## Richard A. Shore : Publications

Curriculum Vitae
Most of the documents below with electronic versions have been compiled for optimum viewing in PDF format. However, some papers (for various reasons) look grainy as PDF files. All the papers with electronic versions are, however, also available in postscript and DVI format.

1. On large cardinals and partition relations, Journal of Symbolic Logic 36 (1971), 305-308 (with E.M. Kleinberg).

2. Weak compactness and square bracket partition relations, Journal of Symbolic Logic 37 (1972), 673-676 (with E.M. Kleinberg).

3. Square bracket partition relations in L, Fundamenta Mathematica 34 (1974), 101-106.

4. Minimal alpha-degrees, Annals of Mathematical Logic 4 (1972), 383-414.

5. Cohesive sets: countable and uncountable, Proceedings of the American Mathematical Society 44 (1974), 442-445.

6. Sigman sets which are Deltan-incomparable (uniformly), Journal of Symbolic Logic 39 (1974), 295-304.

7. Splitting an alpha-recursively enumerable set, Transactions of the American Mathematical Society 204 (1975), 65-77.

8. The recursively enumerable alpha-degrees are dense, Annals of Mathematical Logic 9 (1976), 123-155.

9. The irregular and non-hyperregular alpha-r.e. degrees, Israel Journal of Mathematics 22 (1975), 28-41.

10. On the jump of an alpha-recursively enumerable set, Transactions of the American Mathematical Society 217 (1976), 351-363.

11. Types of simple alpha-recursively enumerable sets, Journal of Symbolic Logic 41 (1976), 681-693 (with A. Leggett).

12. alpha-Recursion theory, in Handbook of Mathematical Logic, J.Barwise ed., North-Holland, 1977, 653-680.

13. Determining automorphisms of the recursively enumerable sets, Proceedings of the American Mathematical Society 65 (1977), 318-325.

14. Controlling the dependence degree of a recursively enumerable vector space, Journal of Symbolic Logic 43 (1978), 13-22.

15. Some more minimal pairs of alpha-recursively enumerable degrees, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik 24 (1978), 409-418.

16. Nowhere simple sets and the lattice of recursively enumerable sets, Journal of Symbolic Logic 43 (1978), 322-330.

17. On the AE-sentences of alpha-recursion theory, in Generalized Recursion Theory II, Fenstad, Gandy and Sacks, eds., North-Holland 1978, 331-354.

18. r-Maximal major subsets, Israel Journal of Mathematics 31 (1978), 1-18 (with M. Lerman and R.I. Soare).

19. Second order logic and first order theories of reducibility orderings in The Kleene Symposium, J. Barwise, H.J. Keisler and K. Kunen, eds., North- Holland, 1980, 181-200 (with A. Nerode).

20. Reducibility orderings: theories, definability and automorphisms, Annals of Mathematical Logic 18 (1980), 61-89 (with A. Nerode).

21. The homogeneity conjecture, Proceedings of the National Academy of Sciences 76 (1979), 4218-4219.

22. On homogeneity and definability in the first order theory of the Turing degrees, Journal of Symbolic Logic 47 (1982), 8-16.

23. L*(K) and other lattices of recursively enumerable sets, Proceedings of the American Mathematical Society 80 (1980), 143-146.

24. Some constructions in alpha-recursion theory, in Recursion Theory: Its Generalizations and Applications, F.R. Drake and S.J. Wainer, eds., London Mathematical Society Lecture Notes Series, no. 45, Cambridge University Press, Cambridge, England, 1980, 158-170.

25. The theory of the degrees below 0', Journal of the London Mathematical Society 24 (1981), 1-14.

26. The elementary theory of the recursively enumerable degrees is not aleph0-categorical, Advances in Mathematics 53 (1984), 301-320 (with M. Lerman and R.I. Soare).

27. Splitting properties and jump classes, Israel Journal of Mathematics 39 (1981), 210-224 (with W. Maass and M. Stob).

28. Definable degrees and automorphisms of D, Bulletin of the American Mathematical Society (NS) 4 (1981), 97-100 (with L. Harrington).

