arXiv subjects

Maciej Malicki

Publications and source records attributed to Maciej Malicki.

At least 19 recordsLinked to original sources

arXiv2026

Rokhlin properties in homeomorphism groups of compact metric spaces

We develop a general framework for studying generic homeomorphisms of compact metric spaces using combinatorial amalgamation methods inspired by Fraïssé theory. Our approach combines Rosendal's criterion for comeager conjugacy classes with combinatorial codings of compact spaces arising from the Bartoš-Bice-Vignati duality. We introduce a new amalgamation property, called weak minimal amalgamation, formulated in suitable paracategories encoding local dynamical data of homeomorphisms. This yields characterizations of the Rokhlin and the strong Rokhlin properties, i.e., the existence of dense and comeager conjugacy classes in homeomorphism groups of compact metric spaces. As applications, we reprove Hjorth's theorem that $\mbox{Homeo}_+([0,1])$ admits a generic homeomorphism, and establish the existence of generic homeomorphisms for the Cantor fan and the Lelek fan. Moreover, we show that generic homeomorphisms of these spaces have no Li--Yorke pairs, and therefore generically have zero topological entropy. More broadly, the paper develops new connections between combinatorial amalgamation properties and generic phenomena in topological group theory and topological dynamics.

math.DS↗

arXiv2026

Universal homogeneous two-sorted ultrametric spaces

We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining isometries and linear order embeddings. We show that the class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé, and that the limit is the countable rational Urysohn ultrametric space $\mathbb{U}$. The space $\mathbb{U}$ is dc-universal for all countable ultrametric spaces, and its Cauchy completion $\overline{\mathbb{U}}$ is dc-universal for all separable ultrametric spaces, which is in contrast with the situation of classical ultrametric spaces and isometric embeddings, where no such universal space can exist. We study further properties of $\mathbb{U}$, of its variants, and of its automorphism group, which is richer than its group of isometries. In particular, we provide two types of tree representations of the two-sorted ultrametric spaces, discuss connections to valued fields, and characterize the automorphism group of $\mathbb{U}$ as the semidirect product of a group of order preserving bijections and a group of isometries. Furthermore, we show universality of $\operatorname{Aut}(\mathbb{U})$ and identify its universal minimal flow.

math.LO↗

arXiv2026

Generic dc-automorphisms of two-sorted ultrametric spaces

We continue to study ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances, together with the dc-embeddings, which we introduced in our earlier paper "Universal homogeneous two-sorted ultrametric spaces". The class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé whose limit we denote by $\mathbb{U}$. The main result of the article is that $\operatorname{Aut}(\mathbb{U})$ has a comeager conjugacy class. For that we show the cofinal amalgamation property of partial automorphisms and characterize amalgamation bases. In fact we develop a general strategy for showing cofinal amalgamation property for a broad class of categories. Furthermore, we show that there is no generic pair of automorphisms, we provide a detailed description of single orbits under dc-automorphisms, and we prove that any finite partial dc-automorphism, even in the presence of other orbits, can be extended to one that is closed or monotone.

math.LO↗

arXiv2026

Hybrid Models for Natural Language Reasoning: The Case of Syllogistic Logic

Despite the remarkable progress in neural models, their ability to generalize, a cornerstone for applications such as logical reasoning, remains a critical challenge. We delineate two fundamental aspects of this ability: compositionality, the capacity to abstract atomic logical rules underlying complex inferences, and recursiveness, the aptitude to build intricate representations through iterative application of inference rules. In the literature, these two aspects are often conflated under the umbrella term of generalization. To sharpen this distinction, we investigate the logical generalization capabilities of LLMs using the syllogistic fragment as a benchmark for natural language reasoning. We extend classical syllogistic forms to construct more complex structures, yielding a foundational yet expressive subset of formal logic that supports controlled evaluation of essential reasoning abilities. Our findings on this non-trivial benchmark show that, while LLMs demonstrate reasonable proficiency in recursiveness, they struggle with compositionality. This disparity is not uniform, as a more detailed analysis reveals substantial variability in generalization performance across individual syllogistic types, ranging from near-perfect accuracy to significantly lower performance. To overcome these limitations and establish a reliable logical prover, we propose a hybrid architecture integrating symbolic reasoning with neural computation. This synergistic interaction enables robust and efficient inference, neural components accelerate processing, while symbolic reasoning guarantees completeness. Our experiments further show that high efficiency is preserved even when using relatively small neural components. Overall, our analysis provides both a rationale for hybrid neuro-symbolic approaches and evidence of their potential to address key generalization barriers in neural reasoning systems.

