Hypercomputation or
superTuring 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
Turingcomputable functions, following
superrecursive algorithms (see also
supertask). The term "superTuring 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: "superTuring 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.
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 (nonrecursive) 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 ChurchTuring 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 "trialanderror 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 higherorder inductive
inference performed collectively by an evergrowing 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 realvalued 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 PSPACEcomplete problems, a complexity class which while Turingdecidable is generally considered computationally intractable.^{[11]}^{[12]}
 According to a 1992 paper,^{[13]} a computer operating in a MalamentHogarth spacetime or in orbit around a rotating black hole^{[14]} could theoretically perform nonTuring 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 otherwiseordinary Turing machine for which nonhalting
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 nonhalting 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, nonhalting 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 noncomputable function.^{[22]} This is not possible using the standard qubitmodel quantum computer, because it is proven that a regular quantum computer is PSPACEreducible (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 logicbased "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 nonfinitistic 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 secondorder 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
truthtable degree containing
or
. Limitingrecursion, by contrast, can compute any predicate or function in the corresponding
Turing degree, which is known to be
. Gold further showed that limiting partial recursion would allow the computation of precisely the
predicates.
supertasking 
tt() 
dependent on outside observer 
^{[28]} 
limiting/trialanderror 


^{[5]} 
iterated limiting (k times) 


^{[7]} 
BlumShubSmale machine 

incomparable with traditional computable real functions. 
^{[29]} 
MalamentHogarth spacetime 
HYP 
Dependent on spacetime structure 
^{[30]} 
Analog recurrent neural network 

f is an advice function giving connection weights; size is bounded by runtime 
^{[31]}^{[32]} 
Infinite time Turing machine 


^{[33]} 
Classical fuzzy Turing machine 

For any computable tnorm 
^{[34]} 
Increasing function oracle 

For the onesequence model; are r.e. 
^{[27]} 
Taxonomy of "superrecursive" computation methodologies
Burgin has collected a list of what he calls "superrecursive 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)
 trialanderror 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":
 Turing machines with arbitrary oracles (Alan Turing)
 Transrecursive operators (Borodyanskii and Burgin^{[38]})
 machines that compute with real numbers (L. Blum, F. Cucker, M. Shub, and S. Smale)
 neural networks based on real numbers (Hava Siegelmann)
Criticism
Martin Davis, in his writings on hypercomputation
^{[39]} ^{[40]}
refers to this subject as "a myth" and offers counterarguments 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
 ^ ^{a} ^{b} Copeland and Proudfoot, Alan Turing's forgotten ideas in computer science. Scientific American, April 1999
 ^ Alan Turing, 1939, Systems of Logic Based on Ordinals Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–228.[1]
 ^ "Let
us suppose that we are supplied with some unspecified means of solving
numbertheoretic 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)
 ^ 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). "Supertasks, accelerating Turing machines and uncomputability". Theor. Comput. Sci. 317, 13 317: 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.
 ^ ^{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/S00199958(67)911655.
 ^ ^{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.
 ^ ^{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.
 ^ Arnold Schönhage, "On the power of random access machines", in Proc. Intl. Colloquium on Automata, Languages, and Programming (ICALP), pages 520529, 1979. Source of citation: Scott Aaronson, "NPcomplete Problems and Physical Reality"[2] p. 12
 ^ Edith Spaan, Leen Torenvliet and Peter van Emde Boas (1989). "Nondeterminism, Fairness and a Fundamental Analogy". EATCS bulletin 37: 186–193.
 ^ Hajnal Andréka, István Németi and Gergely Székely, Closed Timelike Curves in Relativistic Computation, 2011.[3]
 ^ Todd A. Brun, Computers with closed timelike curves can solve hard problems, Found.Phys.Lett. 16 (2003) 245253.[4]
 ^ S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent [5]
 ^ Hogarth,
M., 1992, ‘Does General Relativity Allow an Observer to View an
Eternity in a Finite Time?’, Foundations of Physics Letters, 5, 173–181.
 ^ 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 30July 5, 2006.
Proceedings. Lecture Notes in Computer Science. 3988. Springer. doi:10.1007/11780342.
 ^ Etesi,
G., and Nemeti, I., 2002 'NonTuring computations via MalamentHogarth
spacetimes', Int.J.Theor.Phys. 41 (2002) 341–370, NonTuring Computations via MalamentHogarth SpaceTimes:.
 ^ Earman,
J. and Norton, J., 1993, ‘Forever is a Day: Supertasks in Pitowsky and
MalamentHogarth Spacetimes’, Philosophy of Science, 5, 22–42.
 ^ Verifying Properties of Neural Networks p.6
 ^ Joel David Hamkins and Andy Lewis, Infinite time Turing machines, Journal of Symbolic Logic, 65(2):567604, 2000.[6]
 ^ ^{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. SpringerVerlag.
 ^ 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 ():587612 (). Section 6 in:
the Speed Prior: A New Simplicity Measure Yielding NearOptimal
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 216228. Springer, . 13 (4): 1–5. arXiv:quantph/0011122.
 ^ ^{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.
 ^ 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:quantph/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
 ^ Bernstein and Vazirani, Quantum complexity theory, SIAM Journal on Computing, 26(5):14111473, 1997. [7]
 ^ Santos, Eugene S. (1970). "Fuzzy Algorithms". Information and Control 17 (4): 326–339. doi:10.1016/S00199958(70)800328.
 ^ Biacino, L.; Gerla, G. (2002). "Fuzzy logic, continuity and effectiveness". Archive for Mathematical Logic 41 (7): 643–667. doi:10.1007/s001530100128. ISSN 09335846.
 ^ ^{a} ^{b} Wiedermann, Jiří (2004). "Characterizing the superTuring 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.
 ^ ^{a} ^{b} Dmytro Taranovsky (July 17, 2005). "Finitism and Hypercomputation". Retrieved Apr 26, 2011.
 ^ Petrus H. Potgieter (July 2006). "Zeno machines and hypercomputation". Theoretical Computer Science 358 (1): 23–33. doi:10.1016/j.tcs.2005.11.040.
 ^ Lenore Blum, Felipe Cucker, Michael Shub, and Stephen Smale. Complexity and Real Computation. ISBN 0387982817.
 ^ P. D. Welch (10Sept2006). The extent of computation in MalamentHogarth spacetimes. arXiv:grqc/0609035.
 ^ Hava Siegelmann (April 1995). "Computation Beyond the Turing Limit". Science 268 (5210): 545–548. doi:10.1126/science.268.5210.545. PMID 17756722.
 ^ Hava Siegelmann; Eduardo Sontag (1994). "Analog Computation via Neural Networks". Theoretical Computer Science 131 (2): 331–360. doi:10.1016/03043975(94)901783.
 ^ Joel David Hamkins; Andy Lewis (2000). "Infinite Time Turing machines". Journal of Symbolic Logic 65 (2): 567=604.
 ^ Jiří
Wiedermann (June 4, 2004). "Characterizing the superTuring computing
power and efficiency of classical fuzzy Turing machines". Theoretical Computer Science (Elsevier Science Publishers Ltd. Essex, UK) 317 (1–3).
 ^ Hintikka, Ja; Mutanen, A. (1998). "An Alternative Concept of Computability". Language, Truth, and Logic in Mathematics. Dordrecht. pp. 174–188.
 ^ Darko Roglic (24–Jul–2007). "The universal evolutionary computer based on superrecursive algorithms of evolvability". arXiv:0708.2686 [cs.NE].
 ^ Eugene Eberbach (2002). "On expressiveness of evolutionary computation: is EC algorithmic?". Computational Intelligence, WCCI 1: 564–569. doi:10.1109/CEC.2002.1006988.
 ^ 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.
 ^ 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
 ^ Davis, Martin (2004). "The Myth of Hypercomputation". Alan Turing: Life and Legacy of a Great Thinker. Springer.
 ^ Andrew Hodges (retrieved 23 September 2011). "The Professors and the Brainstorms". The Alan Turing Home Page.
Further reading
 Hava Siegelmann (April 1995). "Computation Beyond the Turing Limit". Science 268 (5210): 545–548. doi:10.1126/science.268.5210.545. PMID 17756722.
 Turing, Alan (1939). "Systems of logic based on ordinals". Proc. London math. Soc. 45.
 Hava Siegelmann and Eduardo Sontag, “Analog Computation via Neural Networks,” Theoretical Computer Science 131, 1994: 331360.
 Hava Siegelmann. Neural Networks and Analog Computation: Beyond the Turing Limit 1998 Boston: Birkhäuser (Book).
 Mike Stannett, The case for hypercomputation, Applied Mathematics and Computation, Volume 178, Issue 1, 1 July 2006, Pages 8–24, Special Issue on Hypercomputation
 Keith Douglas. SuperTuring Computation: a Case Study Analysis (PDF), M.S. Thesis, Carnegie Mellon University, 2003.
 L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation, SpringerVerlag 1997. General development of complexity theory for abstract machines that compute on real numbers instead of bits.
 On the computational power of neural nets
 Toby Ord. Hypercomputation: Computing more than the Turing machine can compute: A survey article on various forms of hypercomputation.
 Apostolos Syropoulos (2008), Hypercomputation: Computing Beyond the ChurchTuring Barrier (preview), Springer. ISBN 9780387308869
 Burgin, M. S. (1983) Inductive Turing Machines, Notices of the Academy of Sciences of the USSR, v. 270, No. 6, pp. 1289–1293
 Mark Burgin (2005), Superrecursive algorithms, Monographs in computer science, Springer. ISBN 0387955690
 Cockshott, P. and Michaelson, G. Are there new Models of Computation? Reply to Wegner and Eberbach, The computer Journal, 2007
 Cooper, S. B. (2006). "Definability as hypercomputational effect". Applied Mathematics and Computation 178: 72–82. doi:10.1016/j.amc.2005.09.072.
 Cooper, S. B.; Odifreddi, P. (2003). "Incomputability in Nature". In S. B. Cooper and S. S. Goncharov. Computability and Models: Perspectives East and West. Plenum Publishers, New York, Boston, Dordrecht, London, Moscow. pp. 137–160.
 Copeland, J. (2002) Hypercomputation, Minds and machines, v. 12, pp. 461–502
 Martin Davis (2006), "The Church–Turing Thesis: Consensus and opposition". Proceedings, Computability in Europe 2006. Lecture notes in computer science, 3988 pp. 125–132
 Hagar, A. and Korolev, A., Quantum Hypercomputation—Hype or Computation?, (2007)
 Rogers, H. (1987) Theory of Recursive Functions and Effective Computability, MIT Press, Cambridge Massachusetts
 Volkmar Putz and Karl Svozil, Can a computer be "pushed" to perform fasterthanlight?, (2010)
External links