Reals which Compute Little André

Reals which compute little

We investigate combinatorial lowness properties of sets of natural numbers (reals). The real A is super-low if A′ ≤tt ∅′, and A is jump-traceable if the values of {e}A(e) can be effectively approximated in a sense to be specified. We investigate those properties, in particular showing that super-lowness and jump-traceability coincide within the r.e. sets but none of the properties implies the o...

Trivial Reals

Solovay showed that there are noncomputable reals α such that H(α ↾ n) 6 H(1) + O(1), where H is prefix-free Kolmogorov complexity. Such H-trivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an H-trivial real. We also analyze various computability-theoretic properties of the H-trivial reals, showing f...

Lowness Properties of Reals and Randomness

Relative (non-)formality of the Little Cubes Operads and the Algebraic Cerf Lemma

It is shown that the operad maps En → En+k are formal over the reals for k ≥ 2 and non-formal for k = 1. Furthermore we compute the homology of the deformation complex of the operad maps En → En+1, proving an algebraic version of the Cerf lemma.

1 5 A pr 1 99 8 Ordinal Computers

Can a computer which runs for time ω 2 compute more than one which runs for time ω? No. Not, at least, for the infinite computer we describe. Our computer gets more powerful when the set of its steps gets larger. We prove that they theory of second order arithmetic cannot be decided by computers running to countable time. Our motivation is to build a computer that will store and manipulate surr...