is indeed shorter, a truth assignment that makes the formula true, a divisor greater than one) that is limited in sizeand for which we can decide in polynomial time whether it solves the problem.
		Whether this is really true or not, is still one of the bigopen questions in computer science (see Complexity classes P and NP for an in-depthdiscussion).
	www.therfcc.org /RFCC/np-complexity--45967.html (280 words)

	Approaches to Complexity Engineering (1986)
		In both of these cases, much more complex behaviour could be obtained from the basic components, whether mechanical or logical, but the principles necessary to make use of such behaviour are not yet known.
		Complexity in natural systems typically arises from the collective effect of a very large number of components.
		There are many NP complete problems, all equivalent in the computational difficulty of their exact solution [7].
	www.stephenwolfram.com /publications/articles/ca/86-approaches/2/text.html (7522 words)

	Greedy Algorithms
		The class NP correspond to the decision problems that have an efficient proof system, which means that each yes-instance must have at least one certificate whose validity can be verified quickly.
		NP is the class of decision problems X that admit a proof system F subset X × Q such that there exists a polynomial p(n) and a polynomial-time algorithm A such that
		Complexity class co-NP This is the set of languages L such that L belongs to NP.
	www.personal.kent.edu /~rmuhamma/Algorithms/MyAlgorithms/Complexity/npComplete.htm (3320 words)

	PlanetMath: counting complexity class
		is a complexity class associated with non-deterministic machines then
		is the class of counting problems associated with
		This is version 1 of counting complexity class, born on 2002-09-07.
	planetmath.org /encyclopedia/CountingComplexityClass.html (65 words)

	COMPLEXITY CLASS
		Complexity class A collection of algorithms or computable functions with the same complexity.
		Any of a set of computational problems with the same bounds ((n)) on time and space, for deterministic and nondeterministic machines.
		Specialty definitions using "COMPLEXITY CLASS": NP-complete, NP-hard &diams; polynomial-time Church-Turing thesis ♦ subadditive ergodic theorem.
	www.websters-online-dictionary.org /co/complexity+class.html (222 words)

	P, NP, CO-NP, NP-complete, NP-hard (Site not responding. Last check: 2007-10-14)
		NP is the set of decision problems solvable in polynomial time on a nondeterministic Turing machine.
		The complexity class NP-complete is the set of problems that are the hardest problems in NP, in the sense that they are the ones most likely not to be in P. If you can find a way to solve an NP-complete problem quickly, then you can use that algorithm to solve all NP problems quickly.
		If you could reduce an NP problem to an NP-hard problem and then solve it in polynomial time, you could solve all NP problems.
	www.cc.gatech.edu /~howardz/micellaneous/gre_cs_sub/np_complete.htm (210 words)

	Complexity Theory 00-01
		In classes we spoke on the Halting problem, on Universal Turing machines and RE sets in general.
		In the b-question approximately 0.1 for spelling out the situation for example by using a diagram, 0.2 for a serious attempt with a typical complexity theory structure (suppose it were...., now consider...., apply this to itself and behold).
		In class on tuesday a hisorical treatment has been given from Leibniz's dream to the solution of the "Entscheidungsproblem" by A. Church.
	www.phil.uu.nl /~jjoosten/compl/compl.html (1527 words)

	INI Programme LAA Workshop - New Diretions in Proof Complexity
		Proof complexity is an area of mathematics (and mathematical logic and computational complexity theory in particular) centered around the problem whether the complexity class NP is closed under complementation.
		The ultimate goal of proof complexity is to show that there is no such proof system; that is, to demonstrate superpolynomial lower bounds for all proof systems.
		Our ambition is to expose, through invited and contributed lectures, current developments in proof complexity as well as new ideas and directions of research pursued most recently.
	www.newton.cam.ac.uk /programmes/LAA/laaw04.html (308 words)

	Canon Np 3050 (Site not responding. Last check: 2007-10-14)
		A rule adopted by an ecumenical council of the Catholic or EasternOrthodox churches.
		NP was the AAR reporting mark for the Northern PacificRailway.
		NP is also a book byJapanese author Yoshimoto Banana.
	www.witchware.com /File/44221-Canon.Np.3050.Html (371 words)

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