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Hypercomputation

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Hypercomputation or super-Turing computation is a set of hypothetical <a href="Model_of_computation" title="Model of computation">models of computation</a> that can provide outputs that are not <a href="Turing-computable" class="mw-redirect" title="Turing-computable">Turing-computable</a>. For example, a machine that could solve the <a href="Halting_problem" title="Halting problem">halting problem</a> would be a hypercomputer; so too would one that could <a href="Entscheidungsproblem" title="Entscheidungsproblem">correctly evaluate every statement</a> in <a href="Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>.

The <a href="Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a> 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 <a href="Turing_machine" title="Turing machine">Turing machine</a> cannot and which are, hence, not computable in the Church–Turing sense.

Technically, the output of a <a href="Random_Turing_machine" class="mw-redirect" title="Random Turing machine">random Turing machine</a> is uncomputable; however, most hypercomputing literature focuses instead on the <a href="Computation" title="Computation">computation</a> of deterministic, rather than random, uncomputable functions.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

A computational model going beyond Turing machines was introduced by <a href="Alan_Turing" title="Alan Turing">Alan Turing</a> in his 1938 PhD dissertation <a href="Systems_of_Logic_Based_on_Ordinals" title="Systems of Logic Based on Ordinals">Systems of Logic Based on Ordinals</a>.<a href="#cite_note-1">[1]</a> This paper investigated mathematical systems in which an <a href="Oracle_machine" title="Oracle machine">oracle</a> was available, which could compute a single arbitrary (non-recursive) function from <a href="Natural_number" title="Natural number">naturals</a> to naturals. He used this device to prove that even in those more powerful systems, <a href="Undecidable_problem" title="Undecidable problem">undecidability</a> is still present. Turing's oracle machines are mathematical abstractions, and are not physically realizable.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="State_space">State space</h2></div>

In a sense, most functions are uncomputable: there are <a href="Aleph_0" class="mw-redirect" title="Aleph 0"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation>

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</math><img src="./721cd7f8c15a2e72ad162bdfa5baea8eef98aab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}" loading="lazy"></a> computable functions, but there are an <a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> number (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\aleph _{0}}}">

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<annotation encoding="application/x-tex">{\displaystyle 2^{\aleph _{0}}}</annotation>

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</math><img src="./779da5db4ed54fa334dd92089cdf1c284e45febb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.231ex; height:2.676ex;" alt="{\displaystyle 2^{\aleph _{0}}}" loading="lazy">) of possible super-Turing functions.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Models">Models</h2></div>

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 <a href="Random_Turing_machine" class="mw-redirect" title="Random Turing machine">random Turing machine</a>).

<div class="mw-heading mw-heading3"><h3 id="Uncomputable_inputs_or_black-box_components">Uncomputable inputs or black-box components</h3></div>

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A system granted knowledge of the uncomputable, oracular <a href="Chaitin's_constant" title="Chaitin's constant">Chaitin's constant</a> (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:

<ul><li>Turing's original oracle machines, defined by Turing in 1939.</li>

<li>A <a href="Real_computer" class="mw-redirect" title="Real computer">real computer</a> (a sort of idealized <a href="Analog_computer" title="Analog computer">analog computer</a>) can perform hypercomputation<a href="#cite_note-4">[4]</a> if physics admits general <a href="Real_number" title="Real number">real</a> variables (not just <a href="Computable_number" title="Computable number">computable reals</a>), 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 <a href="Physical_constant" title="Physical constant">physical constant</a> with an oracular value, such as <a href="Chaitin's_constant" title="Chaitin's constant">Chaitin's constant</a>), 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.<a href="#cite_note-HodgesSCIAM-5">[5]</a>

<ul><li>Similarly, a neural net that somehow had Chaitin's constant exactly embedded in its weight function would be able to solve the halting problem,<a href="#cite_note-6">[6]</a> but is subject to the same physical difficulties as other models of hypercomputation based on real computation.</li></ul></li>

<li>Certain <a href="Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a>-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.<a href="#cite_note-7">[7]</a><a href="#cite_note-ClassicalFuzzy-8">[8]</a>

<ul><li>Similarly, a proposed model known as <a href="Fair_nondeterminism" class="mw-redirect" title="Fair nondeterminism">fair nondeterminism</a> can accidentally allow the oracular computation of noncomputable functions, because some such systems, by definition, have the oracular ability to identify and reject inputs that would "unfairly" cause a subsystem to run forever.<a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a></li></ul></li>

