Applications of Homological Algebra to Equational Theories
Name
Ikebuchi-ikebuchi-PhD-EECS-2022-thesis.pdf
Description
Thesis PDF
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808.41 KB
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Adobe PDF
Checksum (MD5)
4177aed4137f74d1b5ccfb15dfad0fec
Author(s)
Ikebuchi, Mirai
Advisor(s)
Chlipala, Adam
Date Issued
February 2022
Publisher
Massachusetts Institute of Technology
Abstract
It is well-known that some equational theories such as groups or Boolean algebras can be defined by fewer equational axioms than the original axioms. However, it is not easy to determine if a given set of axioms is the smallest or not. Malbos and Mimram investigated a general method to find a lower bound of the cardinality of the set of equational axioms (or rewrite rules) that is equivalent to a given equational theory (or term rewriting system), using homological algebra. Their method is an analog of Squier’s homology theory on string rewriting systems. In this dissertation, I develop the homology theory for term rewriting systems more and provide a better lower bound under a stronger notion of equivalence than their equivalence.
Also, the same methodology applies to equational unification, the problem of solving an equation modulo equational axioms. I provide a relationship between equational unification and homological algebra for equational theories. I will construct abelian groups associated with equational theories. Then, the main theorem gives a necessary condition of equational unifiability that is described in terms of the abelian groups and homomorphisms between them.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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