Limits to
Computability

In the late
1930’s, the UK mathematician Alan Turing established fundamental limits to what
we can reasonably expect to know about computations. Turing invented a very simple
model of computation called a Turing Machine (TM), which uses a set of rules
(quintuplets) to process a sequence of data symbols (tape). Turing explored the
seemingly simple question of whether it’s possible to tell if an arbitrary TM
will ever stop processing an arbitrary tape, and concluded that it is not
possible: the Halting Problem for TMs is undecidabl*e*.

TMs have been
shown to be equivalent to contemporary computer systems and programming
languages, so this result has deep implications for what we can and can’t do
with computers. For example, if we can’t
tell whether or not an arbitrary program ever terminates then we can’t know in
advance how long it will run for or how much memory it will need or how much
power it will consume. We also can’t tell whether or not two arbitrary programs
actually carry out the same computation. Of course it is possible to solve
these problems for particular programs, it just isn’t
possible in general.

People still
question Turing’s results and try to come up with ways round them. For example,
attempts are made to develop new mathematical characterisations of computation
(e.g. “hypercomputation” or “superTuring”)
that are more powerful than TMs, or to design hardware systems based on novel
ideas from physics that can overcome apparent physical limitations to TMs. On the other hand, people also misuse
Turing’s results to argue that human beings are necessarily more powerful than
computers because they don’t share these limitations, or that there are
fundamental limits to particular forms of rule based human activity, such as
economic planning.

We are very
interested in exploring and challenging such claims. While we are certainly
open to new ways of looking at the world, we think that all the evidence shows
that Turing’s results say something deep and fundamental about the limits to
computability just as the laws of physics place fundamental constraints on how
material reality works. Our publications include:

P.Cockshott, L. Mackenzie and G. Michaelson, `Physical constraints on hypercomputation',

A. Cottrell, P. Cockshott and G. J. Michaelson, `Is economic planning hypercomputational? The argument from Cantor diagonalisation',

P.
Cockshott, L. Mackenzie and G. Michaelson, Non-classical computing: feasible
versus infeasible, Proceedings of ACM-BCS Visions of Computer Science 2010,
University of Edinburgh, April, 2010, to appear PDF

P.
Cockshott, L. Mackenzie and G. Michaelson, `*Computation
and its limits*’, Oxford University Press, 2012

Paul
Cockshott, `Turing’s Universal Digital Computer’. British Mathematical
Colloquium, Kent, April 2012 prezi

Greg Michaelson, `Limits to
Computation, British Mathematical Colloquium’, Kent, April
2012 ppt

Greg
Michaelson, `A Visit to the Turing Machine’

·
CS4Fun,
April 2012 here

·
Take Tea With Turing,
January 2013 – spoken
word here

Paul Cockshott and Greg Michaelson, `Tangled Tapes:
Infinity, Interaction and Turing Machines’, Turing Centenary Conference: CIE
2012 – How the World Computes, Cambridge, June 2012 PDF of full paper

Paul Cockshott,’Turing: the
irruption of materialism into thought’, OUPBlog, July 2012 – also at Nature:
soapboxscience

J. Davidson and G. Michaelson, `Brute force is not
ignorance', Computability in Europe 2013, Milan, June, 2013 PDF

G. Michaelson, ‘SKI combinators
(really) are Turing complete’, Glasgow, November 2014 - PPTX

Paul Cockshott, University of Glasgow

Allin Cottrell, Wake-Forest University

Lewis Mackenzie, University of
Glasgow

Greg Michaelson, Heriot-Watt
University