Henryk Michalewski

Small Valdivia compact spaces

Abstract We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of retractions of a certain kind. Consequently, a compact space of weight ⩽ ℵ 1 is Valdivia compact iff it is the limit of an inverse sequence of metric compacta whose bonding maps are retractions. As a corollary, we show that the class of Valdivia compacta of weight ⩽ ℵ 1 is preserved both under retractions and under open 0-dimensional images. Finally, we characterize the class of all Valdivia compacta in the language of category theory, which implies that this class is preserved under all continuous weight preserving functors.

Measure Properties of Game Tree Languages

We introduce a general method for proving measurability of topologically complex sets by establishing a correspondence between the notion of game tree languages from automata theory and the σ-algebra of \(\mathcal{R}\)-sets, introduced by A. Kolmogorov as a foundation for measure theory. We apply the method to answer positively to an open problem regarding the game interpretation of the probabilistic μ-calculus.

An example of a topological group

In this paper three classes of topological groups are considered: the groups which, in the sense of their topological structure, are Lindelof P{spaces, and the groups which are o{bounded or strictly o{ bounded. In the sense of covering properties these classes are close to {compactness. We construct an example of a topological group G which is a Lindelof P{space but is not strictly o{bounded. The example is generated by a ladder system. We answer some questions of C. Hernandez and M. Tkachenko. 2000 Mathematical Subject Classication. Primary 54H11, 22A05, 03E05.

On the Decidability of MSO+U on Infinite Trees

This paper is about MSO+U, an extension of monadic second-order logic, which has a quantifier that can express that a property of sets is true for arbitrarily large sets. We conjecture that the MSO+U theory of the complete binary tree is undecidable. We prove a weaker statement: there is no algorithm which decides this theory and has a correctness proof in zfc. This is because the theory is undecidable, under a set-theoretic assumption consistent with zfc, namely that there exists of projective well-ordering of 2 ω of type ω 1. We use Shelah’s undecidability proof of the MSO theory of the real numbers.

On the Borel Inseparability of Game Tree Languages.

The game tree languages can be viewed as an automata-theoretic counterpart of parity games on graphs. They witness the strictness of the index hierarchy of alternating tree automata, as well as the fixed-point hierarchy over binary trees. We consider a game tree language of the first non-trivial level, where Eve can force that 0 repeats from some moment on, and its dual, where Adam can force that 1 repeats from some moment on. Both these sets (which amount to one up to an obvious renaming) are complete in the class of co-analytic sets. We show that they cannot be separated by any Borel set, hence {\em a fortiori\/} by any weakly definable set of trees. This settles a case left open by L. Santocanale and A. Arnold, who have thoroughly investigated the separation property within the

On the Separation Question for Tree Languages

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On the separation question for tree languages

-calculus and the automata index hierarchies. They showed that separability fails in general for non-deterministic automata of type

On topological completeness of regular tree languages

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Learning to Run Challenge Solutions: Adapting Reinforcement Learning Methods for Neuromusculoskeletal Environments

, starting from level

Measure properties of regular sets of trees

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