Editing Hypercomputation (section)

===Uncomputable inputs or black-box components===
{{prose|section|date=July 2023}}
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:

<!-- Please include only well-cited examples, we don't need to hear about a penguin whose eyeblinks encode Chaitin's constant -->
*Turing's original oracle machines, defined by Turing in 1939.
*A [[real computer]] (a sort of idealized [[analog computer]]) can perform hypercomputation<ref>[[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"[http://www.scottaaronson.com/papers/npcomplete.pdf] p.&nbsp;12</ref> if physics admits general [[real number|real]] variables (not just [[computable number|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.<ref name=HodgesSCIAM>{{cite web |url=http://www.turing.org.uk/philosophy/sciam.html |title=The Professors and the Brainstorms |author=Andrew Hodges |access-date=23 September 2011 |work=The Alan Turing Home Page }}</ref>
**Similarly, a neural net that somehow had Chaitin's constant exactly embedded in its weight function would be able to solve the halting problem,<ref>{{cite journal |author1=H.T. Siegelmann |author2=E.D. Sontag | title=Analog Computation via Neural Networks | journal=Theoretical Computer Science | volume=131 |issue=2 | pages=331–360 | year=1994 |doi=10.1016/0304-3975(94)90178-3 | doi-access=free }}</ref> 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.<ref>{{Cite journal|last=Biacino|first=L.|author2=Gerla, G.|year=2002|title=Fuzzy logic, continuity and effectiveness|journal=Archive for Mathematical Logic|issn=0933-5846|volume=41|issue=7|pages=643–667|doi=10.1007/s001530100128|citeseerx=10.1.1.2.8029|s2cid=12513452}}</ref><ref name=ClassicalFuzzy>{{Cite journal|last=Wiedermann|first=Jiří |year=2004|title=Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines|journal= Theoretical Computer Science|volume=317|issue=1–3|pages=61–69|doi=10.1016/j.tcs.2003.12.004|quote=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.|doi-access=free}}</ref>
**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 and reject inputs that would "unfairly" cause a subsystem to run forever.<ref>{{cite journal|title=Nondeterminism, Fairness and a Fundamental Analogy|journal=EATCS Bulletin|volume=37|pages=186–193|year=1989|author1=Edith Spaan |author2=Leen Torenvliet |author3=Peter van Emde Boas }}</ref><ref>{{Cite journal|doi=10.1016/j.amc.2005.09.076|title=The many forms of hypercomputation|year=2006|last1=Ord|first1=Toby|journal=Applied Mathematics and Computation|volume=178|pages=143–153}}</ref>
*Dmytro Taranovsky has proposed a [[finitism|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.<ref name=Taranovsky>{{cite web | author=Dmytro Taranovsky | date=July 17, 2005 | title=Finitism and Hypercomputation | url=http://web.mit.edu/dmytro/www/FinitismPaper.htm | access-date=Apr 26, 2011}}</ref>
**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.<ref>Hewitt, Carl. "What Is Commitment." Physical, Organizational, and Social (Revised), Coordination, Organizations, Institutions, and Norms in Agent Systems II: AAMAS (2006).</ref><!-- There is also an argument in Hewitt 2006 that this model is justified in the physical world, but this appears to be unbased [[WP:FRINGE]] speculation that I can't find anyone else endorsing or even explicitly acknowledging. -->