Showing posts with label Computers. Show all posts
Showing posts with label Computers. Show all posts

Thursday, October 09, 2014

Majorana Fermions Discovered


A Majorana fermion (/məˈrɒnə ˈfɛərmɒn/[1]), also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesized by Ettore Majorana in 1937. The term is sometimes used in opposition to a Dirac fermion, which describes fermions that are not their own antiparticles.
All of the Standard Model fermions except the neutrino behave as Dirac fermions at low energy (after electroweak symmetry breaking), but the nature of the neutrino is not settled and it may be either Dirac or Majorana. In condensed matter physics, Majorana fermions exist as quasiparticle excitations in superconductors and can be used to form Majorana bound states governed by non-abelian statistics.

*** 

Princeton University physicists built a powerful imaging device called a scanning-tunneling microscope and used it to capture an image of an elusive particle that behaves simultaneously like matter and antimatter. To avoid vibration, the microscope is cooled close to absolute zero and is suspended like a floating island in the floor above. The setup includes a 40-ton block of concrete, which is visible above the researchers. The research team includes, from left, graduate student Ilya Drozdov, postdoctoral researcher Sangjun Jeon, and professors of physics B. Andrei Bernevig and Ali Yazdani. (Photo by Denise Applewhite, Office of Communications)
Princeton University scientists have observed an exotic particle that behaves simultaneously like matter and antimatter, a feat of math and engineering that could eventually enable powerful computers based on quantum mechanics. Capping decades of searching, Princeton scientists observe elusive particle that is its own antiparticle.
***

 Majorana fermions are predicted to localize at the edge of a topological superconductor, a state of matter that can form when a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin-orbit interaction. With the goal of realizing a one-dimensional topological superconductor, we have fabricated ferromagnetic iron (Fe) atomic chains on the surface of superconducting lead (Pb). Using high-resolution spectroscopic imaging techniques, we show that the onset of superconductivity, which gaps the electronic density of states in the bulk of the Fe chains, is accompanied by the appearance of zero energy end states. This spatially resolved signature provides strong evidence, corroborated by other observations, for the formation of a topological phase and edge-bound Majorana fermions in our atomic chains. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor

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Thursday, October 18, 2012

Google Data Center


See inside one of Google's data centers in this guided tour. See what powers our products, and then explore on your own in Street View: http://www.google.com/about/datacenters/streetview


In the early seventies my own Father was a accountant who made use of this system of data storage and while having the equipment for card production and sorting to list his clients, he made use of the rooms in larger office tower that were in hermetically sealed rooms to prevent corruption.

Above left: The row of tape drives for the UNIVAC I computer. Above right: The IBM 3410 Magnetic Tape Subsystem, introduced in 1971.


The progressive feature of computation  data development is always an interesting one to me. The age of its users may be of interest in judging how far back certain features of the development mechanical type set is used helps to portray the age of its users. It's evolutionary history.

The history of computer data storage, in pictures


I mean using Magnetic tapes in terms of data storage although used in type set cards punches as 1's and O's was the precursor to what we see today in the google data centre. Can you imagine how large the data room would be needed in order to compress what you see today in our historical development of the seventies?

How science data storage is effected by what would constrain the amount of information given from experimental procedures? We could not do much with efficiency given that constraint.

Monday, May 28, 2012

Embodied Cognition and iCub

An iCub robot mounted on a supporting frame. The robot is 104 cm high and weighs around 22 kg
An iCub is a 1 metre high humanoid robot testbed for research into human cognition and artificial intelligence.

