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Dan Marsden

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4 papers

2 author rows

MFCS Conference 2022 Conference Paper

Comonadic semantics for hybrid logic

Samson Abramsky
Dan Marsden

Hybrid logic is a widely-studied extension of basic modal logic, which corresponds to the bounded fragment of first-order logic. We study it from two novel perspectives: (1) We apply the recently introduced paradigm of comonadic semantics, which provides a new set of tools drawing on ideas from categorical semantics which can be applied to finite model theory, descriptive complexity and combinatorics. (2) We give a novel semantic characterization of hybrid logic in terms of invariance under disjoint extensions, a minimal form of locality. A notable feature of this result is that we give a uniform proof, valid for both the finite and infinite cases.

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FLAP Journal 2020 Journal Article

Semantic Spaces at the Intersection of NLP, Physics, and Cognitive Science.

Martha Lewis
Dan Marsden
Mehrnoosh Sadrzadeh

The ability to compose parts to form a more complex whole, and to analyze a whole as a combination of elements, is desirable across disciplines. Semantic Spaces at the Intersection of Natural Language Processing (NLP), Physics, and Cognitive Science brought together researchers applying similar compositional approaches within the three disciplines. The categorical model of [6], inspired by quantum protocols, has provided a convincing account of compositionality in vector space models of NLP. Similar category-theoretic approaches have been applied in cognitive science, in the context of conceptual spaces. The interplay between the three disciplines fostered theoretically motivated approaches to understanding how meanings of words interact in sentences and discourse, and how concepts develop in a cognitive space. This volume sees commonalities between the compositional mechanisms employed extracted, and applications and phenomena traditionally thought of as ‘non-compositional’ being shown to be compositional.

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TCS Journal 2018 Journal Article

Generalized relations in linguistics & cognition

Bob Coecke
Fabrizio Genovese
Martha Lewis
Dan Marsden
Alex Toumi

Categorical compositional models of natural language exploit grammatical structure to calculate the meaning of phrases and sentences from the meanings of individual words. More recently, similar compositional techniques have been applied to conceptual space models of cognition. Compact closed categories, particularly the category of finite dimensional vector spaces, have been the most common setting for categorical compositional models. When addressing a new problem domain, such as conceptual space models of meaning, a key problem is finding a compact closed category that captures the features of interest. We propose categories of generalized relations as a source of new, practical models for cognition and NLP. We demonstrate using detailed examples that phenomena such as fuzziness, metrics, convexity, semantic ambiguity can all be described by relational models. Crucially, by exploiting a technical framework described in previous work of the authors, we also show how the above-mentioned phenomena can be combined into a single model, providing a flexible family of new categories for categorical compositional modelling.

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CSL Conference 2018 Conference Paper

Quantitative Foundations for Resource Theories

Dan Marsden
Maaike Zwart

Considering resource usage is a powerful insight in the analysis of many phenomena in the sciences. Much of the current research on these resource theories focuses on the analysis of specific resources such quantum entanglement, purity, randomness or asymmetry. However, the mathematical foundations of resource theories are at a much earlier stage, and there has been no satisfactory account of quantitative aspects such as costs, rates or probabilities. We present a categorical foundation for quantitative resource theories, derived from enriched category theory. Our approach is compositional, with rich algebraic structure facilitating calculations. The resulting theory is parameterized, both in the quantities under consideration, for example costs or probabilities, and in the structural features of the resources such as whether they can be freely copied or deleted. We also achieve a clear separation of concerns between the resource conversions that are freely available, and the costly resources that are typically the object of study. By using an abstract categorical approach, our framework is naturally open to extension. We provide many examples throughout, emphasising the resource theoretic intuitions for each of the mathematical objects under consideration.

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