String Diagrams for Quantum Foundations, Computing and Natural Language Processing

TL;DR

This thesis employs string diagrams from applied category theory to explore quantum foundations, computing, and natural language processing, demonstrating formalizations, conflict analyses, and language circuit mappings.

Contribution

It introduces novel formalizations of constructor theory, wave-based logic circuits, and language circuit frameworks using string diagrams, advancing interdisciplinary applications.

Findings

Categorical quantum mechanics can be viewed as a constructor theory of quantum physics.

Wave-based logic circuits with phase encoding are effective for Boolean circuit design.

Urdu language circuits can be mapped to DisCoCirc frameworks, aligning Urdu and English circuits.

Abstract

Applied category theory provides powerful mathematical tools for modelling processes and their composition. Symmetric monoidal categories, which involve series and parallel composition, are particularly well-suited for describing the composition of processes in space and time. Also called process theories, they admit string diagrams, which constitute a visually intuitive, mathematically rigorous, expressive and flexible syntax that is applicable to wide-ranging scientific domains. In this thesis, we employ string diagrams to investigate a selection of topics in the areas of quantum foundations, computing, and natural language processing: (1) We formalise constructor theory as a process theory. In the context of quantum physics, we also demonstrate the conflict between constructor-theoretic principles of locality and composition. Moreover, we argue that if the principle of locality is rejected, categorical quantum mechanics (CQM) can be conceived as a constructor theory of quantum physics. (2) We develop a formalism for wave-based logic circuits with phase encoding. We motivate the formalism using the example of spin-wave circuits, and then demonstrate its utility in design, analysis and optimisation of Boolean logic circuits. (3) We investigate the elimination of inter-language grammatical bureaucracy in the distributional compositional circuits (DisCoCirc) framework. In particular, we develop a hybrid grammar for a restricted fragment of the Urdu language, and show that Urdu text endowed with this hybrid grammar maps surjectively to DisCoCirc text circuits. Furthermore, we show that for the same language fragment, Urdu and English text circuits become the same up to gate-level translation. The aforementioned work supports the view that a process-relational outlook in science is well-supported by applied category-theoretic tools, particularly string diagrams.