S. Barry Cooper (1943

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Afshari,

Barmpalias,

S.B.

Cooper and

Stephan, Post’s programme for the Ershov hierarchy, J. Log. Comput.17(6) (2007), 1025–1040. doi:10.1093/logcom/exm032.

Ahmad, Some results on enumeration reducibility, PhD thesis, Simon Fraser University, 1989.

Ahmad, Embedding the diamond in the

Σ_{2}

enumeration degrees, J. Symb. Log.56(1) (1991), 195–212. doi:10.2307/2274914.

Ahmad and

A.H.

Lachlan, Some special pairs of

Σ_{2}

e-degrees, Math. Log. Q.44(4) (1998), 431–439. doi:10.1002/malq.19980440402.

Arslanov,

S.B.

Cooper and

Li, There is no low maximal d.c.e. degree, Math. Log. Q.46(3) (2000), 409–416. doi:10.1002/1521-3870(200008)46:3<409::AID-MALQ409>3.0.CO;2-P.

M.M.

Arslanov,

S.B.

Cooper and

I.S.

Kalimullin, Splitting properties of total enumeration degrees, Algebra Log.42(1) (2003), 1–13. doi:10.1023/A:1022660222520.

M.M.

Arslanov and

Sorbi, Relative splittings of

0_{e}^{'}

in the

Δ_{2}^{0}

enumeration degrees, in: Logic Colloquium ’98. Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, University of Economics, Prague, August 9–15, 1998, S.R. Buss, P. Hájek and P. Pudlák, eds, Lecture Notes in Logic, Vol. 13, Association for Symbolic Logic, 1999, pp. 44–56.

Badillo Sánchez, Genericity in the enumeration degrees, PhD thesis, University of Leeds, 2013.

Cai,

H.A.

Ganchev,

Lempp,

J.M.

Miller and

M.I.

Soskova, Defining totality in the enumeration degrees, J. Amer. Math. Soc.29(4) (2016), 1051–1067. doi:10.1090/jams/848.

10.

W.C.

Calhoun and

T.A.

Slaman, The

Π_{2}^{0}

e-degrees are not dense, J. Symb. Log.61(4) (1996), 1364–1379. doi:10.2307/2275821.

11.

S.B.

Cooper, Degrees of unsolvability, PhD thesis, University of Leicester, 1971.

12.

S.B.

Cooper, Degrees of unsolvability complementary between recursively enumerable degrees. I, Ann. Math. Log.4 (1972), 31–73. doi:10.1016/0003-4843(72)90011-3.

13.

S.B.

Cooper, Minimal upper bounds for sequences of recursively enumerable degrees, J. London Math. Soc. (2)5 (1972), 445–450. doi:10.1112/jlms/s2-5.3.445.

14.

S.B.

Cooper, Distinguishing the arithmetical hierarchy, Preprint, 1972.

15.

S.B.

Cooper, Minimal degrees and the jump operator, J. Symb. Log.38 (1973), 249–271. doi:10.2307/2272061.

16.

S.B.

Cooper, Minimal pairs and high recursively enumerable degrees, J. Symb. Log.39 (1974), 655–660. doi:10.2307/2272849.

17.

S.B.

Cooper, An annotated bibliography for the structure of the degrees below

0^{'}

, with special reference to that of the recursively enumerable degrees, Recursive Function Theory Newsletter8(Suppl. 3) (1974).

18.

S.B.

Cooper, Partial degrees and the density problem, J. Symb. Log.47(4) (1982), 854–859. doi:10.2307/2273104.

19.

S.B.

Cooper, Partial degrees and the density problem. Part 2: The enumeration degrees of the

Σ_{2}

sets are dense, J. Symb. Log.49(2) (1984), 503–513. doi:10.2307/2274181.

20.

S.B.

Cooper, Enumeration reducibility using bounded information: Counting minimal covers, Z. Math. Log. Grundlag. Math.33 (1987), 537–560. doi:10.1002/malq.19870330608.

21.

S.B.