<li>Dmytro Taranovsky has proposed a <a href="Finitism" title="Finitism">finitistic</a> 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.<a href="#cite_note-Taranovsky-11">[11]</a>

<ul><li>Similarly, one unorthodox interpretation of a model of <a href="Unbounded_nondeterminism" title="Unbounded nondeterminism">unbounded nondeterminism</a> 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.<a href="#cite_note-12">[12]</a></li></ul></li></ul>

<div class="mw-heading mw-heading3"><h3 id=""Infinite_computational_steps"_models">"Infinite computational steps" models</h3></div>

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 <a href="Supertask" title="Supertask">supertask</a>. Simply being able to run for an unbounded number of steps does not suffice. One mathematical model is the <a href="Zeno_machine" title="Zeno machine">Zeno machine</a> (inspired by <a href="Zeno's_paradox" class="mw-redirect" title="Zeno's paradox">Zeno's paradox</a>). 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 <a href="1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF" title="1/2 + 1/4 + 1/8 + 1/16 + ⋯">1 + ½ + ¼ + ...</a> (a <a href="Geometric_series" title="Geometric series">geometric series</a>) we see that the machine performs infinitely many steps in a total of 2 minutes. According to <a href="Oron_Shagrir" title="Oron Shagrir">Oron Shagrir</a>, 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.<a href="#cite_note-13">[13]</a>

It seems natural that the possibility of time travel (existence of <a href="Closed_timelike_curve" title="Closed timelike curve">closed timelike curves</a> (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.<a href="#cite_note-14">[14]</a> According to a 1992 paper,<a href="#cite_note-15">[15]</a> a computer operating in a <a href="Malament%E2%80%93Hogarth_spacetime" title="Malament–Hogarth spacetime">Malament–Hogarth spacetime</a> or in orbit around a rotating <a href="Black_hole" title="Black hole">black hole</a><a href="#cite_note-16">[16]</a> could theoretically perform non-Turing computations for an observer inside the black hole.<a href="#cite_note-17">[17]</a><a href="#cite_note-18">[18]</a> Access to a CTC may allow the rapid solution to <a href="PSPACE-complete" title="PSPACE-complete">PSPACE-complete</a> problems, a complexity class which, while Turing-decidable, is generally considered computationally intractable.<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a>

<div class="mw-heading mw-heading3"><h3 id="Quantum_models">Quantum models</h3></div>

Some scholars conjecture that a <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> system which somehow uses an infinite superposition of states could compute a non-<a href="Computable_function" title="Computable function">computable function</a>.<a href="#cite_note-21">[21]</a> This is not possible using the standard <a href="Qubit" title="Qubit">qubit</a>-model <a href="Quantum_computer" class="mw-redirect" title="Quantum computer">quantum computer</a>, because it is proven that a regular quantum computer is <a href="PSPACE" title="PSPACE">PSPACE</a>-<a href="Reduction_(complexity)" title="Reduction (complexity)">reducible</a> (a quantum computer running in <a href="Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a> can be simulated by a classical computer running in <a href="Polynomial_space" class="mw-redirect" title="Polynomial space">polynomial space</a>).<a href="#cite_note-22">[22]</a>

<div class="mw-heading mw-heading3"><h3 id=""Eventually_correct"_systems">"Eventually correct" systems</h3></div>

