Fixed-point tile sets and their applications

v4: added references to a paper by Nicolas Ollinger and several historical commentsAn aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). We present a new construction of an aperiodic tile set that is based on Kleene's fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann self-reproducing automata; similar ideas were also used by P. Gacs in the context of error-correcting computations. This construction it rather flexible, so it can be used in many ways: we show how it can be used to implement substitution rules, to construct strongly aperiodic tile sets (any tiling is far from any periodic tiling), to give a new proof for the undecidability of the domino problem and related results, characterize effectively closed 1D subshift it terms of 2D shifts of finite type (improvement of a result by M. Hochman), to construct a tile set which has only complex tilings, and to construct a "robust" aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. For the latter we develop a hierarchical classification of points in random sets into islands of different ranks. Finally, we combine and modify our tools to prove our main result: there exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed

K-trivial, K-low and MLR-low sequences: a tutorial

A remarkable achievement in algorithmic randomness and algorithmic information theory was the discovery of the notions of K-trivial, K-low and Martin-Lof-random-low sets: three different definitions turns out to be equivalent for very non-trivial reasons. This paper, based on the course taught by one of the authors (L.B.) in Poncelet laboratory (CNRS, Moscow) in 2014, provides an exposition of the proof of this equivalence and some related results. We assume that the reader is familiar with basic notions of algorithmic information theory.Comment: 25 page

Characterizing the strongly jump-traceable sets via randomness

We show that if a set $A$ is computable from every superlow 1-random set, then $A$ is strongly jump-traceable. This theorem shows that the computably enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets computable from every superlow 1-random set. We also prove the analogous result for superhighness: a c.e.\ set is strongly jump-traceable if and only if it is computable from every superhigh 1-random set. Finally, we show that for each cost function $c$ with the limit condition there is a 1-random $\Delta^0_2$ set $Y$ such that every c.e.\ set $A \le_T Y$ obeys $c$. To do so, we connect cost function strength and the strength of randomness notions. This result gives a full correspondence between obedience of cost functions and being computable from $\Delta^0_2$ 1-random sets.Comment: 41 page