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. 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.
State space In a sense, most functions are uncomputable: there are ℵ 0 {\displaystyle \aleph _{0}} computable functions, but there are an uncountable number ( 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} ) of possible super-Turing functions.
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 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. Similarly, a neural net that somehow had Chaitin's constant exactly embedded in its weight function would be able to solve the halting problem, 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. 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. 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. 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.
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