Hypercomputation - Hypercomputation Hypercomputation is the theory of methods for the computation of non-recursive functions. The classes of functions which they can compute is studied in the field known as recursion theory. A similar recent term is super-Turing computation, which has been used in the neural network literature to describe machines with various expanded abilities, including the ability to compute directly on real numbers, the ability to carry out infinitely many computations simultaneously, or the ability to carry out computations with exponentially lower complexity than standard Turing machines. Hypercomputation was first introduced by Alan Turing in his 1939 paper Systems of logic based on ordinals, which investigated mathematical systems in which an oracle was available to compute a single arbitrary (non-recursive) function from naturalss to naturals. Other posited kinds of hypercomputer include:.
Centre for Quantum Computation - Centre for Quantum Computation The Centre for Quantum Computation part of the Universities of Oxford and Cambridge, conducts theoretical and experimental research into all aspects of quantum information processing, and into the implications of the quantum theory of computation for physics itself. The Oxford base is the Clarendon Laboratory (Lindemann building), which is located opposite the T-junction of Keble Road and Parks Road. Researchers include Daniel Oi and David Deutsch.
Computation - Computation Computation can be actually defined as finding a solution to a problem from given inputs by means of an algorithm. This is what the theory of computation, a subfield of computer science and mathematics, deals with. For thousands of years, computing was done with pen and paper, or chalk and slate, or mentally, sometimes with the aid of tables. The theory of computation began early in the twentieth century, before modern electronic computers had been invented. At that time, mathematicians were trying to find which math problems can be solved by simple methods and which cannot. The first step was to define what they meant by a "simple method" for solving a problem. In other words, they needed a formal model of computation. Several different.
Computational complexity theory - 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 of the prime factors of that number..
Computational geometry - plane, find two with the smallest distance from each other. Sure, any computer geek will tell you no big deal, compute the distances between all pairs of points (there are "only" N(N-1)/2 of them) and pick the smallest one. This "brute force" algorithm is said to have time complexity O(N2), i.e., its execution time is proportional to the squared number of the points. One of milestones in Computational geometry was an algorithm for the closest-pair problem of time complexity O(N log N). For modern GIS, computer graphics, and integrated circuit design systems routinely handling tens and hundreds of million points the difference between N2 and N log N boils down to the difference between seconds and days of computation. Hence the emphasis on computational complexity in computational geometry. Some core algorithms:.
Computational learning theory - 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. A list of important COLT papers Surveys [Angluin, 92] Angluin, D. 1992. Computational learning theory: Survey and selected bibliography. In Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing (May 1992), pp..
Super-Turing computation - Super-Turing computation Super-Turing computation is any form of computation that cannot be performed by a finite Turing machine. This includes, but is not limited to: Solving problems that can be solved on a Turing machine, in a lower time complexity class than they can be solved in on a Turing machine. Solving an uncountable number of problems simultaneously. Working with irrational numbers with the same efficiency that a finite Turing machine works with rational numbers. No physical examples of Super-Turing computers are currently known. Classes of computers that might have Super-Turing capabilities in some physical models include: Pulse computers Analog computers Quantum computers See also: hypercomputation.
Real computation - Real computation In computability theory, the theory of real computation deals with hypothetical computing machines using infinite-precision real numbers. Within this theory, it is possible to prove interesting statements such as "the complement of the Mandelbrot set is only partially decidable". For other such powerful machines, see super-Turing computation and hypercomputation. These hypothetical computing machines can be viewed as idealised analog computers which operate on real numbers and are differential, whereas digital computers are limited to computable numbers and are algebraic. This means that idealised analog computers have a larger information dimension rate (see Information Theory), or potential computing domain, than do digital computers. This in theory enables analog computers to solve problems that are inextricable on digital computers. Computer theorists often refer to these idealised analog.
Kleene's recursion theorem - functions that refer back to their own description. The theorem can be formulated for any of the equivalent models of computation listed in the computability theory article. Here we will use Pascal programs. If p is a string that describes a Pascal program which reads a single input string and produces a single output string, then we will denote the corresponding partial function from input strings to output strings by fp. (This is a partial function because for some input strings the program described by p may run into an infinite loop and never produce an output.) The statement of the theorem is as follows: If Q is a Pascal program which takes two input strings x and y and produces a single output string Q(x,y) (which again may be undefined.
Jacquard loom - the most amateur weavers to weave complex designs. It was the first machine to use punch cards to control a sequence of operations. Although it did no computation based on them, it is considered an important step in the history of computing hardware. The ability to change the pattern of the loom's weave by simply changing cards was an important conceptual precursor to the development of computer programming. The term "Jacquard loom" is a misnomer. It is the "Jacquard head" that adapts to a great many dobbie looms such as the "Dornier" brand that allows the weaving machine to then create the intricate patterns. Based on the FOLDOC entry.
Joseph Weizenbaum - technology and lays out his case: while Artificial Intelligence may be possible, we should never allow computers to make important decisions because computers will always lack human qualities such as compassion and wisdom. This he sees as a consequence of their not having been raised in the emotional environment of a human family. Works "ELIZA - A Computer Program for the Study of Natural Language Communication between Man and Machine," Communications of the Association for Computing Machinery 9 (1966): 36-45. Computer power and human reason: from judgment to computation (San Francisco: W. H. Freeman, 1976) External Links Joseph Weizenbaum homepage Joseph Weizenbaum: 1988 Winner of CPSR's Norbert Wiener Award for Professional and Social Responsibility A Java applet faithfully recreating the original ELIZA.
