arXiv subjects

David Gonzalez

Publications and source records attributed to David Gonzalez.

At least 19 recordsLinked to original sources

arXiv2026

Tennenbaum-like theorems for cohesive powers

We investigate the encoding ability of the cohesive power construction. We compute a graph $\mathcal{G}$ where the cohesive power $\prod_C \mathcal{G}$ of $\mathcal{G}$ by any $\Delta_2$ cohesive set $C$ has degree $0''$. That is, $0''$ computes a presentation of $\prod_C \mathcal{G}$, and every presentation of $\prod_C \mathcal{G}$ computes $0''$. We also compute a linear order $\mathcal{L}$ where no cohesive power of $\mathcal{L}$ has a computable presentation. We accomplish this by ensuring that if $\mathcal{P}$ is a presentation of a cohesive power of $\mathcal{L}$, then $\mathcal{P}''$ has $\mathrm{PA}$-degree relative to $0''$.

math.LO↗

arXiv2026

Data-free neural PDE solvers based on Graph Neural Networks and weak forms

We present a physics-informed, data-free neural solver for partial differential equations, built on a graph neural network architecture that utilises message passing. By relying on the weak form of the problem, we use gradients of finite-element shape functions (which are therefore polynomials) rather than automatic differentiation operators to compute the residuals of the equation from the displacements predicted by the network itself. Our approach generalises to previously unseen load cases and geometries, achieving easily convergence errors in the residuals of less than 1% and being capable of scaling up to models of considerable size and arbitrary geometries. To ensure compliance with the laws of physics and provide guarantees regarding the inference, it is possible to use the residual itself as an error indicator for the inference, and thus perform a refinement at the testing stage if the residual tolerance set in advance by the user is not met. Examples are provided to demonstrate the performance of the proposed method. This results in a method that avoids the costly process of obtaining, curating and storing high-fidelity synthetic data for training the neural network. Whilst this is not unique to our method, it is the first time it has been combined with a geometric machine learning technique capable of providing the necessary geometric bias to overcome the well-known difficulties of physics-informed neural networks.

cs.CE↗

arXiv2026

Infinite Belligerent Jump Inversion and Computable Scott Analysis

Scott analysis provides two fundamental tools for studying countable structures: Scott sentences, which characterize structures up to isomorphism, and back-and-forth relations, which measure structural similarity. A recurring phenomenon in computable structure theory is that many notions naturally associated with level $\alpha$ of Scott analysis have effective complexity at approximately $2\alpha$ jumps. This discrepancy appears both in the complexity of the back-and-forth relations and in the passage from arbitrary infinitary formulas to computable infinitary formulas. We develop two new coding tools, the Belligerent Pairs Theorem and Belligerent Jump Inversion Theorem, which allow information at complexity level $2\alpha$ to be reflected in computable structures whose distinguishing features already appear at level $\alpha$. These results extend Harrison-Trainor's finite unfriendly jump inversion uniformly throughout the computable ordinals. As applications, we determine the optimal interaction between syntactic complexity and oracle complexity for computable Scott sentences and for formulas distinguishing computable structures. For every computable infinite ordinal $\alpha$, we determine the oracle needed to compute a $\Pi_\alpha$ Scott sentence for a computable structure which has a $\Pi_\alpha$ Scott sentence. Any computable structure with a $\Pi_\alpha$ Scott sentence has a computable $\Pi_{2\alpha}$ Scott sentence. We show that both of these bounds are sharp. We prove analogous optimal results for formulas witnessing failure of the $\alpha$-back-and-forth relation. We also obtain further applications, including a resolution of a question of Chen, Gonzalez, and Harrison-Trainor concerning the complexity of back-and-forth classes.

math.LO↗

arXiv2026

Scott Analysis below the Vaught Ordinal

We develop new tools for determining the existence of models of specific Scott ranks under countability conditions. Using these, we improve a result of Sacks by showing that any counterexample to Vaught's conjecture must have at least two models of every parameterized Scott rank -- a result that contrasts with the unparameterized case, where minimal counterexamples have only one model at many ranks. We further prove that theories with fewer than continuum many models have trivial Scott spectra and provide a general, systematic classification of low Scott rank models when only countably many $\Sigma_\alpha$-types are realized. Additionally, we classify the Scott complexity spectra for many Ehrenfeucht theories, and prove the $\omega$-Vaught's conjecture in this setting, answering an infinitary strengthening of a question of Pillay and Tanovi\'c. We demonstrate that the Scott complexity of prime models for $\omega$-stable first-order theories is commensurate with the complexity of the theory itself. Along the way, we apply our methods to concrete theories like p-groups, trees, and Boolean algebras, answering questions of Harris--Montalb\'an and Alvir--Csima--MacLean regarding specific structures.

