arXiv subjects

Nathanael Ackerman

Publications and source records attributed to Nathanael Ackerman.

At least 19 recordsLinked to original sources

arXiv2026

Forcing with Invariant Measures

This paper introduces a model-theoretic generalization of the notion of forcing with random reals, in which forcing gives rise to random generic structures. Specifically, we consider forcing with $κ$-Borel probability measures on the space of $L$-structures with a (possibly uncountable) infinite set $X$, focusing on those that are invariant under the action of the symmetric group $Sym(X)$. We demonstrate how any $Sym(X)$-invariant measure where $X$ is countable can be uniquely extended to a $Sym(Y)$-invariant measure where $Y$ is uncountable, and prove that forcing with such measures satisfies the countable chain condition. We also show that we can uniformly distinguish between these random generic structures and the Cohen generic structures that arise from forcing with a strong Fraïssé class: There is a $κ$-Borel set of low complexity that contains every Cohen generic structure that is not highly homogeneous but contains no random generic structure, implying that a structure that is not highly homogeneous cannot be both Cohen generic and random generic. Finally, we answer an open question of Kostana in the case of $ω_1$, by establishing a connection between forcing with a strong Fraïssé class and Cohen forcing.

math.LO↗

arXiv2026

On the computability of cofinal Fraïssé limits

For any collection of finite structures closed under isomorphism (i.e., an age) which has the Hereditary Property (HP), the Joint Embedding Property (JEP), and the Cofinal Amalgamation Property (CAP), there is a unique (up to isomorphism) countable structure which is cofinally ultrahomogeneous with the given age. Such a structure is called the cofinal Fraïssé limit of the age. In this paper, we consider the computational strength needed to construct the cofinal Fraïssé limit of a computable age. We show that this construction can always be done using the oracle 0''', and that there are ages that require 0''. In contrast, we show that if one assumes the strengthening of (CAP) known as the Amalgamation Property (AP), then the resulting limit, called the Fraïssé limit, can be constructed from the age using 0'. Our results therefore show that the more general case of cofinal Fraïssé limits requires greater computational strength than Fraïssé limits.

math.LO↗

arXiv2025

Generic sampling and invariant measures on the space of $k$-uniform hypergraphs

We prove a model-theoretic representation theorem for the distribution of an ergodic exchangeable $k$-uniform hypergraph: every such measure arises as the pushforward of the countably-iterated Morley product of a global Borel-definable Keisler measure over the countable universal homogeneous $k$-uniform hypergraph. We show this by starting with a Borel $k$-hypergraphon $W$ and constructing a Keisler measure $μ_{W}$ such that generic sampling with respect to $μ_{W}$ yields the same invariant measure as does the standard hypergraphon sampling procedure with respect to $W$. When $k = 2$, our results give a new representation theorem for ergodic exchangeable graphs via Keisler measures over a monster model of the Rado graph.

math.CO↗

arXiv2025

Structured Sunflowers

We call an infinite structure $\mathcal{M}$ sunflowerable if whenever $\mathcal{M}'$ is isomorphic to $\mathcal{M}$ with underlying set $M'$, consisting of finite sets of bounded size, there is an $M_0 \subseteq M'$ such that $M_0$ is a sunflower and $\mathcal{M}'\!\!\upharpoonright[M_0]$ is isomorphic to $\mathcal{M}$. We give sufficient conditions on $\mathcal{M}$ to show that $\mathcal{M}$ is sunflowerable. These conditions allow us to show that several well-known structures are sunflowerable and give a complete characterization of the countable linear orderings which are sunflowerable. We show that a sunflowerable structure must be indivisible. This allows us to show that any Fraïssé limit which has the 3-disjoint amalgamation property and a single unary type must be indivisible. In addition to studying sunflowerability of infinite structures, we also consider an analogous property of an age which we call the sunflower property. We show that any sunflowerable structure must have an age with the sunflower property. We also give concrete bounds in the case that the age has the hereditary property, the 3-disjoint amalgamation property, and is indivisible.

math.CO↗

arXiv2025

Rainbow Threshold Graphs

We define a generalization of threshold graphs which we call $k$-rainbow threshold graphs. We show that the collection of $k$-rainbow threshold graphs do not satisfy the $0$-$1$ law for first order logic and that asymptotically almost surely all $(k+1)$-rainbow threshold graphs are not isomorphic to a $k$-rainbow threshold graph.

math.CO↗

arXiv2025

Absoluteness of Fixed Points

We characterize those complete commutative positive linear ordered monoids $W$ such that whenever $f$ is a map from a Cauchy complete $W$-metric space to itself, the existence of a fixed point of $f$ is independent of the background model of set theory.

math.GN↗

arXiv2024

Cohen Generic Structures with Functions

Suppose $\mathscr{L}^-\subseteq \mathscr{L}$ are languages where $\mathscr{L} \setminus\mathscr{L}^-$ is relational. Additionally, let $\mathbf{K}$ be a strong $\textrm{Fraïssé}$ class in $\mathscr{L}$. We consider the partial ordering, under substructure, of those elements in $\mathbf{K}$ whose reduct to $\mathscr{L}^-$ are substructures of a fixed $\mathscr{L}^-$-structure $\mathcal{M}^-$. In this paper, we establish that, under general conditions, this partial order satisfies the $|\mathcal{M}^-|$-chain condition. Furthermore, under these conditions, we demonstrate that any generic for such a partial order satisfies the theory of the \Fraisse\ limit of $\mathbf{K}$, provided $\mathcal{M}^-$ satisfies the theory of$\textrm{Fraïssé}$ limit of its age. We also provide general conditions that guarantee all such generics to be rigid, as well as conditions ensuring that these generics possess large automorphism groups.

