Many-Sorted Equivalence of Shiny and Strongly Polite Theories

Abstract

Herein we close the question of the equivalence of shiny and strongly polite theories by establishing that, for theories with a decidable quantifier-free satisfiability problem, the set of many-sorted shiny theories coincides with the set of many-sorted strongly polite theories. Capitalizing on this equivalence, we obtain a Nelson–Oppen combination theorem for many-sorted shiny theories.

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Notes

  1. 1.

    A correctness proof of the method was presented by Tinelli and Harandi [15].

  2. 2.

    As in [10], we do not restrict Y to be the set of variables in \({\mathsf {s}}\textsf {-}{\mathsf {witness}}\) since this generality is needed to show Lemma A.2 and Theorem 3.7 of [10].

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Acknowledgements

We would like to thank the anonymous reviewers for their helpful comments. This work was partially supported by Fundação para a Ciência e a Tecnologia by way of Grant UID/MAT/04561/2013 to Centro de Matemática, Aplicações Fundamentais e Investigação Operacional of Universidade de Lisboa (CMAF-CIO). Furthermore, FC acknowledges the support from the DP-PMI and FCT (Portugal) through scholarship SRFH/BD/52243/2013.

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Correspondence to Filipe Casal.

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Casal, F., Rasga, J. Many-Sorted Equivalence of Shiny and Strongly Polite Theories. J Autom Reasoning 60, 221–236 (2018). https://doi.org/10.1007/s10817-017-9411-y

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Keywords

  • Nelson–Oppen method
  • Combination of satisfiability procedures
  • Shiny theories
  • Polite theories
  • Strongly polite theories
  • First-order logic
  • Many-sorted logic