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This thesis introduces quantum natural language processing algorithms based on a simple yet powerful analogy between computational linguistics and quantum mechanics: grammar as entanglement. The grammatical system of text and sentences connects the word meaning in the same way that entanglement system links the states of quantum systems. The introduction of QNLP models fueled the creation of DisCoPy, the toolkit for applied category theory of which the first chapter provides a comprehensive review. DisCopy is used by DisCopy to create QNLP models as parameterised functors ranging from grammar to quantum circuits. It provides the first proof-of-concept for the more general concept of functorial learning by mapping diagram-like data. We introduce the notion of diagrammatic differentiation: a numerical method for determining parameter gradients of parameterised diagrams in order to determine optimal functor parameters by gradient descent.
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Integrated disease epidemiological modeling of infectious disease on a large scale is vital, but it is also difficult. Some of these difficulties can be alleviated by an approach that treats diagrams seriously as mathematical formalisms in their own right. We first describe categorical stock & flow diagrams, as well as the apparent distinction between stock & flow diagrams and their semantics, along with three examples of semantics already present in the software: ODEs, causal loop diagrams, and system structure diagrams. These frameworks can produce large diagrams from smaller ones in a modular fashion, according to category theory.
On September 2021, the University of Cambridge's Computer Laboratory hosted the fourth International Conference on Applied Category Theory. The contributions to ACT 2021 ranged from pure to applied, including: graph theory; game theory; numerical classification; ethnographics; and finite model theory. About half of the papers that were released as part of ACT 2021's talks include this proceedings volume.
In category theory, we propose a domain-specific model for buildings and evidences. The Yoneda lemma and Co-yoneda lemma are among the first type theoretic evidences establishing unit, tensor and function types, and can be seen as ordered refinements of theorems in predicate logic. Although the results in our type theory seem to be standard set-based arguments, the syntactic discipline guarantees that all proofs and constructions are also valid and internal settings.
Does X compute Y in principle given two mathematical units X and Y? Computability theory is a branch of computer science and mathematical logic where the key issue is: given two mathematical entities X and Y, does X compute Y in principle? Kleene introduced his S1-S9 computing methods to formalize estimation involving abstract objects. In turn, Dag Normann and the author have published a version of the lambda calculus involving fixed point operators that precisely captures S1-S9 and accommodates partial objects.
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