arXiv subjects

Alex Kruckman

Publications and source records attributed to Alex Kruckman.

18 recordsLinked to original sources

arXiv2024

A New Kim's Lemma

Kim's Lemma is a key ingredient in the theory of forking independence in simple theories. It asserts that if a formula divides, then it divides along every Morley sequence in type of the parameters. Variants of Kim's Lemma have formed the core of the theories of independence in two orthogonal generalizations of simplicity - namely, the classes of NTP2 and NSOP1 theories. We introduce a new variant of Kim's Lemma that simultaneously generalizes the NTP2 and NSOP1 variants. We explore examples and non-examples in which this lemma holds, discuss implications with syntactic properties of theories, and ask several questions.

math.LO↗

arXiv2024

Three surprising instances of dividing

We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type $p$ over a set $B$ does not divide over $C\subseteq B$, then no extension of $p$ to a complete type over $\text{acl}(B)$ divides over $C$. Two of our examples are also the first known theories where all sets are extension bases for nonforking, but forking and dividing differ for complete types (answering a question of Adler). One example is an NSOP$_1$ theory with a complete type that forks, but does not divide, over a model (answering a question of d'Elbée). Moreover, dividing independence fails to imply M-independence in this example (which refutes another folklore claim). In addition to these counterexamples, we summarize various related properties of dividing that are still true. We also address consequences for previous literature, including an earlier unpublished result about forking and dividing in free amalgamation theories, and some claims about dividing in the theory of generic $K_{m,n}$-free incidence structures.

math.LO↗

arXiv2022

Invariant measures in simple and in small theories

We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable group G in a simple theory such that G is not definably amenable, i.e. there is no translation invariant Keisler measure on G We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups, and nontriviality of the graded Grothendieck ring.

math.LO↗

arXiv2022

The almost sure theory of finite metric spaces

We establish an approximate zero-one law for sentences of continuous logic over finite metric spaces of diameter at most $1$. More precisely, we axiomatize a complete metric theory $T_{\mathrm{as}}$ such that, given any sentence $σ$ in the language of pure metric spaces and any $ε>0$, the probability that the difference of the value of $σ$ in a random metric space of size $n$ and the value of $σ$ in any model of $T_{\mathrm{as}}$ is less than $ε$ approaches $1$ as $n$ approaches infinity. We also establish some model-theoretic properties of the theory $T_{\mathrm{as}}$.

math.LO↗

arXiv2022

Interpolative fusions II: Preservation results

We study interpolative fusion, a method of combining theories $T_1$ and $T_2$ in distinct languages in a "generic" way over a common reduct $T_\cap$, to obtain a theory $T_\cup^*$. When each $T_i$ is model-complete, $T_\cup^*$ is the model companion of the union $T_1\cup T_2$. Our goal is to prove preservation results, i.e., to find sufficient conditions under which model-theoretic properties of $T_1$ and $T_2$ are inherited by $T_\cup^*$. We first prove preservation results for quantifier elimination, model-completeness, and related properties. We then apply these tools to show that, under mild hypotheses, including stability of $T_\cap$, the property $\mathrm{NSOP}_1$ is preserved. We also show that simplicity is preserved under stronger hypotheses on algebraic closure in $T_1$ and $T_2$. This generalizes many previous results; for example, simplicity of $\mathrm{ACFA}$ and the random $n$-hypergraph are both non-obvious corollaries. We also address preservation of stability, $\mathrm{NIP}$, and $\aleph_0$-categoricity, and we describe examples which witness that these results are sharp.

math.LO↗

arXiv2021

Exploring the Landscape of Relational Syllogistic Logics

This paper explores relational syllogistic logics, a family of logical systems related to reasoning about relations in extensions of the classical syllogistic. These are all decidable logical systems. We prove completeness theorems and complexity results for a natural subfamily of relational syllogistic logics, parametrized by constructors for terms and for sentences.

