Sheaf theory is a mathematical framework consisting of two or more layers: a topology endowed with data assignments, which can further be assigned more information such as probabilities. Inherently, it offers a computational model of syntax, semantics, and statistics all in one framework. Local and global quantitative measures can be defined over sheaves to calculate the distances between different interpretations at two or more of its layers. These properties have made sheaves desirable goto destinations for developing unified computational models, originally to unify different fields of mathematics (e.g. differential equations and logic), but also later of experimental data obtained form different sources (e.g. from structure and statistics). Different communities from afar fields have adopted them, examples range from analysis of engineering, e.g. for signal processing and in robotics; in science, most notably when analysing paradoxes of quantum theory, in the analysis of human behaviour in neuroscience, in social networks and also psycholinguistics, as well as recently even to coherent reasoning in large language models. The aim of this workshop we is to bring these different communities together.
The slides are now on available (see corresponding abstracts)
We are planning to go for a workshop dinner on Thursday 7th at Levarosa (Herner Str. 36, 44787 Bochum, Germany)
Schedule
(abstracts below)
Samson Abramsky, University College London
Anton Ayzenberg, Noeon Research
Daisuke Bekki, Ochanomizu University
Steve Huntsman, Cynnovative
14:00 - 14:30: Noam Zeilberger [slides]
Title: Generalized context-free grammars over categories and operads
Over recent years, in joint work with Paul-André Melliès, we have developed some fibrational perspectives on context-free grammars and finite-state automata, under which a generalized context-free grammar is just a functor from a finitely generated free operad into an arbitrary base operad. The classical case is recovered by taking the base to be a certain operad whose operations are "strings with gaps". One benefit of this perspective is that it interacts well with an analogous perspective on finite-state automata as certain kinds of functors, providing conceptual explanations for some of the properties of context-free and regular languages. Moreover, by taking different base operads, one naturally recovers some well-known extensions of the context-free regime including multiple context-free grammars.
Reference:
* The categorical contours of the Chomsky-Schützenberger representation theorem, LMCS 21:2, https://doi.org/10.46298/lmcs-21(2:12)2025
14:30 - 15:00: Colin Zwanziger [slides]
Title: Sheaf semantics for inquisitive logic
(joint work with Vít Punčochář)
Inquisitive logic is a logic intended to model questions, just as traditional logic models declarations. This logic has many interesting linguistic applications. First-order inquisitive logic was studied in, e.g., Grilletti (2020), and intuitionistic inquisitive logic was introduced in Punčochář (2017, 2019).
We give a sheaf-theoretic semantics for (higher-order, intuitionistic) inquisitive logic. This subsumes as special cases the classical possible-worlds model of inquisitive logic (Roelofsen 2013), a refinement of this based on a topological space of worlds, as well as other models with a topological flavor.
Our analysis extends the observation by Holliday (2020) that the language of (propositional, intuitionistic) inquisitive logic can be identified with that of (propositional, intuitionistic) logic, together with a geometric modality in the sense of Goldblatt (1981), also known as a lax modality. From the inquisitive perspective, the geometric modality is understood as extracting the presupposition behind a question. In the sheaf semantics, this modality is interpreted by a Lawvere-Tierney sheafification operator.
15:00 - 15:30: Anton Ayzenberg [slides]
Title: Sheaves and Grothendieck topologies in core CS
A Grothendieck topology is a structure defined over a category that makes it look like a topological space and opens a box of algebro-geometrical methods and tools - sheaves, global sections, cohomology - that found various geometrical applications. In logic, the category of sheaves over a Grothendieck topology provides the most important examples of topoi. Roughly speaking, Grothendieck sites are "a good place" to do logic.
May be hard to believe, but such abstract nonsense as Grothendieck topologies found a very concrete application in computer science, namely in the field of pattern matching. It was invented in the early 90's by Yellamraju Srinivas who extended the idea of the classical Knuth-Morris-Pratt string-searching algorithm to arbitrary data-structures. A bold claim: whatever you understand by a pattern, the problem of pattern-matching can be formulated as the problem of finding a global section of an appropriate sheaf over an appropriate Grothendieck topology. A global section is merged from local sections piece by piece - this is the main idea of sheaf theory.
Versions of this idea appear in many modern works on applications of sheaf theory, varying from logic to neural networks, from the study of natural language to cognitive studies. I will try to explain the basics of Grothendieck topologies and sheaves using concrete examples. Some modern ideas and philosophy will be discussed if time permits.
14:00 - 14:30: Steve Huntsman [slides]
Title: Motivating coherence-driven inference via sheaves
Coherence-driven inference (CDI) is a versatile model for many aspects of cognition. Pioneered by Paul Thagard in the nineties, CDI has been widely applied in cognitive and social sciences. We discuss how sheaf-theoretical considerations resulted in our simultaneous rediscovery and generalization of CDI from first principles (joint with Michael Robinson and Ludmilla Huntsman). We then briefly discuss how large language models enable automated CDI and an algorithmic benchmark indicating strong performance (joint with Jewell Thomas). This integration of LLMs and CDI is a promising neurosymbolic approach to artificial intelligence that uses LLMs for "fast" or "system 1" reasoning and combinatorial optimization for "slow" or "system 2" reasoning. To conclude, we will provide illustrative real-world examples of the approach.
