The need of formalizing a satisfactory notion of relative computability of partial functions leads to enumeration reducibility, which can be viewed as computing with nondeterministic Turing machines using positive information. This paper is dedicated to certain reducibilities that are stronger than enumeration reducibility, with emphasis given to s-reducibility,which appears often in computability theory and applications. We review some of the most notable properties of s-reducibility, together with the main differences distinguishing the s-degrees from the e-degrees, both at the global and local level.


Partial Function Positive Information Enumeration Reducibility Turing Degree Degree Structure 
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  1. 1.
    Affatato, M.L., Kent, T.F., Sorbi, A.: Undecidability of local structures of s-degrees and Q-degrees. Tbilisi Mathematical Journal 1, 15–32 (2008)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Arslanov, M.M.: On a class of hypersimple incomplete sets. Mat. Zametki 38, 872–874, 984–985 (1985) (English translation)MathSciNetGoogle Scholar
  3. 3.
    Belegradek, O.: On algebraically closed groups. Algebra i Logika 13(3), 813–816 (1974)MathSciNetGoogle Scholar
  4. 4.
    Calhoun, W.C., Slaman, T.A.: The \(\Pi^0_2\) e-degrees are not dense. J. Symbolic Logic 61, 1364–1379 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Casalegno, P.: On the T-degrees of partial functions. JSL 50, 580–588 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cooper, S.B.: Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ 2 sets are dense. J. Symbolic Logic 49, 503–513 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cooper, S.B.: Enumeration reducibility, nondeterministic computations and relative computability of partial functions. In: Ambos-Spies, K., Müller, G., Sacks, G.E. (eds.) Recursion Theory Week, Oberwolfach 1989. Lecture Notes in Mathematics, vol. 1432, pp. 57–110. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  8. 8.
    Cooper, S.B.: Computability Theory. Chapman & Hall/CRC Mathematics, Boca Raton/London (2003)Google Scholar
  9. 9.
    Cooper, S.B., Copestake, C.S.: Properly Σ 2 enumeration degrees. Z. Math. Logik Grundlag. Math. 34, 491–522 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Davis, M.: Computability and Unsolvability. Dover, New York (1982)zbMATHGoogle Scholar
  11. 11.
    Downey, R.G., Laforte, G., Nies, A.: Computably enumerable sets and quasi-reducibility. Ann. Pure Appl. Logic 95, 1–35 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fischer, P., Ambos-Spies, K.: Q-degrees of r.e. sets. J. Symbolic Logic 52(1), 317 (1985)Google Scholar
  13. 13.
    Gill III, J.T., Morris, P.H.: On subcreative sets and S-reducibility. J. Symbolic Logic 39(4), 669–677 (1974)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Gutteridge, L.: Some Results on Enumeration Reducibility. PhD thesis, Simon Fraser University (1971)Google Scholar
  15. 15.
    Harris, C.M.: Good \(\Sigma^0_2\) singleton degrees and density (to appear)Google Scholar
  16. 16.
    Herrmann, E.: The undecidability of the elementary theory of the lattice of recursively enumerable sets. In: Frege conference, 1984, Schwerin, pp. 66–72. Akademie-Verlag, Berlin (1984)Google Scholar
  17. 17.
    Jockusch Jr., C.G.: Semirecursive sets and positive reducibility. Trans. Amer. Math. Soc. 131, 420–436 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kent, T.F.: s-degrees within e-degrees. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 579–587. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Lachlan, A.H., Shore, R.A.: The n-rea enumeration degrees are dense. Arch. Math. Logic 31, 277–285 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Marsibilio, D., Sorbi, A.: Global properties of strong enumeration reducibilities (to appear)Google Scholar
  21. 21.
    McEvoy, K.: The Structure of the Enumeration Degrees. PhD thesis, School of Mathematics, University of Leeds (1984)Google Scholar
  22. 22.
    McEvoy, K.: Jumps of quasi–minimal enumeration degrees. J. Symbolic Logic 50, 839–848 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Myhill, J.: A note on degrees of partial functions. Proc. Amer. Math. Soc. 12, 519–521 (1961)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Nies, A.: A uniformity of degree structures. In: Sorbi, A. (ed.) Complexity, Logic and Recursion Theory, pp. 261–276. Marcel Dekker, New York (1997)Google Scholar
  25. 25.
    Omanadze, R.S., Sorbi, A.: Strong enumeration reducibilities. Arch. Math. Logic 45(7), 869–912 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Omanadze, R.S., Sorbi, A.: A characterization of the \(\Delta^0_2\) hyperhyperimmune sets. J. Symbolic Logic 73(4), 1407–1415 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Omanadze, R.S., Sorbi, A.: Immunity properties of s-degrees (to appear)Google Scholar
  28. 28.
    Polyakov, E.A., Rozinas, M.G.: Enumeration reducibilities. Siberian Math. J. 18(4), 594–599 (1977)CrossRefGoogle Scholar
  29. 29.
    Rogers Jr., H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)zbMATHGoogle Scholar
  30. 30.
    Rozinas, M.G.: Partial degrees of immune and hyperimmune sets. Siberian Math. J. 19, 613–616 (1978)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Sasso, L.P.: Degrees of Unsolvability of Partial Functions. PhD thesis, University of California, Berkeley (1971)Google Scholar
  32. 32.
    Sasso, L.P.: A survey of partial degrees. J. Symbolic Logic 40, 130–140 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Slaman, T., Woodin, W.: Definability in the enumeration degrees. Arch. Math. Logic 36, 225–267 (1997)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Solov’ev, V.D.: Q-reducibility and hyperhypersimple sets. Veroyatn. Metod. i Kibern. 10-11, 121–128 (1974)Google Scholar
  35. 35.
    Sorbi, A.: Sets of generators and automorphism bases for the enumeration degrees. Ann. Pure Appl. Logic 94(3), 263–272 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Turing, A.M.: Systems of logic based on ordinals. Proc. London Math. Soc. 45, 161–228 (1939)zbMATHCrossRefGoogle Scholar
  37. 37.
    Watson, P.: On restricted forms of enumeration reducibility. Ann. Pure Appl. Logic 49, 75–96 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Zacharov, S.D.: e- and s- degrees. Algebra and Logic 23, 273–281 (1984)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Andrea Sorbi
    • 1
  1. 1.University of SienaSienaItaly

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