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Håstad's Switching Lemma
In computational complexity theory, Håstad's switching lemma is a key tool for proving lower bounds on the size of constant-depth Boolean circuits. It was first introduced by Johan Håstad to prove that AC0 Boolean circuits of depth ''k'' require size \exp(\Omega(n^)) to compute the parity function on n bits. He was later awarded the Gödel Prize for this work in 1994. The switching lemma describes the behavior of a depth-2 circuit under ''random restriction'', i.e. when randomly fixing most of the coordinates to 0 or 1. Specifically, from the lemma, it follows that a formula in conjunctive normal form (that is, an AND of ORs) becomes a formula in disjunctive normal form (an OR of ANDs) under random restriction, and vice versa. This "switching" gives the lemma its name. Statement Consider a width-w formula in disjunctive normal form F = C_1 \vee C_2 \vee \cdots \vee C_m , the OR of clauses C_\ell which are the AND of ''w'' literals (x_i or its negation \neg x_i ). For exampl ... [...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] |
Merrick L
Merrick may refer to: Places America * Merrick, New York, a hamlet and census-designated place * Merrick, West Springfield, a neighborhood in western Massachusetts * Merrick County, Nebraska * Merrick State Park, Wisconsin Antarctica * Merrick Mountains, Palmer Land, Antarctica * Merrick Glacier, Oates Land, Antarctica * Merrick Point, Marie Byrd Land, Antarctica Britain * Merrick (Galloway), a mountain in southern Scotland People * Merrick (surname) * Merrick (given name) * Chris Hughes (musician) (born 1954), also known as Merrick, British record producer and musician Arts and entertainment * Merrick Mayfair, a character in The Vampire Chronicles series by Anne Rice ** ''Merrick'' (novel), by Anne Rice * Merrick Baliton, one of the Wild Force Power Rangers in the Power Rangers universe * Merrick, Buffy's mentor in the 1992 film ''Buffy the Vampire Slayer'' * Antoc Merrick, a Rebel pilot and general in the film '' Rogue One: A Star Wars Story'' * Bob Merrick, the ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circuit Value Problem
The circuit value problem (or circuit evaluation problem) is the computational problem of computing the output of a given Boolean circuit on a given input. The problem is complete for P under uniform AC reductions. Note that, in terms of time complexity, it can be solved in linear time simply by a topological sort. The Boolean formula value problem (or Boolean formula evaluation problem) is the special case of the problem when the circuit is a tree. The Boolean formula value problem is complete for NC.Author's draft The problem is closely related to the Boolean satisfiability problem which is complete for NP and its complement, the propositional tautology problem, which is complete for co-NP. See also * Circuit satisfiability * Switching lemma In computational complexity theory, Håstad's switching lemma is a key tool for proving lower bounds on the size of constant-depth Boolean circuits. It was first introduced by Johan Håstad to prove that AC0, AC0 Boolean circuits ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circuit Satisfiability
In theoretical computer science, the circuit satisfiability problem (also known as CIRCUIT-SAT, CircuitSAT, CSAT, etc.) is the decision problem of determining whether a given Boolean circuit has an assignment of its inputs that makes the output true. In other words, it asks whether the inputs to a given Boolean circuit can be consistently set to 1 or 0 such that the circuit outputs 1. If that is the case, the circuit is called ''satisfiable''. Otherwise, the circuit is called ''unsatisfiable.'' In the figure to the right, the left circuit can be satisfied by setting both inputs to be 1, but the right circuit is unsatisfiable. CircuitSAT is closely related to Boolean satisfiability problem (SAT), and likewise, has been proven to be NP-complete. It is a prototypical NP-complete problem; the Cook–Levin theorem is sometimes proved on CircuitSAT instead of on the SAT, and then CircuitSAT can be reduced to the other satisfiability problems to prove their NP-completeness. The satisfiab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Circuit
In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible input length. Boolean circuits are defined in terms of the logic gates they contain. For example, a circuit might contain binary AND and OR gates and unary NOT gates, or be entirely described by binary NAND gates. Each gate corresponds to some Boolean function that takes a fixed number of bits as input and outputs a single bit. Boolean circuits provide a model for many digital components used in computer engineering, including multiplexers, adders, and arithmetic logic units, but they exclude sequential logic. They are an abstraction that omits many aspects relevant to designing real digital logic circuits, such as metastability, fanout, glitches, power consumption, and propagation delay variability. Formal definition In givi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Pure And Applied Logic
