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On degrees of unsolvability and complexity properties

Published online by Cambridge University Press:  12 March 2014

Ivan Marques
Ivan Marques*
Affiliation:
University Of California, Berkeley, California 94720
*
Universidade Federal Do Rio De Janeiro Rio De Janeiro, Brazil
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Extract

In this paper we present two theorems concerning relationships between degrees of unsolvability of recursively enumerable sets and their complexity properties.

The first theorem asserts that there are nonspeedable recursively enumerable sets in every recursively enumerable Turing degree. This theorem disproves the conjecture that all Turing complete sets are speedable, which arose from the fact that a rather inclusive subclass of the Turing complete sets, namely, the subcreative sets, consists solely of effectively speedable sets [2]. Furthermore, the natural construction to produce a nonspeedable set seems to lower the degree of the resulting set.

The second theorem says that every speedable set has jump strictly above the jump of the recursive sets. This theorem is an expected one in view of the fact that all sets which are known to be speedable jump to the double jump of the recursive sets [4].

After this paper was written, R. Soare [8] found a very useful characterization of the speedable sets which greatly facilitated the proofs of the results presented here. In addition his characterization implies that an r.e. degree a contains a speed-able set iff a′ > 0′.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 40 , Issue 4 , December 1975 , pp. 529 - 540
Copyright
Copyright © Association for Symbolic Logic 1975

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Footnotes

1

This research was partially sponsored by National Science Foundation grants GJ-708 and GP-35604X and grant no. 3011/69 of Conselho Nacional de Pesquisas, Brazil. The author is indebted to his thesis advisor, Professor Manuel Blum, for suggesting this research and, together with Dick Epstein, John Gill and Ion Filotti, for important discussions on the subject and to the referee for numerous suggestions and comments.

References

REFERENCES

[1] Blum, M., A machine-independent theory of the complexity of recursive function, Journal of the Association for Computing Machinery, vol. 14 (1967), pp. 322–336.CrossRefGoogle Scholar
[2] Blum, M. and Marques, I., On complexity properties of recursively enumerable sets, this Journal, vol. 38 (1973), pp. 579–593.Google Scholar
[3] Friedberg, R., Three theorems on recursive enumeration, this Journal, vol. 23, (1958), pp. 309–316.Google Scholar
[4] Marques, I., On speedability of recursively enumerable sets (unpublished).Google Scholar
[5] Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[6] Sacks, G., On the degrees less than 0′, Annals of Mathematics, vol. 77 (1963), pp. 211–231.CrossRefGoogle Scholar
[7] Shoenfield, J., On degrees of unsolvability, Annals of Mathematics, vol. 69 (1959), pp. 644–653.CrossRefGoogle Scholar
[8] Soare, R., Degrees and structure of speedable sets, Notices of the American Mathematical Society, vol. 21 (1974), p. A–380. Abstract 74T-E40.Google Scholar