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Turing Computability: Structural Theory

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In this work, we review results of the last years related to the development of the structural theory of n-c.e. Turing degrees for n > 1. We also discuss possible approaches to solution of the open problems.

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References

  1. K. Ambos-Spies, C. G. Jockusch Jr., R. A. Shore, and R. I. Soare, “An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees,” Trans. Am. Math. Soc., 281, 109–128 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  2. K. Ambos-Spies and M. Lerman, “Lattice embeddings into the recursively enumerable degrees,” J. Symb. Logic, 51, 257–272 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  3. K. Ambos-Spies and M. Lerman, “Lattice embeddings into the recursively enumerable degrees, II,” J. Symb. Logic, 54, 735–760 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  4. U. Andrews, R. Kuyper, S. Lempp, M. Soskova, and M. M. Yamaleev, “Nondensity of double bubbles in the D.C.E. degrees,” Lect. Notes Comput. Sci., 10010, 547–562 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. M. Arslanov, “Structural properties of degrees below 0,” Dokl. Akad. Nauk SSSR, 283, 270–273 (1985).

    MathSciNet  Google Scholar 

  6. M. M. Arslanov, “The lattice of the degrees below 0,” Izv. Vyssh. Ucheb. Zaved. Mat., 7, 27–33 (1988).

    MathSciNet  MATH  Google Scholar 

  7. M. M. Arslanov, “Definable relations in Turing degree structures,” J. Logic Comput., 23, No. 6, 1145–1154 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  8. M. M. Arslanov, “Structural theory of degrees of unsolvability: Achievements and open problems,” Algebra Logika, 54, No. 4, 529–535 (2015).

  9. M. M. Arslanov, S. B. Cooper, and A. Li, “There is no low maximal d.c.e. degree,” Math. Logic Quart., 46, 409–416 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  10. M. M. Arslanov, S. B. Cooper, and A. Li, “There is no low maximal d.c.e. degree. Corrigendum,” Math. Logic Quart., 50, 628–636 (2004).

    Article  Google Scholar 

  11. M. M. Arslanov, I. Sh. Kalimullin, and S. Lempp, “On Downey’s conjecture,” J. Symb. Logic, 75, 401–441 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  12. M. M. Arslanov, G. L. LaForte, and T. A. Slaman, “Relative recursive enumerability in the difference hierarchy,” J. Symb. Logic, 63, 411–420 (1998).

    Article  MATH  Google Scholar 

  13. M. M. Arslanov, S. Lempp, and R. A. Shore, “On isolating r.e. and isolated d-r.e. degrees,” London Math. Soc. Lect. Notes, 224, 61–80 (1996).

    MathSciNet  MATH  Google Scholar 

  14. M. M. Arslanov, S. Lempp, and R. A. Shore, “Interpolating d-r.e. and REA degrees between r.e. degrees,” Ann. Pure Appl. Logic, 78, 29–56 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  15. G. Barmpalias, M. Cai, S. Lempp, and T. A. Slaman, “On the existence of a strong minimal pair,” J. Math. Logic, 15, No. 1, 1550003 (2015).

  16. M. Cai, R. A. Shore, and T. A. Slaman, “The n-r.e. degrees: undecidability and Σ1 substructures,” J. Math. Logic, 12, 1–30 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Cholak, “Automorphisms of the lattice of recursively enumerable sets,” Mem. Am. Math. Soc., 541 (1995).

  18. P. Cholak and L. Harrington, “On the definability of the double jump in the computably enumerable sets,” J. Math. Logic, 2, 261–296 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  19. S. B. Cooper, Degrees of Unsolvability, Ph.D. Thesis, Leicester Univ., Leicester (1971).

  20. S. B. Cooper, “The jump is definable in the structure of the degrees of unsolvability,” Bull. Am. Math. Soc. New Ser., 23, No. 1, 151–158 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  21. S. B. Cooper, “The density of the low2 n-r.e. degrees,” Arch. Math. Logic, 31, 19–24 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  22. S. B. Cooper, “A splitting theorem for the n-r.e. degrees,” Proc. Am. Math. Soc., 115, 461–471 (1992).

