Computational complexity theory - Computational complexity theory Complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps does it take to solve a problem) and space (how much memory does it take to solve a problem). Other resources can also be considered, such as how many parallel processors are needed to solve a problem in parallel. Complexity theory differs from computability theory, which deals with whether a problem can be solved at all, regardless of the resources required. Overview A single "problem" is an entire set of related questions, where each question is a finite-length string. For example, the problem FACTORIZE is: given an integer written in binary, return all.
Complexity theory - Complexity theory Complexity theory can refer to more than one thing: Computational complexity theory: a field in theoretical computer science and mathematics dealing with the resources required during computation to solve a given problem Systems theory (or systemics or general systems theory): an interdisciplinary field including engineering, biology and philosophy that incorporates science to study large systems Complexity theory is sometimes used as a broad term addressing the study of complex systems, including subjects such as chaos theory, artificial life, genetic algorithms, and the topics studied by the Santa Fe Institute [1] This is a disambiguation page; that is, one that just points to other pages that might otherwise have the same name. If you followed a link here, you might want to go back and.
Computational learning theory - Computational learning theory COLT (COmputational Learning Theory) is a field of machine learning. In COLT the main interest is the computational aspects of learning, e.g., the time complexity of learning problem. The computational aspects are considered in a learning framework, like the very common one of Probably approximately correct learning. Note that a computation is considered feasible if it can be done in polynomial time. There are two kind of results in COLT. Possitive results - Showing the a certain class of function is learnable in polynomial time. Negative results - Showing that certain classes cannot be learned in polynomial time. Negative results were proven only by assumption. The assumptions the are common in negative results are: Computational - P<>NP Cryptographic - One way functions exits..
Effective results in number theory - Effective results in number theory For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable. This for example brings into question any use of big O notation and its implied constants: are assertions pure existence theorems for such constants, or can one recover a version in which 1000 (say) takes the place of the implied constant? Many of the principal results of analytic number theory proved in the period 1900-1950 were in fact ineffective. These included lower bounds for class numbers (ideal class groups for some families of number fields grow); and bounds for the best rational approximations to algebraic numbers in.
Complexity classes P and NP - Complexity classes P and NP Computational complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps does it take to solve a problem) and space (how much memory does it take to solve a problem). In this theory, the class P consists of all those decision problems that can be solved on a deterministic sequential machine in an amount of time that is polynomial in the size of the input; the class NP consists of all those decision problems whose positive solutions can be verified in polynomial time given the right information, or equivalently, whose solution can be found in polynomial time on a non-deterministic machine..
Complexity - Complexity There are different senses of complexity: In computer science, the study of how much time and memory a computer algorithm may take is the field of computational complexity theory. Complexity is often used as a shorthand for the field that developed in the late 1980s around the use of mathematical and computational modeling of biological, economic and technological systems known as "complex systems" (sometimes complex adaptive systems)..
Computational physics - Computational physics Computational physics is the study and implementation of numerical algorithms in order to solve problems in physics for which a quantitative theory already exists. Physicists often have a very precise mathematical theory describing exactly how a system will operate. Unfortunately, it is often the case that solving these equations in order to produce a useful prediction is a computationally difficult problem. This is especially true with quantum mechanics, where only a handful of simple models can be solved exactly. Even apparently simple problems, such as calculating the wavefunction of an electron orbiting an atom in a strong electric field, may require great effort to formulate a practical algorithm. In addition, quantum mechanical problems are generally exponential in the size of the system (see computational.
Computational geometry - Computational geometry Table of contents showTocToggle("show","hide") 1 General 2 Combinatorial Computational Geometry 3 Numerical Computational Geometry / Geometric Modeling 4 Related Links General In computer science, computational geometry is the study of algorithms to solve problems stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and the study of such problems is also considered to be part of computational geometry. Main driving force for the computational geometry as a discipline was progress in computer graphics and in computer-aided design and manufacturing CAD /CAM, although in the hindsight some problems dealt by this discipline may be traced ages back. Other important "customers" of computational geometry inlude Robotics (motion planning and visibility problems), Geographic Information Systems (GIS) (geometrical location.
Space complexity - Space complexity In computer science, the space complexity of a computer program is the amount of memory space required for its proper execution. The term is an approximation to the original meaning of the term in computational complexity theory, where it has a formal mathematical definition. See also time complexity..
NP (complexity) - NP (complexity) In computational complexity theory, NP ("non-deterministic polynomial-time") is the set of decision problems solvable in polynomial time on a non-deterministic Turing machine. Or, equivalently, YES answers are checkable in polynomial time on a deterministic Turing machine given the right information. A language L belongs to NP if there exists a two input polynomial time algorithm A and a constant c such that L = {x in {0,1}* ∃ certificate, y with y = O(xc) such that A(x,y) = 1} Algorithm A verifies L in polynomial time. See also: Complexity classes P and NP and NP-Complete..
List of computability and complexity topics - List of computability and complexity topics This is a list of computability and complexity topics, by Wikipedia page. Computability theory is the part of the theory of computation that deals with what can be computed, in principle. Computational complexity theory deals with how hard computations are, in quantitative terms, both with upper bounds (algorithms whose complexity in the worst cases, as use of computing resources, can be estimated), and from below (proofs that no procedure to carry out some task can be very fast). For more abstract foundational matters, see the list of mathematical logic topics. Table of contents showTocToggle("show","hide") 1 Calculation 2 Computability theory: models of computation 3 Decision problems 4 Definability questions 5 Complexity theory 6 Complexity classes 7 Named problems 8 Extensions Calculation Mathematical expression.
