An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
Kolmogorov Complexity for Possibly Infinite Computations
Journal of Logic, Language and Information
Another Example of Higher Order Randomness
Fundamenta Informaticae
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We introduce a machine free mathematical framework to get a natural formalization of some general notions of infinite computation in the context of Kolmogorov complexity. Namely, the classes MaxP RX → D and MaxRecX → D of functions X → D which are pointwise maximum of partial or total computable sequences of functions where D = (D, P RX → D, leading to a variant KMaxD of Kolmogorov complexity. We characterize the orders D such that the enumeration theorem (resp. the invariance theorem) also holds for MaxRecX → D. It turns out that MaxRecX → D may satisfy the invariance theorem but not the enumeration theorem. Also, when MaxRecX → D satisfies the invariance theorem then the Kolmogorov complexities associated to MaxRecX → D and MaxP RX → D are equal (up to constant).Letting KminD = KmaxDrev where Drev is the reverse order, we prove that either KminD = ctKmaxD = ctKD (=ct is equality up to a constant) or KminD, KmaxDare ≤ ct incomparable and ct KD and ct K0',D. We characterize the orders leading to each case. We also show that KminD, KmaxD cannot be both much smaller than KD at any point.These results are proved in a more general setting with two orders on D, one extending the other.