Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-20T04:33:15.093Z Has data issue: false hasContentIssue false
  • English
  • Français

Noncomplex sequences: characterizations and examples1

Published online by Cambridge University Press:  12 March 2014

Robert P. Daley
Robert P. Daley*
Affiliation:
University of Pittsburgh, Pittsburg, Pennsylvania 15260
Get access Rights & Permissions [Opens in a new window]

Extract

We are concerned in this paper with infinite binary sequences which are noncomplex in the sense that their minimal-program complexity (i.e., the lengths of shortest programs for computing their finite initial segments), as a function of the lengths of their initial segments, grows arbitrarily (in an effective sense) slowly. Indeed, the existence of such sequences which are also nonrecursive raises some interesting questions concerning the notion of computability itself. Our first task is to give a characterization of these sequences which does not directly involve the notions of complexity theory. Although these characterizations do involve the primitives of recursive function theory, they do not involve the types of properties which one usually encounters there. While a closer connection is still hoped for, the lack of one is not entirely unexpected. For example, except for the trivial case of degree 0, there is no general correlation between program complexity and degrees of unsolvability. The reason, roughly, is this: even though the inequality deg (f) ≤ deg (g) expresses the relative information between f and g (in the sense that if one knows g then one can compute f), it is a qualitative sort of information. The minimal-program complexity is used as a measure of algorithmic information content and as such is quantitative.

The second task of this paper is to present an example of one of these sequences which to its credit possesses a fairly long list of interesting properties, not the least attractive of which is that, though nonrecursive, it is very simple to describe. In fact this sequence has occurred previously in the literature. It is the characteristic sequence of the set of busy beaver numbers first studied by T. Rado [13]. Of particular interest to the discussion in the last section of this paper concerning the notion of computability is the fact that all the initial segments of this sequence are computable by arbitrarily short programs each of which is defined on all inputs and runs very quickly.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 3 , September 1976 , pp. 626 - 638
Copyright
Copyright © Association for Symbolic Logic 1976

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.)

Footnotes

1

The research reported upon here was supported by NSF grant GJ–43223. The results were first presented at the 15th Annual Symposium on Switching and Automata Theory, New Orleans, Oct. 14·16, 1974.

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] Blum, M., On the size of machines, Information and control, vol. 11 (1967), pp. 257–265.CrossRefGoogle Scholar
[3] Chaitin, G., On the length of programs for computing finite binary sequences, Journal of the Association for Computing Machinery, vol. 13 (1966), pp. 547–569.CrossRefGoogle Scholar
[4] Chaitin, G., Information- theoretic characterizations of recursive infinite strings, Theoretical computer science (to appear).Google Scholar
[5] Daley, R., The extent and density of sequences within the minimal-program complexity hierarchies, Journal of computer and system sciences, vol. 9 (1974), pp. 151–163.CrossRefGoogle Scholar
[6] Daley, R., On the inference of optimal descriptions, Theoretical computer science (to appear).Google Scholar
[7] Daley, R., On the learning of non-recursive sequences, Proceedings of the 1th Annual Princeton Conference on Information Sciences and Systems, 1973, pp. 552–556.Google Scholar
[8] Kanovič, M., On the decision complexity of algorithms, Soviet Mathematics — Doklady, vol. 10 (1969), pp. 700–701.Google Scholar
[9] Kolmogorov, A., Three approaches for defining the concept of information quantity, Information transmission, vol. 1 (1965), pp. 3–11.Google Scholar
[10] Levin, L., On the notion of random sequence, Soviet Mathematics — Doklady, vol. 14 (1973), pp. 1413–1416.Google Scholar
[11] Levin, L. and Zvonkin, A., Complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms, Russian Mathematical Surveys, vol. 25 (1970), pp. 83–124.Google Scholar
[12] Loveland, D., A variant of the Kolmogorov concept of complexity, Information and control, vol. 15 (1969), pp. 510–526.CrossRefGoogle Scholar
[13] Rado, T., On a simple source for non-computable functions, Proceedings of the Symposium of Mathematical Theory of Automata (MRI Symposia Series, vol. 12) Polytechnic Press, Brooklyn, 1962, pp. 75–81.Google Scholar
[14] Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, 1967.Google Scholar
[15] Schnorr, C., Zufälligkeit und Wahrscheinlichkeit, Lecture notes in mathematics, No. 218, Springer-Verlag, Berlin, 1971.Google Scholar
[16] Schnorr, C., and Fuchs, H., General random sequences and the concept of learnable sequences, Preprint, Fachbereich Mathematik, Universität Frankfort, 1975.Google Scholar