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A strong law of computationally weak subsets

A strong law of computationally weak subsets

Journal of Mathematical Logic 11 (2011) no. 1, 1—10.
DOI: 10.1142/S0219061311000980
Electronic Colloquium on Computational Complexity, Report No. 150 (2010).

This paper establishes a new statistical law, namely that for each random sequence
it is possible to replace some of the 1s by 0s (in other words, form a subset of 1s) in such a way that no random sequence can be recovered by computational means.

To illustrate, imagine that the new sequence looks like

Technically the result is that each 2-random set has an infinite subset computing no 1-random set. It is perhaps the main result obtained under Prof. Kjos-Hanssen’s grant NSF DMS-0901020 (2009-2013).
Joseph S. Miller at U. of Wisconsin has established a strengthening of this result replacing 2-random by 1-random, but that result is so far unpublished.

Dr. Bjørn Kjos-Hanssen is a professor at the University of Hawai‘i at Manoa in the Department of Mathematics. His research deals with the abstract theory of computation, computability, randomness and compression algorithms.

Kjos-Hanssen is the author of more than 20 papers in journals including the prestigious Mathematical Research Letters and Transactions of the American Mathematical Society, and has a PhD from UC Berkeley in the subject Logic and the Methodology of Science.

Effective dimension of points visited by Brownian motion

Effective dimension of points visited by Brownian motion
(with Anil Nerode). Theoretical Computer Science 410 (2009), no. 4-5, 347—354.

Preliminary version: The law of the iterated logarithm for algorithmically random paths of Brownian motion.
Logical Foundations of Computer Science, Lecture Notes in Computer Science, vol. 4514, Springer, Berlin, 2007, pp. 310—317.

In this paper we answered a question of Fouché regarding algorithmically random Brownian motion. The key idea was to use Carathéodory’s measure algebra isomorphism theorem.

Finding paths through narrow and wide trees

History of the paper


The paper was written at UConn.


Paper appeared in print:

Finding paths through narrow and wide trees
(with Stephen E. Binns). Journal of Symbolic Logic 74 (2009), no. 1, 349—360.


Laurent Bienvenu and Paul Shafer discovered an apparent error in the proof of Lemma 4.6. There it is stated that A wtt-reduces to B; however, it seems that the reduction of A to B also requires oracle access to f. Corollary 4.7 also seems false (take f to be a ML random of hyperimmune-free degree, and A=f; then A is complex but not f-complex).