math.LO↗

arXiv2021

Higher dimensional obstructions for star reductions

A $*$-reduction between two equivalence relations is a Baire measurable reduction which preserves generic notions, i.e., preimages of meager sets are meager. We show that a $*$-reduction between orbit equivalence relations induces generically an embedding between the associated Becker graphs. We introduce a notion of dimension for Polish $G$-spaces which is generically preserved under $*$-reductions. For every natural number $n$ we define a free action of $S_{\infty}$ whose dimension is $n$ on every invariant Baire measurable non-meager set. We also show that the $S_{\infty}$-space which induces the equivalence relation $=^{+}$ of countable sets of reals is $\infty$-dimensional on every invariant Baire measurable non-meager set. We conclude that the orbit equivalence relations associated to all these actions are pairwise incomparable with respect to $*$-reductions.

math.LO↗

arXiv2021

Interpolative Fusions I

We define the interpolative fusion $T^*_\cup$ of a family $(T_i)_{i \in I}$ of first-order theories over a common reduct $T_\cap$, a notion that generalizes many examples of random or generic structures in the model-theoretic literature. When each $T_i$ is model-complete, $T^*_\cup$ coincides with the model companion of $T_\cup = \bigcup_{i \in I} T_i$. By obtaining sufficient conditions for the existence of $T^*_\cup$, we develop new tools to show that theories of interest have model companions.

math.LO↗

arXiv2021

Examples of weak amalgamation classes

We present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum-sized family of classes of finite undirected graphs, as well as an example due to Pouzet with countably categorical generic limit.

math.LO↗

arXiv2021

Tameness in least fixed-point logic and McColm's conjecture

We investigate four model-theoretic tameness properties in the context of least fixed-point logic over a family of finite structures. We find that each of these properties depends only on the elementary (i.e., first-order) limit theory, and we completely determine the valid entailments among them. In contrast to the context of first-order logic on arbitrary structures, the order property and independence property are equivalent in this setting. McColm conjectured that least fixed-point definability collapses to first-order definability exactly when proficiency fails. McColm's conjecture is known to be false in general. However, we show that McColm's conjecture is true for any family of finite structures whose limit theory is model-theoretically tame.

math.LO↗

arXiv2019

Independence in generic incidence structures

We study the theory $T_{m,n}$ of existentially closed incidence structures omitting the complete incidence structure $K_{m,n}$, which can also be viewed as existentially closed $K_{m,n}$-free bipartite graphs. In the case $m = n = 2$, this is the theory of existentially closed projective planes. We give an $\forall\exists$-axiomatization of $T_{m,n}$, show that $T_{m,n}$ does not have a countable saturated model when $m,n\geq 2$, and show that the existence of a prime model for $T_{2,2}$ is equivalent to a longstanding open question about finite projective planes. Finally, we analyze model theoretic notions of complexity for $T_{m,n}$. We show that $T_{m,n}$ is NSOP$_1$, but not simple when $m,n\geq 2$, and we show that $T_{m,n}$ has weak elimination of imaginaries but not full elimination of imaginaries. These results rely on combinatorial characterizations of various notions of independence, including algebraic independence, Kim independence, and forking independence.

math.LO↗

arXiv2018

Generic expansion and Skolemization in NSOP$_1$ theories

We study expansions of NSOP$_1$ theories that preserve NSOP$_1$. We prove that if $T$ is a model complete NSOP$_1$ theory eliminating the quantifier $\exists^{\infty}$, then the generic expansion of $T$ by arbitrary constant, function, and relation symbols is still NSOP$_1$. We give a detailed analysis of the special case of the theory of the generic $L$-structure, the model companion of the empty theory in an arbitrary language $L$. Under the same hypotheses, we show that $T$ may be generically expanded to an NSOP$_1$ theory with built-in Skolem functions. In order to obtain these results, we establish strengthenings of several properties of Kim-independence in NSOP$_1$ theories, adding instances of algebraic independence to their conclusions.

math.LO↗

arXiv2018

Disjoint $n$-amalgamation and pseudofinite countably categorical theories

Disjoint $n$-amalgamation is a condition on a complete first-order theory specifying that certain locally consistent families of types are also globally consistent. In this paper, we show that if a countably categorical theory $T$ admits an expansion with disjoint $n$-amalgamation for all $n$, then $T$ is pseudofinite. All theories which admit an expansion with disjoint $n$-amalgamation for all $n$ are simple, but the method can be extended, using filtrations of Fraïssé classes, to show that certain non-simple theories are pseudofinite. As case studies, we examine two generic theories of equivalence relations, $T^*_{\text{feq}}$ and $T_{\text{CPZ}}$, and show that both are pseudofinite. The theories $T^*_{\text{feq}}$ and $T_{\text{CPZ}}$ are not simple, but they are NSOP$_1$. This is established here for $T_{\text{CPZ}}$ for the first time.

