{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/143377"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/143377","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Applications of Homological Algebra to Equational Theories","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Ikebuchi, Mirai"],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. 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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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Applications of Homological Algebra to Equational Theories"]}]}],"canonical_facts":{"dc:contributor.advisor":["Chlipala, Adam"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science"],"dc:creator":["Ikebuchi, Mirai"],"dc:date.accessioned":["2022-06-15T13:16:27Z"],"dc:date.available":["2022-06-15T13:16:27Z"],"dc:date.issued":["2022-02"],"dc:description.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."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/143377"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"dc:rights.uri":["http://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["Applications of Homological Algebra to Equational Theories"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral","Doctor of Philosophy"]},"updated_at":"2026-07-22T22:21:16Z"}