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Low And High Hierarchies
In the 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 ..., the low hierarchy and high hierarchy of complexity levels were introduced in 1983 by Uwe Schöning to describe the internal structure of the complexity class NP. The low hierarchy starts from complexity class P and grows "upwards", while the high hierarchy starts from class NP and grows "downwards". Later these hierarchies were extended to sets outside NP. The framework of high/low hierarchies makes sense only under the assumption that P is not NP. On the other hand, if the low hierarchy consists of at least two levels, then P is not NP.Lane A. Hemaspaandra, "Lowness: a yardstick for NP-P", ''ACM SIGACT News'', 1993, vol. 24, no.2, pp. 11-14. It is not known whether these hierarchies co ... [...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] |
Uwe Schöning
Uwe Schöning (born 28 December 1955) is a retired German computer scientist, known for his research in computational complexity theory. Education and career Schöning earned his Ph.D. from the University of Stuttgart in 1981, under the supervision of Wolfram Schwabhäuser. He was a professor in the Institute for Theoretical Informatics at the University of Ulm until his retirement in 2021. Contributions Schöning introduced the low and high hierarchies to structural complexity theory in 1983. As Schöning later showed in a 1988 paper, these hierarchies play an important role in the complexity of the graph isomorphism problem, which Schöning further developed in a 1993 monograph with Köbler and Toran. In a 1999 FOCS paper, Schöning showed that WalkSAT, a randomized algorithm previously analyzed for 2-satisfiability by Papadimitriou, had good expected time complexity (although still exponential) when applied to worst-case instances of 3-satisfiability and other NP-complete c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexity Class NP
In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.''Polynomial time'' refers to how quickly the number of operations needed by an algorithm, relative to the size of the problem, grows. It is therefore a measure of efficiency of an algorithm. * NP is the set of decision problems ''solvable'' in polynomial time by a nondeterministic Turing machine. * NP is the set of decision problems ''verifiable'' in polynomial time by a deterministic Turing machine. The first definition is the basis for the abbreviation NP; " nondeterministic, polynomial time". These two definitions are equivalent because the algorithm based on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexity Class P
In computational complexity theory, P, also known as PTIME or DTIME(''n''O(1)), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. Cobham's thesis holds that P is the class of computational problems that are "efficiently solvable" or " tractable". This is inexact: in practice, some problems not known to be in P have practical solutions, and some that are in P do not, but this is a useful rule of thumb. Definition A language ''L'' is in P if and only if there exists a deterministic Turing machine ''M'', such that * ''M'' runs for polynomial time on all inputs * For all ''x'' in ''L'', ''M'' outputs 1 * For all ''x'' not in ''L'', ''M'' outputs 0 P can also be viewed as a uniform family of Boolean circuits. A language ''L'' is in P if and only if there exists a polynomial-time uniform family of Boolean circuits \, such that * For all n \in \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P Is Not NP
P, or p, is the sixteenth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''pee'' (pronounced ), plural ''pees''. History The Semitic Pê (mouth), as well as the Greek Π or π ( Pi), and the Etruscan and Latin letters that developed from the former alphabet all symbolized , a voiceless bilabial plosive. Use in writing systems English In English orthography, represents the sound . A common digraph in English is , which represents the sound , and can be used to transliterate ''phi'' in loanwords from Greek. In German, the digraph is common, representing a labial affricate . Most English words beginning with are of foreign origin, primarily French, Latin and Greek; these languages preserve the Proto-Indo-European initial *p. Native English cognates of such words often start with , since English is a Germanic language and thus has undergone Grimm's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
