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Geometric Complexity Theory
Geometric complexity theory (GCT), is a research program in computational complexity theory proposed by Ketan Mulmuley and Milind Sohoni. The goal of the program is to answer the most famous open problem in computer science – whether P = NP – by showing that the complexity class P is not equal to the complexity class NP. The idea behind the approach is to adopt and develop advanced tools in algebraic geometry and representation theory (i.e., geometric invariant theory) to prove lower bounds for problems. Currently the main focus of the program is on algebraic complexity classes. Proving that computing the permanent cannot be efficiently reduced to computing determinants is considered to be a major milestone for the program. These computational problems can be characterized by their symmetries. The program aims at utilizing these symmetries for proving lower bounds. The approach is considered by some to be the only viable currently active program to separate P fr ... [...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 relating these classes to each other. 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 gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction (complexity)
In computability theory and computational complexity theory, a reduction is an algorithm for transforming one problem into another problem. A sufficiently efficient reduction from one problem to another may be used to show that the second problem is at least as difficult as the first. Intuitively, problem ''A'' is reducible to problem ''B'', if an algorithm for solving problem ''B'' efficiently (if it existed) could also be used as a subroutine to solve problem ''A'' efficiently. When this is true, solving ''A'' cannot be harder than solving ''B''. "Harder" means having a higher estimate of the required computational resources in a given context (e.g., higher time complexity, greater memory requirement, expensive need for extra hardware processor cores for a parallel solution compared to a single-threaded solution, etc.). The existence of a reduction from ''A'' to ''B'', can be written in the shorthand notation ''A'' ≤m ''B'', usually with a subscript on the ≤ to indicate the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1209
Year 1209 ( MCCIX) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. Events By place Europe * May – The First Parliament of Ravennika, convened by Emperor Henry of Flanders, is held in the town of Ravennika in Greece, in an attempt to resolve the rebellion of the Lombard nobles of the Kingdom of Thessalonica. Henry pardons Lord Amédée Pofey (or Buffois), and reinvests with his fief, while the other nobles persist in their rebellion, and keep to their castles. After receiving imperial recognition, Geoffrey I of Villehardouin becomes Henry's vassal, thereby subordinating Achaea directly to Constantinople. * June – Treaty of Sapienza: The Republic of Venice recognizes the possession of the Peloponnese by Geoffrey I of Villehardouin – and keeps only the fortresses of Modon and Coron. Venice also acquires an exemption of her merchants from all tariffs, and the right to establish "a church, a market a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Proof
In computational complexity theory, a natural proof is a certain kind of proof establishing that one complexity class differs from another one. While these proofs are in some sense "natural", it can be shown (assuming a widely believed conjecture on the existence of pseudorandom functions) that no such proof can possibly be used to solve the P vs. NP problem. Overview The notion of natural proofs was introduced by Alexander Razborov and Steven Rudich in their article "Natural Proofs", first presented in 1994, and later published in 1997, for which they received the 2007 Gödel Prize. Specifically, natural proofs prove lower bounds on the circuit complexity of boolean functions. A natural proof shows, either directly or indirectly, that a boolean function has a certain natural combinatorial property. Under the assumption that pseudorandom functions exist with "exponential hardness" as specified in their main theorem, Razborov and Rudich show that these proofs cannot separate certa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oracle Machine
In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with 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 strin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P Vs
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 In English orthography and most other European languages, 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 Proto-Indo-European initial *p. Native English cognates of such words often start with , since English is a Germanic language an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry (mathematics)
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. Given a structured object ''X'' of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This can occur in many ways; for example, if ''X'' is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. If the object ''X'' is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (i.e., an isometry). In general, every kind of structure in mathematics will have its own kind of symmetry, many of which are listed in the given points mentioned above. Symmetry in geometry The types of symmetry considered in basic geometry include ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computing The Permanent
In linear algebra, the computation of the permanent of a matrix is a problem that is thought to be more difficult than the computation of the determinant of a matrix despite the apparent similarity of the definitions. The permanent is defined similarly to the determinant, as a sum of products of sets of matrix entries that lie in distinct rows and columns. However, where the determinant weights each of these products with a ±1 sign based on the parity of the set, the permanent weights them all with a +1 sign. While the determinant can be computed in polynomial time by Gaussian elimination, it is generally believed that the permanent cannot be computed in polynomial time. In computational complexity theory, a theorem of Valiant states that computing permanents is #P-hard, and even #P-complete for matrices in which all entries are 0 or 1 . This puts the computation of the permanent in a class of problems believed to be even more difficult to compute than NP. It is known that c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ketan Mulmuley
Ketan Mulmuley is a professor in the Department of Computer Science at the University of Chicago, and a sometime visiting professor at IIT Bombay. He specializes in theoretical computer science, especially computational complexity theory, and in recent years has been working on "geometric complexity theory", an approach to the P versus NP problem through the techniques of algebraic geometry, with Milind Sohoni of IIT Bombay. He is also known for his result with Umesh Vazirani and Vijay Vazirani that showed that "Matching is as easy as matrix inversion", in a paper that introduced the isolation lemma. He earned his PhD in computer science from Carnegie Mellon University in 1985 under Dana Scott, winning the 1986 ACM Doctoral Dissertation Award for his thesis ''Full Abstraction and Semantic Equivalence''. He also won a Miller fellowship at the University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Circuit Complexity
In computational complexity theory, ''arithmetic circuits'' are the standard model for computing polynomials. Informally, an arithmetic circuit takes as inputs either variables or numbers, and is allowed to either add or multiply two expressions it has already computed. Arithmetic circuits provide a formal way to understand the complexity of computing polynomials. The basic type of question in this line of research is "what is the most efficient way to compute a given polynomial f?" Definitions An ''arithmetic circuit'' C over the field (mathematics), field F and the set of variables x_1, \ldots, x_n is a directed acyclic graph as follows. Every node in it with indegree zero is called an ''input gate'' and is labeled by either a variable x_i or a field element in F. Every other gate is labeled by either + or \times; in the first case it is a ''sum'' gate and in the second a ''product'' gate. An ''arithmetic formula'' is a circuit in which every gate has outdegree one (and so the u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Invariant Theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an action of a group on an algebraic variety (or scheme) and provides techniques for forming the 'quotient' of by as a scheme with reasonable properties. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopoles. Background Invariant theory is concerned with a group action of a group on an algebraic variety (or a scheme) . Classical invariant theory addresses the situation when is a vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