29. The degrees of unsolvability: global results, in Logic Year 1980-81: The University of Connecticut, M. Lerman, J. Schmerl and R. Soare, eds., Springer- Verlag, 1981, 283-301.

30. Finitely generated codings and the degrees r.e. in a degree d, Proceedings of the American Mathematical Society 84 (1982), 256-263.

31. The Turing and truth-table degrees are not elementarily equivalent, in Logic Colloquium 1980, D. van Dalen, D. Lascar and T.J. Smiley, eds., North-Holland, Amsterdam 1982, 231-237.

32. An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees, Transactions of the American Mathematical Society 281 (1984), 109-128 (with K. Ambos-Spies, C. Jockusch and R. Soare).

33. Pseudo-jump operators I: the r.e. case, Transactions of the American Mathematical Society 275 (1983), 599-610 (with C. Jockusch).

34. Pseudo-jump operators II: transfinite iterations hierarchies and minimal covers, Journal of Symbolic Logic 49 (1984), 1205-1236 (with C. Jockusch).

35. The arithmetical degrees are not elementarily equivalent to the Turing degrees, Archiv fur Mathematische Logik und Grundlagenforschung 24 (1984), 137-139.

36. The degrees of unsolvability: the ordering of functions by relative computability, in Proceedings of the International Congress of Mathematicians (Warsaw) 1983, PWN-Polish Scientific Publishers, Warsaw 1984, Vol. 1, 337-346.

37. Recursion Theory, Proceedings of the Symposia in Pure Mathematics, Vol. 42 (Proceedings of the AMS-ASL Summer Institute on Recursion Theory, Cornell, 1982), American Mathematical Society, Providence, R.I., 1985 (edited with A. Nerode).

38. The structure of the degrees of unsolvability, in Recursion Theory, Proceedings of the Symposia in Pure Mathematics 42, A. Nerode and R.A. Shore, eds., American Mathematical Society,Providence, R.I., 1985, 33-51.

39. REA operators, r.e. degrees and minimal covers, in Recursion Theory, Proceedings of the Symposia in Pure Mathematics 42, A. Nerode and R.A. Shore, eds., American Mathematical Society,Providence, R.I., 1985, 3-11 (with C. Jockusch).

40. Embeddings and extensions of embeddings in the r.e. tt and wtt degrees, in Proceedings of the Oberwolfach "Recursion Theoretic Week 1984", H.D. Ebbinghaus, G.H. Müller and G.E. Sacks, eds., Springer-Verlag, Berlin, 1985, 121-140 (with P. Fejer).

41. Recursive limits on the Hahn-Banach theorem, Contemporary Mathematics 39 (1985), 85-91 (with A. Nerode and G. Metakides).

42. Initial segments of the Turing degrees of size aleph1, Israel Journal of Mathematics 55 (1986), 1-51 (with U. Abraham).

43. The degrees of constructibility below a Cohen real, Journal of the London Mathematical Society (3) 53 (1986), 193-208 (with U. Abraham).

44. Infima of recursively enumerable truth table degrees, Notre Dame Journal of Formal Logic 29 (1988), 420-437 (with P. Fejer).

45. A non-inversion theorem for the jump operator, Annals of Pure and Applied Logic 40 (1988), 277-303.

46. Decidability and invariant classes for degree structures, Transactions of the American Mathematical Society 310 (1988), 669-692 (with M. Lerman).

47. Defining jump classes in the degrees below 0', Proceedings of the American Mathematical Society 104 (1988), 287-292.

48. Initial segments of the degrees of constructibility, Israel Journal of Mathematics, Israel Journal of Mathematics 63 (1988), 149-177 (with M. Groszek).

49. The p-T-degrees of the recursive sets: lattice embeddings, extension of embeddings and the two quantifier theory (extended abstract), in Proceedings: Structures in Complexity Theory, 4th Annual Conference, Comp. Soc. IEEE, 1989 (with T. Slaman).

