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Topic: Complexity classes P and NP

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	Computational complexity theory - Wikipedia, the free encyclopedia
		A complexity class is the set of all of the computational problems which can be solved using a certain amount of a certain computational resource.
		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.
	en.wikipedia.org /wiki/Computational_complexity_theory (1589 words)

	P - Biocrawler (Site not responding. Last check: 2007-09-18)
		In biochemistry, P is the symbol for proline.
		In chemistry, P is the symbol for the element phosphorus, or sometimes for phosphate.
		In complexity theory, P is the set of decision problems solvable in polynomial time on a deterministic Turing machine; see complexity classes P and NP.
	www.biocrawler.com /encyclopedia/P (601 words)

	Complexity classes P and NP - Wikipedia, the free encyclopedia
		Computational complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem.
		However, it should be noted that the definition of P and NP are in terms of classical computing models like Turing machines.
		All languages in P can be expressed in first-order logic with the addition of a least fixed point operator (effectively, this allows the definition of recursive functions).
	en.wikipedia.org /wiki/Complexity_classes_P_and_NP (2868 words)

	P (complexity) - (Site not responding. Last check: 2007-09-18)
		In computational complexity theory, P is the complexity class containing 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.
		P is also known to be at least as large as L, the class of problems decidable in a logarithmic amount of memory space.
	psychcentral.com /psypsych/P_(complexity) (682 words)

	P - Wikipedia
		Semitic Pê (mouth) as well as Greek Π or π (Pi) and the Etruscan and Latin letters that developed from the former alphabet all symbolized /p/.
		In chemistry, P is the symbol for the element phosphorus.
		In complexity theory, P is the set of decision problems solvable in polynomial time on a deterministic Turing machine.
	wikipedia.findthelinks.com /pa/P.html (162 words)

	Math Thesis: P versus NP
		Most theorists and mathematicians who have studied P and NP believe that there exist problems that could be checked in polynomial time (in NP), but requires beyond polynomial time to solve (not in P).
		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)

	NP-hard - Wikipedia, the free encyclopedia
		Informally, this class can be described as containing the decision problems that are at least as hard as any problem in NP.
		Furthermore, one should also remember that polynomial complexity problems are contained in the complexity class NP (though are not NP-hard unless P=NP).
		For example the Boolean satisfiability problem can be reduced to the halting problem by transforming it to the description of a Turing machine that tries all truth value assignments and when it finds one that satisfies the formula it halts and otherwise it goes into an infinite loop.
	en.wikipedia.org /wiki/NP-hard (529 words)

	CS385, NP Completeness
		P denotes the set of all languages that are solvable in polynomial time.
		NP denotes the set of all languages that are solvable in nondeterministic polynomial time.
		Verifier viewpoint for NP verifier for a language L is a deterministic Turing machine V such that, for any string w, w is in L if and only if there is a string z (known as a certificate for w) such that V accepts on input .
	www.cs.bc.edu /~alvarez/Theory/npcompleteness.html (1541 words)

	RP
		P is a subset of RP, which is a subset of NP.
		Similarly, P is a subset of Co-RP which is a subset of Co-NP.
		The class BQP is based on another machine with randomness: the quantum computer.
	www.askfactmaster.com /RP (384 words)

	NP-hard
		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.
		It is also easy to see that the halting problem is not in NP since all problems in NP are decidable and the halting problem is not.
	www.ebroadcast.com.au /lookup/encyclopedia/np/NP-hard.html (597 words)

	Complexity classes P and NP (Site not responding. Last check: 2007-09-18)
		The biggest open question in theoretical computer science concerns the relationship between those two classes: :Is Most people think that the answer is probably "no"; some people believe the question may be undecidable from the currently accepted axioms.
		Informally, the NP -complete problems are the "toughest" problems in NP in the sense that they are the ones most likely not to be in P.
		This means that if a single NP -complete problem could be shown to be in P, then it would follow that P = NP Unfortunately, many important problems have been shown to be NP -complete and not a single fast algorithm for any of them is known.
	www.serebella.com /encyclopedia/article-Complexity_classes_P_and_NP.html (2110 words)

	Computational complexity theory From Wikipedia (Site not responding. Last check: 2007-09-18)
		Complexity Theory is a part of the theory of computation dealing with the resources required during computation to solve a given problem.
		The time complexity of a problem is the number of steps that it takes to solve an instance, as a function of the size of the instance.
		The set P is the set of decision problems that can be solved in polynomial time.
	www.xtrj.org /ssm4/complexity_theory.htm (765 words)

	P (Site not responding. Last check: 2007-09-18)
		In physics, p is the symbol for momentum, also for pressure; P for power and polarisation; p is also the symbol for the proton.
		In units the Pico SI Prefix p stands for pico, P for Peta.
		In statistics, the p-value of a given result in an experiment the is result's significance; that is, the probability of observing this result or a more extreme result, by chance alone.
	www.icyclopedia.com /encyclopedia/p/pp/p.html (278 words)

	Science Fair Projects - Complexity classes P and NP
		Complexity classes P and NP (Redirected from P=NP)
		(See NP-complete for the exact definition.) Theoretical computer scientists currently believe that the relationship among the classes P, NP, and NPC is as shown in the picture, with the P and NPC classes disjoint.
		The P = NP question has also been addressed using oracles.
	all-science-fair-projects.com /science_fair_projects_encyclopedia/P=NP (1969 words)

	[12pt,letterpaper] A note on the power of the counting class \#P
		P represents the class of languages (problems with variables) that can be solved by a polynomial time bounded Turing machine (computer program.) NP represents the class of languages whose solutions that can verified by a polynomial time Turing machine.
		The problems in P are such that their solutions are easily verifiable, yet proof that there exists some solution to the problem is considered a hard task.
		In fact, if P is not equal to NP, and NP is not equal to PSPACE, then checking that a solution exists to an instance of a probelm in PSPACE is inherently easier than verifying a solution to a particular instance (i.e.
	www.cs.cmu.edu /~ryanw/project.html (2259 words)

