Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-15T05:26:37.883Z Has data issue: false hasContentIssue false
  • English
  • Français

On computational complexity in weakly admissible structures

Published online by Cambridge University Press:  12 March 2014

Viggo Stoltenberg-Hansen
Viggo Stoltenberg-Hansen*
Affiliation:
University of Oslo, Oslo, Norway
Get accessRights & Permissions [Opens in a new window]

Extract

The theory of abstract computational complexity in ordinary recursion theory (ORT) was initiated by Rabin [7] and Blum [1]. Jacobs [5] generalized the notions to recursion theory on an admissible ordinal α(α-recursion theory) and proved in this setting certain main theorems such as Blum's Theorem and the Compression Theorem.

In ORT the notions of “finite” and “bounded” coincide. Thus when lifting a theorem from ORT to α-recursion theory, “finite” may be translated to either “bounded below α” or to the stronger and more natural “α-finite”. Jacobs conjectured that “bounded below α” was the best possible in the theorems mentioned above. In this paper we shall prove that the stronger version of these theorems are, in fact, true. Furthermore our constructions are uniform for all admissible α.

Rather than restricting ourselves to α-recursion theory we shall consider transitive rudimentarily closed structures M = 〈M, ∈, R〉 which admit what we call an acceptable prewellordering. The special case M = 〈Lα, ∈〉 and M = 〈Sβ, ∈〉 where α is admissible and β a limit ordinal constitute, respectively, α-recursion theory and β-recursion theory (see Friedman [4]).

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 2 , June 1980, pp. 353 - 358
Copyright
Copyright © Association for Symbolic Logic 1980

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.)

Article purchase

Temporarily unavailable

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.CrossRefGoogle Scholar
[2]Devlin, K. J., Aspects of constructibility, Lecture Notes in Mathematics, no. 354, Springer, Berlin and New York, 1973.Google Scholar
[3]Fenstad, J. E., Computation theories: An axiomatic approach to recursion on general structures, Logic Conference, Kiel, 1974, Lecture Notes in Mathematics, no. 499, Springer, Berlin and New York, 1975, pp. 143–168.Google Scholar
[4]Friedman, S., An introduction to β-recursion theory, Generalized recursion theory. II (Fenstad, J.E., Gandy, R.O. and Sacks, G.E., Editors), North-Holland, Amsterdam, 1978, pp. 111–126.Google Scholar
[5]Jacobs, B. E., On generalized computational complexity, this Journal, vol. 42 (1977), pp. 47–58.Google Scholar
[6]Moschovakis, Y. N., Axioms for computation theories—first draft, Logic Colloquium '69 (Gandy, R. O. and Yates, C. E. M., Editors), North-Holland, Amsterdam, 1971, pp. 199–255.CrossRefGoogle Scholar
[7]Rabin, M. O., Degrees of difficulty of computing a function and a partial ordering of recursive sets, Technical Report, 2, Hebrew University, Jerusalem, 1960.Google Scholar
[8]Simpson, S. G., Post's problem for admissible sets, Generalized recursion theory (Fenstad, J. E. and Hinman, P. G., Editors), North-Holland, Amsterdam, 1974, pp. 437–441.Google Scholar
[9]Stoltenberg-Hansen, V., Weakly inadmissible recursion theory, Generalized recursion theory. II (Fenstad, J. E., Gandy, R. O. and Sacks, G. E., Editors), North-Holland, Amsterdam, 1978, pp. 391–405.Google Scholar
[10]Stoltenberg-Hansen, V., Finite injury arguments in infinite computation theories, Annals of Mathematical Logic, vol. 16 (1979), pp. 57–80.CrossRefGoogle Scholar