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Computation and Hypercomputation
 MINDS AND MACHINES
, 2003
"... Does Nature permit the implementation of behaviours that cannot be simulated computationally? We consider the meaning of physical computationality in some detail, and present arguments in favour of physical hypercomputation: for example, modern scientific method does not allow the specification o ..."
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Does Nature permit the implementation of behaviours that cannot be simulated computationally? We consider the meaning of physical computationality in some detail, and present arguments in favour of physical hypercomputation: for example, modern scientific method does not allow the specification of any experiment capable of refuting hypercomputation. We consider the implications of relativistic algorithms capable of solving the (Turing) Halting Problem. We also reject as a fallacy the argument that hypercomputation has no relevance because noncomputable values are indistinguishable from sufficiently close computable approximations. In addition to
On Alan Turing's Anticipation Of Connectionism
, 1996
"... It is not widely realised that Turing was probably the first person to consider building computing machines out of simple, neuronlike elements connected together into networks in a largely random manner. Turing called his networks `unorganised machines'. By the application of what he described as ' ..."
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It is not widely realised that Turing was probably the first person to consider building computing machines out of simple, neuronlike elements connected together into networks in a largely random manner. Turing called his networks `unorganised machines'. By the application of what he described as 'appropriate interference, mimicking education' an unorganised machine can be trained to perform any task that a Turing machine can carry out, provided the number of 'neurons' is sufficient. Turing proposed simulating both the behaviour of the network and the training process by means of a computer program. We outline Turing's connectionist project of 1948.
Alan Turing and the Mathematical Objection
 Minds and Machines 13(1
, 2003
"... Abstract. This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet accord ..."
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Abstract. This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the “mathematical objection ” to his view that machines can think. Logicomathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human mathematicians presumably do. Key words: artificial intelligence, ChurchTuring thesis, computability, effective procedure, incompleteness, machine, mathematical objection, ordinal logics, Turing, undecidability The ‘skin of an onion ’ analogy is also helpful. In considering the functions of the mind or the brain we find certain operations which we can express in purely mechanical terms. This we say does not correspond to the real mind: it is a sort of skin which we must strip off if we are to find the real mind. But then in what remains, we find a further skin to be stripped off, and so on. Proceeding in this way, do we ever come to the ‘real ’ mind, or do we eventually come to the skin which has nothing in it? In the latter case, the whole mind is mechanical (Turing, 1950, p. 454–455). 1.
The Colossus
 History of Computing in the Twentieth Century
, 1980
"... In October 1975, after an official silence lasting thirtytwo years, the British Government made a set of captioned photographs of COLOSSUS available at the Public Record Office, These confirm that a series of programmable electronic digital computers was built in Britain during World War II, the fi ..."
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In October 1975, after an official silence lasting thirtytwo years, the British Government made a set of captioned photographs of COLOSSUS available at the Public Record Office, These confirm that a series of programmable electronic digital computers was built in Britain during World War II, the first being operational in 1943. It is stated that COLOSSUS incorporated 1500 valves, and operated in parallel arithmetic mode at 5000 pulses per second. A number of its features are disclosed, including the fact that it had 5000 character per second punched paper tape inputs, electronic circuits for counting, binary arithmetic and Boolean logic operations, "electronic storage registers changeable by an automatically controlled. sequence of operations", "conditional (branching) logic", "logic functions preset by patchpanels or switches, or conditionally selected by telephone relays", and typewriter output. Professor M.H.A. Newman is named as being responsible for formulating the requirement for COLOSSUS, and Mr. T.H. Flowers as leading the team which developed the machine. An indication is given that the design of COLOSSUS was influenced by the prewar work on computability by Alan Turing, who was employed in the same department of the British Government as Newman. The partial relaxation of the official secrecy surrounding COLOSSUS has made it possible to obtain interviews with a number of people involved in the project. The present paper is, in the main, based on these interviews, but supplemented by material already in the public domain. It attempts to document as fully as is presently permissible the story of the development of COLOSSUS. Particular attention is paid to interactions between the COLOSSUS project and other work carried out elsewhere on digital techniques and c...
Towards a theory of intelligence
 Theoretical Computer Science
"... In 1950, Turing suggested that intelligent behavior might require “a departure from the completely disciplined behaviour involved in computation”, but nothing that a digital computer could not do. In this paper, I want to explore Turing’s suggestion by asking what it is, beyond computation, that int ..."
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In 1950, Turing suggested that intelligent behavior might require “a departure from the completely disciplined behaviour involved in computation”, but nothing that a digital computer could not do. In this paper, I want to explore Turing’s suggestion by asking what it is, beyond computation, that intelligence might require, why it might require it and what knowing the answers to the first two questions might do to help us understand artificial and natural intelligence.
Logic and learning: Turing's legacy
 In
, 1994
"... Turing's best known work is concerned with whether universal machines can decide the truth value of arbitrary logic formulae. However, in this paper it is shown that there is a direct evolution in Turing's ideas from his earlier investigations of computability to his later interests in machine intel ..."
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Turing's best known work is concerned with whether universal machines can decide the truth value of arbitrary logic formulae. However, in this paper it is shown that there is a direct evolution in Turing's ideas from his earlier investigations of computability to his later interests in machine intelligence and machine learning. Turing realised that machines which could learn would be able to avoid some of the consequences of Godel's and his results on incompleteness and undecidability. Machines which learned could continuously add new axioms to their repertoire. Inspired by a radio talk given by Turing in 1951, Christopher Strachey went on to implement the world's first machine learning program. This particular first is usually attributed to A.L. Samuel. Strachey's program, which did rote learning in the game of Nim, preceded Samuel's checker playing program by four years. Neither Strachey's nor Samuel's system took up Turing's suggestion of learning logical formulae. Developments in t...
Comparative Analysis of Hypercomputational Systems Submitted in partial fulfilment
"... In the 1930s, Turing suggested his abstract model for a practical computer, hypothetically visualizing the digital programmable computer long before it was actually invented. His model formed the foundation for every computer made today. The past few years have seen a change in ideas where philosoph ..."
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In the 1930s, Turing suggested his abstract model for a practical computer, hypothetically visualizing the digital programmable computer long before it was actually invented. His model formed the foundation for every computer made today. The past few years have seen a change in ideas where philosophers and scientists are suggesting models of hypothetical computing devices which can outperform the Turing machine, performing some calculations the latter is unable to. The ChurchTuring Thesis, which the Turing machine model embodies, has raised discussions on whether it could be possible to solve undecidable problems which Turing’s model is unable to. Models which could solve these problems, have gone further to claim abilities relating to quantum computing, relativity theory, even the modeling of natural biological laws themselves. These so called ‘hypermachines ’ use hypercomputational abilities to make the impossible possible. Various models belonging to different disciplines of physics, mathematics and philosophy, have been suggested for these theories. My (primarily researchoriented) project is based on the study and review of these different hypercomputational models and attempts to compare the different models in terms of computational power. The project focuses on the ability to compare these models of different disciplines on similar grounds and
The Sources of Certainty in Computation and Formal Systems
, 1999
"... In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical deniti ..."
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In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical denitions and inferences in such a way that they are unshakable." Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not dened in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs,...
The Sources of Certainty in Computation and Formal Systems
"... In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical deniti ..."
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In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical denitions and inferences in such a way that they are unshakable." Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not dened in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs,...
Edsger Dijkstra
, 2003
"... this paper  now I fear there are generations who have never heard of it. If you are a member of one of these generations let me urge you to go find it in the CACM, and read it ..."
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this paper  now I fear there are generations who have never heard of it. If you are a member of one of these generations let me urge you to go find it in the CACM, and read it