Church, Cardinal and Ordinal Representations of Integers and Kolmogorov complexity

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

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

We consider classical representations of integers: Church's function iterators, cardinal equivalence classes of sets, ordinal equivalence classes of totally ordered sets. Since programs do not work on abstract entities and require formal representations of objects, we effectivize these abstract notions in order to allow them to be computed by programs. To any such effectivized representation is then associated a notion of Kolmogorov complexity. We prove that these Kolmogorov complexities form a strict hierarchy which coincides with that obtained by relativization to jump oracles and/or allowance of infinite computations.

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. Church, Cardinal and Ordinal Representations of Integers and Kolmogorov complexity. Denis Richard's 60th Biirthday Conference, 2002, France. pp.1-16. ⟨hal-00201627⟩