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Structural Complexity Theory
In computational complexity theory of computer science, the structural complexity theory or simply structural complexity is the study of complexity classes, rather than computational complexity of individual problems and algorithms. It involves the research of both internal structures of various complexity classes and the relations between different complexity classes.Juris Hartmanis, "New Developments in Structural Complexity Theory" (invited lecture), Proc. 15th International Colloquium on Automata, Languages and Programming, 1988 (ICALP 88), ''Lecture Notes in Computer Science'', vol. 317 (1988), pp. 271-286. History The theory has emerged as a result of (still failing) attempts to resolve the first and still the most important question of this kind, the P = NP problem. Most of the research is done basing on the assumption of P not being equal to NP and on a more far-reaching conjecture that the polynomial time hierarchy of complexity classes is infinite. Important results ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Polynomial Time Hierarchy
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH. Classes within the hierarchy have complete problems (with respect to polynomial-time reductions) that ask if quantified Boolean formulae hold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level. Definitions There are multiple equivalent definitions of the classes of the polynomial hierarchy. Oracle definition For the oracle def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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P (complexity)
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Neil Immerman
Neil Immerman (born 24 November 1953, Manhasset, New York) is an American theoretical computer science, theoretical computer scientist, a professor of computer science at the University of Massachusetts Amherst.Faculty directory: Neil Immerman Computer Science Department, University of Massachusetts Amherst, retrieved 2010-01-23. He is one of the key developers of descriptive complexity, an approach he is currently applying to research in model checking, database theory, and computational complexity theory. Professor Immerman is an editor of the ''SIAM Journal on Computing'' and of ''Logical Methods in Computer Science''. He received B.S. and M.S. degrees from Yale University in 1974 and his Ph.D. from Cornell University in 1980 under the supervision of Juris Hartmanis, a Turing Award winner at Cornell. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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PH (complexity)
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH. Classes within the hierarchy have complete problems (with respect to polynomial-time reductions) that ask if quantified Boolean formulae hold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level. Definitions There are multiple equivalent definitions of the classes of the polynomial hierarchy. Oracle definition For the oracle de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Gödel Prize
The Gödel Prize is an annual prize for outstanding papers in the area of theoretical computer science, given jointly by the European Association for Theoretical Computer Science (EATCS) and the Association for Computing Machinery Special Interest Group on Algorithms and Computational Theory ( ACM SIGACT). The award is named in honor of Kurt Gödel. Gödel's connection to theoretical computer science is that he was the first to mention the "P versus NP" question, in a 1956 letter to John von Neumann in which Gödel asked whether a certain NP-complete problem could be solved in quadratic or linear time. The Gödel Prize has been awarded since 1993. The prize is awarded alternately at ICALP (even years) and STOC (odd years). STOC is the ACM Symposium on Theory of Computing, one of the main North American conferences in theoretical computer science, whereas ICALP is the International Colloquium on Automata, Languages and Programming, one of the main Europe Europe is a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Seinosuke Toda
is a computer scientist working at the Nihon University in Tokyo. Toda earned his Ph.D. from the Tokyo Institute of Technology in 1992, under the supervision of Kojiro Kobayashi. He was a recipient of the 1998 Gödel Prize for proving Toda's theorem in computational complexity theory, which states that every problem in the polynomial hierarchy In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. ... has a polynomial-time Turing reduction to a counting problem. Notes Japanese computer scientists 20th-century Japanese mathematicians 21st-century Japanese mathematicians Theoretical computer scientists Gödel Prize laureates 1959 births Living people Academic staff of Nihon University {{Compu-scientist-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Space Complexity
The space complexity of an algorithm or a data structure is the amount of memory space required to solve an instance of the computational problem as a function of characteristics of the input. It is the memory required by an algorithm until it executes completely. This includes the memory space used by its inputs, called input space, and any other (auxiliary) memory it uses during execution, which is called auxiliary space. Similar to time complexity, space complexity is often expressed asymptotically in big ''O'' notation, such as O(n), O(n\log n), O(n^\alpha), O(2^n), etc., where is a characteristic of the input influencing space complexity. Space complexity classes Analogously to time complexity classes DTIME(f(n)) and NTIME(f(n)), the complexity classes DSPACE(f(n)) and NSPACE(f(n)) are the sets of languages that are decidable by deterministic (respectively, non-deterministic) Turing machines that use O(f(n)) space. The complexity classes PSPACE and NPSPACE allow f to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Walter Savitch
Walter John Savitch (February 21, 1943 – February 1, 2021) was best known for defining the complexity class NL (nondeterministic logarithmic space), and for Savitch's theorem, which defines a relationship between the NSPACE and DSPACE complexity classes. His work in establishing complexity classes has helped to create the background against which non-deterministic and probabilistic reasoning can be performed. He also did extensive work in the field of natural language processing and mathematical linguistics. He was focused on computational complexity as it applies to genetics and biology for over 10 years. Aside from his work in theoretical computer science, Savitch wrote a number of textbooks for learning to program in C/C++, Java, Ada, Pascal and others. Savitch received his PhD in mathematics from University of California, Berkeley in 1969 under the supervision of Stephen Cook. Since then he was a professor at University of California, San Diego The University o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Polynomial Hierarchy
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH. Classes within the hierarchy have complete problems (with respect to polynomial-time reductions) that ask if quantified Boolean formulae hold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level. Definitions There are multiple equivalent definitions of the classes of the polynomial hierarchy. Oracle definition For the oracle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bounded-error Probabilistic Polynomial
computational complexity theory, a branch of computer science, bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded by 1/3 for all instances. BPP is one of the largest ''practical'' classes of problems, meaning most problems of interest in BPP have efficient probabilistic algorithms that can be run quickly on real modern machines. BPP also contains P, the class of problems solvable in polynomial time with a deterministic machine, since a deterministic machine is a special case of a probabilistic machine. Informally, a problem is in BPP if there is an algorithm for it that has the following properties: *It is allowed to flip coins and make random decisions *It is guaranteed to run in polynomial time *On any given run of the algorithm, it has a probability of at most 1/3 of giving the wrong answer, whether the answer is YES or NO. Definition A languag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with A Mathematical Theory of Communication, a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Isolation Lemma
In theoretical computer science, the term isolation lemma (or isolating lemma) refers to randomized algorithms that reduce the number of solutions to a problem to one, should a solution exist. This is achieved by constructing random constraints such that, with non-negligible probability, exactly one solution satisfies these additional constraints if the solution space is not empty. Isolation lemmas have important applications in computer science, such as the Valiant–Vazirani theorem and Toda's theorem in computational complexity theory. The first isolation lemma was introduced by , albeit not under that name. Their isolation lemma chooses a random number of random hyperplanes, and has the property that, with non-negligible probability, the intersection of any fixed non-empty solution space with the chosen hyperplanes contains exactly one element. This suffices to show the Valiant–Vazirani theorem: there exists a randomized polynomial-time reduction from the satisfiability prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |