arXiv subjects

Valeria de Paiva

Publications and source records attributed to Valeria de Paiva.

At least 19 recordsLinked to original sources

arXiv2026

KnowTeX: Visualizing Mathematical Dependencies

Dependency graphs that show how definitions, theorems, and proofs relate to each other are valuable for understanding the structure of mathematical texts. Existing tools such as Lean Blueprint and plasTeXdepgraph generate such graphs within formal proof ecosystems, but they require familiarity with proof assistants or specific compilation pipelines. We present KnowTeX, a standalone Python tool that extracts dependency graphs directly from LaTeX sources without requiring any external framework. KnowTeX supports two complementary modes: a manual mode where authors annotate their source with lightweight commands compatible with Lean Blueprint, and an infer mode that automatically discovers dependencies through a layered system of deterministic and heuristic rules. The tool handles multi-file projects, detects cycles, applies transitive reduction, and exports graphs in DOT, TikZ, and PNG formats with an interactive preview. We evaluate KnowTeX on several mathematical texts and discuss how it complements recent tools such as LeanArchitect, which operates from the Lean side, while KnowTeX works entirely on the LaTeX side without requiring any formalization.

cs.HC↗

arXiv2026

Dialectica Categories over Heyting Algebras

Categorification---the process of constructing a categorical model of a piece of mathematics---often identifies a common abstraction that connects formerly unrelated but known structures. In the case of de Paiva's categorification of Gödel's Dialectica interpretation, we find that its specialization to partial orders produces (functorial) embeddings of Heyting algebras into residuated lattices that appear to have been overlooked. For the non-categorical audience, we present this specialization and take care to reproduce the original proofs in the algebraic setting. Along the way we obtain results particular to this algebraic setting: an embedding lacking an evident adjoint in de Paiva's general construction acquires a definable one here; a single Dialectica tensor validates contraction in the intuitionistic construction D yet refutes it in the classical variant G; and, over ZF, the poset reflection PD(Set) collapses onto the four-element algebra PD(2) exactly when the Axiom of Choice holds.

math.LO↗

arXiv2026

Proceedings 21st International Symposium on Logical and Semantic Frameworks with Applications

This volume contains the proceedings of the 21st Workshop on Logical and Semantic Frameworks with Applications (LSFA 2026), held in Lisbon, Portugal, from July 18 to July 19, 2026. LSFA aims to promote techniques and results from the theoretical side, ranging from well-established ones such as lambda calculus and type theory to state-of-the-art ones such as machine learning, and provide feedback on integrating, implementing, and using such methods and results from the practical side.

cs.LO↗

arXiv2025

Dialectica Petri Nets

The categorical modeling of Petri nets has received much attention recently. The Dialectica construction has also had its fair share of attention. We revisit the use of the Dialectica construction as a categorical model for Petri nets generalising the original application to suggest that Petri nets with different kinds of transitions can be modelled in the same categorical framework. Transitions representing truth-values, probabilities, rates or multiplicities, evaluated in different algebraic structures called lineales are useful and are modelled here in the same category. We investigate (categorical instances of) this generalised model and its connections to more recent models of categorical nets.

math.CT↗

arXiv2025

Math Natural Language Inference: this should be easy!

We ask whether contemporary LLMs are able to perform natural language inference (NLI) tasks on mathematical texts. We call this the Math NLI problem. We construct a corpus of Math NLI pairs whose premises are from extant mathematical text and whose hypotheses and gold labels were provided by people with experience in both research-level mathematics and also in the NLI field. We also investigate the quality of corpora using the same premises but whose hypotheses are provided by LLMs themselves. We not only investigate the performance but also the inter-group consistency of the diverse group of LLMs. We have both positive and negative findings. Among our positive findings: in some settings, using a majority vote of LLMs is approximately equivalent to using human-labeled data in the Math NLI area. On the negative side: LLMs still struggle with mathematical language. They occasionally fail at even basic inferences. Current models are not as prone to hypothesis-only "inference" in our data the way the previous generation had been. In addition to our findings, we also provide our corpora as data to support future work on Math NLI.

cs.CL↗

arXiv2025

Categorifying computable reducibilities

This paper presents categorical formulations of Turing, Medvedev, Muchnik, and Weihrauch reducibilities in Computability Theory, utilizing Lawvere doctrines. While the first notions lend themselves to a smooth categorical presentation, essentially dualizing the traditional idea of realizability doctrines, Weihrauch reducibility and its extensions to represented and multi-represented spaces require a separate investigation. Our abstract analysis of these concepts highlights a shared characteristic among all these reducibilities. Specifically, we demonstrate that all these doctrines stemming from computability concepts can be proven to be instances of completions of quantifiers for doctrines, analogous to what occurs for doctrines for realizability. As a corollary of these results, we will be able to formally compare Weihrauch reducibility with the dialectica doctrine constructed from a doctrine representing Turing degrees.

math.LO↗

arXiv2024

Skolem, Gödel, and Hilbert fibrations

Grothendieck fibrations are fundamental in capturing the concept of dependency, notably in categorical semantics of type theory and programming languages. A relevant instance are Dialectica fibrations which generalise Gödel's Dialectica proof interpretation and have been widely studied in recent years. We characterise when a given fibration is a generalised, dependent Dialectica fibration, namely an iterated completion of a fibration by dependent products and sums (along a given class of display maps). From a technical perspective, we complement the work of Hofstra on Dialectica fibrations by an internal viewpoint, categorifying the classical notion of quantifier-freeness. We also generalise both Hofstra's and Trotta et al.'s work on Gödel fibrations to the dependent case, replacing the class of cartesian projections in the base category by arbitrary display maps. We discuss how this recovers a range of relevant examples in categorical logic and proof theory. Moreover, as another instance, we introduce Hilbert fibrations, providing a categorical understanding of Hilbert's $ε$- and $τ$-operators well-known from proof theory.

math.CT↗

arXiv2024

Mathematical Entities: Corpora and Benchmarks

Mathematics is a highly specialized domain with its own unique set of challenges. Despite this, there has been relatively little research on natural language processing for mathematical texts, and there are few mathematical language resources aimed at NLP. In this paper, we aim to provide annotated corpora that can be used to study the language of mathematics in different contexts, ranging from fundamental concepts found in textbooks to advanced research mathematics. We preprocess the corpora with a neural parsing model and some manual intervention to provide part-of-speech tags, lemmas, and dependency trees. In total, we provide 182397 sentences across three corpora. We then aim to test and evaluate several noteworthy natural language processing models using these corpora, to show how well they can adapt to the domain of mathematics and provide useful tools for exploring mathematical language. We evaluate several neural and symbolic models against benchmarks that we extract from the corpus metadata to show that terminology extraction and definition extraction do not easily generalize to mathematics, and that additional work is needed to achieve good performance on these metrics. Finally, we provide a learning assistant that grants access to the content of these corpora in a context-sensitive manner, utilizing text search and entity linking. Though our corpora and benchmarks provide useful metrics for evaluating mathematical language processing, further work is necessary to adapt models to mathematics in order to provide more effective learning assistants and apply NLP methods to different mathematical domains.

cs.CL↗

arXiv2024

Towards a Brazilian History Knowledge Graph

This short paper describes the first steps in a project to construct a knowledge graph for Brazilian history based on the Brazilian Dictionary of Historical Biographies (DHBB) and Wikipedia/Wikidata. We contend that large repositories of Brazilian-named entities (people, places, organizations, and political events and movements) would be beneficial for extracting information from Portuguese texts. We show that many of the terms/entities described in the DHBB do not have corresponding concepts (or Q items) in Wikidata, the largest structured database of entities associated with Wikipedia. We describe previous work on extracting information from the DHBB and outline the steps to construct a Wikidata-based historical knowledge graph.

cs.AI↗

arXiv2023

MathGloss: Building mathematical glossaries from text

MathGloss is a project to create a knowledge graph (KG) for undergraduate mathematics from text, automatically, using modern natural language processing (NLP) tools and resources already available on the web. MathGloss is a linked database of undergraduate concepts in mathematics. So far, it combines five resources: (i) Wikidata, a collaboratively edited, multilingual knowledge graph hosted by the Wikimedia Foundation, (ii) terms covered in mathematics courses at the University of Chicago, (iii) the syllabus of the French undergraduate mathematics curriculum which includes hyperlinks to the automated theorem prover Lean 4, (iv) MuLiMa, a multilingual dictionary of mathematics curated by mathematicians, and (v) the nLab, a wiki for category theory also curated by mathematicians. MathGloss's goal is to bring together resources for learning mathematics and to allow every mathematician to tailor their learning to their own preferences. Moreover, by organizing different resources for learning undergraduate mathematics alongside those for learning formal mathematics, we hope to make it easier for mathematicians and formal tools (theorem provers, computer algebra systems, etc) experts to "understand" each other and break down some of the barriers to formal math.

cs.CL↗

arXiv2023

Extracting Mathematical Concepts with Large Language Models

We extract mathematical concepts from mathematical text using generative large language models (LLMs) like ChatGPT, contributing to the field of automatic term extraction (ATE) and mathematical text processing, and also to the study of LLMs themselves. Our work builds on that of others in that we aim for automatic extraction of terms (keywords) in one mathematical field, category theory, using as a corpus the 755 abstracts from a snapshot of the online journal "Theory and Applications of Categories", circa 2020. Where our study diverges from previous work is in (1) providing a more thorough analysis of what makes mathematical term extraction a difficult problem to begin with; (2) paying close attention to inter-annotator disagreements; (3) providing a set of guidelines which both human and machine annotators could use to standardize the extraction process; (4) introducing a new annotation tool to help humans with ATE, applicable to any mathematical field and even beyond mathematics; (5) using prompts to ChatGPT as part of the extraction process, and proposing best practices for such prompts; and (6) raising the question of whether ChatGPT could be used as an annotator on the same level as human experts. Our overall findings are that the matter of mathematical ATE is an interesting field which can benefit from participation by LLMs, but LLMs themselves cannot at this time surpass human performance on it.

cs.CL↗

arXiv2023

Parmesan: mathematical concept extraction for education

Mathematics is a highly specialized domain with its own unique set of challenges that has seen limited study in natural language processing. However, mathematics is used in a wide variety of fields and multidisciplinary research in many different domains often relies on an understanding of mathematical concepts. To aid researchers coming from other fields, we develop a prototype system for searching for and defining mathematical concepts in context, focusing on the field of category theory. This system, Parmesan, depends on natural language processing components including concept extraction, relation extraction, definition extraction, and entity linking. In developing this system, we show that existing techniques cannot be applied directly to the category theory domain, and suggest hybrid techniques that do perform well, though we expect the system to evolve over time. We also provide two cleaned mathematical corpora that power the prototype system, which are based on journal articles and wiki pages, respectively. The corpora have been annotated with dependency trees, lemmas, and part-of-speech tags.

cs.CL↗

arXiv2023

Extracting Blockchain Concepts from Text

Blockchains provide a mechanism through which mutually distrustful remote parties can reach consensus on the state of a ledger of information. With the great acceleration with which this space is developed, the demand for those seeking to learn about blockchain also grows. Being a technical subject, it can be quite intimidating to start learning. For this reason, the main objective of this project was to apply machine learning models to extract information from whitepapers and academic articles focused on the blockchain area to organize this information and aid users to navigate the space.

cs.IR↗

arXiv2022

Extracting Mathematical Concepts from Text

We investigate different systems for extracting mathematical entities from English texts in the mathematical field of category theory as a first step for constructing a mathematical knowledge graph. We consider four different term extractors and compare their results. This small experiment showcases some of the issues with the construction and evaluation of terms extracted from noisy domain text. We also make available two open corpora in research mathematics, in particular in category theory: a small corpus of 755 abstracts from the journal TAC (3188 sentences), and a larger corpus from the nLab community wiki (15,000 sentences).