50. A direct construction of a minimal recursively enumerable truth-table degree, in Recursion Theory Week, Proceedings Oberwolfach 1989, K. Ambos-Spies, G. H. Muller and G. E. Sacks, eds. Springer-Verlag, LNMS 1432 (1990), 187-204 (with P. Fejer).

51. Undecidability and initial segments of the (r.e.) tt-degrees, Journal of Symbolic Logic 55 (1990), 987-1006 (with C. Haught).

52. Working below a low2 recursively enumerable degrees, Archive for Mathematical Logic 29 (1990), 201-211 (with T. Slaman).

53. Undecidability and initial segments of the wtt-degrees below 0', in Recursion Theory Week, Proceedings Oberwolfach 1989, K. Ambos-Spies, G. H. Muller and G. E. Sacks, eds. Springer-Verlag, LNMS 1432 (1990), 223-244 (with C. Haught).

54. Global properties of local structures of degrees, Bul. U. Mat. Ital. 7 (1991), 97-120 (with G. Odifreddi).

55. On the strength of König's duality theorem for infinite bipartite graphs, Journal of Combinatorial Theory (B) 54 (1992), 257-290 (with R. Aharoni and M. Magidor).

56. The p-T-degrees of the recursive sets: lattice embeddings, extension of embeddings and the two quantifier theory, Theoretical Computer Science 97 (1992), 263-284 (with T. Slaman).

57. Working below a high recursively enumerable degree, Journal of Symbolic Logic 58 (1993), 824-859 (with T. Slaman).
58. Degrees of constructibility, in Set theory of the Continuum, H. Judah, W. Just and H. Woodin eds., MSRI Publications 26, Springer-Verlag, Berlin, 1992, 123-135.

59. The n-r.e.a. enumeration degrees are dense, Archive for Mathematical Logic 31(1992), 227-285 (with A. Lachlan).

60. The theory of the recursively enumerable weak truth table degrees is undecidable, Journal of Symbolic Logic 57 (1992), 864-874 (with Ambos-Spies and Nies).

61. Countable thin Pi10 classes, Annals of Pure and Applied Logic 59 (1993) 79-139 (with D. Cenzer, R. Downey and C. Jockusch).
Also available in postscript or DVI format.

62. Undecidability and 1-types in the r.e. degrees, Annals of Pure and Applied Logic 63 (1993), 3-37 (with K. Ambos-Spies).

63. On the strength of Fraïssé's conjecture, in Logical Methods, J. N. C. Crossley, J. Remmel, R. A. Shore and M. Sweedler, eds., Birkhäuser, Boston, 1993, 782-813.

64. Interpreting true arithmetic in the theory of the r.e. truth table degrees, Annals of Pure and Applied Logic, 75 (1995), 269-311 (with A. Nies).

65. Degree theoretic definitions of the low2 r. e. sets, Journal of Symbolic Logic 60 (1995), 727-756 (with R. Downey).
Also available in postscript or DVI format.

66. Jumps of minimal degrees below 0', Journal of the London Mathematical Society 60 (1995), 727-756 (with R. Downey and S. Lempp).
Also available in postscript or DVI format.

67. Highness and bounding minimal pairs, Mathematical Logic Quarterly 39 (1993), 475-491 (with R. Downey and S. Lempp)
Also available in postscript or DVI format.

68. The theories of the T, tt and wtt r.e. degrees: undecidability and beyond, in Proc. IX Latin American Symposium on Mathematical Logic (part 1), Notas de Lo'gica Mathema'tica 38, Universidad Nacional del Sur, Bahia Blanca, Argentina, 1993, 61-70.
Also available in postscript or DVI format.

69. Logical Methods: In Honor of Anil Nerode's Sixtieth Birthday, Birkäuser, Boston, Mass., 1993 (edited with J. N. C. Crossley, J. Remmel and M. Sweedler).

70. Logic for Applications, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1993 (with A. Nerode); 2nd ed., Graduate Texts in Computer Science, Springer-Verlag, New York, 1997 (with A. Nerode).

71. Conjectures and questions from Gerald Sacks's Degrees of Unsolvability, Archive for Mathematical Logic 36 (1997), 233-253.
Also available in postscript or DVI format.