    MathSciNet  MATH  Google Scholar 

  23. S. B. Cooper, L. Harrington, A. H. Lachlan, S. Lempp, and R. I. Soare, “The d-r.e. degrees are not dense,” Ann. Pure Appl. Logic, 55, 125–151 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  24. S. B. Cooper, L. Harrington, A. H. Lachlan, S. Lempp, and R. I. Soare, “Corrigendum to “The d.r.e. degrees are not dense,” Ann. Pure Appl. Logic, 168, 2164–2165 (2017).

  25. S. B. Cooper and A. Li, “Non-uniformity and generalised Sacks splitting,” Acta Math. Sinica. Engl. Ser., 18, 327–334 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  26. S. B. Cooper and A. Li, “Splitting and cone avoidance in the d.c.e. degrees,” Sci. China. Ser. A., 45, 1135–1146 (2002).

    MathSciNet  MATH  Google Scholar 

  27. S. B. Cooper and A. Li, “Turing definability in the Ershov hierarchy,” J. London Math. Soc. (2), 66, 513–528 (2002).

  28. S. B. Cooper and X. Yi, Isolated d.r.e. degrees, Univ. of Leeds (1995).

  29. R. G. Downey, “D.r.e. degrees and the nondiamond theorem,” Bull. London Math. Soc., 21, 43–50 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  30. R. Downey and D. Hirschfeldt, Algorithmic Randomness and Complexity, Springer-Verlag (2010).

  31. R. G. Downey, G. A. Laforte, and R. I. Shore, “Decomposition and infima in the computably enumerable degrees,” J. Symb. Logic, 68, 551–579 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  32. R. G. Downey and M. Stob, “Splitting theorems in recursion theory,” Ann. Pure Appl. Logic, 65, 1–106 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  33. R. Epstein, “The nonlow computably enumerable degrees are not definable in 𝜀,” Trans. Am. Math. Soc., 365, 1305–1345 (2013).

    Article  MATH  Google Scholar 

  34. R. L. Epstein, Degrees of Unsolvability: Structure and Theory, Lect. Notes Math., 759, Springer-Verlag, Berlin–Heidelberg–New York (1979).

  35. Yu. L. Ershov, “On a hierarchy of sets, I,” Algebra Logika, 7, 47–73 (1968).

    MathSciNet  MATH  Google Scholar 

  36. Yu. L. Ershov, “On a hierarchy of sets, II,” Algebra Logika, 7, 15–47 (1968).

    MathSciNet  MATH  Google Scholar 

  37. Yu. L. Ershov, “On a hierarchy of sets, III,” Algebra Logika, 9, 34–51 (1970).

    Article  MathSciNet  Google Scholar 

  38. Yu. L. Ershov and E. A. Palyutin, Mathematical Logic [in Russian], Nauka, Moscow (1987).

    MATH  Google Scholar 

  39. C. Fang, J. Liu, G. Wu, and M. M. Yamaleev, “Nonexistence of minimal pairs in L[d],” in: Lect. Notes Comput. Sci., 9136 (2015), pp. 177–185.

  40. C. Fang, G. Wu, and M. M. Yamaleev, “On a problem of Ishmukhametov,” Arch. Math. Logic, 52, 733–741 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  41. M. Kh. Faizrakhmanov, “Decomposability of low 2-computably enumerable degrees and Turing jumps in the Ershov hierarchy,” Izv. Vyssh. Ucheb. Zaved. Mat., 12, 58–66 (2010).

    MathSciNet  MATH  Google Scholar 

  42. L. Feiner, “Hierarchies of Boolean algebras,” J. Symb. Logic, 35, 305–373 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  43. L. Harrington, Understanding Lachlan’s monster paper, manuscript (1980).

    Google Scholar 

  44. L. Harrington and S. Shelah, “The undecidability of the recursively enumerable degrees (research announcement),” Bull. Am. Math. Soc., 6, No. 1, 79–80 (1982).

