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Hypercomputer
Hypercomputation or super-Turing computation refers to models of computation that can provide outputs that are not Turing-computable. Super-Turing computing, introduced at the early 1990's by Hava Siegelmann, refers to such neurological inspired, biological and physical realizable computing; It became the mathematical foundations of Lifelong Machine Learning. Hypercomputation, introduced as a field of science in the late 1990s, is said to be based on the Super Turing but it also includes constructs which are philosophical. For example, a machine that could solve the halting problem would be a hypercomputer; so too would one that can 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 computa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Church–Turing Thesis
In 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 thesis about the nature of computable functions. It states that a 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 formalize the notion of computability: * In 1933, Kurt Gödel, with Jacques Herbrand, formalized the definition of the class of general recursive functions: the smallest class of functions (with arbitrarily many arguments) that is cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halting Problem
In 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. Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program–input pairs cannot exist. For any program that might determine whether programs halt, a "pathological" program , called with some input, can pass its own source and its input to ''f'' and then specifically do the opposite of what ''f'' predicts ''g'' will do. No ''f'' can exist that handles this case. A key part of the proof is a mathematical definition of a computer and program, which is known as a Turing machine; the halting problem is '' undecidable'' over Turing machines. It is one of the first cases of decision problems proven to be unsolvable. This proof is significant to practical computing efforts, defining a class of applications which no programming inventi ... [...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. Models 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 Functional models Functional models include: * Abstract re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Schönhage
Arnold Schönhage (born 1 December 1934 in Lockhausen, now Bad Salzuflen) is a German mathematician and computer scientist. Schönhage was professor at the Rheinische Friedrich-Wilhelms-Universität, Bonn, and also in Tübingen and Konstanz. He now lives near Bonn. Together with Volker Strassen he developed the Schönhage–Strassen algorithm for fast integer multiplication that has a run-time of '' O''(''N'' log ''N'' log log ''N''). Schönhage designed and implemented together with Andreas F. W. Grotefeld and Ekkehart Vetter a multitape Turing machine, called TP, in software. The machine is programmed in TPAL, an assembler language In computer programming, assembly language (or assembler language, or symbolic machine code), often referred to simply as Assembly and commonly abbreviated as ASM or asm, is any low-level programming language with a very strong correspondence be .... They implemented numerous numerical algorithms including the Sch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeno's Paradox
Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. It is usually assumed, based on Plato's ''Parmenides'' (128a–d), that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides' view. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one." Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point. Some of Zeno's nine surviving paradoxes (preserved in Aristotle's ''Physics'' [...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 a infinite number of steps, and then continue take more steps. This can be thought of as a supertask where 1/2^n units of time are taken to perform the n-th step; thus, the first step takes 0.5 units of time, the second takes ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FRINGE
Fringe may refer to: Arts * Edinburgh Festival Fringe, the world's largest arts festival, known as "the Fringe" * Adelaide Fringe, the world's second-largest annual arts festival * Fringe theatre, a name for alternative theatre * The Fringe, the setting for the 2000 computer game '' Tachyon: The Fringe'' * "The Fringe" (short story), a short story by Orson Scott Card * ''Fringe'' (TV series), an American science fiction television series * "The Fringe" (''Smash''), a television episode * Fringe Product, a defunct Canadian record label * Purple fringing, an unfocused purple or magenta "ghost" image on a photograph Science * Fringe science, scientific inquiry in an established field of study that departs significantly from mainstream or orthodox theories * 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 kind of heterogeneous relation in mathematics * Interference fringe, a pattern in wav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unbounded Nondeterminism
In computer science, unbounded nondeterminism or unbounded indeterminacy is a property of concurrency by which the amount of delay in servicing a request can become unbounded as a result of arbitration of contention for shared resources ''while still guaranteeing that the request will eventually be serviced''. Unbounded nondeterminism became an important issue in the development of the denotational semantics of concurrency, and later became part of research into the theoretical concept of hypercomputation. Fairness Discussion of unbounded nondeterminism tends to get involved with discussions of ''fairness''. The basic concept is that all computation paths must be "fair" in the sense that if the machine enters a state infinitely often, it must take every possible transition from that state. This amounts to requiring that the machine be guaranteed to service a request if it can, since an infinite sequence of states will only be allowed if there is no transition that leads to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitism
Finitism is a philosophy of mathematics that accepts the existence only of 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 legitimate. 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 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 what is now called naive set t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Nondeterminism
In computer science, unbounded nondeterminism or unbounded indeterminacy is a property of concurrency by which the amount of delay in servicing a request can become unbounded as a result of arbitration of contention for shared resources ''while still guaranteeing that the request will eventually be serviced''. Unbounded nondeterminism became an important issue in the development of the denotational semantics of concurrency, and later became part of research into the theoretical concept of hypercomputation. Fairness Discussion of unbounded nondeterminism tends to get involved with discussions of ''fairness''. The basic concept is that all computation paths must be "fair" in the sense that if the machine enters a state infinitely often, it must take every possible transition from that state. This amounts to requiring that the machine be guaranteed to service a request if it can, since an infinite sequence of states will only be allowed if there is no transition that leads to the ... [...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 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 Iranian Azerbaijani mathematician Lotfi Zadeh. Fuzzy logic had, however, been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz and Tarski. Fuzzy logic is based on the observation that people make decisions based on imprecise and non-numerical information. Fuzzy models or sets are mathematical means of representing vagueness and imprecise information (hence the term fuzzy). These models have the capability of recognising, representing, manipulating, interpreting, and using data and information that are vague and lack ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