	NP
		In complexity theory, NP ("Non-deterministic Polynomial-time") is the set of decision problems solvable in polynomial time on a non-deterministic Turing machine.
		See Complexity classes P and NP and NP-Complete.
		Alternate use: NP is also the ISO country code for Nepal.
	www.ebroadcast.com.au /lookup/encyclopedia/np/NP.html (80 words)

	PvsNp - PineWiki
		The P vs NP problem is the most famous open problem in Computer Science (specifically in ComputationalComplexityTheory), and one of the most famous outstanding problems in mathematics in general.
		The classes P and NP See ComputationalComplexityTheory for definitions of languages, complexity classes, etc. The definitions of the complexity classes P and NP are:
		Intuitively, a problem is in NP if there is a verifiable proof that the answer is yes for any yes instance of the problem.
	pine.cs.yale.edu /pinewiki/PvsNp (909 words)

	Complexity classes P and NP Info - Encyclopedia WikiWhat.com (Site not responding. Last check: 2007-09-18)
		This means that if a single NP-complete problem could be shown to be in P, then it would follow that P = NP.
		This means it requires exponential time, and so is outside P and NP.
		is not P (in fact, it's exponential time), but is very tractable for values of n up into the thousands.
	www.wikiwhat.com /encyclopedia/c/co/complexity_classes_p_and_np.html (1480 words)

	[No title]
		My work is mainly centered around complexity theory, and inside this area around algebraic models of computation and their analysis.
		It thus can be used in order to define analogues of well known discrete complexity classes like P and NP over general structures and deal with complete problems as well.
		Optimization problems are of major ``theoretical'' interest within the algebraic approaches towards complexity; their behavior under such models seems to be much different from the one when the Turing machine is used.
	www.imada.sdu.dk /~meer/research.html (692 words)

	Subtractive Reductions and Complete Problems for Counting Complexity Classes - Durand, Hermann, Kolaitis (ResearchIndex) (Site not responding. Last check: 2007-09-18)
		We show that the main counting complexity classes #P, #NP, as well as all higher counting complexity classes # P k, k 2, are closed under subtractive reductions.
		We focus on the class #NP (which is the same as the class # coNP) and show that it contains natural complete problems via...
		On the Complexity of Counting the Hilbert Basis of a..
	citeseer.ist.psu.edu /296914.html (649 words)

	Facts about polynomial time (Site not responding. Last check: 2007-09-18)
		In Complexity theory, Polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n.
		The class of decision problems that can be verified in polynomial time is known as NP.
		Equivalently, NP is the class of decision problems that can be solved in polynomial time on a Non-deterministic Turing machine (NP stands for Nondeterministic Polynomial time).
	www.supercrawler.com /Facts/polynomial_time.html (146 words)

	EXPTIME - TheBestLinks.com - Chess, Checkers, Complexity classes P and NP, Computational complexity theory, ... (Site not responding. Last check: 2007-09-18)
		In computational complexity theory, the complexity class EXPTIME (sometimes called EXP) is the set of all decision problems solvable by a deterministic Turing machine in O(2
		P ⊆ NP &sube; PSPACE &sube; EXPTIME &sube; EXPSPACE
		The complexity class EXPTIME-complete is also a set of decision problems.
	www.thebestlinks.com /EXP.html (273 words)

	NP (Site not responding. Last check: 2007-09-18)
		Definition: The 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" is the class that a Nondeterministic Turing machine accepts in Polynomial time.
		Being in NP doesn't require that a negative answer, e.g.
	www.nist.gov /dads/HTML/np.html (262 words)

	[No title] (Site not responding. Last check: 2007-09-18)
		General overview of complexity classes P and NP, including upper, average, and lower bound analysis.
		Complexity classes P, NP, P-Space; tractable and intractable problems, existence of methods for obtaining approximate solutions to intractable problems
		Stu dents will gain an understanding of complexity classes, and a further appreciation for asymptotic measurements of complexity and the difference between average and worst case analysis.
	cs.wwc.edu /KU/AL5.html (96 words)

	MEMO
		Complexity classes P and NP (tractable and intractable problems) and verification of algorithms by formal methods are also discussed.
		Part of the grade will depend on in-class determination via quizzes and exams that students by the end of semester have the basic skills listed on page 1 of the syllabus.
		Students who seek accommodations during the semester because of disabilities should meet with the instructor after class or during office hours early in the semester.
	www.framingham.edu /faculty/dkeil/alg-syllabus.htm (926 words)

	[No title]
		Version 2: NP is the set of problems such that YES instances have short proofs that the answer is yes.
		It's as least as hard as any problem in NP since in a sense any other problem is a special case of it.
		First of all, it's in NP since we can just guess the inputs and run the circuit in time linear in its size.
	www.cs.cmu.edu /afs/cs/academic/class/15451-f99/www/lectures/lect1014 (1255 words)

	CSCI546+646: Syllabus
		The preparation for two classes requires you to search the web for topics and submit a relevant URL that you have discovered as assigned work/study.
		The will be a short test of the material in the first chapter of the book (math) in the third class -- it contributes 50 points(10%) to the total grade for the class.
		To earn all 5 points you must: (1) turn up and be ready to take part at the start of the class, (2) stay until the class is dismissed, (3) Answer questions, (4) ask questions, and (5) take part in class discussions and exercises.
	www.csci.csusb.edu /dick/cs546/syllabus.html (1036 words)

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