The ''Annals of Pure and Applied Logic'' is a peer-reviewed scientific journal published by Elsevier that publishes papers on applications of mathematical logic in mathematics, in computer science, and in other related disciplines.About Annals of Pure and Applied Logic The editors of ''Annals of Pure and Applied Logic'' include mathematicians Ulrich Kohlenbach at TU Darmstadt in Germany, Thomas Scanlon at |
Miklós Ajtai
Miklós Ajtai (born 2 July 1946) is a computer scientist at the IBM Almaden Research Center, United States. In 2003, he received the Knuth Prize for his numerous contributions to the field, including a classic sorting network algorithm (developed jointly with J. Komlós and Endre Szemerédi), exponential lower bounds, superlinear time-space tradeoffs for branching programs, and other "unique and spectacular" results. He is a member of the U.S. National Academy of Sciences. Selected results One of Ajtai's results states that the length of proofs in propositional logic of the pigeonhole principle for ''n'' items grows faster than any polynomial in ''n''. He also proved that the statement "any two countable structures that are second-order equivalent are also isomorphic" is both consistent with and independent of ZFC. Ajtai and Szemerédi proved the corners theorem, an important step toward higher-dimensional generalizations of the Szemerédi theorem. With Komlós and Sze ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Computing Systems
''Theory of Computing Systems'' is a peer-reviewed scientific journal published by Springer Verlag. Published since 1967 as ''Mathematical Systems Theory'' and since volume 30 in 1997 under its current title, it is devoted to publishing original research from all areas of theoretical computer science, such as computational complexity, algorithms and data structures, or parallel and distributed algorithms and architectures. It is published 8 times per year since 2018, although the frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ... varied in the past. References External links * Computer science journals Theoretical computer science Springer Science+Business Media academic journals 8 times per year journals {{Computer-science-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Sipser
Michael Fredric Sipser (born September 17, 1954) is an American theoretical computer scientist who has made early contributions to computational complexity theory. He is a professor of applied mathematics and was the dean of science at the Massachusetts Institute of Technology. Biography Sipser was born and raised in Brooklyn, New York and moved to Oswego, New York when he was 12 years old. He earned his BA in mathematics from Cornell University in 1974 and his PhD in engineering from the University of California at Berkeley in 1980 under the direction of Manuel Blum. He joined MIT's Laboratory for Computer Science as a research associate in 1979 and then was a Research Staff Member at IBM Research in San Jose. In 1980, he joined the MIT faculty. He spent the 1985–1986 academic year on the faculty of the University of California at Berkeley and then returned to MIT. From 2004 until 2014, he served as head of the MIT Mathematics department. He was appointed Interim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Saxe
James may refer to: People * James (given name) * James (surname) * James (musician), aka Faruq Mahfuz Anam James, (born 1964), Bollywood musician * James, brother of Jesus * King James (other), various kings named James * Prince James (other) * Saint James (other) Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Film and television * ''James'' (2005 film), a Bollywood film * ''James'' (2008 film), an Irish short film * ''James'' (2022 film), an Indian Kannada-language film * "James", a television episode of ''Adventure Time'' Music * James (band), a band from Manchester ** ''James'', US ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity Function
In Boolean algebra, a parity function is a Boolean function whose value is one if and only if the input vector has an odd number of ones. The parity function of two inputs is also known as the XOR function. The parity function is notable for its role in theoretical investigation of circuit complexity of Boolean functions. The output of the parity function is the parity bit. Definition The n-variable parity function is the Boolean function f:\^n\to\ with the property that f(x)=1 if and only if the number of ones in the vector x\in\^n is odd. In other words, f is defined as follows: :f(x)=x_1\oplus x_2 \oplus \dots \oplus x_n where \oplus denotes exclusive or. Properties Parity only depends on the number of ones and is therefore a symmetric Boolean function. The ''n''-variable parity function and its negation are the only Boolean functions for which all disjunctive normal forms have the maximal number of 2 ''n'' − 1 monomials of length ''n'' and al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
