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Prof S Barry Cooper

Professor of Pure Mathematics
Pure Mathematics

Contact details

Room: ECS 7.59
Tel: +44 (0)113 3435165
Email: S.B.Cooper @ leeds.ac.uk

Keywords

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

Current postgraduate students

Ingram Bondin (2013)
James Gay (2012)
James Riley (2012)
David Toth (2014)

Publications

Cooper SB 'Real' information is not flat - And why it matters Journal of Experimental and Theoretical Artificial Intelligence, 27, 3-11, 2015
DOI:10.1080/0952813X.2014.940694
View abstract

Gopal TV; Agrawal M; Li A; Cooper SB A roadmap for TAMC Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 8402 LNCS, 1-6, 2014
DOI:10.1007/978-3-319-06089-7-1

Cooper SB 'Real' information is not flat - and why it matters Journal of Experimental and Theoretical Artificial Intelligence, 2014
DOI:10.1080/0952813X.2014.940694
View abstract

Dumitriu A; Cooper SB The future of art and computing: A post turing centennial perspective 2014

Cooper SB; Soskova MI The incomputable JOURNAL OF LOGIC AND COMPUTATION, 23, 1143-1144, 2013
DOI:10.1093/logcom/ext050

Cooper SB; Kashefi E; Panangaden P Preface to special issue: Developments In Computational Models 2010 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE, 23, 201-203, 2013
DOI:10.1017/S0960129512000084

Cooper SB What Makes a Computation Unconventional? In Computing Nature: Turing Centenary Perspective, 2013

Cooper SB What makes a computation unconventional? Or, there is no such thing as non-turing computation AISB/IACAP World Congress 2012: Natural Computing/Unconventional Computing and Its Philosophical Significance, Part of Alan Turing Year 2012, 36, 2012

Calude CS; Cooper SB Introduction: Computability of the physical Mathematical Structures in Computer Science, 22, 723-728, 2012
DOI:10.1017/S0960129511000491

Cooper SB; Maini PK The mathematics of nature at the Alan Turing centenary INTRODUCTION INTERFACE FOCUS, 2, 393-396, 2012
DOI:10.1098/rsfs.2012.0018

Soskova A; Cooper SB; Sorbi A Computability at Logic Colloquium 2009 Preface JOURNAL OF LOGIC AND COMPUTATION, 22, 667-667, 2012
DOI:10.1093/logcom/exq045

Cooper SB; Abramsky S The foundations of computation, physics and mentality: The turing legacy Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370, 3273-3276, 2012
DOI:10.1098/rsta.2012.0221

Dawar A; Cooper SB; Löwe B Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics): Preface Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 7318 LNCS, IV-IX, 2012

Cooper SB; Dawar A; We BL How the World Computes. Springer-Verlag New York Incorporated 2012
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Cooper SB Incomputability after Alan Turing Notices of the American Mathematical Society, 59, 776-784, 2012
DOI:10.1090/noti866|

Agrawal M; Cooper SB; Li A Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics): Preface Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 7287 LNCS, V-VI, 2012

Cooper SB From Turing machine to morphogenesis: Forming and informing computation Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 7287 LNCS, 3-10, 2012
DOI:10.1007/978-3-642-29952-0_2
View abstract

Agrawal M; Cooper SB; Li A Theory and Applications of Models of Computation: 9th Annual Conference, TAMC 2012, Beijing, China, May 16-21, 2012.. 7287, Heidelberg: Springer-Verlag 2012
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Cooper SB Mathematics, metaphysics and the multiverse Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 7160 LNCS, 252-267, 2012
DOI:10.1007/978-3-642-27654-5_20
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Cooper SB Turing's titanic machine? Communications of the ACM, 55, 74-83, 2012
DOI:10.1145/2093548.2093569
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Cooper SB The Incomputable Reality Nature, 482, 465-465, 2012

Cooper SB From Natural Philosophy to Computation, and Back Again, 616-619 2012
DOI:10.1063/1.4756208

Cooper SB; Mardia K Alan Turing and Enigmatic Statistics Bulletin of the Brasilian Section of the International Society for Bayesian Analysis, 5, 2-7, 2012

Cooper SB; Leeuwen JV Alan Turing: His Work and Impact. Elsevier Science 2012
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Cooper SB The Mathematician's Bias - and the Return to Embodied Computation In A Computable Universe: Understanding and Exploring Nature as Computation, 2012

Cooper SB; Sorbi A eds. Computability in Context: Computation and Logic in the Real World 2011
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Chen J; Cooper SB Algorithms, complexity and computational models Preface THEOR COMPUT SCI, 412, 2457-2458, 2011
DOI:10.1016/j.tcs.2010.10.037

Cooper SB From Descartes to Turing: The Computational Content of Supervenience In Information and Computation, 107-148, 2011