math.LO↗

arXiv2023

Computability of Countable Sunflowers

We provide a characterization of when a countably infinite set of finite sets contains an infinite sunflower. We also show that the collection of such sets is Turing equivalent to the set of programs such that whenever the program converges it returns the code of a program with finite range.

math.LO↗

arXiv2023

Algebraic Sunflowers

We study sunflowers within the context of finitely generated substructures of ultrahomogeneous structures. In particular, we look at bounds on how large a set system is needed to guarantee the existence of sunflowers of a given size. We show that if we fix the size of the sunflower, the function which takes the size of the substructures in our set system and outputs the size of a set system needed to guarantee a sunflower of the desired size can grow arbitrarily slowly.

math.CO↗

arXiv2021

On computable learning of continuous features

We introduce definitions of computable PAC learning for binary classification over computable metric spaces. We provide sufficient conditions for learners that are empirical risk minimizers (ERM) to be computable, and bound the strong Weihrauch degree of an ERM learner under more general conditions. We also give a presentation of a hypothesis class that does not admit any proper computable PAC learner with computable sample function, despite the underlying class being PAC learnable.

cs.LG↗

arXiv2021

Representations of Aut(M)-Invariant Measures

In this paper we generalize the Aldous-Hoover-Kallenberg theorem concerning representations of distributions of exchangeable arrays via collections of measurable maps. We give criteria when such a representation theorem exists for arrays which need only be preserved by a closed subgroup of the symmetric group over $\mathbb{N}$. Specifically, for a countable structure M, with underlying set the $\mathbb{N}$, we introduce the notion of an "Aut(M)-recipe", which is an Aut(M)-invariant array obtained via a collection of measurable functions indexed by the Aut(M)-orbits in M. We further introduce the notion of a "free structure" and then show that if M is free then every Aut(M)-invariant measure on an Aut(M)-space is the distribution of an Aut(M)-recipe. We also show that if a measure is the distribution of an Aut(M)-recipe it must be the restriction of a measure on a free structure.

math.LO↗

arXiv2021

On computable aspects of algebraic and definable closure

We investigate the computability of algebraic closure and definable closure with respect to a collection of formulas. We show that for a computable collection of formulas of quantifier rank at most $n$, in any given computable structure, both algebraic and definable closure with respect to that collection are $Σ^0_{n+2}$ sets. We further show that these bounds are tight.

math.LO↗

arXiv2019

Categoricity in multiuniversal classes

The third author has shown that Shelah's eventual categoricity conjecture holds in universal classes: class of structures closed under isomorphisms, substructures, and unions of chains. We extend this result to the framework of multiuniversal classes. Roughly speaking, these are classes with a closure operator that is essentially algebraic closure (instead of, in the universal case, being essentially definable closure). Along the way, we prove in particular that Galois (orbital) types in multiuniversal classes are determined by their finite restrictions, generalizing a result of the second author.

math.LO↗

arXiv2019

The entropy function of an invariant measure

Given a countable relational language $L$, we consider probability measures on the space of $L$-structures with underlying set $\mathbb{N}$ that are invariant under the logic action. We study the growth rate of the entropy function of such a measure, defined to be the function sending $n \in \mathbb{N}$ to the entropy of the measure induced by restrictions to $L$-structures on $\{0, \ldots, n-1\}$. When $L$ has finitely many relation symbols, all of arity $k\ge 1$, and the measure has a property called non-redundance, we show that the entropy function is of the form $Cn^k+o(n^k)$, generalizing a result of Aldous and Janson. When $k\ge 2$, we show that there are invariant measures whose entropy functions grow arbitrarily fast in $o(n^k)$, extending a result of Hatami-Norine. For possibly infinite languages $L$, we give an explicit upper bound on the entropy functions of non-redundant invariant measures in terms of the number of relation symbols in $L$ of each arity; this implies that finite-valued entropy functions can grow arbitrarily fast.

math.LO↗

arXiv2018

Stable regularity for relational structures

We generalize the stable graph regularity lemma of Malliaris and Shelah to the case of finite structures in finite relational languages, e.g., finite hypergraphs. We show that under the model-theoretic assumption of stability, such a structure has an equitable regularity partition of size polynomial in the reciprocal of the desired accuracy, and such that for each $k$-ary relation and $k$-tuple of parts of the partition, the density is close to either 0 or 1. In addition, we provide regularity results for finite and Borel structures that satisfy a weaker notion that we call almost stability.

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↗

arXiv2017

Countable infinitary theories admitting an invariant measure

Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $Σ$ of $\mathcal{L}_{ω_1, ω}(L)$ for which there exists an $S_\infty$-invariant probability measure on the collection of models of $Σ$ with underlying set $\mathbb{N}$. Restricting to $\mathcal{L}_{ω, ω}(L)$, this answers an open question of Gaifman from 1964, via a translation between $S_\infty$-invariant measures and Gaifman's symmetric measure-models with strict equality. It also extends the known characterization in the case where $Σ$ implies a Scott sentence. To establish our result, we introduce machinery for building invariant measures from a directed system of countable structures with measures.

math.LO↗

arXiv2017

On the computability of graph Turing machines

We consider graph Turing machines, a model of parallel computation on a graph, in which each vertex is only capable of performing one of a finite number of operations. This model of computation is a natural generalization of several well-studied notions of computation, including ordinary Turing machines, cellular automata, and parallel graph dynamical systems. We analyze the power of computations that can take place in this model, both in terms of the degrees of computability of the functions that can be computed, and the time and space resources needed to carry out these computations. We further show that properties of the underlying graph have significant consequences for the power of computation thereby obtained. In particular, we show that every arithmetically definable set can be computed by a graph Turing machine in constant time, and that every computably enumerable Turing degree can be computed in constant time and linear space by a graph Turing machine whose underlying graph has finite degree.

math.LO↗