Hypercomputation

Hypercomputation or super-Turing computation refers to 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 can correctly evaluate every statement in Peano arithmetic.

The Church–Turing thesis states that any "effectively 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 effectively 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 useful, rather than random, uncomputable functions.

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.

In a sense, most functions are uncomputable: there are ${\displaystyle \aleph _{0}}$ computable functions, but there are an uncountable number (${\displaystyle 2^{\aleph _{0}}}$) of possible Super-Turing functions. Most of these uncomputable functions are also "indescribable", and have an output that is random rather than useful.

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