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Hypercomputation
Hypercomputation or super-Turing computation is a set of hypothetical models of computation that can provide outputs that are not Turing-computable. For example, a machine that could solve the halting problem would be a hypercomputer; so too would one that could correctly evaluate every statement in Peano arithmetic. The Church–Turing thesis states that any "computable" function that can be computed by a mathematician with a pen and paper using a finite set of simple algorithms, can be computed by a Turing machine. Hypercomputers compute functions that a Turing machine cannot and which are, hence, not computable in the Church–Turing sense. Technically, the output of a random Turing machine is uncomputable; however, most hypercomputing literature focuses instead on the computation of deterministic, rather than random, uncomputable functions. History A computational model going beyond Turing machines was introduced by Alan Turing in his 1938 PhD dissertation '' Systems of L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing-computable
Computable functions are the basic objects of study in computability theory. Informally, a function is ''computable'' if there is an algorithm that computes the value of the function for every value of its argument. Because of the lack of a precise definition of the concept of algorithm, every formal definition of computability must refer to a specific model of computation. Many such models of computation have been proposed, the major ones being Turing machines, register machines, lambda calculus and general recursive functions. Although these four are of a very different nature, they provide exactly the same class of computable functions, and, for every model of computation that has ever been proposed, the computable functions for such a model are computable for the above four models of computation. The Church–Turing thesis is the unprovable assertion that every notion of computability that can be imagined can compute only functions that are computable in the above sense. Bef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Church–Turing Thesis
In Computability theory (computation), computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a wiktionary:thesis, thesis about the nature of computable functions. It states that a function (mathematics), function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term ''effectively calculable'' to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formal system, formalize the notion of computability: * In 1933, Kurt Gödel, with Jacques Herbrand, formalized the definition of the class of general ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Computer
In computability theory, the theory of real computation deals with hypothetical computing machines using infinite-precision real numbers. They are given this name because they operate on the set of real numbers. Within this theory, it is possible to prove interesting statements such as "The complement of the Mandelbrot set is only partially decidable." These hypothetical computing machines can be viewed as idealised analog computers which operate on real numbers, whereas digital computers are limited to computable numbers. They may be further subdivided into differential and algebraic models (digital computers, in this context, should be thought of as topological, at least insofar as their operation on computable reals is concerned). Depending on the model chosen, this may enable real computers to solve problems that are inextricable on digital computers (For example, Hava Siegelmann's neural nets can have noncomputable real weights, making them able to compute nonrecursive la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computation
A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms. Mechanical or electronic devices (or, historically, people) that perform computations are known as ''computers''. Computer science is an academic field that involves the study of computation. Introduction The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s, but agreement on a suitable definition proved elusive. A candidate definition was proposed independently by several mathematicians in the 1930s. The best-known variant was formalised by the mathematician Alan Turing, who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a Turing machine. Other (mathematically equivalent) definitions include Alonzo Church's '' lambda-defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supertask
In philosophy, a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time. Supertasks are called hypertasks when the number of operations becomes uncountably infinite. A hypertask that includes one task for each ordinal number is called an ultratask. The term "supertask" was coined by the philosopher James F. Thomson, who devised Thomson's lamp. The term "hypertask" derives from Clark and Read in their paper of that name. History Zeno Motion The origin of the interest in supertasks is normally attributed to Zeno of Elea. Zeno claimed that motion was impossible. He argued as follows: suppose our burgeoning "mover", Achilles say, wishes to move from A to B. To achieve this he must traverse half the distance from A to B. To get from the midpoint of AB to B, Achilles must traverse half ''this'' distance, and so on and so forth. However many times he performs one of these "traversing" tasks, there is another one left for him ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unbounded Nondeterminism
In computer science, unbounded nondeterminism or unbounded indeterminacy refers to a behavior in concurrency (multiple tasks running at once) where a process may face unpredictable delays due to competition for shared resources—such as a printer or memory—or have infinitely many options to choose from at a given point. While these delays or choices can be arbitrarily large, the process is typically guaranteed to complete eventually under certain conditions (e.g., fairness in resource allocation). This concept, explored in abstract models rather than practical systems, became significant in developing mathematical descriptions of such systems (denotational semantics) and later contributed to research on advanced computing theories (hypercomputation). Fairness Unbounded nondeterminism is often discussed alongside the concept of ''fairness''. In this context, fairness means that if a system keeps returning to a certain state forever, it must eventually try every possible next s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeno Machine
