Takayuki Kihara

Incomputability of Simply Connected Planar Continua

Le Roux and Ziegler asked whether every simply connected compact nonempty planar co-c.e. closed set always contains a computable point. In this paper, we solve the problem of le Roux and Ziegler by showing that there exists a contractible planar co-c.e. dendroid without computable points. We also provide several pathological examples of tree-like co-c.e. continua fulfilling certain global incomputability properties: there is a computable dendrite which does not *-include a co-c.e. tree; there is a co-c.e. dendrite which does not *-include a computable dendrite; there is a computable dendroid which does not *-include a co-c.e. dendrite. Here, a continuum A *-includes a member of a class P of continua if, for every positive real, A includes a P-continuum B such that the Hausdorff distance between A and B is smaller than the real.

Inside the Muchnik degrees I: Discontinuity, learnability and constructivism

Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify nonuniformly computable functions on Baire space from the viewpoint of learning theory and piecewise computability. For instance, we show that mind-change-bounded-learnability is equivalent to finite

The ∀∃-theory of the effectively closed Medvedev degrees is decidable

(\Pi^0_1)_2

Uniform Kurtz randomness

-piecewise computability (where

On effectively closed sets of effective strong measure zero

(\Pi^0_1)_2

Effective strong nullness and effectively closed sets

denotes the difference of two

Unified characterizations of lowness properties via Kolmogorov complexity

\Pi^0_1

On the strength of marriage theorems and uniformity

sets), error-bounded-learnability is equivalent to finite

A hierarchy of immunity and density for sets of reals

\Delta^0_2

The binary expansion and the intermediate value theorem in constructive reverse mathematics

-piecewise computability, and learnability is equivalent to countable