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♯P
In computational complexity theory, the complexity class #P (pronounced "sharp P" or, sometimes "number P" or "hash P") is the set of the counting problems associated with the decision problems in the set NP. More formally, #P is the class of function problems of the form "compute ''f''(''x'')", where ''f'' is the number of accepting paths of a nondeterministic Turing machine running in polynomial time. Unlike most well-known complexity classes, it is not a class of decision problems but a class of function problems. The most difficult, representative problems of this class are #P-complete. Relation to decision problems An NP decision problem is often of the form "Are there any solutions that satisfy certain constraints?" For example: * Are there any subsets of a list of integers that add up to zero? (subset sum problem) * Are there any Hamiltonian cycles in a given graph with cost less than 100? (traveling salesman problem) * Are there any variable assignments that satisfy a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
♯P-complete
The #P-complete problems (pronounced "sharp P complete", "number P complete", or "hash P complete") form a complexity class in computational complexity theory. The problems in this complexity class are defined by having the following two properties: *The problem is in #P, the class of problems that can be defined as counting the number of accepting paths of a polynomial-time non-deterministic Turing machine. *The problem is #P-hard, meaning that every other problem in #P has a Turing reduction or polynomial-time counting reduction to it. A counting reduction is a pair of polynomial-time transformations from inputs of the other problem to inputs of the given problem and from outputs of the given problem to outputs of the other problem, allowing the other problem to be solved using any subroutine for the given problem. A Turing reduction is an algorithm for the other problem that makes a polynomial number of calls to a subroutine for the given problem and, outside of those calls, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counting Problem (complexity)
In computational complexity theory and computability theory, a counting problem is a type of computational problem. If ''R'' is a search problem then :c_R(x)=\vert\\vert \, is the corresponding counting function and :\#R=\ denotes the corresponding decision problem. Note that ''cR'' is a search problem while #''R'' is a decision problem, however ''cR'' can be ''C'' Cook-reduced to #''R'' (for appropriate ''C'') using a binary search (the reason #''R'' is defined the way it is, rather than being the graph of ''cR'', is to make this binary search possible). Counting complexity class Just as NP has NP-complete In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any ... problems via many-one reductions, #P has #P-complete problems via parsimonious reductions, problem transformatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. 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 logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permanent Is Sharp-P-complete , Buddhist concept
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{{disambiguation ...
Permanent may refer to: Art and entertainment * ''Permanent'' (film), a 2017 American film * ''Permanent'' (Joy Division album) * "Permanent" (song), by David Cook *"Permanent", a song by Alex Lahey from ''The Answer Is Always Yes'', 2023 Other uses *Permanent (mathematics), a concept in linear algebra *Permanent (cycling event) *Permanent wave, a hairstyling process See also *Permanence (other) *''Permanently'', a 2000 album by Mark Wills *Endless (other) *Eternal (other) *Forever (other) *Impermanence Impermanence, also known as the philosophical problem of change, is a philosophical concept addressed in a variety of religions and philosophies. In Eastern philosophy it is notable for its role in the Buddhism, Buddhist three marks of existe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Matrix
In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as Shear mapping, shearing or Rotation (mathematics), rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and \mathbf is a column vector describing the Position (vector), position of a point in space, the product R\mathbf yields another column vector describing the position of that point after that rotation. If \mathbf is a row vector, the same transformation can be obtained using where R^ is the transpose of Main diagonal The entries a_ () form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permanent (mathematics)
In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general function of a matrix called the immanant. Definition The permanent of an matrix is defined as \operatorname(A)=\sum_\prod_^n a_. The sum here extends over all elements σ of the symmetric group ''S''''n''; i.e. over all permutations of the numbers 1, 2, ..., ''n''. For example, \operatorname\begina&b \\ c&d\end=ad+bc, and \operatorname\begina&b&c \\ d&e&f \\ g&h&i \end=aei + bfg + cdh + ceg + bdi + afh. The definition of the permanent of ''A'' differs from that of the determinant of ''A'' in that the signatures of the permutations are not taken into account. The permanent of a matrix A is denoted per ''A'', perm ''A'', or Per ''A'', sometimes with parentheses around the argument. Minc uses Per(''A'') for the permanent of rectangula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leslie Valiant
Leslie Gabriel Valiant (born 28 March 1949) is a British American computer scientist and computational theorist. He was born to a chemical engineer father and a translator mother. He is currently the T. Jefferson Coolidge Professor of Computer Science and Applied Mathematics at Harvard University. Valiant was awarded the Turing Award in 2010, having been described by the Association for Computing Machinery, A.C.M. as a heroic figure in theoretical computer science and a role model for his courage and creativity in addressing some of the deepest unsolved problems in science; in particular for his "striking combination of depth and breadth". Education Valiant was educated at King's College, Cambridge, Imperial College London, and the University of Warwick where he received a PhD in computer science in 1974. Research and career Valiant is world-renowned for his work in Theoretical Computer Science. Among his many contributions to computational complexity theory, Complexity Theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deterministic Turing Machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algorithm. The machine operates on an infinite memory tape divided into discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states. At each step of its operation, the head reads the symbol in its cell. Then, based on the symbol and the machine's own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right, or halts the computation. The choice of which replacement symbol to write, which direction to move the head, and whether to halt is based on a finite table that specif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Certificate (complexity)
In computational complexity theory, a certificate (also called a witness) is a string that certifies the answer to a computation, or certifies the membership of some string in a language. A certificate is often thought of as a solution path within a verification process, which is used to check whether a problem gives the answer "Yes" or "No". In the decision tree model of computation, certificate complexity is the minimum number of the n input variables of a decision tree that need to be assigned a value in order to definitely establish the value of the Boolean function f. Use in definitions The notion of certificate is used to define semi-decidability: a formal language L is semi-decidable if there is a two-place predicate relationR \subseteq \Sigma^* \times \Sigma^* such that R is computable, and such that for all x \in \Sigma^*: x ∈ L ⇔ there exists y such that R(x, y) Certificates also give definitions for some complexity classes which can alternatively be chara ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity P
In computational complexity theory, the complexity class ⊕P (pronounced "parity P") is the class of decision problems solvable by a nondeterministic Turing machine in polynomial time, where the acceptance condition is that the number of accepting computation paths is odd. An example of a ⊕P problem is "does a given graph have an odd number of perfect matchings?" The class was defined by Papadimitriou and Zachos in 1983. An example of a ⊕P-complete problem (under many-one reductions) is ⊕SAT: given a Boolean formula, is the number of its satisfying assignments odd? This follows from the Cook–Levin theorem because the reduction is parsimonious. ⊕P is a counting class, and can be seen as finding the least significant bit of the answer to the corresponding #P problem. The problem of finding the most significant bit is in PP. PP is believed to be a considerably harder class than ⊕P; for example, there is a relativized universe (see oracle machine) where P = ⊕P ≠NP = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PP (complexity)
In complexity theory, PP, or PPT is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of less than 1/2 for all instances. The abbreviation PP refers to probabilistic polynomial time. The complexity class was defined by Gill in 1977. If a decision problem is in PP, then there is an algorithm running in polynomial time that is allowed to make random decisions, such that it returns the correct answer with chance higher than 1/2. In more practical terms, it is the class of problems that can be solved to any fixed degree of accuracy by running a randomized, polynomial-time algorithm a sufficient (but bounded) number of times. Turing machines that are polynomially-bound and probabilistic are characterized as PPT, which stands for probabilistic polynomial-time machines. This characterization of Turing machines does not require a bounded error probability. Hence, PP is the complexity class containing all problems s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