In mathematics and computer science, Zeno machines (abbreviated ZM, and also called accelerated Turing machine, ATM) are a hypothetical computational model related to Turing machines that are capable of carrying out computations involving a countably infinite number of algorithmic steps. These machines are ruled out in most models of computation. The idea of Zeno machines was first discussed by Hermann Weyl in 1927; the name refers to Zeno's paradoxes, attributed to the ancient Greek philosopher Zeno of Elea. Zeno machines play a crucial role in some theories. The theory of the Omega Point devised by physicist Frank J. Tipler, for instance, can only be valid if Zeno machines are possible. Definition A Zeno machine is a Turing machine that can take an infinite number of steps, and then continue take more steps. This can be thought of as a supertask where \frac units of time are taken to perform the n-th step; thus, the first step takes 0.5 units of time, the second take ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Nondeterminism
In computer science, unbounded nondeterminism or unbounded indeterminacy refers to a behavior in concurrency (multiple tasks running at once) where a process may face unpredictable delays due to competition for shared resources—such as a printer or memory—or have infinitely many options to choose from at a given point. While these delays or choices can be arbitrarily large, the process is typically guaranteed to complete eventually under certain conditions (e.g., fairness in resource allocation). This concept, explored in abstract models rather than practical systems, became significant in developing mathematical descriptions of such systems (denotational semantics) and later contributed to research on advanced computing theories (hypercomputation). Fairness Unbounded nondeterminism is often discussed alongside the concept of ''fairness''. In this context, fairness means that if a system keeps returning to a certain state forever, it must eventually try every possible next s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FRINGE
Fringe may refer to: Arts and music * "The Fringe", or Edinburgh Festival Fringe, the world's largest arts festival * Adelaide Fringe, the world's second-largest annual arts festival * Fringe theatre, a name for alternative theatre * Purple fringing, an unfocused purple or magenta "ghost" image on a photograph * Fringe Product, a defunct Canadian record label Television and entertainment * ''Fringe'' (TV series), an American science fiction television series * The Fringe, the setting for the 2000 computer game '' Tachyon: The Fringe'' * "The Fringe" (short story), a short story by Orson Scott Card * "The Fringe" (''Smash''), a television episode Science * Fringe science, scientific inquiry that departs significantly from mainstream or orthodox theories in an established field of study * Fringe search, a graph search algorithm that finds the least-cost path from a given initial node to one goal node * Fringe of a relation, a particular sub-relation of a binary relation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitism
Finitism is a philosophy of mathematics that accepts the existence only of finite set, finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite sets) are accepted as existing. Main idea The main idea of finitistic mathematics is not accepting the existence of infinite objects such as infinite sets. While all natural numbers are accepted as existing, the ''set'' of all natural numbers is not considered to exist as a mathematical object. Therefore quantifier (logic), quantification over infinite domains is not considered meaningful. The mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic. History The introduction of infinite mathematical objects occurred a few centuries ago when the use of infinite objects was already a controversial topic among mathematicians. The issue entered a new phase when Georg Cantor in 1874 introduced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean algebra, Boolean logic, the truth values of variables may only be the integer values 0 or 1. The term ''fuzzy logic'' was introduced with the 1965 proposal of fuzzy set theory by mathematician Lotfi A. Zadeh, Lotfi Zadeh. Fuzzy logic had, however, been studied since the 1920s, as Łukasiewicz logic, infinite-valued logic—notably by Jan Łukasiewicz, Łukasiewicz and Alfred Tarski, Tarski. Fuzzy logic is based on the observation that people make decisions based on imprecise and non-numerical information. Fuzzy models or fuzzy sets are mathematical means of representing vagueness and imprecise information (hence the term fuzzy). These models have the capability of recognising, representing, manipulating, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Constant
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that cannot be explained by a theory and therefore must be measured experimentally. It is distinct from a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement. There are many physical constants in science, some of the most widely recognized being the speed of light in vacuum ''c'', the gravitational constant ''G'', the Planck constant ''h'', the electric constant ''ε''0, and the elementary charge ''e''. Physical constants can take many dimensional forms: the speed of light signifies a maximum speed for any object and its dimension is length divided by time; while the proton-to-electron mass ratio is dimensionless. The term "fundamental physical constant" is sometimes used to refer to universal-but-dimensioned physical constants such as those mentioned above. Increasingly, however, physicists reserve the expressi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
