A Homological Condition on Equational Unifiability

Author Mirai Ikebuchi

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Mirai Ikebuchi
  • Massachuseetts Institute of Technology, Cambridge, MA, USA


I would like to thank Assaf Kfoury and Keisuke Nakano for reading a draft of this paper and for their helpful suggestions.

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Mirai Ikebuchi. A Homological Condition on Equational Unifiability. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 61:1-61:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Equational unification is the problem of solving an equation modulo equational axioms. In this paper, we provide a relationship between equational unification and homological algebra for equational theories. We will construct a functor from the category of sets of equational axioms to the category of abelian groups. Then, our main theorem gives a necessary condition of equational unifiability that is described in terms of abelian groups associated with equational axioms and homomorphisms between them. To construct our functor, we use a ringoid (a category enriched over the category of abelian groups) obtained from the equational axioms and a free resolution of a "good" module over the ringoid, which was developed by Malbos and Mimram.

Subject Classification

ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
  • Equational unification
  • Homological algebra
  • equational theories


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