Algebra Discovery via Free Algebra Relations
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xu-charlsxu-sm-eecs-2026-thesis.pdf
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730.64 KB
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4e32c3bb758480c19aeef849ea211c67
Author(s)
Xu, Charles
Advisor(s)
Dunkel, Jörn
Date Issued
May 2026
Publisher
Massachusetts Institute of Technology
Abstract
Understanding the algebraic structure of a system reveals how its operations compose and when different experimental protocols are equivalent. This thesis studies how such structure can be recovered from data. A collection of experimental operations is modeled by the free algebra generated by those operations, while the represented operator algebra is interpreted as a quotient by the ideal of relations consisting of formal expressions that act identically under evaluation. This viewpoint turns algebra discovery into a relation-recovery problem. By evaluating formal expressions and analyzing the nullspaces of the resulting data matrices, algebraic relations among the underlying operations can be identified. Computational experiments in associative and nonassociative settings demonstrate that these relations can be recovered from evaluations alone, suggesting a route toward the data-driven discovery of new algebraic structures in experimental systems.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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