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Slawomir Solecki

Publications and source records attributed to Slawomir Solecki.

11 recordsLinked to original sources
arXiv2022

Logics for Epistemic Actions: Completeness, Decidability, Expressivity

We consider dynamic versions of epistemic logic as formulated in Baltag and Moss "Logics for epistemic programs" (2004). That paper proposed a logical language (actually families of languages parameterized by action signatures) for dynamic epistemic logic. It had been shown that validity in the language is Pi-1-1-complete, so there are no recursively axiomatized complete logical systems for it. In contrast, this paper proves a weak completeness result for the fragment without action iteration, and a strong completeness result for the fragment without action iteration and common knowledge. Our work involves a detour into term rewriting theory. The argument uses modal filtration, and thus we obtain the finite model property and hence decidability. We also give a translation of our largest language into PDL, thereby obtaining a second proof of decidability. The paper closes with some results on expressive power. These are mostly concerned with comparing the action-iteration-free language with modal logic augmented by transitive closure operators. We answer a natural question about the languages we obtain by varying the action signature: we prove that a logical language with operators for private announcements is more expressive than one for public announcements.

cs.LO↗
arXiv2020

A combinatorial model for the Menger curve

We represent the universal Menger curve as the topological realization $|\mathbb{M}|$ of the projective Fraïssé limit ${\mathbb M}$ of the class of all finite connected graphs. We show that $\mathbb{M}$ satisfies combinatorial analogues of the Mayer-Oversteegen-Tymchatyn homogeneity theorem and the Anderson-Wilson projective universality theorem. Our arguments involve only $0$-dimensional topology and constructions on finite graphs. Using the topological realization $\mathbb{M}\mapsto|\mathbb{M}|$, we transfer some of these properties to the Menger curve: we prove the approximate projective homogeneity theorem, recover Anderson's finite homogeneity theorem, and prove a variant of Anderson-Wilson's theorem. The finite homogeneity theorem is the first instance of an "injective" homogeneity theorem being proved using the projective Fraïssé method. We indicate how our approach to the Menger curve may extend to higher dimensions.

math.LO↗
arXiv2015

A Ramsey theorem for partial orders with linear extensions

We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to Sokić. As a bonus, our proof gives new arguments for these two results.

math.CO↗
arXiv2014

Unitary representations of the groups of measurable and continuous functions with values in the circle

We give a classification of unitary representations of certain Polish, not necessarily locally compact, groups: the groups of all measurable functions with values in the circle and the groups of all continuous functions on compact, second countable, zero-dimensional spaces with values in the circle. In the proofs of our classification results, certain structure theorems and factorization theorems for linear operators are used.

math.RT↗
arXiv2013

Abstract approach to finite Ramsey theory and a self-dual Ramsey theorem

We give an abstract approach to finite Ramsey theory and prove a general Ramsey-type theorem. We deduce from it a self-dual Ramsey theorem, which is a new result naturally generalizing both the classical Ramsey theorem and the dual Ramsey theorem of Graham and Rothschild. In fact, we recover the pure finite Ramsey theory from our general Ramsey-type result in the sense that the classical Ramsey theorem, the Hales--Jewett theorem (with Shelah's bounds), the Graham--Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and the new self-dual Ramsey theorem are all obtained as iterative applications of the general result.

math.CO↗
arXiv2012

Closed subgroups generated by generic measure automorphisms

We prove that for a generic measure preserving transformation $T$, the closed group generated by $T$ is a continuous homomorphic image of a closed linear subspace of $L_0(λ,{\mathbb R})$, where $λ$ is Lebesgue measure, and that the closed group generated by $T$ contains an increasing sequence of finite dimensional toruses whose union is dense.

math.DS↗
arXiv2012

Abstract approach to Ramsey theory and Ramsey theorems for finite trees

I will give a presentation of an abstract approach to finite Ramsey theory found in an earlier paper of mine. I will prove from it a common generalization of Deuber's Ramsey theorem for regular trees and a recent Ramsey theorem of Jasinski for boron tree structures. This generalization appears to be new. I will also show, in exercises, how to deduce from it the Milliken Ramsey theorem for strong subtrees.

math.CO↗
arXiv2012

A Boolean action of C(M,U(1)) without a spatial model and a re-examination of the Cameron-Martin theorem

We will demonstrate that if M is an uncountable compact metric space, then there is an action of the Polish group of all continuous functions from M to U(1) on a separable probability algebra which preserves the measure and yet does not admit a point realization in the sense of Mackey. This is achieved by exhibiting a strong form of ergodicity of the Boolean action known as whirliness. This is in contrast with Mackey's point realization theorem, which asserts that any measure preserving Boolean action of a locally compact second countable group on a separable probability algebra can be realized as an action on the points of the associated probability space. In the course of proving the main theorem, we will prove a result concerning infinite dimensional Gaussian measure space which is in contrast with the Cameron-Martin Theorem.

math.FA↗
arXiv2012

Borel equivalence relations and Lascar strong types

The space of Lascar strong types, on some sort and relative to a given first order theory T, is in general not a compact Hausdorff space. This paper has at least three aims. First to show that spaces of Lascar strong types and other related spaces such as the Lascar group, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well as some new examples. The third is to explore notions of definable map, embedding and isomorphism between these and related quotient objects. The motivation for writing this paper is the recent discovery, via definable groups, of new examples of non G-compact first order theories.

math.LO↗
arXiv2006

Automatic continuity of homomorphisms and fixed points on metric compacta

We prove that arbitrary homomorphisms from one of the groups ${\rm Homeo}(\ca)$, ${\rm Homeo}(\ca)^\N$, ${\rm Aut}(\Q,<)$, ${\rm Homeo}(\R)$, or ${\rm Homeo}(S^1)$ into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination with a result on V.G. Pestov, that any action of the discrete group ${\rm Homeo}_+(\R)$ by homeomorphisms on a compact metric space has a fixed point.

math.LO↗