Publications (23)
Causal and Compositional Abstraction
Robin Lorenz, Sean Tull
Abstracting from a low level to a more explanatory high level of description, and ideally while preserving causal structure, is fundamental to scientific practice, to causal infere…
Deriving Dagger Compactness
Sean Tull
Dagger compact structure is a common assumption in the study of physical process theories, but lacks a clear interpretation. Here we derive dagger compactness from more operational…
Picture-perfect Quantum Key Distribution
Aleks Kissinger, Sean Tull, Bas Westerbaan
We provide a new way to bound the security of quantum key distribution using only two high-level, diagrammatic features of quantum processes: the compositional behavior of compleme…
Causal models in string diagrams
Robin Lorenz, Sean Tull
The framework of causal models provides a principled approach to causal reasoning, applied today across many scientific domains. Here we present this framework in the language of s…
Categories of relations as models of quantum theory
Chris Heunen, Sean Tull
Categories of relations over a regular category form a family of models of quantum theory. Using regular logic, many properties of relations over sets lift to these models, includi…
Towards Compositional Interpretability for XAI
Sean Tull, Robin Lorenz, Stephen Clark +2
Artificial intelligence (AI) is currently based largely on black-box machine learning models which lack interpretability. The field of eXplainable AI (XAI) strives to address this…
The Mathematical Structure of Integrated Information Theory
Johannes Kleiner, Sean Tull
Integrated Information Theory is one of the leading models of consciousness. It aims to describe both the quality and quantity of the conscious experience of a physical system, suc…
Quotient Categories and Phases
Sean Tull
We study properties of a category after quotienting out a suitable chosen group of isomorphisms on each object. Coproducts in the original category are described in its quotient by…
Space in Monoidal Categories
Pau Enrique Moliner, Chris Heunen, Sean Tull
The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which for…
Integrated Information in Process Theories
Sean Tull, Johannes Kleiner
We demonstrate how the key notions of Tononi et al.'s Integrated Information Theory (IIT) can be studied within the simple graphical language of process theories, i.e. symmetric mo…
Monoidal Categories for Formal Concept Analysis
Sean Tull
We investigate monoidal categories of formal contexts, in which states correspond to formal concepts. In particular we examine the category of bonds or Chu correspondences between…
The Conceptual VAE
Razin A. Shaikh, Sara Sabrina Zemljic, Sean Tull +1
In this report we present a new model of concepts, based on the framework of variational autoencoders, which is designed to have attractive properties such as factored conceptual d…
A Categorical Reconstruction of Quantum Theory
Sean Tull
We reconstruct finite-dimensional quantum theory from categorical principles. That is, we provide properties ensuring that a given physical theory described by a dagger compact cat…
Two Roads to Classicality
Bob Coecke, John Selby, Sean Tull
Mixing and decoherence are both manifestations of classicality within quantum theory, each of which admit a very general category-theoretic construction. We show under which condit…
Formalising and Learning a Quantum Model of Concepts
Sean Tull, Razin A. Shaikh, Sara Sabrina Zemljic +1
In this report we present a new modelling framework for concepts based on quantum theory, and demonstrate how the conceptual representations can be learned automatically from data.…
Monoidal characterisation of groupoids and connectors
Marino Gran, Chris Heunen, Sean Tull
We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids…
Operational Theories of Physics as Categories
Sean Tull
We introduce a new approach to the study of operational theories of physics using category theory. We define a generalisation of the (causal) operational-probabilistic theories of…
Categorical Operational Physics
Sean Tull
Many insights into the quantum world can be found by studying it from amongst more general operational theories of physics. In this thesis, we develop an approach to the study of s…
A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces
Sean Tull
We define a symmetric monoidal category modelling fuzzy concepts and fuzzy conceptual reasoning within Gärdenfors' framework of conceptual (convex) spaces. We propose log-concave…
From Conceptual Spaces to Quantum Concepts: Formalising and Learning Structured Conceptual Models
Sean Tull, Razin A. Shaikh, Sara Sabrina Zemljic +1
In this article we present a new modelling framework for structured concepts using a category-theoretic generalisation of conceptual spaces, and show how the conceptual representat…
Tensor Topology
Pau Enrique Moliner, Chris Heunen, Sean Tull
A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in ca…
Quantum-like Cognition in Process Theories: An Analysis
Sean Tull, Masanao Ozawa
Various effects in human cognition, often considered `non-classical', have been argued to be most naturally modelled by quantum-like models of decision making. We extend this appro…
Active Inference in String Diagrams: A Categorical Account of Predictive Processing and Free Energy
Sean Tull, Johannes Kleiner, Toby St Clere Smithe
We present a categorical formulation of the cognitive frameworks of Predictive Processing and Active Inference, expressed in terms of string diagrams interpreted in a monoidal cate…