Automatic Construction and Verification of Isotopy Invariants

Volker Sorge, Andreas Meier, Roy McCasland, Simon Colton

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation. Isotopism is an important generalisation of isomorphism, and is studied by mathematicians in domains such as loop theory. This extension was not straightforward, and we had to solve two major technical problems, namely, generating and verifying isotopy invariants. Concentrating on the domain of loop theory, we have developed three novel techniques for generating isotopic invariants, by using the notion of universal identities and by using constructions based on subblocks. In addition, given the complexity of the theorems that verify that a conjunction of the invariants form an isotopy class, we have developed ways of simplifying the problem of proving these theorems. Our techniques employ an interplay of computer algebra, model generation, theorem proving, and satisfiability-solving methods. To demonstrate the power of the approach, we generate isotopic classification theorems for loops of size 6 and 7, which extend the previously known enumeration results. This work was previously beyond the capabilities of automated reasoning techniques.
Original languageEnglish
Pages (from-to)221-243
Number of pages23
JournalJournal of Automated Reasoning
Issue number2-3
Publication statusPublished - Mar 2008

Keywords / Materials (for Non-textual outputs)

  • Automated mathematics
  • Automated theorem proving
  • SAT solving
  • Computer algebra
  • Model generation
  • Isotopy
  • Invariant generation
  • Classification theorems


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