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Toda's Theorem
Toda's theorem is a result in computational complexity theory that was proven by Seinosuke Toda in his paper "PP is as Hard as the Polynomial-Time Hierarchy" and was given the 1998 Gödel Prize. Statement The theorem states that the entire polynomial hierarchy PH is contained in PPP; this implies a closely related statement, that PH is contained in P#P. Definitions #P is the problem of exactly counting the number of solutions to a polynomially-verifiable question (that is, to a question in NP), while loosely speaking, PP is the problem of giving an answer that is correct more than half the time. The class P#P consists of all the problems that can be solved in polynomial time if you have access to instantaneous answers to any counting problem in #P (polynomial time relative to a #P oracle An oracle is a person or thing considered to provide insight, wise counsel or prophetic predictions, most notably including precognition of the future, inspired by deities. If done throu ... [...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] |
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] |
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] |
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] |
NP (complexity)
In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the Set (mathematics), set of decision problems for which the Computational complexity theory#Problem instances, problem instances, where the answer is "yes", have mathematical proof, 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 alg ... [...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] |
Baker-Gill-Solovay Theorem
In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a black box, called an oracle, which is able to solve certain problems in a single operation. The problem can be of any complexity class. Even undecidable problems, such as the halting problem, can be used. Oracles An oracle machine can be conceived as a Turing machine connected to an oracle. The oracle, in this context, is an entity capable of solving some problem, which for example may be a decision problem or a function problem. The problem does not have to be computable; the oracle is not assumed to be a Turing machine or computer program. The oracle is simply a "black box" that is able to produce a solution for any instance of a given computational problem: * A decision problem is represented as a set ''A'' of natural numbers (or strings). An instance of the problem is an arbitrary natural number (or string). The solution to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Turing 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] |
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] |
Blum–Shub–Smale Machine
In computation theory, the Blum–Shub–Smale machine, or BSS machine, is a model of computation introduced by Lenore Blum, Michael Shub and Stephen Smale, intended to describe computations over the real numbers. Essentially, a BSS machine is a Random Access Machine with registers that can store arbitrary real numbers and that can compute rational functions over reals in a single time step. It is closely related to the Real RAM model. BSS machines are more powerful than Turing machines, because the latter are by definition restricted to a finite set of symbols. A Turing machine can represent a countable set (such as the rational numbers) by strings of symbols, but this does not extend to the uncountable real numbers. Definition A BSS machine M is given by a list I of N+1 instructions (to be described below), indexed 0, 1, \dots, N. A configuration of M is a tuple (k,r,w,x), where k is the index of the instruction to be executed next, r and w are registers holding non-negative int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
