Hypercomputation or superTuring computation refers to models of computation that can provide outputs that are not Turing computable. For example, a machine that could solve the halting problem would be a hypercomputer; so too would one that can correctly evaluate every statement in Peano arithmetic.
The Church–Turing thesis states that any "effectively computable" function that can be computed by a mathematician with a pen and paper using a finite set of simple algorithms, can be computed by a Turing machine. Hypercomputers compute functions that a Turing machine cannot and which are, hence, not effectively computable in the Church–Turing sense.
Technically the output of a random Turing machine is uncomputable; however, most hypercomputing literature focuses instead on the computation of useful, rather than random, uncomputable functions.
History
A computational model going beyond Turing machines was introduced by Alan Turing in his 1938 PhD dissertation Systems of Logic Based on Ordinals.^{[1]} 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 mathematical abstractions, and are not physically realizable.^{[2]}
State space
In a sense, most functions are uncomputable: there are ${displaystyle aleph _{0}}$ computable functions, but there are an uncountable number (${displaystyle 2^{aleph _{0}}}$) of possible SuperTuring functions.^{[3]}
Hypercomputer models
Hypercomputer models range from useful but probably unrealizable (such as Turing's original oracle machines), to lessuseful randomfunction generators that are more plausibly "realizable" (such as a random Turing machine).
Hypercomputers with uncomputable inputs or blackbox components
A system granted knowledge of the uncomputable, oracular Chaitin's constant (a number with an infinite sequence of digits that encode the solution to the halting problem) as an input can solve a large number of useful undecidable problems; a system granted an uncomputable randomnumber generator as an input can create random uncomputable functions, but is generally not believed to be able to meaningfully solve "useful" uncomputable functions such as the halting problem. There are an unlimited number of different types of conceivable hypercomputers, including:
 Turing's original oracle machines, defined by Turing in 1939.
 A real computer (a sort of idealized analog computer) can perform hypercomputation^{[4]} if physics admits general real variables (not just computable reals), and these are in some way "harnessable" for useful (rather than random) 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 require the ability to measure the realvalued physical value to arbitrary precision.
 Similarly, a neural net that somehow had Chaitin's constant exactly embedded in its weight function would be able to solve the halting problem,^{[5]} though constructing such an infinitely precise neural net, even if you somehow know Chaitin's constant beforehand, is impossible under the laws of quantum mechanics.^{[6]}
 Certain fuzzy logicbased "fuzzy Turing machines" can, by definition, accidentally solve the halting problem, but only because their ability to solve the halting problem is indirectly assumed in the specification of the machine; this tends to be viewed as a "bug" in the original specification of the machines.^{[7]}^{[8]}
 Similarly, a proposed model known as fair nondeterminism can accidentally allow the oracular computation of noncomputable functions, because some such systems, by definition, have the oracular ability to identify reject inputs that would "unfairly" cause a subsystem to run forever.^{[9]}^{[10]}
 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. These models require an uncomputable input, such as a physical eventgenerating process where the interval between events grows at an uncomputably large rate.^{[11]}
 Similarly, one unorthodox interpretation of a model of unbounded nondeterminism posits, by definition, that the length of time required for an "Actor" to settle is fundamentally unknowable, and therefore it cannot be proven, within the model, that it does not take an uncomputably long period of time.^{[12]}
"Infinite computational steps" models
In order to work correctly, certain computations by the machines below literally require infinite, rather than merely unlimited but finite, physical space and resources; in contrast, with a Turing machine, any given computation that halts will require only finite physical space and resources.
 A Turing machine that can complete infinitely many steps in finite time, a feat known as a supertask. 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. According to Shagrir, Zeno machines introduce physical paradoxes and its state is logically undefined outside of oneside open period of [0, 2), thus undefined exactly at 2 minutes after beginning of the computation.^{[13]}
 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.^{[14]} According to a 1992 paper,^{[15]} a computer operating in a Malament–Hogarth spacetime or in orbit around a rotating black hole^{[16]} could theoretically perform nonTuring computations.^{[17]}^{[18]} Access to a CTC may allow the rapid solution to PSPACEcomplete problems, a complexity class which, while Turingdecidable, is generally considered computationally intractable.^{[19]}^{[20]}
Quantum models
Some scholars conjecture that a quantum mechanical system which somehow uses an infinite superposition of states could compute a noncomputable function.^{}
"Eventually correct" systems
Some physicallyrealizable systems will always eventually converge to the correct answer, but have the defect that they will often output an incorrect answer and stick with the incorrect answer for an uncomputably large period of time before eventually going back and correcting the mistake.