Systems that perceive, understand and act


It was designed by the RobotCub Consortium, of several European universities and is now supported by other projects such as ITALK.[1] The robot is open-source, with the hardware design, software and documentation all released under the GPL license. The name is a partial acronym, cub standing for Cognitive Universal Body.[2] Initial funding for the project was 8.5 million from Unit E5 – Cognitive Systems and Robotics – of the European Commission's Seventh Framework Programme, and this ran for six years from 1 September 2004 until 1 September 2010.[2]

The motivation behind the strongly humanoid design is the embodied cognition hypothesis, that human-like manipulation plays a vital role in the development of human cognition. A baby learns many cognitive skills by interacting with its environment and other humans using its limbs and senses, and consequently its internal model of the world is largely determined by the form of the human body. The robot was designed to test this hypothesis by allowing cognitive learning scenarios to be acted out by an accurate reproduction of the perceptual system and articulation of a small child so that it could interact with the world in the same way that such a child does.[3]


 See Also: RoboCub




In philosophy, the embodied mind thesis holds that the nature of the human mind is largely determined by the form of the human body. Philosophers, psychologists, cognitive scientists and artificial intelligence researchers who study embodied cognition and the embodied mind argue that all aspects of cognition are shaped by aspects of the body. The aspects of cognition include high level mental constructs (such as concepts and categories) and human performance on various cognitive tasks (such as reasoning or judgement). The aspects of the body include the motor system, the perceptual system, the body's interactions with the environment (situatedness) and the ontological assumptions about the world that are built into the body and the brain.

The embodied mind thesis is opposed to other theories of cognition such as cognitivism, computationalism and Cartesian dualism.[1] The idea has roots in Kant and 20th century continental philosophy (such as Merleau-Ponty). The modern version depends on insights drawn from recent research in psychology, linguistics, cognitive science, artificial intelligence, robotics and neurobiology.

Embodied cognition is a topic of research in social and cognitive psychology, covering issues such as social interaction and decision-making.[2] Embodied cognition reflects the argument that the motor system influences our cognition, just as the mind influences bodily actions. For example, when participants hold a pencil in their teeth engaging the muscles of a smile, they comprehend pleasant sentences faster than unpleasant ones.[3] And it works in reverse: holding a pencil in their teeth to engage the muscles of a frown increases the time it takes to comprehend pleasant sentences.[3]

George Lakoff (a cognitive scientist and linguist) and his collaborators (including Mark Johnson, Mark Turner, and Rafael E. Núñez) have written a series of books promoting and expanding the thesis based on discoveries in cognitive science, such as conceptual metaphor and image schema.[4]
Robotics researchers such as Rodney Brooks, Hans Moravec and Rolf Pfeifer have argued that true artificial intelligence can only be achieved by machines that have sensory and motor skills and are connected to the world through a body.[5] The insights of these robotics researchers have in turn inspired philosophers like Andy Clark and Horst Hendriks-Jansen.[6]

Neuroscientists Gerald Edelman, António Damásio and others have outlined the connection between the body, individual structures in the brain and aspects of the mind such as consciousness, emotion, self-awareness and will.[7] Biology has also inspired Gregory Bateson, Humberto Maturana, Francisco Varela, Eleanor Rosch and Evan Thompson to develop a closely related version of the idea, which they call enactivism.[8] The motor theory of speech perception proposed by Alvin Liberman and colleagues at the Haskins Laboratories argues that the identification of words is embodied in perception of the bodily movements by which spoken words are made.[9][10][11][12][13]



The mind-body problem is a philosophical problem arising in the fields of metaphysics and philosophy of mind.[2] The problem arises because mental phenomena arguably differ, qualitatively or substantially, from the physical body on which they apparently depend. There are a few major theories on the resolution of the problem. Dualism is the theory that the mind and body are two distinct substances,[2] and monism is the theory that they are, in reality, just one substance. Monist materialists (also called physicalists) take the view that they are both matter, and monist idealists take the view that they are both in the mind. Neutral monists take the view that both are reducible to a third, neutral substance.

The problem was identified by René Descartes in the sense known by the modern Western world, although the issue was also addressed by pre-Aristotelian philosophers,[3] in Avicennian philosophy,[4] and in earlier Asian traditions.