    Article  MATH  Google Scholar 

  45. L. Harrington and R. I. Soare, “The \( {\Delta}_3^0 \)-automorphism method and noninvariant classes of degrees,” J. Am. Math. Soc., 9, 617–666 (1996).

    Article  MATH  Google Scholar 

  46. Sh. T. Ishmukhametov, “Downward density of exact degrees,” Arch. Math. Logic, 38, 373–386 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  47. Sh. T. Ishmukhametov, “On relative enumerability of Turing degrees,” Arch. Math. Logic, 39, 145–154 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  48. C. G. Jockusch Jr., “Review of Lerman,” Math. Rev., 45, No. 3200 (1973).

  49. C. Jockusch and R. Shore, “Pseudojump operators, II: Transfinite iterations, hierarchies and minimal covers,” J. Symb. Logic, 49, 1205–1236 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  50. A. H. Lachlan, “Lower bounds for pairs of recursively enumerable degrees,” Proc. London Math. Soc., 16, 537–569 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  51. A. H. Lachlan, “The elementary theory of recursively enumerable sets,” Duke Math. J., 35, 123– 146 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  52. A. H. Lachlan, “Degrees of recursively enumerable sets which have no maximal supersets,” J. Symb. Logic, 33, 431–443 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  53. A. H. Lachlan, “Distrubutive initial segments of the degrees of unsolvability,” Z. Math. Logik, 14, 457–472 (1968).

    Article  MATH  Google Scholar 

  54. A. H. Lachlan, “On the lattice of recursively enumerable sets,” Trans. Am. Math. Soc., 130, 1–37 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  55. A. H. Lachlan, “A recursively enumerable degree which will not split over all lesser ones,” Ann. Math. Logic, 9, 307–365 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  56. S. Lempp, M. Lerman, and D. Solomon, “Embedding finite lattices into the computably enumerable degrees—a status survey,” in: Proc. Ann. Eur. Summer Meeting of the Association for Symbolic Logic “Logic Colloquium’02”, Lect. Notes Logic, 27 (2006), pp. 206–229.

  57. S. Lempp and A. Nies, “Differences of computably enumerable sets,” Math. Logic Quart., 46, 555–561 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  58. S. Lempp, A. Nies, and T. A. Slaman, “The Π3-theory of the computably enumerable Turing degrees is undecidable,” Trans. Am. Math. Soc., 350, No. 7, 2719–2736 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  59. M. Lerman, Degrees of Unsolvability, Springer-Verlag, Berlin–Heidelberg–New York (1983).

    Book  MATH  Google Scholar 

  60. M. Lerman, “Embeddings into the recursively enumerable degrees,” in: Computability, Enumerability, Decidability: Directions in Recursion Theory, London Math. Soc. Lect. Notes, 224 (S. B. Cooper, T. A. Slaman, and S. S. Wainer, eds.), Cambridge Univ. Press, Cambridge (1996), pp. 185–204.

  61. M. Lerman and R. A. Shore, “Decidability and invariant classes for degree structures,” Trans. Am. Math. Soc., 301, No. 2, 669–692 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  62. M. Lerman, R. A. Shore, and R. I. Soare, “The elementary theory of the recursively enumerable degrees is not ℵ0-categorical,” Adv. Math., 53, 301–320 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  63. A. Li in: Lect. Notes Comput. Sci., 3526 (2005), pp. 287–296.

  64. A. Li and X. Yi, “Cupping the recursively enumerable degrees by d.r.e. degrees,” Proc. London Math. Soc. (3), 78, 1–21 (1999).

  65. J. Liu, G. Wu, and M. M. Yamaleev, “Downward density of exact degrees,” Lobachevskii J. Math. (4), 36, 389–398 (2015).

  66. D. A. Martin, “Classes of recursively enumerable sets and degrees of unsolvability,” Z. Math. Logik, 12, 295–310 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  67. D. P. Miller, “High recursively enumerable degrees and the anti-cupping property,” Lect. Notes Math., 859, 230– 245 (1981).

