Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-25T01:57:44.059Z Has data issue: false hasContentIssue false
  • English
  • Français

Implicit measurements of dynamic complexity properties and splittings of speedable sets

Published online by Cambridge University Press:  12 March 2014

Michael A. Jahn
Michael A. Jahn*
Affiliation:
Digipen Institute of Technology, 5001 150th Ave. N.E., Redmond, WA 98052, USA E-mail: mjahn@digipen.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that any speedable computably enumerable set may be split into a disjoint pair of speedable computably enumerable sets. This solves a longstanding question of J.B. Remmel concerning the behavior of computably enumerable sets in Blum's machine independent complexity theory. We specify dynamic requirements and implement a novel way of detecting speedability—by embedding the relevant measurements into the substage structure of the tree construction. Technical difficulties in satisfying the dynamic requirements lead us to implement “local” strategies that only look down the tree. The (obvious) problems with locality are then resolved by placing an isomorphic copy of the entire priority tree below each strategy (yielding a self-similar tree). This part of the construction could be replaced by an application of the Recursion Theorem, but shows how to achieve the same effect with a more direct construction.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 3 , September 1999 , pp. 1037 - 1064
Copyright
Copyright © Association for Symbolic Logic 1999

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] Bäuerle, F. A. and Remmel, J. B., On speedable and levelable vector spaces, Annals of Pure and Applied Logic, vol. 67 (1994), pp. 61–112.CrossRefGoogle Scholar
[2] Bell, D. E. and LaPadula, L. J., Secure computer systems: Mathematical foundations, Technical Report MTR-2547, MITRE Corp., Bedford, Massachusetts, 1973.Google Scholar
[3] 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
[4] Blum, M., On effective procedures for speeding up algorithms, Journal of the Association for Computing Machinery, vol. 18 (1971), pp. 290–305.Google Scholar
[5] Blum, M. and Marques, I., On complexity properties of recursively enumerable sets, this Journal, vol. 38 (1973), pp. 579–593.Google Scholar
[6] Dekker, J. C. E., Two notes on vector spaces with recursive operations, Notre Dame Journal of Formal Logic, vol. 12 (1971), pp. 329–334.Google Scholar
[7] Downey, R. and Shore, R., Splitting theorems and the jump operator, to appear.Google Scholar
[8] Downey, R. and Stob, M., Splitting theorems in recursion theory, Annals of Pure and Applied Logic, vol. 65 (1993), pp. 1–106.Google Scholar
[9] Gurevich, Y., Evolving algebra 1993: Lipari guide, Specification and validation methods (Börger, E., editor), Oxford University Press, Oxford, 1995, pp. 9–36.Google Scholar
[10] Gurevich, Y. and Spielmann, M., Recursive abstract state machines, Springer Journal of Universal Computer Science, vol. 3 (1997), no. 4, pp. 233–246.Google Scholar
[11] Harrington, L. and Soare, R. I., Codable sets and orbits of computably enumerable sets, this Journal, vol. 63 (1998), pp. 1–28.Google Scholar
[12] Knight, J., Effective construction of models, Logic colloquium 1984 (Paris, , Wilkie, , and Wilmers, , editors), North-Holland, Amsterdam, 1986.Google Scholar
[13] Landweber, L. H. and Robertson, E. L., Recursive properties of abstract complexity classes, Journal of the Association for Computing Machinery, vol. 19 (1972), pp. 296–308.Google Scholar
[14] Lempp, S. and Slaman, T. A., A limit on relative genericity in the recursively enumerable sets, this Journal, vol. 54 (1989), pp. 376–395.Google Scholar
[15] Miller, A. W., Descriptive set theory and forcing: Proving theorems about the Borel sets the hard way, Lecture Notes in Logic, vol. 4, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1995.Google Scholar
[16] Papadimitriou, C., Computational complexity, Addison-Wesley, 1994.Google Scholar
[17] Remmel, J. B., personal communication.Google Scholar
[18] Soare, R., Computational complexity, speedable and levelable sets, this Journal, vol. 42 (1977), pp. 545–563.Google Scholar
[19] Soare, R., Computational complexity of recursively enumerable sets, Information and Control, vol. 52 (1982), pp. 8–18.CrossRefGoogle Scholar
[20] Soare, R., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1987.Google Scholar