Infinite computable version of Lovasz Local Lemma

Lov\'asz Local Lemma (LLL) is a probabilistic tool that allows us to prove the existence of combinatorial objects in the cases when standard probabilistic argument does not work (there are many partly independent conditions). LLL can be also used to prove the consistency of an infinite set of conditions, using standard compactness argument (if an infinite set of conditions is inconsistent, then some finite part of it is inconsistent, too, which contradicts LLL). In this way we show that objects satisfying all the conditions do exist (though the probability of this event equals~$0$). However, if we are interested in finding a computable solution that satisfies all the constraints, compactness arguments do not work anymore. Moser and Tardos recently gave a nice constructive proof of LLL. Lance Fortnow asked whether one can apply Moser--Tardos technique to prove the existence of a computable solution. We show that this is indeed possible (under almost the same conditions as used in the non-constructive version)

Compressibility and probabilistic proofs

We consider several examples of probabilistic existence proofs using compressibility arguments, including some results that involve Lov\'asz local lemma.Comment: Invited talk for CiE 2017 (full version

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

06051 Abstracts Collection -- Kolmogorov Complexity and Applications

From 29.01.06 to 03.02.06, the Dagstuhl Seminar 06051 ``Kolmogorov Complexity and Applications\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

μ\mu-Limit Sets of Cellular Automata from a Computational Complexity Perspective

This paper concerns $\mu$-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial $\mu$-random configuration. More precisely, we investigate the computational complexity of these sets and of related decision problems. Main results: first, $\mu$-limit sets can have a $\Sigma\_3^0$-hard language, second, they can contain only $\alpha$-complex configurations, third, any non-trivial property concerning them is at least $\Pi\_3^0$-hard. We prove complexity upper bounds, study restrictions of these questions to particular classes of CA, and different types of (non-)convergence of the measure of a word during the evolution.Comment: 41 page