Max Tegmark

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

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

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

History

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

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 and which are, hence, not computable in the Church-Turing sense.

An example of a problem a Turing machine cannot solve is the halting problem. A Turing machine cannot decide if an arbitrary program halts or runs forever. Some proposed hypercomputers can simulate the program for an infinite number of steps and tell the user whether 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. 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.
• 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" and "trial-and-error predicates", 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.)" L. K. Schubert's 1974 paper "Iterated Limiting Recursion and the Program Minimization Problem" studied the effects of iterating the limiting procedure; this allows any arithmetic predicate to be computed. Schubert wrote, "Intuitively, iterated limiting identification might be regarded as higher-order inductive inference performed collectively by an ever-growing community of lower order inductive inference machines."
• A real computer (a sort of idealized analog computer) can perform hypercomputation if physics admits general real variables (not just computable reals), and these are in some way "harnessable" for computation. This might require quite bizarre laws of physics (for example, a measurable physical constant with an oracular value, such as Chaitin's constant), and would at minimum require the ability to measure a real-valued physical value to arbitrary precision despite thermal noise and quantum effects.
• A proposed technique known as fair nondeterminism or unbounded nondeterminism may allow the computation of noncomputable functions. 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. 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.
• According to a 1992 paper, a computer operating in a Malament-Hogarth spacetime or in orbit around a rotating black hole could theoretically perform non-Turing computations.
• 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.
• The infinite time Turing machine is a generalization of the Zeno machine, that can perform infinitely long computations whose steps are enumerated by potentially transfinite ordinal numbers. It models an otherwise-ordinary Turing machine for which non-halting computations are completed by entering a special state reserved for reaching a limit ordinal and to which the results of the preceding infinite computation are available.
• Jan van Leeuwen and Jiří Wiedermann wrote a paper suggesting that the Internet should be modeled as a nonuniform computing system equipped with an advice function representing the ability of computers to be upgraded.
• A symbol sequence is computable in the limit if there is a finite, possibly non-halting program on a universal Turing machine that incrementally outputs every symbol of the sequence. This includes the dyadic expansion of π and of every other computable real, but still excludes all noncomputable reals. Traditional Turing machines cannot edit their previous outputs; generalized Turing machines, as defined by Jürgen Schmidhuber, can. He defines the constructively describable symbol sequences as those that have a finite, non-halting program running on a generalized Turing machine, such that any output symbol eventually converges; that is, it does not change any more after some finite initial time interval. Due to limitations first exhibited by Kurt Gödel (1931), it may be impossible to predict the convergence time itself by a halting program, otherwise the halting problem could be solved. Schmidhuber () uses this approach to define the set of formally describable or constructively computable universes or constructive theories of everything. Generalized Turing machines can solve the halting problem by evaluating a Specker sequence.
• A quantum mechanical system which somehow uses an infinite superposition of states to compute a non-computable function. 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).
• In 1970, E.S. Santos defined a class of fuzzy logic-based "fuzzy algorithms" and "fuzzy Turing machines". 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. Jiří Wiedermann analyzed the capabilities of Santos' original proposal in 2004.
• 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.

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 ${\mathrm{\Sigma }}_1^0$ or ${\mathrm{\Pi }}_1^0$. Limiting-recursion, by contrast, can compute any predicate or function in the corresponding Turing degree, which is known to be ${\mathrm{\Delta }}_2^0$. Gold further showed that limiting partial recursion would allow the computation of precisely the ${\mathrm{\Sigma }}_2^0$ predicates.

Model	Computable predicates	Notes	Refs
supertasking	tt(${\mathrm{\Sigma }}_1^0,{\mathrm{\Pi }}_1^0$)	dependent on outside observer
limiting/trial-and-error	${\mathrm{\Delta }}_2^0$
iterated limiting (k times)	${\mathrm{\Delta }}_{k+1}^0$
Blum-Shub-Smale machine		incomparable with traditional computable real functions.
Malament-Hogarth spacetime	HYP	Dependent on spacetime structure
Analog recurrent neural network	${\mathrm{\Delta }}_1^0[f]$	f is an advice function giving connection weights; size is bounded by runtime
Infinite time Turing machine	$\ge T({\mathrm{\Sigma }}_1^1)$
Classical fuzzy Turing machine	${\mathrm{\Sigma }}_1^0\cup {\mathrm{\Pi }}_1^0$	For any computable t-norm
Increasing function oracle	${\mathrm{\Delta }}_1^1$	For the one-sequence model; ${\mathrm{\Pi }}_1^1$ are r.e.

Taxonomy of "super-recursive" computation methodologies

Mark Burgin has collected a list of what he calls "super-recursive algorithms" (from Burgin 2005: 132):

• limiting recursive functions and limiting partial recursive functions (E. M. Gold)
• trial and error predicates (Hilary Putnam)
• inductive inference machines (Carl Herbert Smith)
• inductive Turing machines (one of Burgin's own models)
• limit Turing machines (another of Burgin's models)
• trial-and-error machines (Ja. Hintikka and A. Mutanen )
• general Turing machines (J. Schmidhuber)
• Internet machines (van Leeuwen, J. and Wiedermann, J.)
• evolutionary computers, which use DNA to produce the value of a function (Darko Roglic)
• fuzzy computation (Jiří Wiedermann)
• evolutionary Turing machines (Eugene Eberbach)

In the same book, he presents also a list of "algorithmic schemes":

• Turing machines with arbitrary oracles (Alan Turing)
• Transrecursive operators (Borodyanskii and Burgin)
• 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 refers to this subject as "a myth" and offers counter-arguments to the physical realizability of hypercomputation. As for its theory, he argues against the claims that this is a new field founded in 1990s. This point of view relies on the history of computability theory (degrees of unsolvability, computability over functions, real numbers and ordinals), as also mentioned above.

Andrew Hodges wrote a critical commentary on Copeland and Proudfoot's article.

See also

• Computation
• Digital physics
• Supertask

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• Hava Siegelmann and Eduardo Sontag, “Analog Computation via Neural Networks,” Theoretical Computer Science 131, 1994: 331-360.
• 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. Super-Turing 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, Springer-Verlag 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 Church-Turing Barrier (preview), Springer. ISBN 978-0-387-30886-9
• 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), Super-recursive algorithms, Monographs in computer science, Springer. ISBN 0-387-95569-0
• Cockshott, P. and Michaelson, G. Are there new Models of Computation? Reply to Wegner and Eberbach, The computer Journal, 2007
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 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 faster-than-light?, (2010)

External links

• Hypercomputation Research Network
• Hypercomputation
• Toby Ord, Hypercomputation: computing more than the Turing machine
• Toby Ord, The many forms of hypercomputation
• Paolo Cotogno, Hypercomputation and the Physical Church-Turing thesis
• Gualtiero Piccinini, Computation in Physical Systems