## $$\Pi _1^0$$ classes with complex elements.(English)Zbl 1155.03032

Summary: An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a $$\Pi _1^0$$ class $$P$$ contains a complex element if and only if it contains a wtt-cover for the Cantor set. That is, if and only if for every $$Y\subseteq \omega$$ there is an $$X$$ in $$P$$ such that $$X\geqslant _{\text{wtt}} Y$$. We show that this is also equivalent to the $$\Pi_1^0$$ class’s being large in some sense. We give an example of how this result can be used in the study of scattered linear orders.

### MSC:

 03D80 Applications of computability and recursion theory 68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
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### References:

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