Current Work

Projects

Directed Type Theory
Paranatural Category Theory
Categorical Logic in Lean

Preprints

Talks

Type Theory

Modal Logic


Master's Thesis

J. Neumann. Semantics of Nondeterministic Construction. Master's thesis, Carnegie Mellon University, 2020.
Slides from thesis defense

My master's thesis combines insights from modal logic, formal epistemology, and theoretical computer science into a mathematically-ambitious analysis of a very simple activity: flipping a coin. Based on an insight by my thesis advisor, Adam Bjorndahl, I develop dynamic topological logic as a suitable language for formally articulating and reasoning about the capacities of knowledgeable agents in situations where nondeterministic actions are possible. I focus on situations where our agent is able to flip a coin to choose between two possible courses of action, and rigorously examine how the uncertainty created by the coin interacts with the agent's capacity for knowledge.

This work is primarily focused on the model theory of dynamic topological logic: "equipping the agent with a coin to flip" corresponds mathematically to a kind of transformation of dynamic topological structures, which I call a program constructor. To fully elucidate the theory of program constructors requires extensive mathematical development, including the introduction of several novel notions in the theory of dynamic topological models and frames. Along the way, I develop a deep connection between program construction in the dynamic topological setting, and analogous processes in the relational model theory of propositional dynamic logic. Though this thesis leaves much work to be done, it establishes a robust framework for future exploration of this very rich topic.