Stephen Binns

A splitting theorem for the Medvedev and Muchnik lattices

This is a contribution to the study of the Muchnik and Medvedev lattices of non-empty Π01 subsets of 2ω. In both these lattices, any non-minimum element can be split, i. e. it is the non-trivial join of two other elements. In fact, in the Medvedev case, ifP > MQ, then P can be split above Q. Both of these facts are then generalised to the embedding of arbitrary finite distributive lattices. A consequence of this is that both lattices have decidible ∃-theories.

Embeddings into the Medvedev and Muchnik lattices of Π0 1 classes

Abstract.Let w and M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty Π10 subsets of 2ω, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of w. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of M.

Compressibility and Kolmogorov Complexity

We continue the investigation of the path-connected geometry on the Cantor space and the related notions of dilution and compressibility described in [1]. These ideas are closely related to the notions of effective Hausdorff and packing dimensions of reals, and we argue that this geometry provides the natural context in which to study them. In particular we show that every regular real can be maximally compressed that is every regular real is a dilution of some real of maximum effective Hausdorff dimension.

Mass problems and density

Recall that ℰw is the lattice of Muchnik degrees of nonempty effectively compact sets in Euclidean space. We solve a long-standing open problem by proving that ℰw is dense, i.e. satisfies ∀x∀y(x < y ⇒∃z(x < z < y)). Our proof combines an oracle construction with hyperarithmetical theory.

Completeness, Compactness, Effective Dimensions

We investigate a directed metric on the space of infinite binary sequences defined by where C(X↾n‖Y↾n) is the Kolmogorov complexity of X↾n given Y↾n. In particular we focus on the topological aspects of the associated metric space—proving that it is complete though very far from being compact. This is a continuation of earlier work investigating other geometrical and toplogical aspects of this metric.