# School of Mathematics

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## Prof S Barry Cooper

Professor of Pure Mathematics

Pure Mathematics

### Contact details

Room: 9.314

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** Incomputability Emergent, and Higher Type Computation In *Turing’s Revolution. The Impact of His ideas about Computability*, 2015

**Cooper SB** The machine as data: a computational view of emergence and definability *Synthese*, 2015

DOI:10.1007/s11229-015-0803-4

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**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

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**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

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**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 **7287 LNCS**, 3-10, 2012

DOI:10.1007/978-3-642-29952-0_2

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**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 **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** The Mathematician's Bias - and the Return to Embodied Computation In *A Computable Universe: Understanding and Exploring Nature as Computation*, 2012

**Cooper SB; Leeuwen JV ** *Alan Turing: His Work and Impact*. Elsevier Science 2012

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

**Cooper SB ** *Extending and interpreting Post's programme*, 775-788 2010

DOI:10.1016/j.apal.2009.06.007

**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 ** *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, 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, 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 **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

**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, 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, 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 **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

**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, Li A** On Lachlan's major subdegree problem *Archive for Mathematical Logic*, **47**, 341-434, 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; 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

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**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, 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

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

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

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

**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

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**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

**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 arithmetic*17 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

**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, 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

**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 ofΣ^{0}_{2}sets *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 Theory*26

**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** Alan Turing In *International Encyclopedia of Communication Theory and Philosophy*

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