 In mid 1960s, E Mark Gold and Hilary Putnam independently proposed models of inductive inference (the "limiting recursive functionals"^{} 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 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 (^{[26]}^{[27]}) uses this approach to define the set of formally describable or constructively computable universes or constructive theories of everything. Generalized Turing machines can eventually converge to a correct solution of the halting problem by evaluating a Specker sequence.
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 ${displaystyle Sigma _{1}^{0}}$ or ${displaystyle Pi _{1}^{0}}$. Limitingrecursion, by contrast, can compute any predicate or function in the corresponding Turing degree, which is known to be ${displaystyle Delta _{2}^{0}}$. Gold further showed that limiting partial recursion would allow the computation of precisely the ${displaystyle Sigma _{2}^{0}}$ predicates.
Model 
Computable predicates 
Notes 
Refs 
supertasking 
tt(${displaystyle Sigma _{1}^{0},Pi _{1}^{0}}$) 
dependent on outside observer 
^{[28]} 
limiting/trialanderror 
${displaystyle Delta _{2}^{0}}$ 

^{[23]} 
iterated limiting (k times) 
${displaystyle Delta _{k+1}^{0}}$ 

^{[25]} 
BlumShubSmale machine 

incomparable with traditional computable real functions 
^{[29]} 
MalamentHogarth spacetime 
HYP 
dependent on spacetime structure 
^{[30]} 
analog recurrent neural network 
${displaystyle Delta _{1}^{0}[f]}$ 
f is an advice function giving connection weights; size is bounded by runtime 
^{[31]}^{[32]} 
infinite time Turing machine 
${displaystyle AQI}$ 
Arithmetical QuasiInductive sets 
^{[33]} 
classical fuzzy Turing machine 
${displaystyle Sigma _{1}^{0}cup Pi _{1}^{0}}$ 
for any computable tnorm 
^{[8]} 
increasing function oracle 
${displaystyle Delta _{1}^{1}}$ 
for the onesequence model; ${displaystyle Pi _{1}^{1}}$ are r.e. 
^{[11]} 
Criticism
Martin Davis, in his writings on hypercomputation^{[34]}^{[35]} 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 the 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. In his argument he makes a remark that all of hypercomputation is little more than: " if noncomputable inputs are permitted then non computable outputs are attainable."^{[36]}
See also
References
 ^ 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)
 ^ J. Cabessa; H.T. Siegelmann (Apr 2012). "The Computational Power of Interactive Recurrent Neural Networks" (PDF). Neural Computation. 24 (4): 996–1019. doi:10.1162/neco_a_00263.
 ^ 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, "NPcomplete Problems and Physical Reality"[2] p. 12
 ^ H.T. Siegelmann; E.D. Sontag (1994). "Analog Computation via Neural Networks" (PDF). Theoretical Computer Science. 131: 331—360. doi:10.1016/03043975(94)901783.
 ^ Andrew Hodges. "The Professors and the Brainstorms". The Alan Turing Home Page. Retrieved 23 September 2011.
 ^ 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.
Their (ability to solve the halting problem) is due to their acceptance criterion in which the ability to solve the halting problem is indirectly assumed.
 ^ Edith Spaan; Leen Torenvliet; Peter van Emde Boas (1989). "Nondeterminism, Fairness and a Fundamental Analogy". EATCS bulletin. 37: 186–193.
 ^ Ord, Toby. "The many forms of hypercomputation." Applied mathematics and computation 178.1 (2006): 143–153.
 ^ ^{a} ^{b} Dmytro Taranovsky (July 17, 2005). "Finitism and Hypercomputation". Retrieved Apr 26, 2011.
 ^ Hewitt, Carl. "What Is Commitment." Physical, Organizational, and Social (Revised), Coordination, Organizations, Institutions, and Norms in Agent Systems II: AAMAS (2006).
 ^ 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" (PDF). Theor. Comput. Sci. 317: 105–114. doi:10.1016/j.tcs.2003.12.007. Archived from the original (PDF) on 20070709. , Petrus H. Potgieter (July 2006). "Zeno machines and hypercomputation". Theoretical Computer Science. 358 (1): 23–33. doi:10.1016/j.tcs.2005.11.040. and Vincent C. Müller (2011). "On the possibilities of hypercomputing supertasks". Minds and Machines. 21 (1): 83–96. doi:10.1007/s1102301192226.
 ^ Hajnal Andréka, István Németi and Gergely Székely, Closed Timelike Curves in Relativistic Computation Parallel Process. Lett. 22, 1240010 (2012).[3]
 ^ 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.
