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Recursively enumerable m- and tt-degrees. I: The quantity of m-degrees

Published online by Cambridge University Press:  12 March 2014

R. G. Downey
R. G. Downey*
Affiliation:
Department of Mathematics, Victoria University, Wellington, New Zealand
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Extract

In [1], Degtëv constructed a nonzero r.e. tt-degree containing a single r.e. m-degree. It is not difficult to construct an r.e. tt-degree containing infinitely many r.e. m-degrees (Fischer [6]); indeed, in [3], the author constructed an r.e. tt-degree with no greatest r.e. m-degree. Odifreddi [12, Problem 10] asked if every r.e. tt-degree contains either one or infinitely many r.e. m-degrees. The goal of this paper is to solve Odifreddi's question by showing:

Theorem. There exists a nonzero r.e. tt-degree containing exactly 3 r.e. m-degrees.

This theorem can be extended to show that there exist r.e. tt-degrees with arbitrarily large finite numbers of r.e. m-degrees.

We remark that save for the aforementioned results, very little is known about the structures that can be realized as the collection of r.e. m-degrees within an r.e. tt-degree. It seems conceivable that the methods of the present paper may be useful in, for example, embedding distributive (semi) lattices into such structures.

In part II of this paper [4], we continue our analysis of r.e. m- and tt-degrees. We define an r.e. tt-degree to be singular if it contains a single r.e. m-degree, and an r.e. T-degree a to be singular if a contains a singular r.e. tt-degree.

In [4] we study the distribution (in the r.e. T-degrees) of singular tt-degrees. We show that 0′T is singular (solving a question of Odifreddi [11]), and that the singular T-degrees are dense, but also we construct a nonsingular T-degree. The techniques used for the first results extend those of §2 of the present paper.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 2 , June 1989 , pp. 553 - 567
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

[1]Degtëv, A. N., tt- and m-degrees, Algebra i Logika, vol. 12 (1973), pp. 143–161; English translation, Algebra and Logic, vol. 12 (1973), pp. 78–89.Google Scholar
[2]Degtëv, A. N., Minimal l-degrees and truth-table reducibility, Sibirskiǐ Matematičeskiǐ Žurnal, vol. 17 (1976), pp. 1014–1022; English translation, Siberian Mathematical Journal, vol. 17 (1976), pp. 751–757.Google Scholar
[2]Degtëv, A. N., Three theorems on tt-degrees, Algebra i Logika, vol. 17 (1978), pp. 270–281; English translation, Algebra and Logic, vol. 17 (1978), pp. 187–194.Google Scholar
[3]Downey, R. G., Two theorems on truth table degrees, Proceedings of the American Mathematical Society, vol. 103 (1988), pp. 281–287.CrossRefGoogle Scholar
[4]Downey, R. G., Recursively enumerable m- and tt-degrees. II: The distribution of singular degrees, Archive for Mathematical Logic, vol. 27 (1988), pp. 135–148.CrossRefGoogle Scholar
[5]Downey, R. G. and Jockusch, C. G., T-degrees, jump classes and strong reducibilities, Transactions of the American Mathematical Society, vol. 301 (1987), pp. 103–136.CrossRefGoogle Scholar
[6]Fischer, P. C., A note on bounded-truth-table reducibility, Proceedings of the American Mathematical Society, vol. 14 (1963), pp. 875–877.CrossRefGoogle Scholar
[7]Jockusch, C. G., Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420–436.CrossRefGoogle Scholar
[8]Kallibekov, S., On degrees of recursively enumerable sets, Sibirskiǐ Matematičeskiǐ Žurnal, vol. 14 (1973), pp. 421–426; English translation, Siberian Mathematical Journal, vol. 14 (1973), pp. 200–203.Google Scholar
[9]Kobzev, G. N., Relationships between recursively enumerable tt- and w-degrees, Soobščenija Akademii Nauk Gruzinskoĭ SSR, vol. 84 (1976), pp. 585–587. (Russian)Google Scholar
[10]Odifreddi, P., Strong reducibilities, Bulletin (New Series) of the American Mathematical Society, vol. 4 (1981), pp. 37–86.CrossRefGoogle Scholar
[11]Odifreddi, P., Questions on tt-degrees, handwritten notes, 1985.Google Scholar
[12]Odifreddi, P., Classical recursion theory, North-Holland, Amsterdam (to appear).Google Scholar
[13]Soare, R. I., Tree arguments in recursion theory and the 0′″-priority method, Recursion theory (Nerode, A. and Shore, R., editors), Proceedings of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, Providence, Rhode Island, 1985, pp. 53–106.CrossRefGoogle Scholar
[14]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar