### Abstract

In this paper we construct a structure R that is a “finite version” of the semi-lattice of Turing degrees. Its elements are strings (technically, sequences of strings) and x `<=` y means that K(x|y)=(conditional Kolmogorov complexity of x relative to y) is small. We construct two elements in R that do not have greatest lower bound. We give a series of examples that show how natural algebraic constructions give two elements that have lower bound 0 (minimal element) but significant mutual information. (A first example of that kind was constructed by Gács–Körner(Problems Control Inform. Theory 2 (1973) 149) using a completely different technique.) Wede4ne a notion of “complexity profile” of the pair of elements of R and give (exact) upper andlower bounds for it in a particular case.

Original language | English |
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Pages (from-to) | 69-95 |

Number of pages | 27 |

Journal | Theoretical Computer Science |

Volume | 271 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 28 Jan 2002 |

### Keywords

- Kolmogorov complexity
- common information
- conditional complexity

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## Cite this

Chernov, A., Muchnik, A., Romashchenko, A., Shen, A., & Vereshchagin, N. (2002). Upper semi-lattice of binary strings with the relation "x is simple conditional to y".

*Theoretical Computer Science*,*271*(1-2), 69-95. https://doi.org/10.1016/S0304-3975(01)00032-9