## Upper semilattice of recursively enumerable sQ-degrees.(English. Russian original)Zbl 0788.03060

Algebra Logic 30, No. 4, 265-271 (1991); translation from Algebra Logika 30, No. 4, 405-413 (1991).
A set $$A$$ is called $$sQ$$-reducible to $$B$$ if there exist recursive functions $$f$$ and $$g$$ such that, for all $$x$$, $$x \in A$$ iff $$W_{f(x)} \subseteq B$$ (i.e. $$A \leq_ QB)$$ and, for all $$y$$, $$y \in W_{f(x)}$$ implies $$y \leq g(x)$$. The author studies various properties of the upper semilattice of recursively enumerable $$sQ$$-degrees and relationships to abstract complexity properties such as speedability in the sense of M. Blum and I. Marques [J. Symb. Logic 38, 579-593 (1973; Zbl 0335.02024)]. For instance, a density theorem is proven, and relationships with $$wtt$$- and $$T$$-degrees are discussed.

### MSC:

 03D25 Recursively (computably) enumerable sets and degrees 03D30 Other degrees and reducibilities in computability and recursion theory 03D15 Complexity of computation (including implicit computational complexity)

Zbl 0335.02024
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### References:

 [1] H. Rogers, Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York (1967). · Zbl 0183.01401 [2] R. Sh. Omanadze, ”On the upper semilattice of recursively enumerable Q-degrees,” Algebra Logika,23, No. 2, 175–184 (1984). · Zbl 0586.03034 [3] A. N. Degtev, Enumerable Sets and Reducibility [in Russian], Tyumen’ (1988). [4] R. A. Shore, ”Nowhere simple sets and the lattice of recursively enumerable sets,” J. Symb. Logic,43, No. 2, 322–330 (1978). · Zbl 0398.03029 [5] R. Sh. Omanadze, ”Q-reducibility and nowhere simple sets,” Soobshch. Akad. Nauk Gruz. SSR,127, No. 1, 29–32 (1987). · Zbl 0649.03030 [6] C. E. M. Yates, ”Three theorems on the degree of recursively enumerable sets,” Duke Math. J.,32, 461–468 (1965). · Zbl 0134.00805 [7] D. Miller and J. R. Remmel, ”Effectively nowhere simple sets,” J. Symb. Logic,49, 129–136 (1984). · Zbl 0598.03036 [8] M. Blum and I. Marques, ”On complexity properties of recursively enumerable sets,” J. Symb. Logic,38, No. 4, 579–593 (1973). · Zbl 0335.02024 [9] R. I. Soare, ”Computational complexity, speedable and levelable sets,” J. Symb. Logic,42, No. 4, 545–562 (1977). · Zbl 0401.68020 [10] R. I. Soare, Recursively Enumerable Sets and Degrees, Springer, Berlin (1987). · Zbl 0667.03030 [11] S. S. Marchenko, ”On truth-table degrees of maximal sets,” Mat. Zametki,20, No. 3, 373–381 (1976). [12] S. D. Denisov, ”On m-degrees of recursively enumerable sets,” Algebra Logika,9, No. 4, 422–427 (1970). · Zbl 0246.17007
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