School of Mathematics

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Prof S Barry Cooper

Pure Mathematics

Contact details

Room: 8.20f
Tel: +44 (0)113 3435165
Email: S.B.Cooper @


Mathematical logic
Computability theory
Foundations of science
Definability in the real world
Physics and computability
Computational informatics

Research interests

I am interested in the computational content of causality in the real world, particularly in the physical, mental and social contexts. Einstein said in 1950: "When we say that we understand a group of natural phenomena, we mean that we have found a constructive theory which embraces them".

Of the various ways of computably modelling causality in science, the best known is that of Turing reducibility. Allowing computation between real numbers using oracles, this is sufficient to capture the basics of familiar physical frameworks, such as that of Newtonian dynamics. One can generalise the notion to deal with computation relative to incomplete information via non-deterministic Turing machines, leading to a notion of enumeration reducibility. These reducibilities lead to quite beautiful and richly complex mathematical structures, termed the Turing and enumeration degrees, respectively. A main research interest is the ongoing project of characterising the structure of the Turing and enumeration degrees, locating basic real-world information within the structures, and furthering our understanding of the real-world consequences, at both the local and global levels.

I am President of the Association Computability in Europe, author of the standard text 'Computability Theory', Chair of the Turing Centenary Advisory Committee, a Managing Editor of the journal 'Computability', and an Advisor to the John Templeton Foundation.

Useful links

Personal homepage for S Barry Cooper
Association Computability in Europe
Alan Turing Year
COMPUTABILITY, journal of the Association CiE

Postgraduate students

Ingram Bondin (2013)
Ahmet Cevik (2010)
Benedict Durrant (2010)
James Gay (2012)
James Riley (2012)
Daoud Siniora (2013)

Selected publications

S. Barry Cooper: The Incomputable Reality, Nature, 482 (23 February 2012), 465.

S. Barry Cooper: Turing's Titanic Machine? Communications of the ACM, 55 (3) (March 2012), 74-83.

S. Barry Cooper and Jan van Leeuwen: Alan Turing - His Work and Impact, Elsevier, Amsterdam, London, New York, 2013.

S. Barry Cooper: Incomputability after Alan Turing, Notices of the American Mathematical Society, 59(6) (June/July 2012), 776-784.