72. Lattice embeddings below a nonlow2 recursively enumerable degree, Israel Journal of Mathematics 94 (1996), 221-246 (with R. Downey).
Also available in postscript or DVI format.

73. Interpolating d-r.e. and REA degrees between r.e. degrees, Annals of Pure and Applied Logic 78 (1996), 29-56 (with M. Arslanov and S. Lempp).
Also available in postscript or DVI format.

74. On isolating r.e. and isolated d-r.e. degrees, in Computability, Enumerability, Unsolvability: Directions in Recursion Theory, S. B. Cooper, T. A. Slaman and S. S. Wainer eds., LMSLN 224, Cambridge University Press, Cambridge, England, 1996, 61-80 (with M. Arslanov and S. Lempp).
Also available in postscript or DVI format.

75. The recursively enumerable degrees, in Handbook of Recursion Theory, E. Griffor ed., North-Holland, Amsterdam, 1999, 169-197.
Also available in postscript or DVI format.

76. Intervals without critical triples, in Logic Colloquium '95, J. A. Makowsky and E. V. Ravve, eds., Lecture Notes in Logic 11, Springer-Verlag, Heidelberg, 1998, 17-43 (with P. Cholak and R. Downey).
Also available in postscript or DVI format.

77. Definability in the recursively enumerable degrees, Bulletin of Symbolic Logic 2 (1996), 392-404, (with A. Nies and T. Slaman).
Also available in postscript or DVI format.

78. Computably categorical structures and extensions by constants, Journal of Symbolic Logic, 64 (1999), 13-37 (with P. Cholak, S. Goncharov and B. Khoussainov).
Also available in postscript or DVI format.

79. Computable models of theories with few models, Notre Dame Journal of Formal Logic 38 (1997), 165-178 (with B. Khoussainov and A. Nies).
Also available in postscript or DVI format.

80. Splitting theorems and the jump operator, Annals of Pure and Applied Logic 94 (1997), 45-52 (with R. Downey).
Also available in postscript or DVI format.

81. There is no degree invariant half jump, Proc. Am Math. Soc. 125 (1997), 3033-3037 (with R. Downey).
Also available in postscript or DVI format.

82. Categoricity and Scott families, extended abstract, Combinatorics, Complexity and Logic, Proceedings of DMTCS '96, D. Bridges et al. eds., Springer-Verlag, Singapore, 1997, 299-308 (with B. Khoussainov).
Also available in postscript or DVI format.

83. Interpretability and definability in the recursively enumerable degrees, Proc. Lon. Math. Soc. (3) 77 (1998), 241-291 (with A. Nies and T. Slaman).
Also available in postscript or DVI format.

84. Computable Isomorphisms, Degree Spectra of Relations, and Scott Families, Annals of Pure and Applied Logic 93 (1998), 153-193 (with B. Khoussainov).
Also available in postscript or DVI format.
Revised version in postscript or DVI format.

85. Undecidability and 1-types in intervals of the c.e. degrees, Annals of Pure and Applied Logic 106 (2000), 1-48 (with K. Ambos-Spies and D. Hirschfeldt).
Also available in postscript or DVI format.

86. Jumps of Sigma2-high e-degrees and properly Sigma2 e-degrees, in Recursion Theory and Complexity, M. Arslanov and S. Lempp eds., de Gruyter Series in Logic and its Applications 2, de Gruyter, Berlin, 1999, 157-172 (with A. Sorbi).
Also available in postscript or DVI format.

87. Effective model theory: the number of models and their compelxity, in Models and Computability, Invited papers from Logic Colloquium '97, S. B. Cooper and J. K. Truss eds., LMSLNS 259, Cambridge University Press, Cambridge, England, 1999, 193-240 (with B. Khoussainov).
Also available in postscript or DVI format.

88. Every incomplete computably enumerable truth-table degree is branching, Archive for Mathematical Logic 40 (2001), 113-123 (with P. Fejer).
Also available in postscript or DVI format.

