PP-2010-25:
Baartse, Martijn and Barmpalias, George
(2010)
*On the gap between trivial and nontrivial initial segment prefix-free complexity.*
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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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