Cooper, The strong anticupping property for recursively enumerable degrees, J. Symb. Log.54(2) (1989), 527–539. doi:10.2307/2274867.

22.

S.B.

Cooper, Enumeration reducibility, nondeterministic computations and relative computability of partial functions, in: Recursion Theory Week. Proceedings of the Conference Held at the Mathematisches Forschungsinstitut, Oberwolfach, March 19–25, 1989,

Ambos-Spies,

G.H.

Müller and

G.E.

Sacks, eds, Lecture Notes in Mathematics, Vol. 1432, Springer-Verlag, 1990, pp. 57–110. doi:10.1007/BFb0086114.

23.

S.B.

Cooper, The density of the

{low}_{2} n

-r.e. degrees, Arch. Math. Log.31(1) (1991), 19–24. doi:10.1007/BF01370692.

24.

S.B.

Cooper, A splitting theorem for the n-r.e. degrees, Proc. Amer. Math. Soc.115(2) (1992), 461–471.

25.

S.B.

Cooper, Discontinuous phenomena and Turing definability, in: Algebra and Analysis. Proceedings of the International Centennial Chebotarev Conference, Kazan, June 5–11, 1994,

M.M.

Arslanov,

A.N.

Parshin and

I.R.

Shafarevich, eds, Walter de Gruyter, 1996, pp. 41–55.

26.

S.B.

Cooper, Local degree theory, in: Handbook of Computability Theory,

E.R.

Griffor, ed., Studies in Logic and the Foundations of Mathematics, Vol. 140, North-Holland, 1999, pp. 121–153. doi:10.1016/S0049-237X(99)80020-2.

27.

S.B.

Cooper, Computability Theory, Chapman & Hall/CRC Mathematics, 2004.

28.

S.B.

Cooper, A characterisation of the jumps of minimal degrees below

0^{'}

, in: Computability, Enumerability, Unsolvability. Directions in Recursion Theory, S.B. Cooper, T.A. Slaman and S.S. Wainer, eds, London Mathematical Society Lecture Note Series, Vol. 224, Cambridge University Press, 1996, pp. 81–92. doi:10.1017/CBO9780511629167.004.

29.

S.B.

Cooper, Recursive function theory newsletter 1972–1997. Webpage hosted on the author’s website by Leeds University, 2010.

30.

S.B.

Cooper and

R.L.

Epstein, Complementing below recursively enumerable degrees, Ann. Pure Appl. Log.34(1) (1987), 15–32. doi:10.1016/0168-0072(87)90039-X.

31.

S.B.

Cooper,

Harrington,

A.H.

Lachlan,

Lempp and

R.I.

Soare, The d.r.e. degrees are not dense, Ann. Pure Appl. Log.55(2) (1991), 125–151. doi:10.1016/0168-0072(91)90005-7.

32.

S.B.

Cooper,

Lempp and

Watson, Weak density and cupping in the d-r.e. degrees, Isr. J. Math.67(2) (1989), 137–152. doi:10.1007/BF02937291.

33.

S.B.

Cooper and

Li, Non-uniformity and generalised Sacks splitting, Acta Math. Sin.18(2) (2002), 327–334. doi:10.1007/s101140100150.

34.

S.B.

Cooper and

Li, Splitting and cone avoidance in the d.c.e. degrees, Sci. China Ser. A45(9) (2002), 1135–1146.

35.

S.B.

Cooper and

Li, Splitting and nonsplitting. II. A

{low}_{2}

C.E. degree about which

0^{'}

is not splittable, J. Symb. Log.67(4) (2002), 1391–1430. doi:10.2178/jsl/1190150292.

36.

S.B.

Cooper and

Li, Turing definability in the Ershov hierarchy, J. London Math. Soc. (2)66(3) (2002), 513–528. doi:10.1112/S0024610702003691.

37.

S.B.

Cooper and

Li, On Lachlan’s major sub-degree problem, Arch. Math. Log.47(4) (2008), 341–434. doi:10.1007/s00153-008-0083-5.

38.

S.B.

