Date of Thesis

Spring 2026

Description

Code refactoring is a fundamental practice in software engineering, in which a program is restructured without changing the actions it performs and the results it produces. To carry out refactoring with confidence, one requires a formal method for verifying that two programs are equivalent. Guarded Kleene Algebra with Tests (GKAT) provides such a framework, an algebraic system designed to reason about a natural class of programs, namely those in which every branch and loop is governed by a Boolean condition, such as if–else and while statements. Central to GKAT is a finite set of algebraic axioms for deriving program equivalences. These axioms are known to be sound, meaning that any equivalence derivable from the axioms holds in every model of program behavior. The converse property, completeness — that every true equivalence between programs can be derived from the axioms — has remained an open problem since the system was introduced by Smolka et al. in 2019.

This thesis works toward a completeness theorem for the finite axiom system of GKAT and makes two contributions. First, we show that the three loop axioms of Smolka et al. are derivable from a single axiom, reducing the system to a smaller and more tractable core. Second, we establish a uniqueness theorem for solutions of the equation systems associated with Thompson-generated automata, showing that any two solutions of the same automaton are provably equal within the axiom system. This result opens a new direction toward the completeness theorem, reducing the remaining work to showing that a solution exists for the relevant automaton. It also yields additional insight into the equational characterization of G-automata.

Keywords

Guarded Kleene Algebra with Tests, completeness theorem, automata, bisimulation, program equivalence, equational reasoning

Access Type

Honors Thesis

Degree Type

Bachelor of Science in Computer Science and Engineering

Major

Mathematics

First Advisor

Peter Brooksbank

Second Advisor

Todd Schmid

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