SIAM Journal on Computing
Visualization 2001 Conference (Acm
Visualization 2001 Conference (Acm
An Introduction to Kolmogorov Complexity and Its Applications
An Introduction to Kolmogorov Complexity and Its Applications
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
Derandomizing from Random Strings
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Limits on the computational power of random strings
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Reductions to the set of random strings: the resource-bounded case
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
| Hi-index | 0.00 |
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.