14:30 - 14:50: Lachlan McPheat & Daisuke Bekki [slides1]
Title: Sheaves for Dependent Types and Natural Language Semantics
Dependent type theory (DTT) has proven a rich alternative formalism for natural language semantics, where an extension of DTT, namely Dependent Type Semantics (DTS), has shown utility in handling semantics of delicate linguistic phenomena. Sheaves, on the other hand have proven fruitful models for simpler type theories, whereas both the presheaves and stacks model dependent type theories. Recent work tries to condense stack models of dependent types into sheaf models of dependent types; however it remains an open question if there is a natural sheaf model of DTS for natural language semantics.
14:50 - 15:10: Tilen Limbäk-Stockin [slides]
Title: Quantum Phenomena in Natural Language Semantics
In recent times a new field of linguistics has emerged taking inspiration from physics, which has been dubbed Quantum Natural Language Processing (QNLP). So far in QNLP, the focus has been on variational quantum circuits (VQCs) and a succinct translation between linguistic structures and quantum circuits, primarily based on syntax. However, the particular relationship of quantum resources in QNLP is not yet well understood. Two resources of particular interest are the degree of entanglement of the semantic tensors, and the degree of contextuality of the quantum system. We focus on a downstream language task: pronominal anaphora resolution and develop a procedure for probing the presence of quantum resources in the circuits of these sentences. We observe that whenever there is information flow between a pronoun and a noun, the degrees of entanglement and contextuality of the corresponding circuits increase. This suggests that semantically informative linguistic relations, such as anaphors, are associated with measures of correlations in quantum systems.
15:10 - 15:30: Daphne Wang [slides]
Title: Improving surprisal-based models of parsing using a sheaf-theoretical approach
Prediction and reanalysis are considered two key processes that underly humans’ capacity to comprehend language in real time. Computational models capture it using Large Language Models (LLMs) and a statistical measure known as ‘surprisal’. Despite successes of LLMs, surprisal-based models face challenges when it comes to sentences requiring reanalysis due to pervasive temporary structural ambiguities, such as garden path sentences. We ask whether structural information can be extracted from LLM’s and develop a model that integrates it with their learnt statistics. When applied to a dataset of garden path sentences, the model achieved a significantly higher correlation with human reading times than surprisal. It also provided a better prediction of the garden path effect and could distinguish between sentence types with different levels of difficulty.
14:00 - 14:30: Nicola Pinzani [slides]
Title: The Topology of Causality
Quantum theory compels a generalization of the classical notion of causality, one in which conditional probability distributions no longer arise solely from ignorance about latent variables. Instead, their stochastic character frequently reflects the dependency of observed data on the structure of the contexts that define the experimental set-up. To meaningfully articulate these departures from determinism, one requires a generalized semantics for contextual correlations.
We present a framework for understanding conditional probability distributions associated in a contextual way with a broad class of causal structures. Drawing upon the sheaf-theoretic account of contextuality and non-locality, our approach offers a unified language for characterizing the non-classical features of quantum protocols. In doing so, we provide a foundation for quantifying and understanding the departure from classical causality in quantum mechanics.
14:30 - 15:00: Samson Abramsky [slides]
Title: Why Sheaves?
Sheaves were originally introduced as a tool in pure mathematics (differential equations and algebraic geometry), as the mathematics of the passage between local and global information.
Over the past few decades, they have increasingly been used as a tool in mathematical modelling of a diverse range of applications, including logic, semantics of computation, quantum foundations, quantum information and computation, computational linguistics, data networks, artificial intelligence and machine learning, and more.
We will try to identify some key features of sheaf theory which makes it amenable to and useful for these applications, and to raise some questions which may motivate further developments.
15:00 - 15:30: Farid Shahandeh [slides]
Title: Generalized Contextuality via equirank NMF
Generalized contextuality is a hallmark of nonclassical theories like quantum mechanics. It characterizes the fact that describing nonclassical phenomena using classical probabilistic models requires an overhead in the number of parameters. A theory is thus noncontextual if it admits a probabilistic model without an overhead. We present a reformulation of generalized contextuality as a matrix decomposition problem with a specific condition on the rank of its components, which we call equirank nonnegative matrix factorization (ENMF). We provide examples of theories which admit and do not admit such a decomposition. We briefly discuss the complexity of finding ENMFs.
This talk is based on the following two papers:
Characterizing Contextuality via Rank Separation with Applications to Cloning [2406.19382]
Complexity of Contextuality [2506.09133]