cs.CL↗

arXiv2022

Dialectica Principles via Gödel Doctrines

Gödel's Dialectica interpretation was conceived as a tool to obtain the consistency of Peano arithmetic via a proof of consistency of Heyting arithmetic in the 40s. In recent years, several proof-theoretic transformations, based on Gödel's Dialectica interpretation, have been used systematically to extract new content from classical proofs, following a suggestion of Kreisel. Thus, the interpretation has found new relevant applications in several areas of mathematics and computer science. Several authors have explained the Dialectica interpretation in categorical terms. In our previous work, we introduced an intrinsic categorical presentation of the Dialectica construction via a generalisation of Hofstra's work, using the notion of Gödel fibration and its proof-irrelevant version, a Gödel doctrine. The key idea is that Gödel fibrations can be thought of as fibrations generated by some basic elements playing the role of quantifier-free elements. This categorification of quantifier-free elements is crucial not only to show that our notion of Gödel fibration is equivalent to Hofstra's Dialectica fibration in the appropriate way, but also to show how Gödel doctrines embody the main logical features of the Dialectica Interpretation. To show that, we derive the soundness of the interpretation of the implication connective, as expounded by Troelstra, in the categorical model. This requires extra logical principles, going beyond intuitionistic logic, namely Markov Principle and the Independence of Premise principle, as well as some choice. We show how these principles are satisfied in the categorical setting, establishing a tight correspondence between the logical system and the categorical framework. Finally, to complete our analysis, we characterise categories obtained as results of the tripos-to-topos of Hyland, Johnstone and Pitts applied to Gödel doctrines.

math.CT↗

arXiv2022

Kolmogorov-Veloso Problems and Dialectica Categories

We investigate the categorical connection between Dialectica constructions, Kolmogorov problems, Veloso problems and Blass problems. We show that the work of Kolmogorov can be regarded as a bridge between Veloso's abstract notion of a problem and the conceptual problems Blass discussed in his questions-and-answers framework. This bridge can be seen by means of the categorical Dialectica constructions introduced in de Paiva's dissertation and reformulated by da Silva to account for set-theoretical foundational assumptions. The use of categorical concepts allows us to provide several examples, connecting extremely different areas of mathematics, and using simple methods. This paper also shows that while Blass and Kolmogorov notions of problem can be investigated using the Zermelo-Fraenkel (ZF) set-theoretical framework, Veloso problems require the Axiom of Choice (AC). Moreover, weaker notions of choice (dependent choice and countable choice) can also be accounted for in the problems' framework.

math.LO↗

arXiv2021

Proceedings Second Joint International Workshop on Linearity & Trends in Linear Logic and Applications

This volume contains a selection of papers presented at Linearity&TLLA 2020, namely the Second Joint International Workshop on Linearity & Trends in Linear Logic and Applications, held on June 29-30, 2020 online. (The workshop was supposed to take place in Paris as part of FSCD 2020, but due to the COVID pandemic it was decided not to hold the event live.) Linearity is a central concept in many theoretical and practical approaches to computer science. On the theoretical side, there is much work stemming from linear logic and dealing with resource control, complexity classes, and more recently quantum computation. On the practical side there is certainly work on program analysis, operational semantics, logic programming languages, program transformations, and efficient implementation techniques. Linear logic is not only a theoretical tool for the analysis of resource usage in logic and computation. It is also a corpus of distinct approaches and methodologies (e.g., proof nets, geometry of interaction, coherent spaces, relational models) that were originally developed for the study of linear logic's syntax and semantics have nowadays found applications in several other fields.

cs.LO↗

arXiv2021

Dialectica Logical Principles

Gödel's Dialectica interpretation was designed to obtain a relative consistency proof for Heyting arithmetic, to be used in conjunction with the double negation interpretation to obtain the consistency of Peano arithmetic. In recent years, proof theoretic transformations (so-called proof interpretations) that are based on Gödel's Dialectica interpretation have been used systematically to extract new content from proofs and so the interpretation has found relevant applications in several areas of mathematics and computer science. Following our previous work on Gödel fibrations, we present a (hyper)doctrine characterisation of the Dialectica which corresponds exactly to the logical description of the interpretation. To show that we derive in the category theory the soundness of the interpretation of the implication connective, as expounded on by Spector and Troelstra. This requires extra logical principles, going beyond intuitionistic logic, Markov's Principle (MP) and the Independence of Premise (IP) principle, as well as some choice. We show how these principles are satisfied in the categorical setting, establishing a tight (internal language) correspondence between the logical system and the categorical framework. This tight correspondence should come handy not only when discussing the applications of the Dialectica already known, like its use to extract computational content from (some) classical theorems (proof mining), its use to help to model specific abstract machines, etc. but also to help devise new applications.

math.CT↗