Cooper,

Li,

Sorbi and

Yang, Bounding and nonbounding minimal pairs in the enumeration degrees, J. Symb. Log.70(3) (2005), 741–766. doi:10.2178/jsl/1122038912.

39.

S.B.

Cooper,

Li and

Yi, On the distribution of Lachlan nonsplitting bases, Arch. Math. Log.41(5) (2002), 455–482. doi:10.1007/s001530100095.

40.

S.B.

Cooper,

Löwe and

Torenvliet, Preface, in: New Computational Paradigms. First Conference on Computability in Europe, CiE 2005,

S.B.

Cooper,

Löwe and

Torenvliet, eds, Amsterdam, The Netherlands, June 8–12, 2005, Proceedings, Lecture Notes in Computer Science, Vol. 3526, Springer-Verlag, 2005, pp. V–X.

41.

S.B.

Cooper and

Odifreddi, Incomputability in nature, in: Computability and Models. Perspectives East and West,

S.B.

Cooper and

S.S.

Goncharov, eds, The University Series in Mathematics, Kluwer Academic/Plenum Publishers, 2003, pp. 137–160. doi:10.1007/978-1-4615-0755-0_6.

42.

S.B.

Cooper,

Sorbi and

Yi, Cupping and noncupping in the enumeration degrees of

Σ_{2}^{0}

sets, Ann. Pure Appl. Log.82(3) (1997), 317–342. doi:10.1016/S0168-0072(96)00009-7.

43.

S.B.

Cooper and

Yi, The discontinuity of splitting in the recursively enumerable degrees, Arch. Math. Log.34(4) (1995), 247–256. doi:10.1007/BF01469381.

44.

C.S.

Copestake, The enumeration degrees of the

Σ_{2}^{0}

sets, PhD thesis, University of Leeds, 1987.

45.

C.S.

Copestake and

S.B.

Cooper, Properly

Σ_{2}

enumeration degrees, Z. Math. Log. Grundlag. Math.34 (1988), 491–522. doi:10.1002/malq.19880340603.

46.

Davis, Computability and Unsolvability, Dover, 1982.

47.

R.G.

Downey,

Lempp and

R.A.

Shore, Jumps of minimal degrees below

0^{'}

, J. London Math. Soc. (2)54(3) (1996), 417–439. doi:10.1112/jlms/54.3.417.

48.

Durrant,

Lewis-Pye,

K.M.

Ng and

Riley, Computably enumerable Turing degrees and the meet property, Proc. Amer. Math. Soc.144(4) (2016), 1735–1744. doi:10.1090/proc/12808.

49.

Friedberg, A criterion for completeness of degrees of unsolvability, J. Symb. Log.22 (1957), 159–160. doi:10.2307/2964177.

50.

R.M.

Friedberg and

RogersJr., Reducibility and completeness for sets of integers, Z. Math. Log. Grundlag. Math.5 (1959), 117–125. doi:10.1002/malq.19590050703.

51.

A.H.

Ganchev and

M.I.

Soskova, The high/low hierarchy in the local structure of the

Σ_{2}^{0}

-enumeration degrees, Ann. Pure Appl. Log.163(5) (2012), 547–566. doi:10.1016/j.apal.2010.10.004.

52.

A.H.

Ganchev and

M.I.

Soskova, Definability via Kalimullin pairs in the structure of the enumeration degrees, Trans. Amer. Math. Soc.367(7) (2014), 4873–4893. doi:10.1090/S0002-9947-2014-06157-6.

53.

A.H.

Ganchev and

M.I.

Soskova, The jump hierarchy in the enumeration degrees, Submitted.

54.

E.M.

Gold, Limiting recursion, J. Symb. Log.30 (1965), 28–48. doi:10.2307/2270580.

55.

Gutteridge, Some results on enumeration reducibility, PhD thesis, Simon Fraser University, 1971.

56.

C.M.

Harris, Enumeration reducibility and polynomial time bounds, PhD thesis, University of Leeds, 2006.

57.

C.M.

Harris, On the symmetric enumeration degrees, Notre Dame J. Form. Log.48(2) (2007), 175–204. doi:10.1305/ndjfl/1179323263.