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, <a href="E_Mark_Gold" class="mw-redirect" title="E Mark Gold">E Mark Gold</a> and <a href="Hilary_Putnam" title="Hilary Putnam">Hilary Putnam</a> independently proposed models of <a href="Inductive_inference" class="mw-redirect" title="Inductive inference">inductive inference</a> (the "limiting recursive functionals"<a href="#cite_note-LimRecurs-23">[23]</a> and "trial-and-error predicates",<a href="#cite_note-TrialError-24">[24]</a> respectively). These models enable some nonrecursive sets of numbers or languages (including all <a href="Recursively_enumerable" class="mw-redirect" title="Recursively enumerable">recursively enumerable</a> 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.)"<a href="#cite_note-TrialError-24">[24]</a> L. K. Schubert's 1974 paper "Iterated Limiting Recursion and the Program Minimization Problem"<a href="#cite_note-IterLimRec-25">[25]</a> studied the effects of iterating the limiting procedure; this allows any <a href="Arithmetic_hierarchy" class="mw-redirect" title="Arithmetic hierarchy">arithmetic</a> 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 <a href="Universal_Turing_machine" title="Universal Turing machine">universal Turing machine</a> that incrementally outputs every symbol of the sequence. This includes the dyadic expansion of π and of every other <a href="Computable_real" class="mw-redirect" title="Computable real">computable real</a>, but still excludes all noncomputable reals. The 'Monotone Turing machines' traditionally used in <a href="Minimum_description_length" title="Minimum description length">description size</a> theory cannot edit their previous outputs; generalized Turing machines, as defined by <a href="J%C3%BCrgen_Schmidhuber" title="Jürgen Schmidhuber">Jürgen Schmidhuber</a>, 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 <a href="Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> (1931), it may be impossible to predict the convergence time itself by a halting program, otherwise the <a href="Halting_problem" title="Halting problem">halting problem</a> could be solved. Schmidhuber (<a href="#cite_note-genTuring2000-26">[26]</a><a href="#cite_note-GenKolm-27">[27]</a>) uses this approach to define the set of formally describable or constructively computable universes or constructive <a href="Theory_of_everything" title="Theory of everything">theories of everything</a>. Generalized Turing machines can eventually converge to a correct solution of the halting problem by evaluating a <a href="Specker_sequence" title="Specker sequence">Specker sequence</a>.

<div class="mw-heading mw-heading2"><h2 id="Analysis_of_capabilities">Analysis of capabilities</h2></div>

Many hypercomputation proposals amount to alternative ways to read an <a href="Oracle_machine" title="Oracle machine">oracle</a> or <a href="Advice_(complexity)" title="Advice (complexity)">advice function</a> embedded into an otherwise classical machine. Others allow access to some higher level of the <a href="Arithmetic_hierarchy" class="mw-redirect" title="Arithmetic hierarchy">arithmetic hierarchy</a>. For example, supertasking Turing machines, under the usual assumptions, would be able to compute any predicate in the <a href="Truth-table_reduction" title="Truth-table reduction">truth-table degree</a> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}^{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{0}}</annotation>

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</math><img src="./8130a55f302deb0b733de5a526f2d92c364e4dd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.732ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{0}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{0}}</annotation>

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</math><img src="./b0b3c0e79d8a5db977a9838be477eb3e30348937.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{0}}" loading="lazy">. Limiting-recursion, by contrast, can compute any predicate or function in the corresponding <a href="Turing_degree" title="Turing degree">Turing degree</a>, which is known to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{2}^{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{0}}</annotation>

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</math><img src="./e2647e336f34cacfba9934032137522052e8af0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.99ex; height:3.176ex;" alt="{\displaystyle \Delta _{2}^{0}}" loading="lazy">. Gold further showed that limiting partial recursion would allow the computation of precisely the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{2}^{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma _{2}^{0}}</annotation>

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</math><img src="./09ae5829fb5735899799870a3156c903f0573f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.732ex; height:3.176ex;" alt="{\displaystyle \Sigma _{2}^{0}}" loading="lazy"> predicates.

<th>Model

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<th>Computable predicates

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<th>Notes

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<td>supertasking

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {tt} \left(\Sigma _{1}^{0},\Pi _{1}^{0}\right)}</annotation>

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</math><img src="./916e9507df008c80751ee5c03dcc7b5d8146a79f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.503ex; height:3.176ex;" alt="{\displaystyle \operatorname {tt} \left(\Sigma _{1}^{0},\Pi _{1}^{0}\right)}" loading="lazy">

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<td>dependent on outside observer

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<td>limiting/trial-and-error

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{0}}</annotation>

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<td>iterated limiting (k times)

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{k+1}^{0}}</annotation>

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</math><img src="./7998a80622f98398217de20a2b8ab42463df8cd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.125ex; height:3.343ex;" alt="{\displaystyle \Delta _{k+1}^{0}}" loading="lazy">

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<td>

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<td><a href="Blum%E2%80%93Shub%E2%80%93Smale_machine" title="Blum–Shub–Smale machine">Blum–Shub–Smale machine</a>

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<td>incomparable with traditional <a href="Computable_real" class="mw-redirect" title="Computable real">computable real</a> functions

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<td><a href="Malament%E2%80%93Hogarth_spacetime" title="Malament–Hogarth spacetime">Malament–Hogarth spacetime</a>