89. Reasoning about common knowledge with infinitely many agents, Information and Computation 191 (2004), 1-40 (with J. Halpern).
Also available in postscript as is the Extended abstract from the Proceedeings of LICS Meeting, Trento, Italy, July 1999 or DVI as is the Extended abstract from the Proceedeings of LICS Meeting, Trento, Italy, July 1999.

90. Solutions of the Goncharov-Millar and degree spectra problems in the theory of computable models, Dokl. Akad. Nauk SSSR 371 (2000) 30--31 (Russian), English version: Doklady Mathematics 61 (2000), 178-179 (with B. Khoussainov).
Also available in postscript or DVI format.

91. Decomposition and infima in the computably enumerable degrees, Journal of Symbolic Logic, 68 (2003), 551-579 (with R. Downey and G. Laforte).
Also available in postscript or DVI format.

92. Degree spectra and computable dimension in algebraic structures, Annals of Pure and Applied Logic 115 (2002), 71-113 (with D. Hirschfeldt, B. Khoussainov and A. Slinko).
Also available in postscript or DVI format.

93. Natural definability in degree structures, in Computability Theory and Its Applications: Current Trends and Open Problems, P. Cholak, S. Lempp, M. Lerman and R. A. Shore eds., Contemporary Mathematics, AMS, Providence RI, 2000, 255-272.
Also available in postscript or DVI format.

94. A splitting theorem for n-REA degrees, Proc. American Mathematicl Society 129 (2001), 3721-3728 (with T. Slaman).
Also available in postscript or DVI format.
A follow up paper is A discrete splitting theorem for the 2-REA degrees (with T. Slaman).
Also available in postscript or DVI format.

95. Defining the Turing jump, Math. Research Letters 6 (1999), 711-722 (with T. Slaman).
Also available in postscript or DVI format.

96. Computability Theory and Its Applications: Current Trends and Open Problems, Contemporary Mathematics, AMS, Providence RI, 2000 (edited with P. Cholak, S. Lempp and M. Lerman).

97. Computable Structures: Presentations Matter, in In the scope of logic, methodology and the philosophy of science}, Int. Congress of LMPS, Cracow, August 1999, P. Gardenfors, J. Wolenski and K. Kijania-Placek eds., Synthese Library 315, Kluwer Academic Publishers, Dordrecht, 2002, vol. 1, 81-95.
Also available in postscript or DVI format.

98. A nonlow2 r.e. degree with the extension of embeddings properties of a low2 degree, Mathematical Logic Quarterly 6 (2002), no. 1 (with Y. Yang).
Also available in postscript or DVI

99. Intepreting arithmetic in the r.e. degrees under ISigma-4-induction, in Reverse Mathematics 2001, S. Simpson, ed., Lecture Notes in Logic 21, Association for Symbolic Logic, 2005 (with C. T. Chong and Y. Yang).

100. The prospects for mathematical logic in the twenty-first century, Bulletin of Symbolic Logic 7 (2001), 169-196 (with S. Buss, A. Kechris and A. Pillay).
Also available in postscript or DVI format.

101. The theory of the metarecursively enumerable degrees, Journal of Mathematical Logic, 6 (2006), 49-68 (with T. Slaman and N. Greenberg).
Also available in postscript or DVI format.

102. A computably categorical structure whose expansion by a constant has infinite computable dimension, Journal of Symbolic Logic 68 (2003), 1199-1241 (with D. Hirschfeldt and B. Khoussainov).
Also available in postscript or DVI format.

