Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-18T04:23:52.115Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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