cs.CL↗

arXiv2026

Teaching Small Language Models to Learn Logic through Meta-Learning

Large language models (LLMs) are increasingly evaluated on reasoning tasks, yet their logical abilities remain contested. To address this, we study LLMs' reasoning in a well-defined fragment of logic: syllogistic reasoning. We cast the problem as premise selection and construct controlled datasets to isolate logical competence. Beyond evaluation, an open challenge is enabling LLMs to acquire abstract inference patterns that generalize to novel structures. We propose to apply few-shot meta-learning to this domain, thereby encouraging models to extract rules across tasks rather than memorize patterns within tasks. Although meta-learning has been little explored in the context of logic learnability, our experiments show that it is effective: small models (1.5B-7B) fine-tuned with meta-learning demonstrate strong gains in generalization, with especially pronounced benefits in low-data regimes. These meta-learned models outperform GPT-4o and o3-mini on our syllogistic reasoning task.

cs.CL↗

arXiv2026

A logic of co-valuations

A co-valuation is, essentially, a minimal finite cover. We introduce a logic based on co-valuations, which play the role of valuations of free variables in classical first-order logic, and show that the fundamental tools of model theory -- such as ultraproducts, compactness, and omitting types -- can be developed in this setup. Using a recently discovered duality between certain countable posets and second-countable compact $T_1$ spaces, we show that these spaces are counterparts of countable universes in first-order logic. Thus, although no topology appears in the initial formulation, the logic of co-valuations turns out to be naturally suited for studying compact topological objects. Standard topological notions, such as connectedness and covering dimension, are easily expressible, and model-theoretic properties, such as atomicity, can be effectively analyzed. The framework also interacts well with Fraïssé-type constructions.

math.LO↗

arXiv2025

The rational Gurarii space and its linear isometry group

We show that the classes of partial isometries in finite-dimensional polyhedral spaces and in finite-dimensional rational polyhedral spaces do not have the weak amalgamation property. This implies that the linear isometry group of the rational Gurarii space does not have a comeager conjugacy class. Our methods demonstrate also that the classes of finite-dimensional polyhedral spaces and of finite-dimensional rational polyhedral spaces fail to have the Hrushovski property.

math.LO↗

arXiv2025

Isomorphism of almost locally compact Polish metric structures

A topological space is almost locally compact if it contains a dense locally compact subspace. We generalize a result from \cite{Ma}, showing that isomorphism on Borel classes of almost locally compact Polish metric structures is always classifiable by countable structures. This allows to remove a gap in the proof presented in \cite{Ma} of a positive answer to a question of Gao and Kechris, who asked whether isometry of locally compact Polish metric spaces is classifiable by countable structures.

math.LO↗

arXiv2025

Automorphism groups of measures on the Cantor space. Part I: Good measures and Rokhlin properties

We study criteria for the existence of a dense or comeager conjugacy class in the automorphism group of a given measure on the Cantor space. We concentrate on good measures, defined by Akin [\emph{Trans.\ Amer.\ Math.\ Soc.} \textbf{357} (2005), no. 7, 2681--2722], which we characterize as a particular subclass of ultrahomogeneous measures. We determine good measures with rational values on clopen sets whose automorphism group admits a comeager conjugacy class. Our approach uses the Fraïssé theory.

math.LO↗

arXiv2024

Homeomorphisms of the Pseudoarc

We construct homeomorphisms of compacta from relations between finite graphs representing their open covers. Applied to the pseudoarc, this yields simple Fraïssé theoretic proofs of several important results, both old and new. Specifically, we recover Bing's classic results on the uniqueness and homogeneity of the pseudoarc. We also show that the autohomeomorphism group of the pseudoarc has a dense conjugacy class, thus confirming a conjecture of Kwiatkowska.

math.GN↗

arXiv2022

Isomorphism of locally compact Polish metric structures

We study the isomorphism relation on Borel classes of locally compact Polish metric structures. We prove that isomorphism on such classes is always classifiable by countable structures (equivalently: Borel reducible to graph isomorphism), which implies, in particular, that isometry of locally compact Polish metric spaces is Borel reducible to graph isomorphism. We show that potentially $\pmbΠ^0_{α+1}$ isomorphism relations are Borel reducible to equality on hereditarily countable sets of rank $α$, $α\geq 2$. We also study approximations of the Hjorth-isomorphism game, and formulate a condition ruling out classifiability by countable structures.

math.LO↗

arXiv2022

Remarks on weak amalgamation and large conjugacy classes in non-archimedean groups

We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure $M$, Polish group $G$ of permutations of $M$, and $n \geq 1$, $G$ has a comeager $n$-diagonal conjugacy class iff the family of all $n$-tuples of $G$-extendable bijections between finitely generated substructures of $M$, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fraïssé classes that are not homogenizable. Finally, we investigate $1$- and $2$-diagonal conjugacy classes in groups of ball-preserving bijections of certain ordered ultrametric spaces.

math.LO↗

arXiv2021

Continuous logic and Borel equivalence relations

We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbfΣ^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth--Kechris about discrete structures. As a different application, we also give a new proof of Kechris's theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable.

math.LO↗

arXiv2020

On Polish groups admitting non-essentially countable actions

It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

math.LO↗

arXiv2020

Topologically 2-generated groups

We prove that for a number of ultrahomogeneous structures $M$, including those with the free amalgamation property, the powers of the automorphism group ${\rm{Aut}}(M)^n$, $n=1,2,\ldots$, and the group $L_0({\rm{ Aut}}(M))$ of measurable functions with values in ${\rm{Aut}}(M)$, have a cyclically dense conjugacy class, in particular, are topologically 2-generated. This provides a number of new examples of groups with this property. Moreover, we will show that each of these groups is cyclically generated by a pair generating the free group.

math.LO↗

arXiv2019

Ordered structures and large conjugacy classes

This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraisse limit) that is extremely amenable, and has ample generics. As Fraisse limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraisse limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager $2$-dimensional diagonal conjugacy class. We also provide general conditions implying that there is no comeager conjugacy class, comeager $2$-dimensional diagonal conjugacy class or non-meager $2$-dimensional topological similarity class in the automorphism group of an ordered Fraisse limit. We provide a number of applications of these results.

math.LO↗

arXiv2019

Generic representations of countable groups

The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups $Γ$ in Polish groups $G$, i.e. those elements in the Polish space $\mathrm{Rep}(Γ,G)$ of all representations of $Γ$ in $G$, whose orbit under the conjugation action of $G$ on $\mathrm{Rep}(Γ,G)$ is comeager. We investigate a closely related notion of finite approximability of actions on countable structures such as tournaments or $K_n$-free graphs, and we show its connections with Ribes-Zalesski-like properties of the acting groups. We prove that $\mathbb{N}$ has a generic representation in the automorphism group of the random tournament (i.e., there is a comeager conjugacy class in this group). We formulate a Ribes-Zalesskii-like condition on a group that guarantees finite approximability of its actions on tournaments. We also provide a simpler proof of a result of Glasner, Kitroser and Melleray characterizing groups with a generic permutation representation. We also investigate representations of infinite groups $Γ$ in automorphism groups of metric structures such as the isometry group $\mathrm{Iso}(\mathbb{U})$ of the Urysohn space, isometry group $\mathrm{Iso}(\mathbb{U}_1)$ of the Urysohn sphere, or the linear isometry group $\mbox{LIso}(\mathbb{G})$ of the Gurarii space. We show that the conjugation action of $\mathrm{Iso}(\mathbb{U})$ on $\mathrm{Rep}(Γ,\mathrm{Iso}(\mathbb{U}))$ is generically turbulent answering a question of Kechris and Rosendal.

math.GR↗

arXiv2018

Automorphism groups of countable structures and groups of measurable functions

Let $G$ be a topological group and let $μ$ be the Lebesgue measure on the interval $[0,1]$. We let $L_0(G)$ to be the topological group of all $μ$-equivalence classes of $μ$-measurable functions defined on [0,1] with values in $G$, taken with the pointwise multiplication and the topology of convergence in measure. We show that for a Polish group $G$, if $L_0(G)$ has ample generics, then $G$ has ample generics, thus the converse to a result of Kaïchouh and Le Maître. We further study topological similarity classes and conjugacy classes for many groups ${\rm{Aut}}(M)$ and $L_0({\rm{Aut}}(M))$, where $M$ is a countable structure. We make a connection between the structure of groups generated by tuples, the Hrushovski property, and the structure of their topological similarity classes. In particular, we prove the trichotomy that for every tuple $ \bar{f}$ of ${\rm{Aut}}(M)$, where $M$ is a countable structure such that algebraic closures of finite sets are finite, either the countable group $\langle \bar{f} \rangle$ is precompact, or it is discrete, or the similarity class of $\bar{f}$ is meager, in particular the conjugacy class of $\bar{f}$ is meager. We prove an analogous trichotomy for groups $L_0({\rm{Aut}}(M))$.

math.LO↗