PP-2010-25: Baartse, Martijn and Barmpalias, George (2010) On the gap between trivial and nontrivial initial segment prefix-free complexity. [Report]
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Abstract
An infinite sequence X is said to have trivial (prefix-free) initial
segment complexity if K(X~n) ~+ K(0^n) for all n, where K is the
prefix-free complexity and ~+ denotes inequality modulo a constant. In
other words, if the information in any initial segment of it is merely
the information in a sequence of 0s of the same length. We study the
gap between the trivial com- plexity K(0^n) and the complexity of a
non-trivial sequence, i.e. the functions f such that
(*) K(X~n) ~+ K(0^n) + f (n) for all n
for a non-trivial (in terms of initial segment complexity) sequence
X. We show that given any ~^0_2 unbounded non-decreasing function f
there exist uncountably many sequences X which satisfy (*). On the
other hand there exists a ~^0_3 unbounded non-decreasing function f
which does not satisfy (*) for any X with non-trivial initial segment
complexity. This improves the bound ~^0_4 that was known from
[CM06]. Finally we give some applications of these results.
| Item Type: | Report |
|---|---|
| Report Nr: | PP-2010-25 |
| Series Name: | Prepublication (PP) Series |
| Year: | 2010 |
| Uncontrolled Keywords: | Kolmogorov complexity; Prefix-free machine; Low complexity; Effectively Closed sets |
| Depositing User: | gbarmpa1 |
| Date Deposited: | 12 Oct 2016 14:37 |
| Last Modified: | 12 Oct 2016 14:37 |
| URI: | https://eprints.illc.uva.nl/id/eprint/404 |
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