A dualist view of reality may lead one to consider the corporeal as little valued[3] and trivial. The rejection of the mind–body dichotomy is found in French Structuralism, and is a position that generally characterized post-war French philosophy.[5] The absence of an empirically identifiable meeting point between the non-physical mind and its physical extension has proven problematic to dualism and many modern philosophers of mind maintain that the mind is not something separate from the body.[6] These approaches have been particularly influential in the sciences, particularly in the fields of sociobiology, computer science, evolutionary psychology and the various neurosciences.[7][8][9][10]

Wednesday, May 23, 2012

Hypercomputation

Hypercomputation or super-Turing computation refers to models of computation that go beyond, or are incomparable to, Turing computability. This includes various hypothetical methods for the computation of non-Turing-computable functions, following super-recursive algorithms (see also supertask). The term "super-Turing computation" appeared in a 1995 Science paper by Hava Siegelmann. The term "hypercomputation" was introduced in 1999 by Jack Copeland and Diane Proudfoot.[1]

The terms are not quite synonymous: "super-Turing computation" usually implies that the proposed model is supposed to be physically realizable, while "hypercomputation" does not.

Technical arguments against the physical realizability of hypercomputations have been presented.

Contents

 

History

 

A computational model going beyond Turing machines was introduced by Alan Turing in his 1938 PhD dissertation Systems of Logic Based on Ordinals.[2] This paper investigated mathematical systems in which an oracle was available, which could compute a single arbitrary (non-recursive) function from naturals to naturals. He used this device to prove that even in those more powerful systems, undecidability is still present. Turing's oracle machines are strictly mathematical abstractions, and are not physically realizable.[3]

Hypercomputation and the Church–Turing thesis

 

The Church–Turing thesis states that any function that is algorithmically computable can be computed by a Turing machine. Hypercomputers compute functions that a Turing machine cannot, hence, not computable in the Church-Turing sense.
An example of a problem a Turing machine cannot solve is the halting problem. A Turing machine cannot decide if an arbitrary program halts or runs forever. Some proposed hypercomputers can simulate the program for an infinite number of steps and tell the user whether or not the program halted.

Hypercomputer proposals

 

  • A Turing machine that can complete infinitely many steps. Simply being able to run for an unbounded number of steps does not suffice. One mathematical model is the Zeno machine (inspired by Zeno's paradox). The Zeno machine performs its first computation step in (say) 1 minute, the second step in ½ minute, the third step in ¼ minute, etc. By summing 1+½+¼+... (a geometric series) we see that the machine performs infinitely many steps in a total of 2 minutes. However, some[who?] people claim that, following the reasoning from Zeno's paradox, Zeno machines are not just physically impossible, but logically impossible.[4]
  • Turing's original oracle machines, defined by Turing in 1939.
  • In mid 1960s, E Mark Gold and Hilary Putnam independently proposed models of inductive inference (the "limiting recursive functionals"[5] and "trial-and-error predicates",[6] respectively). These models enable some nonrecursive sets of numbers or languages (including all recursively enumerable sets of languages) to be "learned in the limit"; whereas, by definition, only recursive sets of numbers or languages could be identified by a Turing machine. While the machine will stabilize to the correct answer on any learnable set in some finite time, it can only identify it as correct if it is recursive; otherwise, the correctness is established only by running the machine forever and noting that it never revises its answer. Putnam identified this new interpretation as the class of "empirical" predicates, stating: "if we always 'posit' that the most recently generated answer is correct, we will make a finite number of mistakes, but we will eventually get the correct answer. (Note, however, that even if we have gotten to the correct answer (the end of the finite sequence) we are never sure that we have the correct answer.)"[6] L. K. Schubert's 1974 paper "Iterated Limiting Recursion and the Program Minimization Problem" [7] studied the effects of iterating the limiting procedure; this allows any arithmetic predicate to be computed. Schubert wrote, "Intuitively, iterated limiting identification might be regarded as higher-order inductive inference performed collectively by an ever-growing community of lower order inductive inference machines."
  • A real computer (a sort of idealized analog computer) can perform hypercomputation[8] if physics admits general real variables (not just computable reals), and these are in some way "harnessable" for computation. This might require quite bizarre laws of physics (for example, a measurable physical constant with an oracular value, such as Chaitin's constant), and would at minimum require the ability to measure a real-valued physical value to arbitrary precision despite thermal noise and quantum effects.
  • A proposed technique known as fair nondeterminism or unbounded nondeterminism may allow the computation of noncomputable functions.[9] There is dispute in the literature over whether this technique is coherent, and whether it actually allows noncomputable functions to be "computed".
  • It seems natural that the possibility of time travel (existence of closed timelike curves (CTCs)) makes hypercomputation possible by itself. However, this is not so since a CTC does not provide (by itself) the unbounded amount of storage that an infinite computation would require. Nevertheless, there are spacetimes in which the CTC region can be used for relativistic hypercomputation.[10] Access to a CTC may allow the rapid solution to PSPACE-complete problems, a complexity class which while Turing-decidable is generally considered computationally intractable.[11][12]
  • According to a 1992 paper,[13] a computer operating in a Malament-Hogarth spacetime or in orbit around a rotating black hole[14] could theoretically perform non-Turing computations.[15][16]
  • In 1994, Hava Siegelmann proved that her new (1991) computational model, the Artificial Recurrent Neural Network (ARNN), could perform hypercomputation (using infinite precision real weights for the synapses). It is based on evolving an artificial neural network through a discrete, infinite succession of states.[17]
  • The infinite time Turing machine is a generalization of the Zeno machine, that can perform infinitely long computations whose steps are enumerated by potentially transfinite ordinal numbers. It models an otherwise-ordinary Turing machine for which non-halting computations are completed by entering a special state reserved for reaching a limit ordinal and to which the results of the preceding infinite computation are available.[18]
  • Jan van Leeuwen and Jiří Wiedermann wrote a 2000 paper[19] suggesting that the Internet should be modeled as a nonuniform computing system equipped with an advice function representing the ability of computers to be upgraded.
  • A symbol sequence is computable in the limit if there is a finite, possibly non-halting program on a universal Turing machine that incrementally outputs every symbol of the sequence. This includes the dyadic expansion of π and of every other computable real, but still excludes all noncomputable reals. Traditional Turing machines cannot edit their previous outputs; generalized Turing machines, as defined by Jürgen Schmidhuber, can. He defines the constructively describable symbol sequences as those that have a finite, non-halting program running on a generalized Turing machine, such that any output symbol eventually converges, that is, it does not change any more after some finite initial time interval. Due to limitations first exhibited by Kurt Gödel (1931), it may be impossible to predict the convergence time itself by a halting program, otherwise the halting problem could be solved. Schmidhuber ([20][21]) uses this approach to define the set of formally describable or constructively computable universes or constructive theories of everything. Generalized Turing machines can solve the halting problem by evaluating a Specker sequence.
  • A quantum mechanical system which somehow uses an infinite superposition of states to compute a non-computable function.[22] This is not possible using the standard qubit-model quantum computer, because it is proven that a regular quantum computer is PSPACE-reducible (a quantum computer running in polynomial time can be simulated by a classical computer running in polynomial space).[23]
  • In 1970, E.S. Santos defined a class of fuzzy logic-based "fuzzy algorithms" and "fuzzy Turing machines".[24] Subsequently, L. Biacino and G. Gerla showed that such a definition would allow the computation of nonrecursive languages; they suggested an alternative set of definitions without this difficulty.[25] Jiří Wiedermann analyzed the capabilities of Santos' original proposal in 2004.[26]
  • Dmytro Taranovsky has proposed a finitistic model of traditionally non-finitistic branches of analysis, built around a Turing machine equipped with a rapidly increasing function as its oracle. By this and more complicated models he was able to give an interpretation of second-order arithmetic.[27]

 

Analysis of capabilities


Many hypercomputation proposals amount to alternative ways to read an oracle or advice function embedded into an otherwise classical machine. Others allow access to some higher level of the arithmetic hierarchy. For example, supertasking Turing machines, under the usual assumptions, would be able to compute any predicate in the truth-table degree containing \Sigma^0_1 or \Pi^0_1. Limiting-recursion, by contrast, can compute any predicate or function in the corresponding Turing degree, which is known to be \Delta^0_2. Gold further showed that limiting partial recursion would allow the computation of precisely the \Sigma^0_2 predicates.
Model Computable predicates Notes Refs
supertasking tt(\Sigma^0_1, \Pi^0_1) dependent on outside observer [28]
limiting/trial-and-error  \Delta^0_2
[5]
iterated limiting (k times)  \Delta^0_{k+1}
[7]
Blum-Shub-Smale machine
incomparable with traditional computable real functions. [29]
Malament-Hogarth spacetime HYP Dependent on spacetime structure [30]
Analog recurrent neural network  \Delta^0_1[f] f is an advice function giving connection weights; size is bounded by runtime [31][32]
Infinite time Turing machine  \ge T(\Sigma^1_1)
[33]
Classical fuzzy Turing machine  \Sigma^0_1 \cup \Pi^0_1 For any computable t-norm [34]
Increasing function oracle  \Delta^1_1 For the one-sequence model;  \Pi^1_1 are r.e. [27]

 

Taxonomy of "super-recursive" computation methodologies


Burgin has collected a list of what he calls "super-recursive algorithms" (from Burgin 2005: 132):
  • limiting recursive functions and limiting partial recursive functions (E. M. Gold[5])
  • trial and error predicates (Hilary Putnam[6])
  • inductive inference machines (Carl Herbert Smith)
  • inductive Turing machines (one of Burgin's own models)
  • limit Turing machines (another of Burgin's models)
  • trial-and-error machines (Ja. Hintikka and A. Mutanen [35])
  • general Turing machines (J. Schmidhuber[21])
  • Internet machines (van Leeuwen, J. and Wiedermann, J.[19])
  • evolutionary computers, which use DNA to produce the value of a function (Darko Roglic[36])
  • fuzzy computation (Jiří Wiedermann[26])
  • evolutionary Turing machines (Eugene Eberbach[37])
In the same book, he presents also a list of "algorithmic schemes":

 

Criticism


Martin Davis, in his writings on hypercomputation [39] [40] refers to this subject as "a myth" and offers counter-arguments to the physical realizability of hypercomputation. As for its theory, he argues against the claims that this is a new field founded in 1990s. This point of view relies on the history of computability theory (degrees of unsolvability, computability over functions, real numbers and ordinals), as also mentioned above.
Andrew Hodges wrote a critical commentary[41] on Copeland and Proudfoot's article[1].

 

See also

 

References

  1. ^ a b Copeland and Proudfoot, Alan Turing's forgotten ideas in computer science. Scientific American, April 1999
  2. ^ Alan Turing, 1939, Systems of Logic Based on Ordinals Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–228.[1]
  3. ^ "Let us suppose that we are supplied with some unspecified means of solving number-theoretic problems; a kind of oracle as it were. We shall not go any further into the nature of this oracle apart from saying that it cannot be a machine" (Undecidable p. 167, a reprint of Turing's paper Systems of Logic Based On Ordinals)
  4. ^ These models have been independently developed by many different authors, including Hermann Weyl (1927). Philosophie der Mathematik und Naturwissenschaft.; the model is discussed in Shagrir, O. (June 2004). "Super-tasks, accelerating Turing machines and uncomputability". Theor. Comput. Sci. 317, 1-3 317: 105–114. doi:10.1016/j.tcs.2003.12.007. and in Petrus H. Potgieter (July 2006). "Zeno machines and hypercomputation". Theoretical Computer Science 358 (1): 23–33. doi:10.1016/j.tcs.2005.11.040.
  5. ^ a b c E. M. Gold (1965). "Limiting Recursion". Journal of Symbolic Logic 30 (1): 28–48. doi:10.2307/2270580. JSTOR 2270580., E. Mark Gold (1967). "Language identification in the limit". Information and Control 10 (5): 447–474. doi:10.1016/S0019-9958(67)91165-5.
  6. ^ a b c Hilary Putnam (1965). "Trial and Error Predicates and the Solution to a Problem of Mostowksi". Journal of Symbolic Logic 30 (1): 49–57. doi:10.2307/2270581. JSTOR 2270581.
  7. ^ a b L. K. Schubert (July 1974). "Iterated Limiting Recursion and the Program Minimization Problem". Journal of the ACM 21 (3): 436–445. doi:10.1145/321832.321841.
  8. ^ Arnold Schönhage, "On the power of random access machines", in Proc. Intl. Colloquium on Automata, Languages, and Programming (ICALP), pages 520-529, 1979. Source of citation: Scott Aaronson, "NP-complete Problems and Physical Reality"[2] p. 12
  9. ^ Edith Spaan, Leen Torenvliet and Peter van Emde Boas (1989). "Nondeterminism, Fairness and a Fundamental Analogy". EATCS bulletin 37: 186–193.
  10. ^ Hajnal Andréka, István Németi and Gergely Székely, Closed Timelike Curves in Relativistic Computation, 2011.[3]
  11. ^ Todd A. Brun, Computers with closed timelike curves can solve hard problems, Found.Phys.Lett. 16 (2003) 245-253.[4]
  12. ^ S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent [5]
  13. ^ Hogarth, M., 1992, ‘Does General Relativity Allow an Observer to View an Eternity in a Finite Time?’, Foundations of Physics Letters, 5, 173–181.
  14. ^ István Neméti; Hajnal Andréka (2006). "Can General Relativistic Computers Break the Turing Barrier?". Logical Approaches to Computational Barriers, Second Conference on Computability in Europe, CiE 2006, Swansea, UK, June 30-July 5, 2006. Proceedings. Lecture Notes in Computer Science. 3988. Springer. doi:10.1007/11780342.
  15. ^ Etesi, G., and Nemeti, I., 2002 'Non-Turing computations via Malament-Hogarth space-times', Int.J.Theor.Phys. 41 (2002) 341–370, Non-Turing Computations via Malament-Hogarth Space-Times:.
  16. ^ Earman, J. and Norton, J., 1993, ‘Forever is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes’, Philosophy of Science, 5, 22–42.
  17. ^ Verifying Properties of Neural Networks p.6
  18. ^ Joel David Hamkins and Andy Lewis, Infinite time Turing machines, Journal of Symbolic Logic, 65(2):567-604, 2000.[6]
  19. ^ a b Jan van Leeuwen; Jiří Wiedermann (September 2000). "On Algorithms and Interaction". MFCS '00: Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science. Springer-Verlag.
  20. ^ Jürgen Schmidhuber (2000). "Algorithmic Theories of Everything". Sections in: Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit. International Journal of Foundations of Computer Science ():587-612 (). Section 6 in: the Speed Prior: A New Simplicity Measure Yielding Near-Optimal Computable Predictions. in J. Kivinen and R. H. Sloan, editors, Proceedings of the 15th Annual Conference on Computational Learning Theory (COLT ), Sydney, Australia, Lecture Notes in Artificial Intelligence, pages 216--228. Springer, . 13 (4): 1–5. arXiv:quant-ph/0011122.
  21. ^ a b J. Schmidhuber (2002). "Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit". International Journal of Foundations of Computer Science 13 (4): 587–612. doi:10.1142/S0129054102001291.
  22. ^ There have been some claims to this effect; see Tien Kieu (2003). "Quantum Algorithm for the Hilbert's Tenth Problem". Int. J. Theor. Phys. 42 (7): 1461–1478. arXiv:quant-ph/0110136. doi:10.1023/A:1025780028846.. & the ensuing literature. Errors have been pointed out in Kieu's approach by Warren D. Smith in Three counterexamples refuting Kieu’s plan for “quantum adiabatic hypercomputation”; and some uncomputable quantum mechanical tasks
  23. ^ Bernstein and Vazirani, Quantum complexity theory, SIAM Journal on Computing, 26(5):1411-1473, 1997. [7]
  24. ^ Santos, Eugene S. (1970). "Fuzzy Algorithms". Information and Control 17 (4): 326–339. doi:10.1016/S0019-9958(70)80032-8.
  25. ^ Biacino, L.; Gerla, G. (2002). "Fuzzy logic, continuity and effectiveness". Archive for Mathematical Logic 41 (7): 643–667. doi:10.1007/s001530100128. ISSN 0933-5846.
  26. ^ a b Wiedermann, Jiří (2004). "Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines". Theor. Comput. Sci. 317 (1–3): 61–69. doi:10.1016/j.tcs.2003.12.004.
  27. ^ a b Dmytro Taranovsky (July 17, 2005). "Finitism and Hypercomputation". Retrieved Apr 26, 2011.
  28. ^ Petrus H. Potgieter (July 2006). "Zeno machines and hypercomputation". Theoretical Computer Science 358 (1): 23–33. doi:10.1016/j.tcs.2005.11.040.
  29. ^ Lenore Blum, Felipe Cucker, Michael Shub, and Stephen Smale. Complexity and Real Computation. ISBN 0-387-98281-7.
  30. ^ P. D. Welch (10-Sept-2006). The extent of computation in Malament-Hogarth spacetimes. arXiv:gr-qc/0609035.
  31. ^ Hava Siegelmann (April 1995). "Computation Beyond the Turing Limit". Science 268 (5210): 545–548. doi:10.1126/science.268.5210.545. PMID 17756722.
  32. ^ Hava Siegelmann; Eduardo Sontag (1994). "Analog Computation via Neural Networks". Theoretical Computer Science 131 (2): 331–360. doi:10.1016/0304-3975(94)90178-3.
  33. ^ Joel David Hamkins; Andy Lewis (2000). "Infinite Time Turing machines". Journal of Symbolic Logic 65 (2): 567=604.
  34. ^ Jiří Wiedermann (June 4, 2004). "Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines". Theoretical Computer Science (Elsevier Science Publishers Ltd. Essex, UK) 317 (1–3).
  35. ^ Hintikka, Ja; Mutanen, A. (1998). "An Alternative Concept of Computability". Language, Truth, and Logic in Mathematics. Dordrecht. pp. 174–188.
  36. ^ Darko Roglic (24–Jul–2007). "The universal evolutionary computer based on super-recursive algorithms of evolvability". arXiv:0708.2686 [cs.NE].
  37. ^ Eugene Eberbach (2002). "On expressiveness of evolutionary computation: is EC algorithmic?". Computational Intelligence, WCCI 1: 564–569. doi:10.1109/CEC.2002.1006988.
  38. ^ Borodyanskii, Yu M; Burgin, M. S. (1994). "Operations and compositions in transrecursive operators". Cybernetics and Systems Analysis 30 (4): 473–478. doi:10.1007/BF02366556.
  39. ^ Davis, Martin, Why there is no such discipline as hypercomputation, Applied Mathematics and Computation, Volume 178, Issue 1, 1 July 2006, Pages 4–7, Special Issue on Hypercomputation
  40. ^ Davis, Martin (2004). "The Myth of Hypercomputation". Alan Turing: Life and Legacy of a Great Thinker. Springer.
  41. ^ Andrew Hodges (retrieved 23 September 2011). "The Professors and the Brainstorms". The Alan Turing Home Page.

 

Further reading

 

External links