103. Undecidability of the AE-theory of $\mathcal{R}(\leq ,\vee ,\wedge ), Transactions of the American Mathematical Society, 356 (2004), 3025-3067 (with A. Nies and R. Miller). Also available in postscript or DVI 104. Minimal degrees which are Sigma_2 but not Delta_2, Proc. American Mathematicl Society 132 (2004), 563-565. Also available in postscript or DVI format. 105. Pi^1_1 relations and paths through O, Journal of Symbolic Logic 69 (2004), 585-611 (with S.S. Goncharov, V. Harizanov and J. Knight). Also available in postscript or DVI format. 106. The AE theory of$\mathcal{D}(\leq ,\vee ,^{\prime })$is undecidable, in Logic Colloquium '03, Viggo Stoltenberg-Hansen and Jouko Vaananen eds., Lecture Notes in Logic 24, ASL, ASL, 2006, 326-344 (with T.A. Slaman). Also available in postscript or DVI format. 107. Generalized high degrees have the complementatation property, Journal of Symbolic Logic 69 (2004), 1200-1220 (with N. Greenberg and A. Montalban). Also available in postscript or DVI format. 108. Invariants, Boolean algebras and ACA_0^+ , Transactions of the American Mathematical Society, 358 (2006), 989-1014. Also available in postscript or DVI format. 109. A computably stable structure with no Scott family of finitary formulas, Archive for Mathematical Logic, 45 (2006), 519-538 (with P. Cholak and R. Solomon). Also available in postscript or DVI format. 110. Boolean Algebras, Tarski Invariants, and Index Sets , Notre Dame Journal of Formal Logic, 47 (2006), 1-23 (with B. Csima and A. Montalban). Also available in postscript or DVI format. 111. The low-n and low-m r.e. degrees are not elementarily equivalent, Science in China A-Mathematics, 47 (2004), 950-956. Also available in postscript or DVI format. 112. Rigidity and biinterpretability in the hyperdegrees, in Recursion Theory Worksphop:Proceedings of the IMS Program, Computational Prospects of Infinity, C. T. Chong , F. Qi and Y. Yang eds., World Scientific Publishing Co., Singapore, 2008, 299-312. Also available in postscript or DVI format. 113. Combinatorial principles weaker than Ramsey's theorem for pairs, Journal of Symbolic Logic 72 (2007), 171-206 (with D. Hirschfeldt). Also available in postscript or DVI format. 114. Degree Structures: Local and Global Investigations, Bulletin of Symbolic Logic, 12 (2006), 369-389. Also available in postscript or DVI format. 115. The settling-time reducibility ordering, Journal of Symbolic Logic, 72 (2007), 1055-71 (with B. Csima). Also available in postscript or DVI format. 116. Direct and local defintions of the Turing jump, Journal of Mathematical Logic 7 (2007), 229-262. Also available in postscript or DVI format. 117. Local definability in degree structures: the Turing jump, hyperdegrees and beyond, Bulletin of Symbolic Logic, 13 (2007), 226-239. Also available in postscript or DVI format. 118. Lattice initial segments of the hyperdegrees, Journal of Symbolic Logic 75 (2010), 103- 130 (with B. Kjos-Hanssen). Also available in postscript or DVI format. 119. The atomic model theorem, Transactions of the American Mathematical Society, 361 (2009), 5805-5837 (with D. Hirschfeldt and T. Slaman). Also available in postscript or DVI format. 120. Domination, forcing, array nonrecursiveness and relative recursive enumerability, Journal of Symbolic Logic, 77 (2012), 226-239 (with M. Cai). Also available in postscript DVI format. 121. Reverse mathematics, countable and uncountable: a computational approach, Effective Mathematics of the Uncountable, D. Hirschfeldt, N. Greenbergm J. D. Hamkins and R. Miller, eds., Lecture Notes in Logic, ASL and Cambridge University Press, New York, 2013, 150-163. Also available in postscript or DVI format. 122. Topological aspects of the Medvedev lattice, Archive for Mathematical Logic 50 (2011), 319-340 (with A. E. M. Lewis and A. Sorbi). Also available in postscript or DVI format. 123. The limits of determinacy in second order arithmetic Proceedings of the London Mathematical Society, 104 (3) (2012), 223-252 (with A. Montalban). Also available in postscript or DVI format. 124. Reverse Mathematics: the Playground of Logic, Bulletin of Symbolic Logic 16 (2010), 378-402. Also available in postscript or DVI format. 125. The n-r.e. degrees: undecidability and$\Sigma _{1}\$ substructures, Journal of Mathematical Logic, 12 (2012), 1-30 (with M. Cai and T. Slaman).
Also available in postscript or DVI format.

126. The maximal linear extension theorem in second order arithmetic, Archive for Mathematical Logic 50 (2011), 543-564 (with A. Marcone).
Also available in postscript or DVI format.