 ^ Todd A. Brun, Computers with closed timelike curves can solve hard problems, Found.Phys.Lett. 16 (2003) 245–253.[4]
 ^ S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent [5]
 ^ There have been some claims to this effect; see Tien Kieu (2003). "Hilbert's Tenth Problem". Int. J. Theor. Phys. 42 (7): 1461–1478. arXiv:quantph/0110136 . doi:10.1023/A:1025780028846. or M. Ziegler (2005). "Computational Power of Infinite Quantum Parallelism". International Journal of Theoretical Physics. 44 (11): 2059–2071. arXiv:quantph/0410141 . Bibcode:2005IJTP...44.2059Z. doi:10.1007/s1077300589840. and the ensuing literature. For a retort see Warren D. Smith. "Three counterexamples refuting Kieu's plan for "quantum adiabatic hypercomputation"; and some uncomputable quantum mechanical tasks". Applied Mathematics and Computation. 178 (1): 184–193. doi:10.1016/j.amc.2005.09.078. .
 ^ Bernstein and Vazirani, Quantum complexity theory, SIAM Journal on Computing, 26(5):1411–1473, 1997. [6]
 ^ ^{a} ^{b} 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} 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.
 ^ 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 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 216–228. Springer, . 13 (4): 1–5. arXiv:quantph/0011122 . Bibcode:2000quant.ph.11122S.
 ^ 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.
 ^ 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 (2008). "The extent of computation in MalamentHogarth spacetimes". British J. for Philosophy of Science. 59: 659–674. arXiv:grqc/0609035 . doi:10.1093/bjps/axn031.
 ^ H.T. Siegelmann (Apr 1995). "Computation Beyond the Turing Limit" (PDF). Science. 268 (5210): 545—548. Bibcode:1995Sci...268..545S. 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.
 ^ P.D. Welch (2009). "Characteristics of discrete transfinite time Turing machine models: Halting times, stabilization times, and Normal Form theorems". Theoretical Computer Science. 410: 426–442. doi:10.1016/j.tcs.2008.09.050.
 ^ Davis, Martin (2006). "Why there is no such discipline as hypercomputation". Applied Mathematics and Computation. 178 (1): 4–7. doi:10.1016/j.amc.2005.09.066.
 ^ Davis, Martin (2004). "The Myth of Hypercomputation". Alan Turing: Life and Legacy of a Great Thinker. Springer.
 ^ Martin Davis (Jan 2003). "The Myth of Hypercomputation". In Alexandra Shlapentokh. Miniworkshop: Hilbert's Tenth Problem, Mazur's Conjecture and Divisibility Sequences (PDF). MFO Report. 3. Mathematisches Forschungsinstitut Oberwolfach. p. 2.
Further reading
 Mario Antoine Aoun, "Advances in Three Hypercomputation Models", (2016)
 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.
 Burgin, M. S. (1983) Inductive Turing Machines, Notices of the Academy of Sciences of the USSR, v. 270, No. 6, pp. 1289–1293
 Keith Douglas. SuperTuring Computation: a Case Study Analysis (PDF), M.S. Thesis, Carnegie Mellon University, 2003.
 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" (PDF). 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 (PDF). Plenum Publishers, New York, Boston, Dordrecht, London, Moscow. pp. 137–160.
 Copeland, J. (2002) Hypercomputation, Minds and machines, v. 12, pp. 461–502
 Davis, Martin (2006), "The Church–Turing Thesis: Consensus and opposition". Proceedings, Computability in Europe 2006. The requested URL /~simon/TEACH/28000/DavisUniversal.pdf was not found on this server. Lecture Notes in Computer Science, 3988 pp. 125–132
 Hagar, A. and Korolev, A., Quantum Hypercomputation—Hype or Computation?, (2007)
 Müller, Vincent C. (2011). "On the possibilities of hypercomputing supertasks". Minds and Machines. 21 (1): 83–96. doi:10.1007/s1102301192226.
 Ord, Toby. Hypercomputation: Computing more than the Turing machine can compute: A survey article on various forms of hypercomputation.
 Piccinini, Gualtiero: Computation in Physical Systems
 Putz, Volkmar and Karl Svozil, Can a computer be "pushed" to perform fasterthanlight?, (2010)
 Rogers, H. (1987) Theory of Recursive Functions and Effective Computability, MIT Press, Cambridge Massachusetts
 Mike Stannett, Mike (1990). "Xmachines and the halting problem: Building a superTuring machine". Formal Aspects of Computing. 2 (1): 331–341. doi:10.1007/BF01888233.
 Mike Stannett, The case for hypercomputation, Applied Mathematics and Computation, Volume 178, Issue 1, 1 July 2006, Pages 8–24, Special Issue on Hypercomputation
 Syropoulos, Apostolos (2008), Hypercomputation: Computing Beyond the Church–Turing Barrier (preview), Springer. ISBN 9780387308869
 Turing, Alan (1939). "Systems of logic based on ordinals". Proceedings of the London Mathematical Society. 45: 161–228. doi:10.1112/plms/s245.1.161.
External links