# Curiouser and Curiouser: The Link between Incompressibility and Complexity

## Abstract

This talk centers around some audacious conjectures that attempt to forge firm links between computational complexity classes and the study of Kolmogorov complexity.

More specifically, let *R* denote the set of Kolmogorov-random strings. Let \(\mbox{\rm BPP}\) denote the class of problems that can be solved with negligible error by probabilistic polynomial-time computations, and let \(\mbox{\rm NEXP}\) denote the class of problems solvable in nondeterministic exponential time.

*Conjecture 1.* \(\mbox{\rm NEXP} = \mbox{\rm NP}^R\).

*Conjecture 2.* \(\mbox{\rm BPP}\) is the class of problems non-adaptively polynomial-time reducible to *R*.

These conjectures are not only audacious; they are obviously false! *R* is not a decidable set, and thus it is absurd to suggest that the class of problems reducible to it constitutes a complexity class.

The absurdity fades if, for example, we interpret “\(\mbox{\rm NP}^R\)” to be “the class of problems that are \(\mbox{\rm NP}\)-Turing reducible to *R*, no matter which universal machine we use in defining Kolmogorov complexity”. The lecture will survey the body of work (some of it quite recent) that suggests that, when interpreted properly, the conjectures may actually be true.

## Keywords

Turing Machine Complexity Class Kolmogorov Complexity Random String Peano Arithmetic## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Allender, E., Buhrman, H., Koucký, M.: What can be efficiently reduced to the Kolmogorov-random strings? Annals of Pure and Applied Logic 138, 2–19 (2006)MathSciNetMATHCrossRefGoogle Scholar
- 2.Allender, E., Buhrman, H., Koucký, M., van Melkebeek, D., Ronneburger, D.: Power from random strings. SIAM Journal on Computing 35, 1467–1493 (2006)MathSciNetMATHCrossRefGoogle Scholar
- 3.Allender, E., Davie, G., Friedman, L., Hopkins, S.B., Tzameret, I.: Kolmogorov complexity, circuits, and the strength of formal theories of arithmetic. Tech. Rep. TR12-028, Electronic Colloquium on Computational Complexity (submitted for publication, 2012)Google Scholar
- 4.Allender, E., Friedman, L., Gasarch, W.: Limits on the Computational Power of Random Strings. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 293–304. Springer, Heidelberg (2011)CrossRefGoogle Scholar
- 5.Arora, S., Barak, B.: Computational Complexity, a modern approach. Cambridge University Press (2009)Google Scholar
- 6.Buhrman, H., Fortnow, L., Koucký, M., Loff, B.: Derandomizing from random strings. In: 25th IEEE Conference on Computational Complexity (CCC), pp. 58–63. IEEE Computer Society Press (2010)Google Scholar
- 7.Buhrman, H., Loff, B.: Personal Communication (2012)Google Scholar
- 8.Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Springer (2010)Google Scholar
- 9.Li, M., Vitanyi, P.: Introduction to Kolmogorov Complexity and its Applications, 3rd edn. Springer (2008)Google Scholar
- 10.Martin, D.A.: Completeness, the recursion theorem and effectively simple sets. Proceedings of the American Mathematical Society 17, 838–842 (1966)MathSciNetMATHCrossRefGoogle Scholar