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<td>dependent on spacetime structure

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<td>analog recurrent neural network

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{1}^{0}[f]}</annotation>

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</math><img src="./37c34712f0c08302bbf749e599f3c41a295484ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.562ex; height:3.176ex;" alt="{\displaystyle \Delta _{1}^{0}[f]}" loading="lazy">

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<td>f is an advice function giving connection weights; size is bounded by runtime

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<td>infinite time Turing machine

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<annotation encoding="application/x-tex">{\displaystyle AQI}</annotation>

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</math><img src="./808e78de30d18e7611476cfc0f523da29e8c18aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.753ex; height:2.509ex;" alt="{\displaystyle AQI}" loading="lazy">

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<td>Arithmetical Quasi-Inductive sets

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<td>classical fuzzy Turing machine

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<annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{0}\cup \Pi _{1}^{0}}</annotation>

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</math><img src="./30d8c3d3fdacbb70af717acd9a9cec752a5982db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.112ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{0}\cup \Pi _{1}^{0}}" loading="lazy">

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<td>for any computable <a href="T-norm_fuzzy_logics" title="T-norm fuzzy logics">t-norm</a>

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<td>increasing function oracle

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{1}^{1}}</annotation>

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</math><img src="./c6e18a003a02e8b704ccc02dbded53054df3eeaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.99ex; height:3.176ex;" alt="{\displaystyle \Delta _{1}^{1}}" loading="lazy">

</td>

<td>for the one-sequence model; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}}</annotation>

</semantics>

</math><img src="./0fd076fb78a7a5ee340920a3d846ddf67455f941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}}" loading="lazy"> are r.e.

</td>

</td></tr>

<tr>

<td>ordinal turing machine

</td>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{1}}</annotation>

</semantics>

</math><img src="./b2bb95191e7af7bd30330e983e82620aa5b2a1b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.99ex; height:3.176ex;" alt="{\displaystyle \Delta _{2}^{1}}" loading="lazy">

</td>

<td>for the parameter-free model

</td>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Criticism">Criticism</h2></div>

<a href="Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Martin Davis</a>, in his writings on hypercomputation,<a href="#cite_note-Davis95-35">[35]</a><a href="#cite_note-36">[36]</a>

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 <a href="Computability_theory" title="Computability theory">computability theory</a> (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."<a href="#cite_note-37">[37]</a>

<ul><li><a href="Digital_physics" title="Digital physics">Digital physics</a></li>

<li><a href="Limits_of_computation" title="Limits of computation">Limits of computation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> "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)

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="Arnold_Sch%C3%B6nhage" title="Arnold Schönhage">Arnold Schönhage</a>, "On the power of random access machines", in Proc. Intl. Colloquium on Automata, Languages, and Programming (ICALP), pages 520–529, 1979. Source of citation: <a href="Scott_Aaronson" title="Scott Aaronson">Scott Aaronson</a>, "NP-complete Problems and Physical Reality"<a rel="nofollow" class="external autonumber" href="http://www.scottaaronson.com/papers/npcomplete.pdf">[1]</a> p. 12

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFBiacinoGerla,_G.2002" class="citation journal cs1">Biacino, L.; Gerla, G. (2002). "Fuzzy logic, continuity and effectiveness". Archive for Mathematical Logic. 41 (7): 643–667. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.2.8029">10.1.1.2.8029</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs001530100128">10.1007/s001530100128</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0933-5846">0933-5846</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12513452">12513452</a>.</cite>

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<li id="cite_note-ClassicalFuzzy-8">^ <a href="#cite_ref-ClassicalFuzzy_8-0">a</a> <a href="#cite_ref-ClassicalFuzzy_8-1">b</a> <cite id="CITEREFWiedermann2004" class="citation journal cs1">Wiedermann, Jiří (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tcs.2003.12.004">"Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines"</a>. Theoretical Computer Science. 317 (1–3): 61–69. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tcs.2003.12.004">10.1016/j.tcs.2003.12.004</a>. <q>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.</q></cite>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFEdith_SpaanLeen_TorenvlietPeter_van_Emde_Boas1989" class="citation journal cs1">Edith Spaan; Leen Torenvliet; Peter van Emde Boas (1989). "Nondeterminism, Fairness and a Fundamental Analogy". EATCS Bulletin. 37: 186–193.</cite>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFOrd2006" class="citation journal cs1">Ord, Toby (2006). "The many forms of hypercomputation". Applied Mathematics and Computation. 178: 143–153. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.amc.2005.09.076">10.1016/j.amc.2005.09.076</a>.</cite>

