PP-2010-25: On the gap between trivial and nontrivial initial segment prefix-free complexity

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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