math.LO↗

arXiv2017

Properly ergodic structures

We consider ergodic $\mathrm{Sym}(\mathbb{N})$-invariant probability measures on the space of $L$-structures with domain $\mathbb{N}$ (for $L$ a countable relational language), and call such a measure a properly ergodic structure when no isomorphism class of structures is assigned measure $1$. We characterize those theories in countable fragments of $\mathcal{L}_{ω_1, ω}$ for which there is a properly ergodic structure concentrated on the models of the theory. We show that for a countable fragment $F$ of $\mathcal{L}_{ω_1, ω}$ the almost-sure $F$-theory of a properly ergodic structure has continuum-many models (an analogue of Vaught's Conjecture in this context), but its full almost-sure $\mathcal{L}_{ω_1, ω}$-theory has no models. We also show that, for an $F$-theory $T$, if there is some properly ergodic structure that concentrates on the class of models of $T$, then there are continuum-many such properly ergodic structures.

math.LO↗

arXiv2016

Actions arising from intersection and union

An action is a pair of sets, $C$ and $S$, and a function $f\colon C\times S \to C$. Rothschild and Yalcin gave a simple axiomatic characterization of those actions arising from set intersection, i.e.\ for which the elements of $C$ and $S$ can be identified with sets in such a way that elements of $S$ act on elements of $C$ by intersection. We introduce and axiomatically characterize two natural classes of actions which arise from set intersection and union. In the first class, the $\uparrow\mathrel{\mspace{-2mu}}\downarrow$-actions, each element of $S$ is identified with a pair of sets $(s^\downarrow,s^\uparrow)$, which act on a set $c$ by intersection with $s^\downarrow$ and union with $s^\uparrow$. In the second class, the $\uparrow\mathrel{\mspace{-2mu}}\downarrow$-biactions, each element of $S$ is labeled as an intersection or a union, and acts accordingly on $C$. We give intuitive examples of these actions, one involving conversations and another a university's changing student body. The examples give some motivation for considering these actions, and also help give intuitive readings of the axioms. The class of $\uparrow\mathrel{\mspace{-2mu}}\downarrow$-actions is closely related to a class of single-sorted algebras, which was previously treated by Margolis et al., albeit in another guise (hyperplane arrangements), and we note this connection. Along the way, we make some useful, though very general, observations about axiomatization and representation problems for classes of algebras.

math.LO↗

arXiv2013

The Ax-Kochen Theorem: an application of model theory to algebra

The Ax-Kochen Theorem is a purely algebraic statement about the zeros of homogeneous polynomials over the p-adic numbers, but it was originally proved using techniques from mathematical logic. This document, the author's undergraduate honors thesis, provides an exposition of the theorem and its proof via model theory, assuming no previous experience with logic.

math.LO↗

arXiv2012

A Myhill-Nerode theorem for automata with advice

An automaton with advice is a finite state automaton which has access to an additional fixed infinite string called an advice tape. We refine the Myhill-Nerode theorem to characterize the languages of finite strings that are accepted by automata with advice. We do the same for tree automata with advice.

cs.FL↗

arXiv2008

Chains of distributions, hierarchical Bayesian models and Benford's Law

Kossovsky recently conjectured that the distribution of leading digits of a chain of probability distributions converges to Benford's law as the length of the chain grows. We prove his conjecture in many cases, and provide an interpretation in terms of products of independent random variables and a central limit theorem. An interesting consequence is that in hierarchical Bayesian models priors tend to satisfy Benford's Law as the number of levels of the hierarchy increases, which allows us to develop some simple tests (based on Benford's law) to test proposed models. We give explicit formulas for the error terms as sums of Mellin transforms, which converges extremely rapidly as the number of terms in the chain grows. We may interpret our results as showing that certain Markov chain Monte Carlo processes are rapidly mixing to Benford's law.

math.PR↗