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On complexity properties of recursively enumerable sets1

Published online by Cambridge University Press:  12 March 2014

M. Blum  and
I. Marques
M. Blum
Affiliation:
University of California, Berkeley, California 94720
I. Marques
Affiliation:
Núcleo de Computaçāo Eletrônica and Coppe, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
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Extract

An important goal of complexity theory, as we see it, is to characterize those partial recursive functions and recursively enumerable sets having some given complexity properties, and to do so in terms which do not involve the notion of complexity.

As a contribution to this goal, we provide characterizations of the effectively speedable, speedable and levelable [2] sets in purely recursive theoretic terms. We introduce the notion of subcreativeness and show that every program for computing a partial recursive function f can be effectively speeded up on infinitely many integers if and only if the graph of f is subcreative.

In addition, in order to cast some light on the concepts of effectively speedable, speedable and levelable sets we show that all maximal sets are levelable (and hence speedable) but not effectively speedable and we exhibit a set which is not levelable in a very strong sense but yet is effectively speedable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 4 , December 1973 , pp. 579 - 593
Copyright
Copyright © Association for Symbolic Logic 1973

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Footnotes

1

This work was supported in part by National Science Foundation grants GJ-708 and GP-35604X and in part by grant 3011/69 of Conselho Nacional de Pesquisas, Rio de Janeiro (Brazil). We are very grateful to Albert Meyer and John Gill for several helpful discussions on the material of this paper, and to the referee for numerous fine comments.

References

REFERENCES

[1] Blum, M., A machine-independent theory of the complexity of recursive functions, Journal of the Association for Computing Machinery, vol. 14 (1967), pp. 322–336.Google Scholar
[2] Blum, M., On effective procedures for speeding up algorithms, Journal of the Association for Computing Machinery, vol. 18 (1971), pp. 290–305.Google Scholar
[3] Gill, J., Personal communication.Google Scholar
[4] Landweber, L. H. and Robertson, E. L., Recursive properties of abstract complexity classes, Journal of the Association for Computing Machinery (to appear).Google Scholar
[5] Marques, I., Ph.D. Dissertation, University of California, Berkeley.Google Scholar
[6] Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar