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Weakly semirecursive sets

  • Carl G. Jockusch (a1) and James C. Owings (a2)

Abstract

We introduce the notion of “semi-r.e.” for subsets of ω, a generalization of “semirecursive” and of “r.e.”, and the notion of “weakly semirecursive”, a generalization of “semi-r.e.”. We show that A is weakly semirecursive iff, for any n numbers x1, …,xn, knowing how many of these numbers belong to A is equivalent to knowing which of these numbers belong to A. It is shown that there exist weakly semirecursive sets that are neither semi-r.e. nor co-semi-r.e. On the other hand, we exhibit nonzero Turing degrees in which every weakly semirecursive set is semirecursive. We characterize the notion “A is weakly semirecursive and recursive in K” in terms of recursive approximations to A. We also show that if a finite Boolean combination of r.e. sets is semirecursive then it must be r.e. or co-r.e. Several open questions are raised.

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References

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[1] Beigel, R., Gasarch, W., Gill, J., and Owings, J., Terse, superterse, and verbose sets, Information and Computation (to appear).
[2] Beigel, R., Gasarch, W., and Owings, J., Nondeterministic bounded query reducibilities, Annals of Pure and Applied Logic, vol. 41 (1989), pp. 107118.
[3] Jockusch, C. G., Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420436.
[4] Owings, J., A cardinality version of Beigel's nonspeedup theorem, this Journal, vol. 54 (1989), pp. 761767.
[5] Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.

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Weakly semirecursive sets

  • Carl G. Jockusch (a1) and James C. Owings (a2)

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