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<li id="cite_note-Taranovsky-11">^ <a href="#cite_ref-Taranovsky_11-0">a</a> <a href="#cite_ref-Taranovsky_11-1">b</a> <cite id="CITEREFDmytro_Taranovsky2005" class="citation web cs1">Dmytro Taranovsky (July 17, 2005). <a rel="nofollow" class="external text" href="http://web.mit.edu/dmytro/www/FinitismPaper.htm">"Finitism and Hypercomputation"</a>. Retrieved Apr 26, 2011.</cite>

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<li id="cite_note-12"><a href="#cite_ref-12">^</a> Hewitt, Carl. "What Is Commitment." Physical, Organizational, and Social (Revised), Coordination, Organizations, Institutions, and Norms in Agent Systems II: AAMAS (2006).

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> These models have been independently developed by many different authors, including <cite id="CITEREFHermann_Weyl1927" class="citation book cs1"><a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> (1927). Philosophie der Mathematik und Naturwissenschaft.</cite>; the model is discussed in <cite id="CITEREFShagrir,_O.2004" class="citation journal cs1"><a href="Oron_Shagrir" title="Oron Shagrir">Shagrir, O.</a> (June 2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tcs.2003.12.007">"Super-tasks, accelerating Turing machines and uncomputability"</a>. Theoretical Computer Science. 317 (1–3): 105–114. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tcs.2003.12.007">10.1016/j.tcs.2003.12.007</a>.</cite>, <cite id="CITEREFPetrus_H._Potgieter2006" class="citation journal cs1">Petrus H. Potgieter (July 2006). "Zeno machines and hypercomputation". Theoretical Computer Science. 358 (1): 23–33. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cs/0412022">cs/0412022</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tcs.2005.11.040">10.1016/j.tcs.2005.11.040</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6749770">6749770</a>.</cite> and <cite id="CITEREFVincent_C._Müller2011" class="citation journal cs1">Vincent C. Müller (2011). <a rel="nofollow" class="external text" href="http://philpapers.org/rec/MLLOTP">"On the possibilities of hypercomputing supertasks"</a>. Minds and Machines. 21 (1): 83–96. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.225.3696">10.1.1.225.3696</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11023-011-9222-6">10.1007/s11023-011-9222-6</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:253434">253434</a>.</cite>

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<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFAndrékaNémetiSzékely2012" class="citation journal cs1">Andréka, Hajnal; Németi, István; Székely, Gergely (2012). "Closed Timelike Curves in Relativistic Computation". Parallel Processing Letters. 22 (3). <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1105.0047">1105.0047</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0129626412400105">10.1142/S0129626412400105</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16816151">16816151</a>.</cite>

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<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFHogarth1992" class="citation journal cs1">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. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992FoPhL...5..173H">1992FoPhL...5..173H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00682813">10.1007/BF00682813</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120917288">120917288</a>.</cite>

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<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFIstván_NemétiHajnal_Andréka2006" class="citation book cs1">István Neméti; <a href="Hajnal_Andr%C3%A9ka" title="Hajnal Andréka">Hajnal Andréka</a> (2006). "Can General Relativistic Computers Break the Turing Barrier?". <a rel="nofollow" class="external text" href="https://archive.org/details/logicalapproache0000conf">Logical Approaches to Computational Barriers, Second Conference on Computability in Europe, CiE 2006, Swansea, UK, June 30-July 5, 2006. Proceedings</a>. Lecture Notes in Computer Science. Vol. 3988. Springer. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F11780342">10.1007/11780342</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-35466-6</bdi>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFEtesiNemeti2002" class="citation journal cs1">Etesi, Gabor; Nemeti, Istvan (2002). "Non-Turing computations via Malament-Hogarth space-times". International Journal of Theoretical Physics. 41 (2): 341–370. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0104023">gr-qc/0104023</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1014019225365">10.1023/A:1014019225365</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17081866">17081866</a>.</cite>

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<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFEarmanNorton1993" class="citation journal cs1">Earman, John; Norton, John D. (1993). "Forever is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes". Philosophy of Science. 60: 22–42. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F289716">10.1086/289716</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122764068">122764068</a>.</cite>

