A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces

Tull, Sean

doi:10.4204/eptcs.372.22

Cited by 1 publication

(7 citation statements)

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“…The conceptual space framework has been formalized in a variety of ways (Aisbett & Gibbon, 2001;Rickard et al, 2007;Lewis & Lawry, 2016;Bechberger & Kühnberger, 2017). Here we draw on the treatment of fuzzy concepts from Tull (2021) which builds on the category-theoretic approach of Bolt et al (2019).…”

Section: Conceptual Spaces Formalisedmentioning

confidence: 99%

“…The value C(z) represents the degree to which z is an instance of the concept, with 0 meaning that the concept is not at all satisfied, and 1 meaning the concept is perfectly satisfied. The membership function should respect the existing convex structure in an appropriate sense, such as in the following definition proposed by one of the authors in Tull (2021).…”

Section: Conceptual Spaces Formalisedmentioning

confidence: 99%

“…Example 8. The category ConSp (Tull, 2021) is defined just like Prob except that the objects X, Y are now convex spaces and the morphisms f : X → Y are those (sub)kernels which are log-concave, meaning that they satisfy…”

Section: Example: Convex Spacesmentioning

confidence: 99%

“…fuzzy concepts. The log-concavity condition for channels (6) comes from the fact that it provides the most general way to form a monoidal category containing all such fuzzy concepts as effects; again we refer to Tull (2021) for a full derivation.…”

Section: Example: Convex Spacesmentioning

confidence: 99%

“…In this report we present a new modelling framework for concepts based on the mathematical formalism used in quantum theory, and demonstrate how the conceptual representations can be learned automatically from data. A contribution of the work is a thorough category-theoretic formalisation of our framework, following Bolt et al (2019) and Tull (2021). Formalisation of conceptual models is not new (Ganter & Obiedkov, 2016), but we claim that the use of category theory (Fong, 2019), and in particular the use of string diagrams to describe quantum processes (Coecke & Kissinger, 2017), helps elucidate some of the most important features of our quantum approach to concept modelling.…”

Section: Introductionmentioning

confidence: 99%

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Formalising and Learning a Quantum Model of Concepts

Tull¹,

Shaikh²,

Zemljič³

et al. 2023

Preprint

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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. A contribution of the work is a thorough category-theoretic formalisation of our framework. We claim that the use of category theory, and in particular the use of string diagrams to describe quantum processes, helps elucidate some of the most important features of our quantum approach to concept modelling. Our approach builds upon Gärdenfors' classical framework of conceptual spaces, in which cognition is modelled geometrically through the use of convex spaces, which in turn factorise in terms of simpler spaces called domains. We show how concepts from the domains of shape, colour, size and position can be learned from images of simple shapes, where individual images are represented as quantum states and concepts as quantum effects. Concepts are learned by a hybrid classical-quantum network trained to perform concept classification, where the classical image processing is carried out by a convolutional neural network and the quantum representations are produced by a parameterised quantum circuit. We also use discarding to produce mixed effects, which can then be used to learn concepts which only apply to a subset of the domains, and show how entanglement (together with discarding) can be used to capture interesting correlations across domains. Finally, we consider the question of whether our quantum models of concepts can be considered conceptual spaces in the Gärdenfors sense.

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