    Article  MATH  Google Scholar 

  68. A. Nies, T. A. Slaman, and R. A. Shore, “Interpretability and definability in the recursively enumerable degrees,” Proc. London Math. Soc. (3), 77, 241–291 (1998).

  69. H. Putnam, “Trial and error predicates and the solution to a problem of Mostowski,” J. Symb. Logic, 30, 49–57 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  70. R. W. Robinson, “Interpolation and embedding in the recursively enumerable degrees,” Ann. Math. (2), 93, 285–314 (1971).

  71. H. Rogers Jr., Theory of Recursive Functions and Effective Computability, McGraw Hill, New York (1967).

    MATH  Google Scholar 

  72. G. E. Sacks, “A minimal degree less than 0,” Bull. Am. Math. Soc., 67, 416–419 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  73. G. E. Sacks, “On the degrees less than 0,” Ann. Math. (2), 77, 211–231 (1963).

  74. G. E. Sacks, “The recursively enumerable degrees are dense,” Ann. Math. (2), 80, 300–312 (1964).

  75. J. R. Shoenfield, “Degrees of classes of RE sets,” J. Symb. Logic, 41, 695–696 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  76. R. A. Shore, “On the ∀∃-sentences of α-recursive theory,” in: Proc. Second Symp. “Generalized Recursion Theory,” Oslo, 1977, Vol. II, Oslo (1978).

  77. R. A. Shore, “Finitely generated codings and the degrees r.e. in a degree d,” Proc. Am. Math. Soc., 84, 256–263 (1982).

    MathSciNet  MATH  Google Scholar 

  78. R. A. Shore and T. A. Slaman, “Working below a low2 recursively enumerable degrees,” Arch. Math. Logic, 29, 201–211 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  79. R. A. Shore and T. A. Slaman, “A splitting theorem for n-REA degrees,” Proc. Am. Math. Soc., 129, 3721–3728 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  80. T. A. Slaman and R. I. Soare, “Algebraic aspects of the computably enumerable degrees,” Proc. Natl. Acad. Sci., 2, 617–621 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  81. T. A. Slaman and R. I. Soare, “Extension of embeddings in the computably enumerable degrees,” Ann. Math. (2), 154, 1–43 (2001).

  82. T. A. Slaman and W. Woodin, “Definability in Turing degrees,” Ill. J. Math. (2), 30, 320–334 (1986).

  83. R. Soare, Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets, Springer-Verlag, Berlin (1987).

  84. L. Welch, A hierarchy of families of recursively enumerable degrees and a theorem on bounding minimal pairs, Ph.D. Thesis, Univ. of Illinois, Urbana (1980).

  85. G. Wu, “Isolation and lattice embedding,” J. Symb. Logic, 67, 1055–1064 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  86. G. Wu and M. M. Yamaleev, “Isolation: motivations and applications,” Uch. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki, 154, 204–217 (2012).

    MathSciNet  Google Scholar 

  87. M. M. Yamaleev, “Splitting of 2-computably enumerable degrees with avoiding cones,” Izv. Vyssh. Ucheb. Zaved. Mat., 6, 76–80 (2009).

    MATH  Google Scholar 

  88. M. M. Yamaleev, Structural properties of Turing degrees of sets from the Ershov hierarchy, Ph.D. thesis, Kazan State Univ., Kazan (2009).

  89. Y. Yang and L. Yu, “On Σ1-structural differences among Ershov hierarchies,” J. Symb. Logic, 71, 1223–1236 (2006).

    Article  MATH  Google Scholar 

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Correspondence to M. M. Arslanov.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 157, Proceedings of the Seminar on Algebra and Mathematical Logic of the Kazan (Volga Region) Federal University, 2018.

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Arslanov, M.M., Yamaleev, M.M. Turing Computability: Structural Theory. J Math Sci 256, 1–33 (2021). https://doi.org/10.1007/s10958-021-05418-y

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