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NE (complexity)
In computational complexity theory, the complexity class NE is the set of decision problems that can be solved by a non-deterministic Turing machine In theoretical computer science, a nondeterministic Turing machine (NTM) is a theoretical model of computation whose governing rules specify more than one possible action when in some given situations. That is, an NTM's next state is ''not'' comp ... in time O(''k''n) for some ''k''. NE, unlike the similar class NEXPTIME, is not closed under polynomial-time many-one reductions. See also * E (complexity). References {{DEFAULTSORT:Ne (Complexity) Complexity classes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complexity Class
In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. counting problems and function problems) and using other models of computation (e.g. probabilistic Turing machines, interactive proof systems, Boolean circuits, and quantum computers). The study of the relationships between complexity class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whether a given natural number is prime. Another is the problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?". The answer is either 'yes' or 'no' depending upon the values of ''x'' and ''y''. A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem. A decision procedure for the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" would give the steps for determining whether ''x'' evenly divides ''y''. One such algorithm is long division. If the remainder is zero the answer is 'yes', otherwise it is 'no'. A decision problem which can be solved by an algorithm is called ''decidable''. Decision problems typically appear in ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Non-deterministic Turing Machine
In theoretical computer science, a nondeterministic Turing machine (NTM) is a theoretical model of computation whose governing rules specify more than one possible action when in some given situations. That is, an NTM's next state is ''not'' completely determined by its action and the current symbol it sees, unlike a deterministic Turing machine. NTMs are sometimes used in thought experiments to examine the abilities and limits of computers. One of the most important open problems in theoretical computer science is the P versus NP problem, which (among other equivalent formulations) concerns the question of how difficult it is to simulate nondeterministic computation with a deterministic computer. Background In essence, a Turing machine is imagined to be a simple computer that reads and writes symbols one at a time on an endless tape by strictly following a set of rules. It determines what action it should perform next according to its internal ''state'' and ''what symbol it cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '' Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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NEXPTIME
In computational complexity theory, the complexity class NEXPTIME (sometimes called NEXP) is the set of decision problems that can be solved by a non-deterministic Turing machine using time 2^. In terms of NTIME, :\mathsf = \bigcup_ \mathsf(2^) Alternatively, NEXPTIME can be defined using deterministic Turing machines as verifiers. A language ''L'' is in NEXPTIME if and only if there exist polynomials ''p'' and ''q'', and a deterministic Turing machine ''M'', such that * For all ''x'' and ''y'', the machine ''M'' runs in time 2^ on input * For all ''x'' in ''L'', there exists a string ''y'' of length 2^ such that * For all ''x'' not in ''L'' and all strings ''y'' of length 2^, We know : and also, by the time hierarchy theorem, that : If , then ( padding argument); more precisely, if and only if there exist sparse languages in NP that are not in P. Alternative characterizations NEXPTIME often arises in the context of interactive proof systems, where there are two maj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Polynomial-time Reduction
In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming or reducing it to inputs for the second problem and calling the subroutine one or more times. If both the time required to transform the first problem to the second, and the number of times the subroutine is called is polynomial, then the first problem is polynomial-time reducible to the second. A polynomial-time reduction proves that the first problem is no more difficult than the second one, because whenever an efficient algorithm exists for the second problem, one exists for the first problem as well. By contraposition, if no efficient algorithm exists for the first problem, none exists for the second either. Polynomial-time reductions are frequently used in complexity theory for defining both complexity classes and complete problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Many-one Reduction
In computability theory and computational complexity theory, a many-one reduction (also called mapping reduction) is a reduction which converts instances of one decision problem L_1 into instances of a second decision problem L_2 where the instance reduced to is in the language L_2 if the initial instance was in its language L_1 and is not in the language L_2 if the initial instance was not in its language L_1. Thus if we can decide whether L_2 instances are in the language L_2, we can decide whether L_1 instances are in its language by applying the reduction and solving L_2. Thus, reductions can be used to measure the relative computational difficulty of two problems. It is said that L_1 reduces to L_2 if, in layman's terms L_2 is harder to solve than L_1. That is to say, any algorithm that solves L_2 can also be used as part of a (otherwise relatively simple) program that solves L_1. Many-one reductions are a special case and stronger form of Turing reductions. With many-one re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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E (complexity)
In computational complexity theory, the complexity class E is the set of decision problems that can be solved by a deterministic Turing machine in time 2 O(''n'') and is therefore equal to the complexity class DTIME(2O(''n'')). E, unlike the similar class EXPTIME In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems that are solvable by a deterministic Turing machine in exponential time, i.e., in O(2''p''(''n'')) time, ..., is not closed under polynomial-time many-one reductions. References *. *. *. *. *. External links * {{comp-sci-theory-stub Complexity classes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |