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| '''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-[[Computable function|Turing-computable function]]s, following [[super-recursive algorithm]]s (see also [[supertask]]). The term "super-Turing computation" appeared in a 1995 ''[[Science (journal)|Science]]'' paper by [[Hava Siegelmann]]. The term "hypercomputation" was introduced in 1999 by [[Jack Copeland]] and [[Diane Proudfoot]].<ref name=CandP>Copeland and Proudfoot, ''[http://www.sciamdigital.com/index.cfm?fa=Products.ViewIssuePreview&ARTICLEID_CHAR=94B166BF-E481-47FA-80C8-112C6BAF404 Alan Turing's forgotten ideas in computer science]''. [[Scientific American]], April 1999</ref>
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| 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.
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| Technical arguments against the physical realizability of hypercomputations have been presented.
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| ==History==
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| A computational model going beyond Turing machines was introduced by [[Alan Turing]] in his 1938 PhD dissertation ''[[Systems of Logic Based on Ordinals]]''.<ref>Alan Turing, 1939, ''Systems of Logic Based on Ordinals'' Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–228.[http://plms.oxfordjournals.org/content/s2-45/1/161.extract]</ref> This paper investigated mathematical systems in which an [[Oracle machine|oracle]] was available, which could compute a single arbitrary (non-recursive) function from [[Natural number|naturals]] to naturals. He used this device to prove that even in those more powerful systems, [[Undecidable problem|undecidability]] is still present. Turing's oracle machines are strictly mathematical abstractions, and are not physically realizable.<ref>"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'')</ref>
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| ==Hypercomputation and the Church–Turing thesis==
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| 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 and which are, hence, not computable in the Church-Turing sense.
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| 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 the program halted.
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| ==Hypercomputer proposals==
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| *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 [[1/2 + 1/4 + 1/8 + 1/16 + · · ·|1+½+¼+...]] (a [[geometric series]]) we see that the machine performs infinitely many steps in a total of 2 minutes. According to Shagrir, Zeno machines introduce physical paradoxes and its state is logically undefined outside of one-side open period of [0, 2), thus undefined exactly at 2 minutes after beginning of the computation.<ref>These models have been independently developed by many different authors, including {{cite book|author=[[Hermann Weyl]]| year=1927 | title=Philosophie der Mathematik und Naturwissenschaft}}; the model is discussed in {{cite journal
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| |author=Shagrir, O.
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| |title=Super-tasks, accelerating Turing machines and uncomputability
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| |journal=Theor. Comput. Sci. 317, 1-3
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| |date=June 2004
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| |pages=105–114
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| |doi=10.1016/j.tcs.2003.12.007
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| |url=http://edelstein.huji.ac.il/staff/shagrir/papers/Supertasks_Accelerating_Turing_Machines_and_Uncomputability.pdf
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| |volume=317}} and in {{cite journal| author=Petrus H. Potgieter| title=Zeno machines and hypercomputation| journal=Theoretical Computer Science| volume=358 | issue=1 |date=July 2006 | pages=23–33| doi=10.1016/j.tcs.2005.11.040}}</ref>
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| *Turing's original oracle machines, defined by Turing in 1939.
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| *In mid 1960s, [[E Mark Gold]] and [[Hilary Putnam]] independently proposed models of [[inductive inference]] (the "limiting recursive functionals"<ref name=LimRecurs>{{cite journal | author=E. M. Gold | title=Limiting Recursion | journal=Journal of Symbolic Logic | volume=30 | issue=1 | pages=28–48 | year=1965 | jstor=2270580 | doi=10.2307/2270580}}, {{cite journal | author=E. Mark Gold | title=Language identification in the limit | journal=Information and Control | volume=10 | pages=447–474 | year=1967 | doi=10.1016/S0019-9958(67)91165-5 | issue=5}}</ref> and "trial-and-error predicates",<ref name=TrialError>{{cite journal | author=Hilary Putnam | title=Trial and Error Predicates and the Solution to a Problem of Mostowksi | journal=Journal of Symbolic Logic | volume=30 | issue=1 | pages=49–57 | year=1965 | jstor=2270581 | doi=10.2307/2270581}}</ref> 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.)"<ref name=TrialError/> [[L. K. Schubert]]'s 1974 paper "Iterated Limiting Recursion and the Program Minimization Problem" <ref name=IterLimRec>{{cite journal| author=L. K. Schubert | title=Iterated Limiting Recursion and the Program Minimization Problem | journal=Journal of the ACM | volume=21 | issue=3 |date=July 1974 | url=http://portal.acm.org/citation.cfm?id=321832.321841 | doi=10.1145/321832.321841| pages=436–445}}</ref> studied the effects of iterating the limiting procedure; this allows any [[arithmetic hierarchy|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."
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| *A [[real computer]] (a sort of idealized [[analog computer]]) can perform hypercomputation<ref>[[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"[http://www.scottaaronson.com/papers/npcomplete.pdf] p. 12</ref> if physics admits general [[real number|real]] variables (not just [[computable number|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 mechanics|quantum]] effects.
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| *A proposed technique known as [[fair nondeterminism]] or [[unbounded nondeterminism]] may allow the computation of noncomputable functions.<ref>{{cite journal|title=Nondeterminism, Fairness and a Fundamental Analogy|journal=EATCS bulletin|volume=37|pages=186–193|year=1989|author=Edith Spaan, Leen Torenvliet and Peter van Emde Boas}}</ref> There is dispute in the literature over whether this technique is coherent, and whether it actually allows noncomputable functions to be "computed".
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| *It seems natural that the possibility of time travel (existence of [[closed timelike curve]]s (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.<ref>Hajnal Andréka, István Németi and Gergely Székely, ''Closed Timelike Curves in Relativistic Computation'', 2011.[http://arxiv.org/abs/1105.0047]</ref> 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.<ref>Todd A. Brun, ''Computers with closed timelike curves can solve hard problems'', Found.Phys.Lett. 16 (2003) 245-253.[http://arxiv.org/abs/gr-qc/0209061]</ref><ref>[[Scott Aaronson|S. Aaronson]] and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent [http://scottaaronson.com/papers/ctc.pdf]</ref>
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| * According to a 1992 paper,<ref>Hogarth, M., 1992, ‘Does General Relativity Allow an Observer to View an Eternity in a Finite Time?’, Foundations of Physics Letters, 5, 173–181.</ref> a computer operating in a [[Malament-Hogarth spacetime]] or in orbit around a rotating [[black hole]]<ref>{{cite book|chapter=Can General Relativistic Computers Break the Turing Barrier? | author=István Neméti | author2=Hajnal Andréka | title=Logical Approaches to Computational Barriers, Second Conference on Computability in Europe, CiE 2006, Swansea, UK, June 30-July 5, 2006. Proceedings | publisher=Springer | series=Lecture Notes in Computer Science | volume=3988 | doi=10.1007/11780342 | year=2006}}</ref> could theoretically perform non-Turing computations.<ref>Etesi, G., and Nemeti, I., 2002 'Non-Turing computations via Malament-Hogarth space-times', Int.J.Theor.Phys. 41 (2002) 341–370, [http://lanl.arxiv.org/abs/gr-qc/0104023 Non-Turing Computations via Malament-Hogarth Space-Times:].</ref><ref>Earman, J. and Norton, J., 1993, ‘Forever is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes’, Philosophy of Science, 5, 22–42.</ref>
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| *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.<ref>[http://www.cs.math.ist.utl.pt/ftp/pub/CostaJF/01-RCS-iwann.pdf Verifying Properties of Neural Networks] p.6</ref>
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| *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 number]]s. 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.<ref>[[Joel David Hamkins]] and [[Andy Lewis (mathematician)|Andy Lewis]], Infinite time Turing machines, ''Journal of Symbolic Logic'', 65(2):567-604, 2000.[http://jdh.hamkins.org/Publications/2000e]</ref>
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| *[[Jan van Leeuwen]] and Jiří Wiedermann wrote a paper<ref name=InternetMachines>{{cite book | author=Jan van Leeuwen | author2=Jiří Wiedermann | chapter=On Algorithms and Interaction | title=MFCS '00: Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science | date=September 2000 | publisher=[[Springer-Verlag]] }}</ref> suggesting that the Internet should be modeled as a nonuniform computing system equipped with an [[advice (complexity)|advice]] function representing the ability of computers to be upgraded.
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| *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 (<ref name=genTuring2000>{{cite journal | author=[[Jürgen Schmidhuber]]| year=2000 | title=Algorithmic Theories of Everything | pages=1–5 | volume=13 | issue=4 | journal=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, . | arxiv=quant-ph/0011122 }}</ref><ref name=GenKolm/>) uses this approach to define the set of formally describable or constructively computable universes or constructive [[theory of everything|theories of everything]]. Generalized Turing machines can solve the halting problem by evaluating a [[Specker sequence]].
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| *A [[Quantum mechanics|quantum mechanical]] system which somehow uses an infinite superposition of states to compute a non-[[computable function]].<ref>There have been some claims to this effect; see {{cite journal | author = Tien Kieu | title = Quantum Algorithm for the [[Hilbert problems|Hilbert's Tenth Problem]] | journal = Int. J. Theor. Phys. | year = 2003 | volume = 42 | arxiv = quant-ph/0110136 | pages = 1461–1478 | doi = 10.1023/A:1025780028846 | issue = 7}}. & the ensuing literature. Errors have been pointed out in Kieu's approach by Warren D. Smith in [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TY8-4JD0GX5-1&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=a63612cc7522010ee340e8ddada13779 Three counterexamples refuting Kieu’s plan for “quantum adiabatic hypercomputation”; and some uncomputable quantum mechanical tasks]
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| </ref> This is not possible using the standard [[qubit]]-model [[quantum computer]], because it is proven that a regular quantum computer is [[PSPACE-reduction|PSPACE-reducible]] (a quantum computer running in [[polynomial time]] can be simulated by a classical computer running in [[polynomial space]]).<ref>Bernstein and Vazirani, Quantum complexity theory, [[SIAM Journal on Computing]], 26(5):1411-1473, 1997. [http://www.cs.berkeley.edu/~vazirani/bv.ps]</ref>
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| *In 1970, E.S. Santos defined a class of [[fuzzy logic]]-based "fuzzy algorithms" and "fuzzy Turing machines".<ref>{{Cite journal|doi=10.1016/S0019-9958(70)80032-8|last=Santos|first=Eugene S.|year=1970|title=Fuzzy Algorithms|journal=Information and Control|volume=17|issue=4|pages=326–339}}</ref> 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.<ref>{{Cite journal|last=Biacino|first=L.|coauthors=Gerla, G.|year=2002|title=Fuzzy logic, continuity and effectiveness|url=|journal=Archive for Mathematical Logic|issn=0933-5846|volume=41|issue=7|pages=643–667|doi=10.1007/s001530100128}}</ref> Jiří Wiedermann analyzed the capabilities of Santos' original proposal in 2004.<ref name=ClassicalFuzzy>{{Cite journal|last=Wiedermann|first=Jiří |year=2004|title=Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines|url=http://portal.acm.org/citation.cfm?id=1011188|journal=Theor. Comput. Sci.|volume=317|issue=1–3|pages=61–69|doi=10.1016/j.tcs.2003.12.004}}</ref>
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| *Dmytro Taranovsky has proposed a [[finitism|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.<ref name=Taranovsky>{{cite web | author=Dmytro Taranovsky | date=July 17, 2005 | title=Finitism and Hypercomputation | url=http://web.mit.edu/dmytro/www/FinitismPaper.htm | accessdate=Apr 26, 2011}}</ref>
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| ==Analysis of capabilities==
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| Many hypercomputation proposals amount to alternative ways to read an [[oracle machine|oracle]] or [[advice (complexity)|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 reduction|truth-table degree]] containing <math>\Sigma^0_1</math> or <math>\Pi^0_1</math>. Limiting-recursion, by contrast, can compute any predicate or function in the corresponding [[Turing degree]], which is known to be <math>\Delta^0_2</math>. Gold further showed that limiting partial recursion would allow the computation of precisely the <math>\Sigma^0_2</math> predicates.
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| {| class="wikitable sortable"
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| |-
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| ! Model
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| ! Computable predicates
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| ! Notes
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| ! Refs
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| | supertasking
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| | tt(<math>\Sigma^0_1, \Pi^0_1</math>)
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| | dependent on outside observer
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| | <ref>{{cite journal| author=Petrus H. Potgieter| title=Zeno machines and hypercomputation| journal=Theoretical Computer Science| volume=358 | issue=1 |date=July 2006 | pages=23–33| doi=10.1016/j.tcs.2005.11.040}}</ref>
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| |-
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| | limiting/trial-and-error
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| | <math> \Delta^0_2 </math>
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| | <ref name=LimRecurs/>
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| |-
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| | iterated limiting (''k'' times)
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| | <math> \Delta^0_{k+1} </math>
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| | <ref name=IterLimRec/>
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| |-
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| | [[Blum-Shub-Smale machine]]
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| | incomparable with traditional [[computable real]] functions.
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| | <ref>{{cite book|author=[[Lenore Blum]], Felipe Cucker, Michael Shub, and [[Stephen Smale]]|title=Complexity and Real Computation|isbn=0-387-98281-7}}</ref>
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| |-
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| | Malament-Hogarth spacetime
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| | '''[[Hyperarithmetic hierarchy|HYP]]'''
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| | Dependent on spacetime structure
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| | <ref>{{cite journal | author=P. D. Welch | title = The extent of computation in Malament-Hogarth spacetimes | date=10 Sep 2006 | arxiv=gr-qc/0609035}}</ref>
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| |-
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| | Analog recurrent neural network
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| | <math> \Delta^0_1[f] </math>
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| | ''f'' is an advice function giving connection weights; size is bounded by runtime
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| | <ref>{{cite journal | author=[[Hava Siegelmann]] | title=Computation Beyond the Turing Limit | journal=Science | volume=268 | issue=5210 |date=April 1995 | pages=545–548 | doi=10.1126/science.268.5210.545 | pmid=17756722}}</ref><ref>{{cite journal | author=[[Hava Siegelmann]] | author2=Eduardo Sontag | title=Analog Computation via Neural Networks | journal=Theoretical Computer Science | volume=131 | year=1994 | pages=331–360 | doi=10.1016/0304-3975(94)90178-3 | issue=2 | authorlink2=Eduardo Sontag}}</ref>
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| |-
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| | Infinite time Turing machine
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| | <math> \ge T(\Sigma^1_1) </math>
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| | <ref>{{cite journal|author=Joel David Hamkins|author2=Andy Lewis|title=Infinite Time Turing machines|journal=Journal of Symbolic Logic|year=2000|volume=65|issue=2|page=567=604|url=http://jdh.hamkins.org/Publications/2000e}}</ref>
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| | Classical fuzzy Turing machine
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| | <math> \Sigma^0_1 \cup \Pi^0_1 </math>
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| | For any computable [[T-norm fuzzy logics|t-norm]]
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| | <ref>{{cite journal | author=Jiří Wiedermann | title=Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines | journal=Theoretical Computer Science | volume=317 | issue=1–3 | date=June 4, 2004 | publisher=Elsevier Science Publishers Ltd. Essex, UK}}</ref>
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| |-
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| | Increasing function oracle
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| | <math> \Delta^1_1 </math>
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| | For the one-sequence model; <math> \Pi^1_1 </math> are r.e.
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| | <ref name=Taranovsky/>
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| |}
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| ==Taxonomy of "super-recursive" computation methodologies==
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| [[Mark Burgin]] has collected a list of what he calls "super-recursive algorithms" (from Burgin 2005: 132):
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| * '''limiting recursive functions''' and '''limiting partial recursive functions''' (E. M. Gold<ref name=LimRecurs/>)
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| * '''trial and error predicates''' ([[Hilary Putnam]]<ref name=TrialError/>)
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| * '''[[inductive inference]] machines''' ([[Carl Herbert Smith]])
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| * '''inductive Turing machines''' (one of Burgin's own models)
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| * '''limit Turing machines''' (another of Burgin's models)
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| * '''trial-and-error machines''' (Ja. Hintikka and A. Mutanen <ref>{{cite book | author=Hintikka, Ja | author2=Mutanen, A. | chapter=An Alternative Concept of Computability | title=Language, Truth, and Logic in Mathematics | publisher=Dordrecht | pages=174–188 | year=1998}}</ref>)
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| * '''general Turing machines''' (J. Schmidhuber<ref name=GenKolm>{{cite journal| author=J. Schmidhuber | title=Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit | journal=International Journal of Foundations of Computer Science | volume=13 | issue=4 | pages=587–612 | year=2002 | url=http://www.idsia.ch/~juergen/kolmogorov.html| doi=10.1142/S0129054102001291}}</ref>)
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| * '''Internet machines''' ([[Jan van Leeuwen|van Leeuwen, J.]] and Wiedermann, J.<ref name="InternetMachines"/>) <!-- should go in the previous section as well. -->
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| * '''evolutionary computers''', which use DNA to produce the value of a function (Darko Roglic<ref>{{cite arXiv | author=Darko Roglic | title=The universal evolutionary computer based on super-recursive algorithms of evolvability | year = 24–Jul–2007 | eprint=0708.2686 | class=cs.NE}}</ref>)
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| * '''fuzzy computation''' (Jiří Wiedermann<ref name=ClassicalFuzzy/>)
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| * '''evolutionary Turing machines''' (Eugene Eberbach<ref>{{cite journal | author=Eugene Eberbach | title=On expressiveness of evolutionary computation: is EC algorithmic? | journal=Computational Intelligence, WCCI | year=2002 | volume=1 | pages=564–569 | url=http://www.computer.org/portal/web/csdl/doi/10.1109/CEC.2002.1006988 | doi=10.1109/CEC.2002.1006988}}</ref>)
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| In the same book, he presents also a list of "algorithmic schemes":
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| * '''Turing machines with arbitrary [[Oracle (computer science)|oracles]]''' (Alan Turing)
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| * '''Transrecursive operators''' (Borodyanskii and Burgin<ref>{{cite journal | author=Borodyanskii, Yu M | author2=Burgin, M. S. | title=Operations and compositions in transrecursive operators | journal=Cybernetics and Systems Analysis |
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| volume=30 | issue=4 | pages=473–478 | doi=10.1007/BF02366556 | url=http://www.springerlink.com/content/a70r23722wqu43t7/ | year=1994 }}
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| </ref>)
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| * '''[[Real computation|machines that compute with real numbers]]''' (L. Blum, F. Cucker, M. Shub, and S. Smale)
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| * '''neural networks based on real numbers''' (Hava Siegelmann)
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| ==Criticism==
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| [[Martin Davis]], in his writings on hypercomputation <ref name=Davis95>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</ref>
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| <ref>{{cite book |last=Davis| first=Martin|title=Alan Turing: Life and Legacy of a Great Thinker|publisher=Springer|year=2004 |chapter=The Myth of Hypercomputation}}</ref>
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| refers to this subject as "a myth" and offers counter-arguments to the
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| physical realizability of hypercomputation. As for its theory, he argues against
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| 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
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| functions, real numbers and ordinals), as also mentioned above.
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| | |
| [[Andrew Hodges]] wrote a critical commentary<ref name=HodgesSCIAM>{{cite web |url=http://www.turing.org.uk/philosophy/sciam.html |title=The Professors and the Brainstorms |author=Andrew Hodges |date=retrieved 23 September 2011 |work=The Alan Turing Home Page }}</ref> on Copeland and Proudfoot's article.<ref name=CandP />
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| ==See also==
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| * [[Computation]]
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| * [[Digital physics]]
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| * [[Supertask]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * {{cite journal | author=[[Hava Siegelmann]] | title=Computation Beyond the Turing Limit | journal=Science | volume=268 | issue=5210 |date=April 1995 | pages=545–548 | doi=10.1126/science.268.5210.545 | pmid=17756722}}
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| *{{cite journal | last1 = Turing | first1 = Alan | authorlink = Alan Turing | title = Systems of logic based on ordinals | url = | journal = Proceedings of the London Mathematical Society | volume = 45 | issue = | year = 1939 }}
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| *[[Hava Siegelmann]] and [[Eduardo Sontag]], “Analog Computation via Neural Networks,” Theoretical Computer Science 131, 1994: 331-360.
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| *[[Hava Siegelmann]]. ''Neural Networks and Analog Computation: Beyond the Turing Limit'' 1998 Boston: Birkhäuser (Book).
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| *[[Mike Stannett]], ''[http://research.cs.queensu.ca/home/akl/cisc879/papers/PAPERS_FROM_APPLIED_MATHEMATICS_AND_COMPUTATION/Special_Issue_on_Hypercomputation/stannett%5b1%5d.pdf The case for hypercomputation]'', Applied Mathematics and Computation, Volume 178, Issue 1, 1 July 2006, Pages 8–24, Special Issue on Hypercomputation
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| *Keith Douglas. ''[http://www.philosopher-animal.com/papers/take6c.PDF Super-Turing Computation: a Case Study Analysis]'' ([[Portable Document Format|PDF]]), M.S. Thesis, Carnegie Mellon University, 2003.
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| *L. Blum, F. Cucker, M. Shub, S. Smale, ''Complexity and Real Computation'', Springer-Verlag 1997. General development of complexity theory for [[abstract machine]]s that compute on [[real computation|real numbers]] instead of bits.
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| <!-- #http://www.nature.com/nsu/010329/010329-8.html A ''[[Nature (journal)|Nature]]'' article on the above. This link doesn't seem to go to the article anymore. -->
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| *[ftp://ftp.cs.cuhk.hk/pub/neuro/papers/jcss1.ps.Z On the computational power of neural nets]
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| *Toby Ord. [http://arxiv.org/abs/math/0209332 ''Hypercomputation: Computing more than the Turing machine can compute'']: A survey article on various forms of hypercomputation.
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| *Apostolos Syropoulos (2008), ''[http://www.springer.com/computer/foundations/book/978-0-387-30886-9 Hypercomputation: Computing Beyond the Church-Turing Barrier]'' ([http://books.google.com/books?id=5gVOf_OQa04C preview]), Springer. ISBN 978-0-387-30886-9
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| * Burgin, M. S. (1983) Inductive Turing Machines, ''Notices of the Academy of Sciences of the USSR'', v. 270, No. 6, pp. 1289–1293
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| * Mark Burgin (2005), ''Super-recursive algorithms'', Monographs in computer science, Springer. ISBN 0-387-95569-0
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| * Cockshott, P. and Michaelson, G. Are there new Models of Computation? Reply to Wegner and Eberbach, ''The computer Journal'', 2007
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| *{{cite journal|last = Cooper|first = S. B.|title = Definability as hypercomputational effect|journal = Applied Mathematics and Computation|volume = 178|year = 2006|pages = 72–82|url = http://www.amsta.leeds.ac.uk/~pmt6sbc/preprints/hyp.comp.eff.pdf|doi = 10.1016/j.amc.2005.09.072}}
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| *{{cite book|last = Cooper|first = S. B.|coauthors = Odifreddi, P.|chapter = Incomputability in Nature|title= Computability and Models: Perspectives East and West| editors = S. B. Cooper and S. S. Goncharov | publisher = Plenum Publishers, New York, Boston, Dordrecht, London, Moscow|year = 2003|pages = 137–160|url = http://www.amsta.leeds.ac.uk/~pmt6sbc/preprints/co.pdf}}
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| * Copeland, J. (2002) ''[http://research.cs.queensu.ca/home/akl/cisc879/papers/PAPERS_FROM_MINDS_AND_MACHINES/VOLUME_12_NO_4/NV6361035557Q678.pdf Hypercomputation]'', Minds and machines, v. 12, pp. 461–502
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| * Martin Davis (2006), "[http://people.cs.uchicago.edu/~simon/TEACH/28000/DavisUniversal.pdf The Church–Turing Thesis: Consensus and opposition]". Proceedings, Computability in Europe 2006. Lecture notes in computer science, 3988 pp. 125–132
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| * Hagar, A. and Korolev, A., ''[http://philsci-archive.pitt.edu/archive/00003180/ Quantum Hypercomputation—Hype or Computation?]'', (2007)
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| * Rogers, H. (1987) Theory of Recursive Functions and Effective Computability, MIT Press, Cambridge Massachusetts
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| * Volkmar Putz and Karl Svozil, ''[http://arxiv.org/abs/1003.1238 Can a computer be "pushed" to perform faster-than-light?]'', (2010)
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| ==External links==
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| *[http://www.hypercomputation.net/ Hypercomputation Research Network]
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| *[http://www.hypercomputation.blogspot.com/ Hypercomputation]
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| *[http://arxiv.org/abs/math/0209332 Toby Ord, ''Hypercomputation: computing more than the Turing machine'']
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| *[http://www.amirrorclear.net/academic/papers/many-forms.pdf Toby Ord, ''The many forms of hypercomputation'']
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| *[http://citeseer.ist.psu.edu/cotogno03hypercomputation.html Paolo Cotogno, ''Hypercomputation and the Physical Church-Turing thesis'']
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| *[[Gualtiero Piccinini]], [http://plato.stanford.edu/entries/computation-physicalsystems/ ''Computation in Physical Systems'']
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| [[Category:Theory of computation]]
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| [[Category:Hypercomputation| ]]
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