# Hypercomputation

**Hypercomputation** is the theory of methods for the computation of

non-recursive functions. The classes of functions which they can compute is studied in the field known as

recursion theory. A similar recent term is

super-Turing computation, which has been used in the

neural network literature to describe machines with various expanded abilities, including the ability to compute directly on

real numbers, the ability to carry out infinitely many computations simultaneously, or the ability to carry out computations with exponentially lower

complexity than standard Turing machines.

Hypercomputation was first introduced by Alan Turing in his 1939 paper *Systems of logic based on ordinals*, which investigated mathematical systems in which an oracle was available to compute a single arbitrary (non-recursive) function from naturalss to naturals.

Other posited kinds of hypercomputer include:

- A quantum mechanical system which somehow uses (for example) an infinite superposition of states to compute a non-recursive function
^{1}. Such a system could not be an ordinary qubit quantum computer.
- A Turing machine which runs for an infinite period of time (perhaps the observer is being dropped into a black hole).
- A Turing machine which increases its speed exponentially over time. In a Newtonian universe, such a gadget might operate by manufacturing a clone of itself which was only half the size and operated at twice the speed.
- A non-deterministic Turing machine which has a preference ordering over its final states.
- A "real computer" (a sort of idealized analog computer) might be able to 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 outlandish laws of physics (for example, a measurable physical constant with an oracular value, such as &Omega), and would at minimum require the ability to measure a real-valued physical value to arbitrary precision despite thermal noise and quantum effects.

At this stage, none of these devices seem physically plausible, and so hypercomputers are likely to remain a mathematical fiction.

### Notes

- There have been some claims to this effect; see Tien Kieu,
*Quantum Algorithm for Hilbert's Tenth Problem* and the ensuing literature. It is very likely that these results will turn out to be erroneous or non-physical. Until this is well established and explained, the possibility of quantum hypercomputation is deserving of investigation.

### References

- Alan Turing,
*Systems of logic based on ordinals*, Proc. London math. soc., **45**, 1939
- Tien Kieu,
*Quantum Algorithm for the Hilbert's Tenth Problem*, Int. J. Theor. Phys., **42** (2003) 1461-1478, e-print archive quant-ph/0110136 (pdf format)
- Toby Ord,
*Hypercomputation: computing more than the Turing machine* (pdf format), Honours Thesis, University of Melbourne, 2002.