Abstract
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.
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References
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)
Arslanov, M.M.: On a class of hypersimple incomplete sets. Mat. Zametki 38, 872–874, 984–985 (1985) (English translation)
Belegradek, O.: On algebraically closed groups. Algebra i Logika 13(3), 813–816 (1974)
Calhoun, W.C., Slaman, T.A.: The \(\Pi^0_2\) e-degrees are not dense. J. Symbolic Logic 61, 1364–1379 (1996)
Casalegno, P.: On the T-degrees of partial functions. JSL 50, 580–588 (1985)
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)
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)
Cooper, S.B.: Computability Theory. Chapman & Hall/CRC Mathematics, Boca Raton/London (2003)
Cooper, S.B., Copestake, C.S.: Properly Σ 2 enumeration degrees. Z. Math. Logik Grundlag. Math. 34, 491–522 (1988)
Davis, M.: Computability and Unsolvability. Dover, New York (1982)
Downey, R.G., Laforte, G., Nies, A.: Computably enumerable sets and quasi-reducibility. Ann. Pure Appl. Logic 95, 1–35 (1998)
Fischer, P., Ambos-Spies, K.: Q-degrees of r.e. sets. J. Symbolic Logic 52(1), 317 (1985)
Gill III, J.T., Morris, P.H.: On subcreative sets and S-reducibility. J. Symbolic Logic 39(4), 669–677 (1974)
Gutteridge, L.: Some Results on Enumeration Reducibility. PhD thesis, Simon Fraser University (1971)
Harris, C.M.: Good \(\Sigma^0_2\) singleton degrees and density (to appear)
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)
Jockusch Jr., C.G.: Semirecursive sets and positive reducibility. Trans. Amer. Math. Soc. 131, 420–436 (1968)
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)
Lachlan, A.H., Shore, R.A.: The n-rea enumeration degrees are dense. Arch. Math. Logic 31, 277–285 (1992)
Marsibilio, D., Sorbi, A.: Global properties of strong enumeration reducibilities (to appear)
McEvoy, K.: The Structure of the Enumeration Degrees. PhD thesis, School of Mathematics, University of Leeds (1984)
McEvoy, K.: Jumps of quasi–minimal enumeration degrees. J. Symbolic Logic 50, 839–848 (1985)
Myhill, J.: A note on degrees of partial functions. Proc. Amer. Math. Soc. 12, 519–521 (1961)
Nies, A.: A uniformity of degree structures. In: Sorbi, A. (ed.) Complexity, Logic and Recursion Theory, pp. 261–276. Marcel Dekker, New York (1997)
Omanadze, R.S., Sorbi, A.: Strong enumeration reducibilities. Arch. Math. Logic 45(7), 869–912 (2006)
Omanadze, R.S., Sorbi, A.: A characterization of the \(\Delta^0_2\) hyperhyperimmune sets. J. Symbolic Logic 73(4), 1407–1415 (2008)
Omanadze, R.S., Sorbi, A.: Immunity properties of s-degrees (to appear)
Polyakov, E.A., Rozinas, M.G.: Enumeration reducibilities. Siberian Math. J. 18(4), 594–599 (1977)
Rogers Jr., H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)
Rozinas, M.G.: Partial degrees of immune and hyperimmune sets. Siberian Math. J. 19, 613–616 (1978)
Sasso, L.P.: Degrees of Unsolvability of Partial Functions. PhD thesis, University of California, Berkeley (1971)
Sasso, L.P.: A survey of partial degrees. J. Symbolic Logic 40, 130–140 (1975)
Slaman, T., Woodin, W.: Definability in the enumeration degrees. Arch. Math. Logic 36, 225–267 (1997)
Solov’ev, V.D.: Q-reducibility and hyperhypersimple sets. Veroyatn. Metod. i Kibern. 10-11, 121–128 (1974)
Sorbi, A.: Sets of generators and automorphism bases for the enumeration degrees. Ann. Pure Appl. Logic 94(3), 263–272 (1998)
Turing, A.M.: Systems of logic based on ordinals. Proc. London Math. Soc. 45, 161–228 (1939)
Watson, P.: On restricted forms of enumeration reducibility. Ann. Pure Appl. Logic 49, 75–96 (1990)
Zacharov, S.D.: e- and s- degrees. Algebra and Logic 23, 273–281 (1984)
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Sorbi, A. (2009). Strong Positive Reducibilities. In: Chen, J., Cooper, S.B. (eds) Theory and Applications of Models of Computation. TAMC 2009. Lecture Notes in Computer Science, vol 5532. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02017-9_6
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DOI: https://doi.org/10.1007/978-3-642-02017-9_6
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