Arslanov MM; Cooper SB; Kalimullin IS; Soskova MI Splitting and nonsplitting in the Sigma(0)(2) enumeration degrees, 1669-1685 2011
DOI:10.1016/j.tcs.2010.12.042

Cooper SB Definability in the Real Universe In Computability in Context: Computation and Logic in the Real World, 131-167, 2011

Cooper SB; Arslanov MM; Kalimullin IS; Soskova MI Splitting and nonsplitting in the Sigma(0)(2) enumeration degrees Theoretical Computer Science, 412, 1669-1685, 2011

Cooper SB Extending and interpreting Post's programme, 775-788 2010
DOI:10.1016/j.apal.2009.06.007

Cooper SB Incomputability, Emergence and the Turing Universe In Causality, Meaningful Complexity and Embodied Cognition, 135-153, 2010
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Cooper SB The Computational Roots of Hyper-Causality, 437-449 2010

Cooper SB; Mayordomo E; Sorbi A Computation and Logic in the Real World: CiE 2007 THEOR COMPUT SYST, 45, 647-649, 2009
DOI:10.1007/s00224-008-9164-4

Cooper SB Emergence as a computability-theoretic phenomenon, 1351-1360 2009
DOI:10.1016/j.amc.2009.04.050

Cooper SB; Geuvers H; Pillay A; Väänänen J Logic Colloquium 2006. Cambridge University Press 2009
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Buss SR; Cooper SB; Lowe B; Sorbi A Special Issue Computation and Logic in the Real World: CiE 2007 Preface ANN PURE APPL LOGIC, 160, 229-230, 2009
DOI:10.1016/j.apal.2009.01.011

Buss SR; Cooper SB; Löwe B; Löwe B; Löwe B; Sorbi A Preface Annals of Pure and Applied Logic, 160, 229-230, 2009
DOI:10.1016/j.apal.2009.01.011

Chen J; Cooper SB Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics): Preface Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 5532 LNCS, V-VI, 2009

Chen J; Cooper SB Theory and Applications of Models of Computation. Springer-Verlag New York Incorporated 2009
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Cooper SB; Zhu H Preface: Algorithms, complexity and models of computation THEOR COMPUT SCI, 410, 1001-1002, 2009
DOI:10.1016/j.tcs.2008.10.026

Bonizzoni P; Cooper SB; Lowe B; Sorbi A Computational Paradigms from Nature Foreword THEOR COMPUT SCI, 410, 283-284, 2009
DOI:10.1016/j.tcs.2008.09.040

Bonizzoni P; Cooper SB; Löwe B; Löwe B; Löwe B; Sorbi A Foreword Theoretical Computer Science, 410, 283-284, 2009
DOI:10.1016/j.tcs.2008.09.040

Cooper SB; Lowe B; Sorbi A Logic and Computation in the Real World: CiE 2007 J LOGIC COMPUT, 19, 1-2, 2009
DOI:10.1093/logcom/exn020

Cai JY; Cooper SB; Li AS Preface to Special Issue: Theory and Applications of Models of Computation (TAMC) MATH STRUCT COMP SCI, 19, 5-7, 2009
DOI:10.1017/S0960129508007214

Cooper SB The Extended Turing Model as Contextual Tool, 18-28 2009

Cooper SB Emergence as a computability-theoretic phenomenon Applied Mathematics and Computation, 215, 1351-1360, 2009

Arslanov MM; Cooper SB; Kalimullin IS; Soskova MI Total degrees and nonsplitting properties of∑2 0 enumeration degrees Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 4978 LNCS, 568-578, 2008
DOI:10.1007/978-3-540-79228-4-49
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Cooper SB Special Issue Logic Colloquium 2006 Preface ANN PURE APPL LOGIC, 156, 1-2, 2008
DOI:10.1016/j.apal.2008.06.001

Soskova MI; Cooper SB How enumeration reducibility yields extended Harrington non-splitting J SYMBOLIC LOGIC, 73, 634-655, 2008

Cooper SB; Salts MC; Wu GH The non-isolating degrees are upwards dense in the computably enumerable degrees, 588-596 2008
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Cooper SB; Li A On Lachlan's major subdegree problem Archive for Mathematical Logic, 47, 341-434, 2008

Cooper SB; Löwe B; Sorbi A New Computational Paradigms. New York: Springer 2008
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Afshari B; Barmpalias G; Cooper SB; Stephan F Post's programme for the Ershov hierarchy, 1025-1040 2007
DOI:10.1093/logcom/exm032

Cooper SB; Lowe B; Boas PV Theory of computation at CiE 2005 THEOR COMPUT SYST, 41, 1-2, 2007
DOI:10.1007/s00224-006-4101-x

Soskova MI; Cooper SB The strongest nonsplitting theorem, 199-211 2007

Cooper SB; Löwe B; Sorbi A Computation and Logic in the Real World. Heidelberg: Springer-Verlag New York Incorporated 2007
View abstract

