Title
Random Variable Spaces: Mathematical Properties and an Extension to Programming Computable Functions
Date of Award
Fall 12-2023
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Computational and Data Sciences
First Advisor
M. Andrew Moshier
Second Advisor
Peter Jipsen
Third Advisor
Alexander Kurz
Abstract
This dissertation aims to extend the boundaries of Programming Computable Functions (PCF) by introducing a novel collection of categories referred to as Random Variable Spaces. Originating as a generalization of Quasi-Borel Spaces, Random Variable Spaces are rigorously defined as categories where objects are sets paired with a collection of random variables from an underlying measurable space. These spaces offer a theoretical foundation for extending PCF to natively handle stochastic elements.
The dissertation is structured into seven chapters that provide a multi-disciplinary background, from PCF and Measure Theory to Category Theory with special attention to Monads and the Giry Monad. The crux of the dissertation lies in the detailed exploration of Random Variable Spaces. Their mathematical properties are studied, including the establishment of relationships between these spaces through functors based on measurable functions. Additionally, the dissertation investigates the possibility of a fibration between the category of Measurable Spaces (Meas) and Random Variable Spaces. While such a fibration is not found, it offers an alternative fibration after particular alterations are made to Meas. Further, the work culminates in an extension to PCF, introducing computational rules to accommodate random variables. The dissertation provides a formal proof of correctness for these new rules, validating the extended framework's reliability.
The dissertation makes contributions to the fields of mathematical logic and computation. It not only extends PCF in a significant manner but also lays the groundwork for future research in Random Variable Spaces and their applicability to the theory of computation. With its focus on allowing PCF to natively handle stochastic elements, this dissertation aims to enrich our understanding of computation in an increasingly complex and probabilistic world.
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.
Recommended Citation
Firstinitial. Lastname, "Random variable spaces: Mathematical properties and an extension to programming computable functions," Ph.D. dissertation, Chapman University, Orange, CA, 2023. https://doi.org/10.36837/chapman.000516
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