Hypercomputation

Hypercomputation or super-Turing computation is a set of hypothetical 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 could correctly evaluate every statement in Peano arithmetic.

The Church–Turing thesis states that any "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 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 deterministic, 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 (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 mathematical abstractions, and are not physically realizable.^[2]

State space

In a sense, most functions are uncomputable: there are $\aleph _{0}$ computable functions, but there are an uncountable number ( $2^{\aleph _{0}}$ ) of possible super-Turing functions.^[3]

Models

Hypercomputer models range from useful but probably unrealizable (such as Turing's original oracle machines), to less-useful random-function generators that are more plausibly "realizable" (such as a random Turing machine).

Uncomputable inputs or black-box 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 random-number 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 real-valued physical value to arbitrary precision, though standard physics makes such arbitrary-precision measurements theoretically infeasible.^[5]
- Similarly, a neural net that somehow had Chaitin's constant exactly embedded in its weight function would be able to solve the halting problem,^[6] but is subject to the same physical difficulties as other models of hypercomputation based on real computation.
Certain fuzzy logic-based "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 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. These models require an uncomputable input, such as a physical event-generating 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 Oron 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.^[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 non-Turing computations for an observer inside the black hole.^[17]^[18] 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.^[19]^[20]

Quantum models

Some scholars conjecture that a quantum mechanical system which somehow uses an infinite superposition of states could compute a non-computable function.^[21] 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).^[22]

"Eventually correct" systems

Some physically realizable 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"^[23] and "trial-and-error predicates",^[24] 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.)"^[24] L. K. Schubert's 1974 paper "Iterated Limiting Recursion and the Program Minimization Problem"^[25] 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 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. The 'Monotone Turing machines' traditionally used in description size theory 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 (^[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 truth-table degree containing $\Sigma _{1}^{0}$ or $\Pi _{1}^{0}$ . Limiting-recursion, by contrast, can compute any predicate or function in the corresponding Turing degree, which is known to be $\Delta _{2}^{0}$ . Gold further showed that limiting partial recursion would allow the computation of precisely the $\Sigma _{2}^{0}$ predicates.

Model	Computable predicates	Notes	Refs
supertasking	$\operatorname {tt} \left(\Sigma _{1}^{0},\Pi _{1}^{0}\right)$	dependent on outside observer	^[28]
limiting/trial-and-error	$\Delta _{2}^{0}$		^[23]
iterated limiting (k times)	$\Delta _{k+1}^{0}$		^[25]
Blum–Shub–Smale machine		incomparable with traditional computable real functions	^[29]
Malament–Hogarth spacetime	HYP	dependent on spacetime structure	^[30]
analog recurrent neural network	$\Delta _{1}^{0}[f]$	f is an advice function giving connection weights; size is bounded by runtime	^[31]^[32]
infinite time Turing machine	$AQI$	Arithmetical Quasi-Inductive sets	^[33]
classical fuzzy Turing machine	$\Sigma _{1}^{0}\cup \Pi _{1}^{0}$	for any computable t-norm	^[8]
increasing function oracle	$\Delta _{1}^{1}$	for the one-sequence model; $\Pi _{1}^{1}$ are r.e.	^[11]

Criticism

Martin Davis, in his writings on hypercomputation,^[34]^[35] refers to this subject as "a myth" and offers 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 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 non-computable inputs are permitted, then non-computable outputs are attainable."^[36]