Johann Franz Encke - in London by presenting the gold medal to him in 1823. In this year Encke married Amalie Becker (1787-1879), daughter of a bookseller. They had three sons and two daughters. Eight masterly treatises on its movements were published by him in the Berlin Abhandlungen (1829-1859). From a fresh discussion of the transitss of Venus in 1761 and 1769 he deduced (1822-1824) a solar parallax of 8.57 arcsecond, long accepted as authoritative. In 1822 he became director of the Seeberg observatory, and in 1825 was promoted to a corresponding position at Berlin, where a new observatory, built under his superintendence and with the support of Alexander von Humboldt and the Prussian king Friedrich Wilhelm III, was inaugurated in 1835. Encke became director of the new observatory. He directed the preparation of the.
IA-64 - contains platform specific heuristics.) In order to support IA-32, the Itanium can switch into 32-bit mode with special jump escape instructions, and then return in an analogous way. The IA-32 instructions have been mapped to the Itanium's functional units. However, since the Itanium is built primarily for speed of its EPIC-style instructions, and because it has not out-of-order execution capabilities, the IA-32 instructions execute at a severe performance penalty compared to either the IA-64 mode, or its Pentium line of processors. For example, the Itanium functional units do not automatically generate integer flags as a side effect of ordinary ALU computation, and does not intrinsically support multiple outstanding unaligned memory loads. There have been reports that Intel is seeking a software emulation based solution (much like Transmeta did) for executing IA-32.
Identical particles - statistics. This phenomenon has been observed in the two-dimensional electron gases that form the inversion layer of MOSFETs. There is also yet another statistic called plektons with braid statistics. The spin-statistics theorem relates the exchange symmetry of identical particles to their spin. It states that bosons have integer spin, and fermions have half-integer spin. Anyons possess fractional spin. Symmetrization and Antisymmetrization Earlier, we stated that the two-particle state ψψ′> is some combination of the single-particle states ψ> and ψ′>. However, we have not stated what the combination is. The natural guess is (since it is the canonical way to define the basis of the Hilbert space of two particles from one-particle states; in what follows we assume all states refer to some basis) One can readily verify that this choice is.
Ideal class group - multiplication defined above turns the set of ideal classes into an abelian group, the ideal class group of R. The ideal class group is trivial (i.e. contains only its identity element) if and only if all ideals of R are principal. In this sense, the ideal class group measures how far R is from being a principal ideal domain, and hence from satisfying unique prime factorization (Dedekind domains are unique factorization domains if and only if they are principal ideal domains). The number of ideal classes (the class number of R) may be infinite in general. But if R is in fact a ring of algebraic integers, then this number is always finite. This is one of the main results of classical algebraic number theory. It turns out that an alternative.
Imperative programming - science, imperative programming, as opposed to declarative programming, is a programming style that describes computation in terms of a program state and statements that change the program state. In much the same way as the imperative mood in natural languages expresses commands to take action, imperative programs are a sequence of commands for the computer to perform. The hardware implementation of almost all computers is imperative; nearly all computer hardware is designed to execute machine code, which is native to the computer, written in the imperative style. From this low-level perspective, the program state is defined by the contents of memory, and the statements are instructions in the native machine language of the computer. Higher-level imperative languages use variables and more complex statements, but still follow the same paradigm. Recipes and.
Indian numerals - Chinese numerals). By the middle of the 1st Millennium AD a base 10 numeral system with 9 glyphs was being used in India. This numeral system spread to the Middle East and is believed to have greatly contributed to the development of the Arabic numeral system. In 662 a Nestorian bishop living in what is now called Iraq said of the numeral system: I will omit all discussion of the science of the Indians ... of their subtle discoveries in astronomy - discoveries that are more ingenious than those of the Greeks and the Babylonians - and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs. If those who believe that because they speak Greek they.
Interactive proof system - 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 by a Turing machine.
Infinitesimal - 2x+dx = 2x, since dx is infinitesimally small. This argument, while intuitively appealing, and producing the correct result, is not mathematically rigorous. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The analyst: or a discourse addressed to an infidel mathematician. The fundamental problem is that dx is first treated as non-zero (because we divide by it), but then later discarded as if it were zero. It was not until the second half of the nineteenth century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit, which obviates the need to use infinitesimals. Nevertheless, the use of infinitesimals continues to be convenient for simplifying notation and calculation. Infinitesimals are legitimate quantities in the non-standard analysis.
Information Processing Language - Problem Solver (1957), and also their chess program NSS (1958). IPL pioneered the concept of list processing. The first application of IPL was to demonstrate that the theorems in Principia Mathematica which were laboriously proven by hand, by Bertrand Russell and Alfred North Whitehead, could in fact be proven by computation. According to Simon's autobiography Models of My Life, this first application was developed first by hand simulation, using his children as the computing elements, while writing on and holding up note cards as the registers which contained the state variables of the program. To this day in the CRC method, object-oriented programmers still use note cards to encapsulate simple attributes of the roles played by the programmed objects. Several versions of IPL were created: IPL-I (never implemented), IPL-II (1957 for.