58.

C.M.

Harris, Goodness in the enumeration and singleton degrees, Arch. Math. Log.49(6) (2010), 673–691. doi:10.1007/s00153-010-0192-9.

59.

C.M.

Harris, On the jump classes of noncuppable enumeration degrees, J. Symb. Log.76(1) (2011), 177–197. doi:10.2178/jsl/1294170994.

60.

C.G.

JockuschJr. and

D.B.

Posner, Double jumps of minimal degrees, J. Symb. Log.43(4) (1978), 715–724. doi:10.2307/2273510.

61.

I.S.

Kalimullin, Definability of the jump operator in the enumeration degrees, J. Math. Log.3(2) (2003), 257–267. doi:10.1142/S0219061303000285.

62.

T.F.

Kent, The

Π_{3}

-theory of the

Σ_{2}^{0}

-enumeration degrees is undecidable, J. Symb. Log.71(4) (2006), 1284–1302. doi:10.2178/jsl/1164060455.

63.

T.F.

Kent, The structure of the s-degrees contained within a single e-degree, Ann. Pure Appl. Log.160(1) (2009), 13–21.

64.

S.C.

Kleene, Introduction to Metamathematics, Van Nostrand, 1952.

65.

A.H.

Lachlan, A recursively enumerable degree which will not split over all lesser ones, Ann. Math. Log.9(4) (1976), 307–365. doi:10.1016/0003-4843(76)90016-4.

66.

A.H.

Lachlan and

R.A.

Shore, The n-r.e.a. enumeration degrees are dense, Arch. Math. Log.31(4) (1992), 277–285. doi:10.1007/BF01794984.

67.

Lempp,

T.A.

Slaman and

Sorbi, On extensions of embeddings into the enumeration degrees of the

Σ_{2}^{0}

sets, J. Math. Log.5(2) (2005), 247–298. doi:10.1142/S0219061305000432.

68.

A.E.M.

Lewis, A note on the join property, Proc. Amer. Math. Soc.140(2) (2012), 707–714. doi:10.1090/S0002-9939-2011-10908-0.

69.

A.E.M.

Lewis, Properties of the jump classes, J. Log. Comput.22(4) (2012), 845–855. doi:10.1093/logcom/exq047.

70.

London Mathematical Society, Barry Cooper, LMS Newsletter454 (2016), 49–50.

71.

Löwe, Barry Cooper (1943–2015): The engine of computability in Europe, Computability5(1) (2016), 3–11. doi:10.3233/COM-160052.

72.

McEvoy, The Structure of the enumeration degrees, PhD thesis, University of Leeds, 1984.

73.

McEvoy and

S.B.

Cooper, On minimal pairs of enumeration degrees, J. Symb. Log.50(4) (1985), 983–1001. doi:10.2307/2273985.

74.

Myhill, A note on degrees of partial functions, Proc. Amer. Math. Soc.12 (1961), 519–521. doi:10.1090/S0002-9939-1961-0125794-X.

75.

Nikolova, Shared thoughts on Ivan Soskov (1954–2013), Computability3(1) (2014), 63–72.

76.

Petkov (ed.), Mathematical Logic: Proceedings of the Summer School and Conference on Mathematical Logic, Honourably Dedicated to the 90th Anniversary of Arend Heyting (1898–1980), Chaika (near Varna), Bulgaria, September 13–23, 1988, Plenum Press, 1990.

77.

E.L.

Post, Recursively enumerable sets of positive integers and their decision problems, Bull. Amer. Math. Soc.50 (1944), 284–316. doi:10.1090/S0002-9904-1944-08111-1.

78.

Putnam, Trial and error predicates and the solution to a problem of Mostowski, J. Symb. Log.30 (1965), 49–57. doi:10.2307/2270581.

79.

RogersJr., Theory of Recursive Functions and Effective Computability, McGraw-Hill, 1967.

80.

G.E.

Sacks, A minimal degree less than

0^{'}

, Bull. Amer. Math. Soc.67 (1961), 416–419. doi:10.1090/S0002-9904-1961-10652-6.

81.

G.E.

Sacks, On the degrees less than

0^{'}

, Ann. Math. (2)77 (1963), 211–231. doi:10.2307/1970214.

82.

G.E.

Sacks, The recursively enumerable degrees are dense, Ann. Math. (2)80 (1964), 300–312. doi:10.2307/1970393.

83.

L.P.

SassoJr., A minimal degree not realizing least possible jump, J. Symb. Log.39 (1974), 571–574. doi:10.2307/2272899.

84.

L.P.

Sasso, Degrees of unsolvability of partial functions, PhD thesis, University of California, Berkeley, 1971.

85.

L.P.

Sasso, A survey of partial degrees, J. Symb. Log.40(2) (1975), 130–140. doi:10.2307/2271892.

86.

J.R.

Shoenfield, On degrees of unsolvability, Ann. Math. (2)69 (1959), 644–653. doi:10.2307/1970028.

87.

J.R.

Shoenfield, A theorem on minimal degrees, J. Symb. Log.31 (1966), 539–544. doi:10.2307/2269688.

88.

J.R.

Shoenfield, MR0294126, Review of “S. B. Cooper, Degrees of unsolvability complementary between recursively enumerable degrees, I”, Math. Rev.45 (1973), #3199.

89.

R.A.

Shore and

T.A.

Slaman, Working below a high recursively enumerable degree, J. Symb. Log.58(3) (1993), 824–859. doi:10.2307/2275099.

90.

T.A.

Slaman and

Sorbi, A note on initial segments of the enumeration degrees, J. Symb. Log.79(2) (2014), 633–643. doi:10.1017/jsl.2013.1.

91.

R.I.

Soare, Automorphisms of the lattice of recursively enumerable sets. I. Maximal sets, Ann. Math. (2)100 (1974), 80–120. doi:10.2307/1970842.

92.

Soskova,

S.B.

Cooper and

Sorbi (eds), Computability at Logic Colloquium 2009, J. Log. Comput.22(4) (2012), 667.

93.

M.I.

Soskova, The local structure of the enumeration degrees, PhD thesis, University of Leeds, 2008.

94.

M.I.

Soskova, A non-splitting theorem in the enumeration degrees, Ann. Pure Appl. Log.160(3) (2009), 400–418. doi:10.1016/j.apal.2009.01.009.

95.

M.I.

Soskova and

S.B.

Cooper, How enumeration reducibility yields extended Harrington non-splitting, J. Symb. Log.73(2) (2008), 634–655. doi:10.2178/jsl/1208359064.

96.

M.I.

Soskova and

I.N.

Soskov, Embedding countable partial orderings in the enumeration degrees and the ω-enumeration degrees, J. Log. Comput.22(4) (2012), 927–952. doi:10.1093/logcom/exq051.

97.

Spector, On degrees of recursive unsolvability, Ann. of Math. (2)64 (1956), 581–592. doi:10.2307/1969604.

98.

Stob,

wtt

-degrees and T-degrees of r.e. sets, J. Symb. Log.48(4) (1983), 921–930. doi:10.2307/2273658.

99.

A.M.

Turing, On computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. Soc.42 (1936), 230–265.

100.

van Emde Boas, Three funerals and a group wedding; CoLoRet, Recolet and Coin; the prehistory of CiE, presentation at CiE 2014, Budapest, Hungary, 23 June 2014.

101.

Watson, On restricted forms of enumeration reducibility, Ann. Pure Appl. Log.49(1) (1990), 75–96. doi:10.1016/0168-0072(90)90051-3.

102.

P.R.

Watson, Concerning d-recursively enumerable sets and s-reducibility of

Σ_{2}

sets, PhD thesis, University of Leeds, 1988.

103.

White, Letter: Barry Cooper obituary, The Guardian, online 27 January 2016.

104.

C.E.M.

Yates, Initial segments of the degrees of unsolvability. II. Minimal degrees, J. Symb. Log.35 (1970), 243–266. doi:10.2307/2270517.

S. Barry Cooper (1943–2015)

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