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Automatic Construction and Verification of Isotopy Invariants

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Abstract

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.

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Authors and Affiliations

  1. School of Computer Science, University of Birmingham, Birmingham, UK

    Volker Sorge

  2. DFKI GmbH, Saarbrücken, Germany

    Andreas Meier

  3. School of Informatics, University of Edinburgh, Edinburgh, UK

    Roy McCasland

  4. Department of Computing, Imperial College London, London, UK

    Simon Colton

Authors
  1. Volker Sorge
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  2. Andreas Meier
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  3. Roy McCasland
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  4. Simon Colton
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Corresponding author

Correspondence to Simon Colton.

Additional information

The author’s work was supported by EPSRC MathFIT grant GR/S31099.

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Sorge, V., Meier, A., McCasland, R. et al. Automatic Construction and Verification of Isotopy Invariants. J Autom Reasoning 40, 221–243 (2008). https://doi.org/10.1007/s10817-007-9093-y

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  • DOI: https://doi.org/10.1007/s10817-007-9093-y

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