math.LO↗

arXiv2026

Computable Scott Sentences and the Friedman-Stanley embedding

Friedman and Stanley developed the notion of Borel reducibility and illustrated its use in comparing classification problems for some familiar classes of countable structures. For many embeddings, the fact that the embedding is $1-1$ on isomorphism types is explained by the existence of simple formulas that, uniformly, interpret the input structure in the output structure. For the embeddings of graphs in trees, and in linear orderings, there is no uniform interpretation. We focus on a version of the Friedman-Stanley embedding introduced by Harrison-Trainor and Montalban that takes each structure $A$ for the language of graphs to a labeled tree $T_A$. Gonzalez and Rossegger showed that this embedding preserves Scott complexity. We refine this result, showing that for an $X$-computable ordinal, if one of $A$, $T_A$ has a computable infinitary Scott sentence, then so does the other, and the complexities match. Let $\mathbb{T}$ be the class of labeled trees isomorphic to those in the range of the embedding, and let $\mathbb{T}^\alpha$ be the subclass consisting of structures of Scott rank at most $\alpha$. It follows from results of Gao that $\mathbb{T}$ is not Borel. We show that for each $\alpha$, $\mathbb{T}^\alpha$ is Borel. In fact, if $\alpha$ is an $X$-computable ordinal, then $\mathbb{T}^\alpha$ is complete $X$-effective $\Pi_{2\alpha+2}$.

math.LO↗

arXiv2026

Enumerative Combinatorics of Homogeneous Linear Orderings

We count the number of countable homogeneous colored linear orderings in $k$ colors. Relatedly, we count the number of countable $C_{n,m}$-homogeneous linear orderings. $C_{n,m}$-homogeneity is a strong homogeneity notion that approximates $sp-$homogeneity, a notion recently uncovered in [2] to have important computability theoretic properties. Explicit formulas are derived for both of the quantities in question, along with asymptotic bounds. The objects being counted are generally infinite, and it is not obvious that there are even only finitely many. This fact, along with the more precise counting, is demonstrated by corresponding the linear orderings with finite objects.

math.CO↗

arXiv2026

Variational Graph Neural Networks for Uncertainty Quantification in Inverse Problems

The increasingly wide use of deep machine learning techniques in computational mechanics has significantly accelerated simulations of problems that were considered unapproachable just a few years ago. However, in critical applications such as Digital Twins for engineering or medicine, fast responses are not enough; reliable results must also be provided. In certain cases, traditional deterministic methods may not be optimal as they do not provide a measure of confidence in their predictions or results, especially in inverse problems where the solution may not be unique or the initial data may not be entirely reliable due to the presence of noise, for instance. Classic deep neural networks also lack a clear measure to quantify the uncertainty of their predictions. In this work, we present a variational graph neural network (VGNN) architecture that integrates variational layers into its architecture to model the probability distribution of weights. Unlike computationally expensive full Bayesian networks, our approach strategically introduces variational layers exclusively in the decoder, allowing us to estimate cognitive uncertainty and statistical uncertainty at a relatively lower cost. In this work, we validate the proposed methodology in two cases of solid mechanics: the identification of the value of the elastic modulus with nonlinear distribution in a 2D elastic problem and the location and quantification of the loads applied to a 3D hyperelastic beam, in both cases using only the displacement field of each test as input data. The results show that the model not only recovers the physical parameters with high precision, but also provides confidence intervals consistent with the physics of the problem, as well as being able to locate the position of the applied load and estimate its value, giving a confidence interval for that experiment.

cs.LG↗

arXiv2026

MeshGraphNet-Transformer: Scalable Mesh-based Learned Simulation for Solid Mechanics

We present MeshGraphNet-Transformer (MGN-T), a novel architecture that combines the global modeling capabilities of Transformers with the geometric inductive bias of MeshGraphNets, while preserving a mesh-based graph representation. MGN-T overcomes a key limitation of standard MGN, the inefficient long-range information propagation caused by iterative message passing on large, high-resolution meshes. A physics-attention Transformer serves as a global processor, updating all nodal states simultaneously while explicitly retaining node and edge attributes. By directly capturing long-range physical interactions, MGN-T eliminates the need for deep message-passing stacks or hierarchical, coarsened meshes, enabling efficient learning on high-resolution meshes with varying geometries, topologies, and boundary conditions at an industrial scale. We demonstrate that MGN-T successfully handles industrial-scale meshes for impact dynamics, a setting in which standard MGN fails due message-passing under-reaching. The method accurately models self-contact, plasticity, and multivariate outputs, including internal, phenomenological plastic variables. Moreover, MGN-T outperforms state-of-the-art approaches on classical benchmarks, achieving higher accuracy while maintaining practical efficiency, using only a fraction of the parameters required by competing baselines.

cs.LG↗

arXiv2025

sp-Homogeneous Linear Orderings

We study linear orderings expanded by functions for successor and predecessor. The successor and predecessor on linear orderings capture the relatively intrinsically computably enumerable information about orderings in much the same way that dependence captures that for vector spaces. In particular, the sp-homogeneous and weakly sp-homogeneous linear orderings are those which are (ultra-)homogeneous or weakly homogeneous with this additional structure. We demonstrate that these orderings are always relatively $\Delta_4$ categorical and determine exactly which ones are (uniformly) relatively $\Delta_3$ categorical. We also provide a classification for sp-homogeneity and weak sp-homogeneity. We establish that this is the best possible classification by showing that the set of sp-homogeneous linear orderings is $\Pi_5^0$ complete, and that the set of weakly sp-homogeneous linear orderings is $\Sigma_6^0$ complete. These results are obtained in two different ways, one using a hands-on computability theoretic approach and another using more abstract descriptive set theory.

math.LO↗

arXiv2025

Computability of Separation Axioms in Countable Second Countable Spaces

We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic completeness result for each of the Tychonoff separation axioms. Beyond this, we prove completeness results for various other topological properties, such as being Polish and having a particular Cantor-Bendixson rank, using tools from computable structure theory. This work contrasts with previous work analyzing countable, second countable spaces which used the framework of reverse mathematics, as reverse mathematics generally lacks the precision to pin down exact arithmetic complexity levels for properties of interest.

math.LO↗

arXiv2025

Classifying the complexity of models of arithmetic

We classify the possible Scott complexities for models of Peano arithmetic. We construct models of particular complexities by first giving a complete Scott analysis of colored linear orderings and constructing models of Peano arithmetic from these colored orderings. We also provide tight connections of certain Scott complexities with notions from the classical theory of models of Peano arithmetic, such as prime, finitely generated, and recursively saturated. This effort provides a powerful set of tools to understand the models of Peano arithmetic.

math.LO↗

arXiv2025

On the under-reaching phenomenon in message-passing neural PDE solvers: revisiting the CFL condition

This paper proposes sharp lower bounds for the number of message passing iterations required in graph neural networks (GNNs) when solving partial differential equations (PDE). This significantly reduces the need for exhaustive hyperparameter tuning. Bounds are derived for the three fundamental classes of PDEs (hyperbolic, parabolic and elliptic) by relating the physical characteristics of the problem in question to the message-passing requirement of GNNs. In particular, we investigate the relationship between the physical constants of the equations governing the problem, the spatial and temporal discretisation and the message passing mechanisms in GNNs. When the number of message passing iterations is below these proposed limits, information does not propagate efficiently through the network, resulting in poor solutions, even for deep GNN architectures. In contrast, when the suggested lower bound is satisfied, the GNN parameterisation allows the model to accurately capture the underlying phenomenology, resulting in solvers of adequate accuracy. Examples are provided for four different examples of equations that show the sharpness of the proposed lower bounds.

cs.LG↗

arXiv2025

Optimal Syntactic Definitions of Back-and-Forth Types

The back-and-forth relations $M\leq_\alpha N$ are central to computable structure theory and countable model theory. It is well-known that the relation $\{(M,N) : M \leq_\alpha N\}$ is (lightface) $\Pi^0_{2\alpha}$. We show that this is optimal as the set is $\mathbf{\Pi}^0_{2\alpha}$-complete. We are also interested in the one-sided relations $\{ N : M \leq_\alpha N\}$ and $\{ N : M \geq_\alpha N\}$ for a fixed $M$, measuring the $\Pi_\alpha$ and $\Sigma_\alpha$ types of $M$. We show that these sets are always $\mathbf{\Pi}^0_{\alpha + 2}$ and $\mathbf{\Pi}^0_{\alpha+3}$ respectively, and that for most $\alpha$ there are structures $M$ for which these relations are complete at that level. In particular, there are structures $M$ such that there is no $\Pi_\alpha$ (or even $\Pi_{\alpha+1})$ sentence $\varphi$ such that $N \models \varphi \Longleftrightarrow M \leq_\alpha N$. This is unfortunate as not all $\Pi_{\alpha+2}$ sentences are preserved under $\leq_\alpha$. We define a new hierarchy of syntactic complexity closely related to the back-and-forth game, which can both define the back-and-forth types as well as be preserved by them. These hierarchies of formulas have already been useful in certain Henkin constructions, one of which we give in this paper, and another previously used by Gonzalez and Harrison-Trainor to show that every $\Pi_\alpha$ theory of linear orders has a model with Scott rank at most $\alpha+3$.

math.LO↗

arXiv2025

Sunrise III: Overview of Observatory and Instruments

In July 2024, Sunrise completed its third successful science flight. The Sunrise III observatory had been upgraded significantly after the two previous successful flights in 2009 and 2013. Three completely new instruments focus on the small-scale physical processes and their complex interaction from the deepest observable layers in the photosphere up to chromospheric heights. Previously poorly explored spectral regions and lines are exploited to paint a three-dimensional picture of the solar atmosphere with unprecedented completeness and level of detail. The full polarimetric information is captured by all three instruments to reveal the interaction between the magnetic fields and the hydrodynamic processes. Two slit-based spectropolarimeters, the Sunrise UV Spectropolarimeter and Imager (SUSI) and the Sunrise Chromospheric Infrared spectro-Polarimeter (SCIP), focus on the near-ultraviolet and the near-infrared regions respectively, and the imaging spectropolarimeter Tunable Magnetograph (TuMag) simultaneously obtains maps of the full field-of-view of $46 \times 46$ Mm$^2$ in the photosphere and the chromosphere in the visible. The instruments are operated in an orchestrated mode, benefiting from a new Image Stabilization and Light Distribution unit (ISLiD), with the Correlating Wavefront Sensor (CWS) providing the autofocus control and an image stability with a root-mean-square value smaller than 0.005''. A new gondola was constructed to significantly improve the telescope pointing stability, required to achieve uninterrupted observations over many hours. Sunrise III was launched successfully on July 10, 2024, from the Esrange Space Center near Kiruna (Sweden). It reached the landing site between the Mackenzie River and the Great Bear Lake in Canada after a flight duration of 6.5 days. In this paper, we give an overview of the Sunrise III observatory and its instruments.

astro-ph.IM↗

arXiv2024

Scott Spectral Gaps are Bounded for Linear Orderings

We demonstrate that any $\Pi_\alpha$ sentence of the infinitary logic $L_{\omega_1 \omega}$ extending the theory of linear orderings has a model with a $\Pi_{\alpha+4}$ Scott sentence and hence of Scott rank at most $\alpha+3$. In other words, the gap between the complexity of the theory and the complexity of the simplest model is always bounded by $4$. This contrasts the situation with general structures where for any $\alpha$ there is a $\Pi_2$ sentence all of whose models have Scott rank $\alpha$. We also give new lower bounds, though there remains a small gap between our lower and upper bounds: For most (but not all) $\alpha$, we construct a $\Pi_\alpha$ sentence extending the theory of linear orderings such that no models have a $\Sigma_{\alpha+2}$ Scott sentence and hence no models have Scott rank less than or equal to $\alpha$.

math.LO↗

arXiv2024

On the feasibility of foundational models for the simulation of physical phenomena

We explore the feasibility of foundation models for the simulation of physical phenomena, with emphasis on continuum (solid and fluid) mechanics. Although so-called learned simulators have shown some success when applied to specific tasks, it remains to be studied to what extent they are able to undergo severe changes in domain shape, boundary conditions and/or constitutive laws and still provide robust (i.e., hallucination-free) and accurate results. In this paper we perform an exhaustive study of these features, put ourselves in the worst-case scenario and study their resistance to such strong changes in their domain of application.

cs.CE↗

arXiv2024

Scott analysis, linear orders and almost periodic functions

For any limit ordinal $\lambda$, we construct a linear order $L_\lambda$ whose Scott complexity is $\Sigma_{\lambda+1}$. This completes the classification of the possible Scott sentence complexities of linear orderings. Previously, there was only one known construction of any structure (of any signature) with Scott complexity $\Sigma_{\lambda+1}$, and our construction gives new examples, e.g., rigid structures, of this complexity. Moreover, we can construct the linear orders $L_\lambda$ so that not only does $L_\lambda$ have Scott complexity $\Sigma_{\lambda+1}$, but there are continuum-many structures $M \equiv_\lambda L_\lambda$ and all such structures also have Scott complexity $\Sigma_{\lambda+1}$. In contrast, we demonstrate that there is no structure (of any signature) with Scott complexity $\Pi_{\lambda+1}$ that is only $\lambda$-equivalent to structures with Scott complexity $\Pi_{\lambda+1}$. Our construction is based on functions $f \colon \mathbb{Z}\to \mathbb{N}\cup \{\infty\}$ which are almost periodic but not periodic, such as those arising from shifts of the $p$-adic valuations.

math.LO↗

arXiv2024

Hybrid Maximal Filter Spaces

We introduce a new way of encoding general topology in second order arithmetic that we call hybrid maximal filter (hybrid MF) spaces. This notion is a modification of the notion of a proper MF space introduced by Montalb\'an. We justify the shift by showing that proper MF spaces are not able to code most topological spaces, while hybrid MF spaces can code any second countable MF space. We then answer Montalb\'an's question about metrization of well-behaved MF spaces to this shifted context. To be specific, we show that in stark contrast to the original MF formalization used by Mummert and Simpson, the metrization theorem can be proven for hybrid MF spaces within $\text{ACA}_0$ instead of needing $\Pi_2^1-\text{CA}_0$.

math.LO↗