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<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFBrun2003" class="citation journal cs1">Brun, Todd A. (2003). "Computers with closed timelike curves can solve hard problems". Found. Phys. Lett. 16 (3): 245–253. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0209061">gr-qc/0209061</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1025967225931">10.1023/A:1025967225931</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16136314">16136314</a>.</cite>

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<li id="cite_note-20"><a href="#cite_ref-20">^</a> <a href="Scott_Aaronson" title="Scott Aaronson">S. Aaronson</a> and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent <a rel="nofollow" class="external autonumber" href="http://scottaaronson.com/papers/ctc.pdf">[2]</a>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> There have been some claims to this effect; see <cite id="CITEREFTien_Kieu2003" class="citation journal cs1">Tien Kieu (2003). <a href="Hilbert_problems" class="mw-redirect" title="Hilbert problems">"Quantum Algorithm for the Hilbert's Tenth Problem"</a>. Int. J. Theor. Phys. 42 (7): 1461–1478. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0110136">quant-ph/0110136</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1025780028846">10.1023/A:1025780028846</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6634980">6634980</a>.</cite> or <cite id="CITEREFM._Ziegler2005" class="citation journal cs1">M. Ziegler (2005). "Computational Power of Infinite Quantum Parallelism". <a href="International_Journal_of_Theoretical_Physics" title="International Journal of Theoretical Physics">International Journal of Theoretical Physics</a>. 44 (11): 2059–2071. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0410141">quant-ph/0410141</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005IJTP...44.2059Z">2005IJTP...44.2059Z</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10773-005-8984-0">10.1007/s10773-005-8984-0</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9879859">9879859</a>.</cite> and the ensuing literature. For a retort see <cite id="CITEREFWarren_D._Smith2006" class="citation journal cs1">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. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.amc.2005.09.078">10.1016/j.amc.2005.09.078</a>.</cite>.

</li>

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<li id="cite_note-Siegelmann.1995-31"><a href="#cite_ref-Siegelmann.1995_31-0">^</a> <cite id="CITEREFH.T._Siegelmann1995" class="citation journal cs1">H.T. Siegelmann (Apr 1995). <a rel="nofollow" class="external text" href="http://binds.cs.umass.edu/papers/1995_Siegelmann_Science.pdf">"Computation Beyond the Turing Limit"</a> (PDF). Science. 268 (5210): 545–548. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995Sci...268..545S">1995Sci...268..545S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.268.5210.545">10.1126/science.268.5210.545</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17756722">17756722</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17495161">17495161</a>.</cite>

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFAoun2016" class="citation journal cs1">Aoun, Mario Antoine (2016). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170206185118/http://www.ejtp.com/articles/ejtpv13i36p169.pdf">"Advances in Three Hypercomputation Models"</a> (PDF). Electronic Journal of Theoretical Physics. 13 (36): 169–182. Archived from <a rel="nofollow" class="external text" href="http://www.ejtp.com/articles/ejtpv13i36p169.pdf">the original</a> (PDF) on 2017-02-06. Retrieved 2023-07-28.</cite></li>

<li><cite id="CITEREFBurgin1983" class="citation journal cs1">Burgin, M. S. (1983). "Inductive Turing Machines". Notices of the Academy of Sciences of the USSR. 270 (6): 1289–1293.</cite></li>

<li><cite id="CITEREFBurgin2005" class="citation book cs1">Burgin, Mark (2005). Super-recursive algorithms. Monographs in computer science. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-95569-0</bdi>.</cite></li>

<li><cite id="CITEREFCockshottMichaelson2007" class="citation journal cs1">Cockshott, P.; Michaelson, G. (2007). "Are there new Models of Computation? Reply to Wegner and Eberbach". The Computer Journal. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fcomjnl%2Fbxl062">10.1093/comjnl/bxl062</a>.</cite></li>

<li><cite id="CITEREFCooperOdifreddi,_P.2003" class="citation book cs1">Cooper, S. B.; Odifreddi, P. (2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110724020055/http://www.amsta.leeds.ac.uk/~pmt6sbc/preprints/co.pdf">"Incomputability in Nature"</a> (PDF). In Cooper, S. B.; Goncharov, S. S. (eds.). Computability and Models: Perspectives East and West. New York, Boston, Dordrecht, London, Moscow: Plenum Publishers. pp. 137–160. Archived from <a rel="nofollow" class="external text" href="http://www.amsta.leeds.ac.uk/~pmt6sbc/preprints/co.pdf">the original</a> (PDF) on 2011-07-24. Retrieved 2011-06-16.</cite></li>