Cooper SB; Lowe B; Normann D Mathematics of computing at CiE 2005 MATH STRUCT COMP SCI, 16, 735-736, 2006
DOI:10.1017/S0960129506005433

Cooper SB Definability as hypercomputational effect APPL MATH COMPUT, 178, 72-82, 2006
DOI:10.1016/j.amc.2005.09.072

Cooper SB How can nature help us compute?, 1-13 2006

Afshari B; Barmpalias G; Cooper SB Immunity properties and the n-CE hierarchy, 694-703 2006

Cooper SB; Li AS; Sorbi A; Yang Y Bounding and nonbounding minimal pairs in the enumeration degrees J SYMBOLIC LOGIC, 70, 741-766, 2005

Cooper SB; Li A; Sorbi A; Yang Y Bounding and nonbounding minimal pairs in the enumeration degrees Journal of Symbolic Logic, 70, pp.741–766, 2005
DOI:10.2178/jsl/1122038912

Cooper SB Introduction: If GiE did not exist, it would be necessary to invent it, 1-7 2005

Arslanov MM; Chong CT; Cooper SB; Yang Y The minimal e-degree problem in fragments of Peano arithmetic ANN PURE APPL LOGIC, 131, 159-175, 2005
DOI:10.1016/j.apal.2004.04.010

Arslanov M; Cooper SB; Li A There is no low maximal d. c. e. degree - Corrigendum Mathematical Logic Quarterly, 50, 628-636, 2004
DOI:10.1002/malq.200410006
View abstract

Arslanov M; Cooper SB; Li AS There is no low maximal d.c.e. degree (vol 46, pg 409, 2000) MATH LOGIC QUART, 50, 628-636, 2004
DOI:10.1002/malq.200410006

Cooper SB Computability Theory. Chapman&Hall 2004
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Arslanov MM; Chong CT; Cooper SB; Yang Y The minimal e-degree problem in fragments of Peano arithmetic17 2003

Cooper SB; Li A; Sorbi A; Yang Y There exists a maximal 3-c.e. enumeration degree Israel Journal of Mathematics, 137, 285-320, 2003

Cooper SB; Odifreddi P Incomputability in Nature In Commputability and Models: Perspectives East and West, 137-160, 2003

Cooper SB; Lewis AEM; Yang Y Properly Sigma2 minimal degrees and 0" complementation 2003

Hardy B; van Raak A; Mur-Veerman I; Steenbergen M; Paulus A Introduction In Integrated care in Europe: description and comparison of integrated care in six EU countries, 9-14, 2003

Arslanov MM; Kalimullin ISH; Cooper SB Splitting properties of total enumeration degrees Algebra and Logic, 42, 1-13, 2003
DOI:10.1023/A:1022660222520

Cooper SB; Li A Turing definability in the Ershov hierarchy J LOND MATH SOC, 66, 513-528, 2002
DOI:10.1112/S0024610702003691

Cooper SB; Li AS Splitting and nonsplitting, II: A low(2) C.E. degree above which 0 ' is not splittable J SYMBOLIC LOGIC, 67, 1391-1430, 2002

Cooper SB; Li AS Splitting and cone avoidance in the DCE degrees SCI CHINA SER A, 45, 1135-1146, 2002

Cooper SB; Li A; Yi X On the distribution of Lachlan nonsplitting bases Archive for Mathematical Logic, 41, 455-482, 2002
DOI:10.1007/s001530100095

Cooper SB; Li A Non-uniformity and generalised Sacks splitting Acta Mathematica Sinica, New Series, 18, 327-334, 2002

Cooper SB On the intensionality of Turing definitions, 40-58 2002

Cooper SB On a conjecture of Kleene and Post MATH LOGIC QUART, 47, 3-33, 2001

Li A; Cooper SB; Arslanov MM There is no low maximal d.c.e degree Mathematical Logic Quarterly, 46, 409-416, 2000

Cooper SB Hartley Rogers' 1965 agenda In Logic Colloquium '98, 13, 154-172, 2000

Cooper SB Clockwork or Turing U/universe? - remarks on causal determinism and computability In Models and Computability, 259, 63-116, 1999

Cooper SB Beyond Gödel's theorem - the failure to capture information content In Complexity, Logic and Recursion Theory, 187, 93-122, 1997

Cooper SB; Sorbi A; Yi X Cupping and noncupping in the enumeration degrees ofΣ02sets Annals of Pure and Applied Logic, 82, 317-342, 1996

Cooper SB; Sorbi A Noncappable enumeration degrees below 0'e Journal of Symbolic Logic, 61, 1347-1363, 1996

Cooper SB Computability Theory26

Cooper SB; Harris CM; Lee KI Automorphisms ofη-like Computable Linear Orderings and Kierstead's Conjecture Mathematical logic quarterly
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Cooper SB The Machine as Data: A Computational View of Emergence and Definability Synthese