127. Degrees of categoricity and the hyperarithmetic hierarchy, Notre Dame Journal of Formal Logic, 54 (2012), 215-232 (with B. F. Csima and J. Y. N. Franklin).
Also available in postscript

128. Computably enumerable partial orders, Computability, 1 (2012), 99-107 (with P. A. Cholak, D. D. Dzhafarov and N. Schweber).
Also available in postscript or DVI format.

129. Biinterpretability up to double jump in the degrees below 0', Proceedings of the American Mathematical Society, 142 (2014), 351-360.
Also available in postscript or DVI format.

130. The Turing Degrees: An Introduction, in Forcing, Iterated Ultrapowers, and Turing Degrees, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore 29, eds. Chong Chi Tat, Feng Qi, Yang Yue, Theodore Slaman and Hugh Woodin, World Scientific Publishing, 2015, 39-121.
Also available in postscript or DVI format.

131. Computing maximal chains, Archive for Mathematical Logic 51 (2012), 651-660 (with A. Marcone and A. Montalban).
Also available in postscript or DVI format.

132. The complexity of ascendant sequences in locally nilpotent groups, International Journal of Algebra and Computation 24 (2014), 189-205 (with C. Conidis).
Also available in DVI format.

133. Low level nondefinability results: domination and recursive enumeration, Journal of Symbolic Logic 78 (2013), 1005-1024 (with M. Cai).
Also available in postscript or DVI format.

134. The Turing degrees below generics and randoms, Journal of Symbolic Logic, 79 (2014), 171-178.
Also available in postscript or DVI format.

135. The Limits of Determinacy in Second Order Arithmetic: Consistency and Complexity Strength, Israel Journal of Mathematics, 204 (2014), 477-508 (with A. Montalban).
Also available in postscript or DVI format.

136. The Strength of Turing Determinacy within Second Order Arithmetic, Fundamenta Mathematica, 232 (2016), 249-268 (with A. Montalban).
Also available in postscript or DVI format.

137. Induction, Bounding, Weak Combinatorial Principles, and The Homogeneous Model Theorem, Memoirs of the American Mathemactial Society 249, American Mathematical Society, Providence, R.I. 2107, vi + 101 (with D. Hirschfeldt and K. Lange). Also available in postscript or DVI format.

138. On the jumps of the degrees below an r.e. degree, Notre Dame Journal of Formal Logic to appear (with D. Belanger).
Also available in postscript or DVI format.

139. The strength of the Graetzer-Schmidt theorem, Archive for Mathematical Logic, 55 (2016), 687-704 (with K. Brodhead, M. Khan, B. Kjos-Hansen, W A. Lampe and K. L. V. Nguyen).
Also available in postscript or DVI format.

140. Mass problems and density, Journal of Mathematical Logic 16 (2016), 1650006 (with S. Binns and S. Simpson).
Also available in postscript or DVI format.

141. Sigma ^1_1 in every real in a Sigma ^1_1 class of reals is Sigma ^1_1, in Computability and Complexity, Essays Dedicated to Rodney Downey on the Occasion of His 60th Birthday, A. Day, M. Fellows, N. Greenberg, B. Khoussainov, A. Melnikov and F. Rosamond eds., Springer-Verlag, 2017, 455-468 (with L. Harrington and T. A. Slaman).
Also available in postscript or DVI format.

142. Conservativity of ultrafilters over subsystems of second order arithmetic, Journal of Symbolic Logic, 83 (2018), 740-765 (with A. Montalban).
Also available in postscript or DVI format.

143. Type omitting theorems for fragments of second order logics: classical and modal, to appear.

144. Halin's infinite ray theorems: complexity and reverse mathmematics, to appear (with J. Barnes and J. L. Goh).

145. Almost theorems of hyperarithmetic analysis, Journal of Symbolic Logic, to appear.

146. Theorems of hyperarithmetic analysis and almost theorems of hyperarithmetic analysis, Bulletin of Symbolic Logic to appear (with J. Barnes and J. L. Goh).