Juris Hartmanis - ACM Turing Award "in recognition of their seminal paper which established the foundations for the field of computational complexity theory". Born in Latvia, he moved to Germany after the Second World War. He received his Bachelor's degree in Physics from the University of Marburg, then emigrated to the United States for his Master's degree in Mathematics at the University of Kansas. He received his Ph.D in Mathematics from Caltech. Thereafter, he worked for the General Electric Research Laboratory, developing and introducing computer science principles. In 1965, he became a professor at Cornell University, where he helped created its computer science department and was its first chairman. This is a stub. Please help to improve this article! References Hartmanis, J., and Stearns, R. E. On the computational complexity of algorithms. Trans. Amer..
Interactive proof system - system The interactive proof system is a concept in computational complexity theory that models computation as the exchange of messages between two parties. The parties, the verifier and the prover, interact by exchanging messages in order to ascertain whether a given string belongs to a language or not. The prover is all-powerful, with unlimited computational resources while the verifier has bounded computation power. The verifier queries the prover a limited number of times and finds out whether the string belongs to the specified language or not. This concept of computation as interaction between parties was suggested by Babai et al and Goldwasser et al. It has also been proven that the set of all languages recognizable by interaction (which is called IP) is equivalent to the set of all languages recognizable.
History of mathematics - study of structure, space and change. The study of structure starts with numbers, firstly the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by the familiar numbers. The physically important concept of vector, generalized to vector spaces and studied in linear algebra, belongs to the two branches of structure and space. The study of space originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space, but later also generalized to non-Euclidean geometries which play a central role in general relativity. Several long.
Exponential time - Exponential time In complexity theory, exponential time refers to the computation time of a problem where the time, m(n), is bounded by an exponential function of the problem size, n. Written mathematically, there exists k > 1 such that m(n) = θ(kn) and there exists c such that m(n) = O(cn). Mathematicians sometimes think of polynomial time as "fast", and anything slower than that as "slow". Exponential time would therefore be considered slow. There are algorithms which take time slower than polynomial time ("super-polynomial time") but faster than exponential time ("sub-exponential time"). These are also considered "slow". One example is the best known algorithm for integer factorization. See also: Computational complexity theory Polynomial time Algorithm Big O notation.
Danny Hillis - hence they can carry out only one operation at a time. The trick is that they’re fast -- a single operation can be done very, very quickly. So a computer can give the illusion of doing many things at once – say, running a game while downloading e-mail while showing an animation – when in fact its processor’s time is swapping back and forth from one task to another rapid-fire. The brain, on the other hand, has around a hundred billion neurons, and in principle they’re all working in parallel, simultaneously. Each one of them acts much more slowly than a computer processor, but what they lack in speed they make up for in bulk and in parallelism. Hillis’s Connection Machine was an elegant compromise, the nature of which changed over.
A New Kind of Science (book) - systems are not isolated from the natural world, but in fact essential for understanding it. Chapters 10, 11, and 12 address our understanding of the world, cutting across semiotics, linguistics, philosophy, as well as more formalized structures such as mathematics and computer science. They build the theoretical underpinnings that explain in fundamental way the discoveries and observations of the earlier chapters. This new kind of science is essentially the study of simple computer programs. Because of undecidability, it is impossible to predict what they will do before actually running them. Wolfram terms this inability to shortcut the program, or otherwise describe its behavior in a simple way, "computational irreducibility". The empirical fact is that the world of simple programs contains a great diversity of behavior, and Wolfram advocates that it be.
Abstract machine - Hermes, LOWL, Christmas, SDL, S-K reduction machine, SECD, Tbl, Tcode, TDF, TL0, WAM. Abstract machine is a procedure for executing a set of instructions in some formal language, possibly also taking in input data and producing output. Such abstract machines are not intended to be constructed as hardware but are used in thought experiments about computability. Abstract machine, or abstract computer is a model of computation in computational complexity theory to analyze the complexity of algorihms. An abstract computer usually postulates its design in terms of the set of allowable operations and time and space complexity of atomic operations. The best known example is Turing machine. See also: Computer Hardware Programming This article (or an earlier version of it) contains material from FOLDOC, used with permission. Modify if needed..
Algorithm - Formalized algorithms 2 Implementing algorithms 3 Example 4 History 5 Classes of algorithms 6 See also 7 References 8 Other Web Reference Formalized algorithms Algorithms are essential to the way computers process information, because a computer program is essentially an algorithm that tells the computer what specific steps to perform (in what specific order) in order to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Typically, when an algorithm is associated with processing information, data is read from an input source or device, written to an output sink or device, and/or stored for further use. Stored data is regarded as part of the internal state of the entity performing the algorithm. For any such computational process, the algorithm must be rigorously defined: specified in.
Amortized analysis - Amortized analysis In computational complexity theory, Amortized analysis is the time per operation averaged over a worst-case sequence of operations. Amortized analysis differs from average-case performance in that probability is not involved; amortized analysis guarantees the time per operation over worst-case performance. There are several techniques used in Amortized analysis: Aggregate analysis - determines upper bound T(n) on total cost of sequence of n operations, then calculates average cost to be T(n)/n Accounting method - determines the individual cost of each operation. Potential method - like Accounting method, but the method overcharges operations early to compensate for undercharges later..
