Kolmogorov complexities Kmax, Kmin on computable partially ordered sets.

Marie Ferbus-Zanda (1) , Serge Grigorieff (1)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

We introduce a machine free mathematical framework to get a natural formaliza- tion of some general notions of infinite computation in the context of Kolmogorov complexity. Namely, the classes MaxX→D;PR and MaxX→D;Rec of functions X → D which are pointwise maximum of partial or total computable sequences of functions where D = (D,<) is some computable partially ordered set. The enumeration theorem and the invariance theorem always hold for MaxX→D;PR, leading to a variant KD;max of Kolmogorov complexity. We characterize the orders D such that the enumera- tion theorem (resp. the invariance theorem) also holds for MaxX→D;Rec . It turns out that MaxX→D;Rec may satisfy the invariance theorem but not the enumeration theo- rem. Also, when MaxX→D ;Rec satisfies the invariance theorem then the Kolmogorov complexities associated to MaxX→D ;Rec and MaxX→D;PR are equal (up to a constant). Letting KD;min = KDrev;max , where Drev is the reverse order, we prove that either KD;min =ct KD;max =ct KD (=ct is equality up to a constant) or KD;min,KD ;max are ≤ct incomparable and ct K∅',D. We characterize the orders leading to each case. We also show that KD;min,KDmax cannot be both much smaller than KD at any point.These results are proved in a more general setting with two orders on D, oneextending the other.

Mots clés: Logic; Computer Science; Algorithmic Information Theory; Kolmogorov Complexity; Recursion Theory

Domaines: Logique [math.LO]; Logique en informatique [cs.LO]; Complexité [cs.CC]

Marie Ferbus-Zanda, Serge Grigorieff. Kolmogorov complexities Kmax, Kmin on computable partially ordered sets.. Theoretical Computer Science, 2006, 352, pp.159-180. ⟨hal-00201621⟩