<li><cite id="CITEREFCooper2006" class="citation journal cs1">Cooper, S. B. (2006). "Definability as hypercomputational effect". Applied Mathematics and Computation. 178: 72–82. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.4088">10.1.1.65.4088</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.amc.2005.09.072">10.1016/j.amc.2005.09.072</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1487739">1487739</a>.</cite></li>

<li><cite id="CITEREFCopeland2002" class="citation journal cs1">Copeland, J. (2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160314161451/http://research.cs.queensu.ca/home/akl/cisc879/papers/PAPERS_FROM_MINDS_AND_MACHINES/VOLUME_12_NO_4/NV6361035557Q678.pdf">"Hypercomputation"</a> (PDF). Minds and Machines. 12 (4): 461–502. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1021105915386">10.1023/A:1021105915386</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:218585685">218585685</a>. Archived from <a rel="nofollow" class="external text" href="http://research.cs.queensu.ca/home/akl/cisc879/papers/PAPERS_FROM_MINDS_AND_MACHINES/VOLUME_12_NO_4/NV6361035557Q678.pdf">the original</a> (PDF) on 2016-03-14.</cite></li>

<li><cite id="CITEREFHagarKorolev2007" class="citation journal cs1">Hagar, A.; Korolev, A. (2007). <a rel="nofollow" class="external text" href="http://philsci-archive.pitt.edu/3180/1/Quantum_Hype.pdf">"Quantum Hypercomputation—Hype or Computation?*"</a> (PDF). Philosophy of Science. 74 (3): 347–363. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F521969">10.1086/521969</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9857468">9857468</a>.</cite></li>

<li><cite id="CITEREFOrd2002" class="citation arxiv cs1">Ord, Toby (2002). "Hypercomputation: Computing more than the Turing machine can compute: A survey article on various forms of hypercomputation". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0209332">math/0209332</a>.</cite></li>

<li><cite id="CITEREFPiccinini2021" class="citation web cs1"><a href="Gualtiero_Piccinini" title="Gualtiero Piccinini">Piccinini, Gualtiero</a> (June 16, 2021). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/computation-physicalsystems/">"Computation in Physical Systems"</a>. Stanford Encyclopedia of Philosophy. Retrieved 2023-07-31.</cite></li>

<li><cite id="CITEREFSharma2022" class="citation journal cs1">Sharma, Ashish (2022). "Nature Inspired Algorithms with Randomized Hypercomputational Perspective". Information Sciences. 608: 670–695. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ins.2022.05.020">10.1016/j.ins.2022.05.020</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:248881264">248881264</a>.</cite></li>

<li><cite id="CITEREFStannett1990" class="citation journal cs1">Stannett, Mike (1990). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01888233">"X-machines and the halting problem: Building a super-Turing machine"</a>. Formal Aspects of Computing. 2 (1): 331–341. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01888233">10.1007/BF01888233</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7406983">7406983</a>.</cite></li>

<li><cite id="CITEREFStannett2006" class="citation journal cs1">Stannett, Mike (2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304053941/http://research.cs.queensu.ca/home/akl/cisc879/papers/PAPERS_FROM_APPLIED_MATHEMATICS_AND_COMPUTATION/Special_Issue_on_Hypercomputation/stannett%5b1%5d.pdf">"The case for hypercomputation"</a> (PDF). Applied Mathematics and Computation. 178 (1): 8–24. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.amc.2005.09.067">10.1016/j.amc.2005.09.067</a>. Archived from <a rel="nofollow" class="external text" href="http://research.cs.queensu.ca/home/akl/cisc879/papers/PAPERS_FROM_APPLIED_MATHEMATICS_AND_COMPUTATION/Special_Issue_on_Hypercomputation/stannett%5b1%5d.pdf">the original</a> (PDF) on 2016-03-04.</cite></li>

<li><cite id="CITEREFSyropoulos2008" class="citation book cs1">Syropoulos, Apostolos (2008). Hypercomputation: Computing Beyond the Church–Turing Barrier. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-30886-9</bdi>.</cite></li></ul>

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