References

^ Turing, A. M. (1939). "Systems of Logic Based on Ordinals†". Proceedings of the London Mathematical Society. 45: 161–228. doi:10.1112/plms/s2-45.1.161. hdl:21.11116/0000-0001-91CE-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)
^ J. Cabessa; H.T. Siegelmann (Apr 2012). "The Computational Power of Interactive Recurrent Neural Networks" (PDF). Neural Computation. 24 (4): 996–1019. CiteSeerX 10.1.1.411.7540. doi:10.1162/neco_a_00263. PMID 22295978. S2CID 5826757.
^ 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"[1] p. 12
^ Andrew Hodges. "The Professors and the Brainstorms". The Alan Turing Home Page. Retrieved 23 September 2011.
^ H.T. Siegelmann; E.D. Sontag (1994). "Analog Computation via Neural Networks". Theoretical Computer Science. 131 (2): 331–360. doi:10.1016/0304-3975(94)90178-3.
^ Biacino, L.; Gerla, G. (2002). "Fuzzy logic, continuity and effectiveness". Archive for Mathematical Logic. 41 (7): 643–667. CiteSeerX 10.1.1.2.8029. doi:10.1007/s001530100128. ISSN 0933-5846. S2CID 12513452.
^ ^a ^b Wiedermann, Jiří (2004). "Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines". Theoretical Computer Science. 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 (2006). "The many forms of hypercomputation". Applied Mathematics and Computation. 178: 143–153. doi:10.1016/j.amc.2005.09.076.
^ ^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). "Super-tasks, accelerating Turing machines and uncomputability". Theoretical Computer Science. 317 (1–3): 105–114. doi:10.1016/j.tcs.2003.12.007., Petrus H. Potgieter (July 2006). "Zeno machines and hypercomputation". Theoretical Computer Science. 358 (1): 23–33. arXiv:cs/0412022. doi:10.1016/j.tcs.2005.11.040. S2CID 6749770. and Vincent C. Müller (2011). "On the possibilities of hypercomputing supertasks". Minds and Machines. 21 (1): 83–96. CiteSeerX 10.1.1.225.3696. doi:10.1007/s11023-011-9222-6. S2CID 253434.
^ Andréka, Hajnal; Németi, István; Székely, Gergely (2012). "Closed Timelike Curves in Relativistic Computation". Parallel Processing Letters. 22 (3). arXiv:1105.0047. doi:10.1142/S0129626412400105. S2CID 16816151.
^ Hogarth, Mark L. (1992). "Does general relativity allow an observer to view an eternity in a finite time?". Foundations of Physics Letters. 5 (2): 173–181. Bibcode:1992FoPhL...5..173H. doi:10.1007/BF00682813. S2CID 120917288.
^ 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. Vol. 3988. Springer. doi:10.1007/11780342. ISBN 978-3-540-35466-6.
^ Etesi, Gabor; Nemeti, Istvan (2002). "Non-Turing computations via Malament-Hogarth space-times". International Journal of Theoretical Physics. 41 (2): 341–370. arXiv:gr-qc/0104023. doi:10.1023/A:1014019225365. S2CID 17081866.
^ Earman, John; Norton, John D. (1993). "Forever is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes". Philosophy of Science. 60: 22–42. doi:10.1086/289716. S2CID 122764068.
^ Brun, Todd A. (2003). "Computers with closed timelike curves can solve hard problems". Found. Phys. Lett. 16 (3): 245–253. arXiv:gr-qc/0209061. doi:10.1023/A:1025967225931. S2CID 16136314.
^ S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent [2]
^ 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. S2CID 6634980. or M. Ziegler (2005). "Computational Power of Infinite Quantum Parallelism". International Journal of Theoretical Physics. 44 (11): 2059–2071. arXiv:quant-ph/0410141. Bibcode:2005IJTP...44.2059Z. doi:10.1007/s10773-005-8984-0. S2CID 9879859. and the ensuing literature. For a retort see Warren D. Smith (2006). "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, Ethan; Vazirani, Umesh (1997). "Quantum Complexity Theory". SIAM Journal on Computing. 26 (5): 1411–1473. doi:10.1137/S0097539796300921.
^ ^a ^b E. M. Gold (1965). "Limiting Recursion". Journal of Symbolic Logic. 30 (1): 28–48. doi:10.2307/2270580. JSTOR 2270580. S2CID 33811657., E. Mark Gold (1967). "Language identification in the limit". Information and Control. 10 (5): 447–474. doi:10.1016/S0019-9958(67)91165-5.
^ ^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. S2CID 44655062.
^ ^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. S2CID 2071951.
^ Schmidhuber, Juergen (2000). "Algorithmic Theories of Everything". arXiv:quant-ph/0011122.
^ 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. arXiv:quant-ph/0011122. Bibcode:2000quant.ph.11122S. doi:10.1142/S0129054102001291.
^ Petrus H. Potgieter (July 2006). "Zeno machines and hypercomputation". Theoretical Computer Science. 358 (1): 23–33. arXiv:cs/0412022. doi:10.1016/j.tcs.2005.11.040. S2CID 6749770.
^ Lenore Blum, Felipe Cucker, Michael Shub, and Stephen Smale (1998). Complexity and Real Computation. Springer. ISBN 978-0-387-98281-6.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ P.D. Welch (2008). "The extent of computation in Malament-Hogarth spacetimes". British Journal for the Philosophy of Science. 59 (4): 659–674. arXiv:gr-qc/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. S2CID 17495161.
^ 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.
^ P.D. Welch (2009). "Characteristics of discrete transfinite time Turing machine models: Halting times, stabilization times, and Normal Form theorems". Theoretical Computer Science. 410 (4–5): 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 (ed.). Miniworkshop: Hilbert's Tenth Problem, Mazur's Conjecture and Divisibility Sequences (PDF). MFO Report. Vol. 3. Mathematisches Forschungsinstitut Oberwolfach. p. 2.