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Hypercomputer
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 Logic ... [...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] |
Halting Problem
In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is ''Undecidable problem, undecidable'', meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs. The problem comes up often in discussions of computability since it demonstrates that some functions are mathematically Definable set, definable but not Computable function, computable. A key part of the formal statement of the problem is a mathematical definition of a computer and program, usually via a Turing machine. The proof then shows, for any program that might determine whether programs halt, that a "pathological" program exists for which makes an incorrect determination. Specifically, is the program that, when called with some input, passes its own s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Of Computation
In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how units of computations, memories, and communications are organized. The computational complexity of an algorithm can be measured given a model of computation. Using a model allows studying the performance of algorithms independently of the variations that are specific to particular implementations and specific technology. Categories Models of computation can be classified into three categories: sequential models, functional models, and concurrent models. Sequential models Sequential models include: * Finite-state machines * Post machines ( Post–Turing machines and tag machines). * Pushdown automata * Register machines ** Random-access machines * Turing machines * Decision tree model * External memory model Functional models Functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analog Computer
An analog computer or analogue computer is a type of computation machine (computer) that uses physical phenomena such as Electrical network, electrical, Mechanics, mechanical, or Hydraulics, hydraulic quantities behaving according to the mathematical principles in question (''analog signals'') to Scientific modelling, model the problem being solved. In contrast, digital computers represent varying quantities symbolically and by discrete values of both time and amplitude (digital signals). Analog computers can have a very wide range of complexity. Slide rules and nomograms are the simplest, while naval gunfire control computers and large hybrid digital/analog computers were among the most complicated. Complex mechanisms for process control and protective relays used analog computation to perform control and protective functions. Analog computers were widely used in scientific and industrial applications even after the advent of digital computers, because at the time they were ... [...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] |
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] |
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] |
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] |
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] |
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] |
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] |
