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Complexity of the Universal Theory of Modal Algebras

  • D. ShkatovEmail author
  • C. J. Van Alten
Article
  • 16 Downloads

Abstract

We apply the theory of partial algebras, following the approach developed by Van Alten (Theor Comput Sci 501:82–92, 2013), to the study of the computational complexity of universal theories of monotonic and normal modal algebras. We show how the theory of partial algebras can be deployed to obtain co-NP and EXPTIME upper bounds for the universal theories of, respectively, monotonic and normal modal algebras. We also obtain the corresponding lower bounds, which means that the universal theory of monotonic modal algebras is co-NP-complete and the universal theory of normal modal algebras is EXPTIME-complete. It also follows that the quasi-equational theory of monotonic modal algebras is co-NP-complete. While the EXPTIME upper bound for the universal theory of normal modal algebras can be obtained in a more straightforward way, as discussed in the paper, due to its close connection to the equational theory of normal modal algebras with the universal modality operator, the technique based on the theory of partial algebras is applicable to the study of universal theories of algebras corresponding to a wide range of logics with intensional operators, where no such connection is available.

Keywords

Universal theory Complexity Monotonic modal algebra Normal modal algebra Partial algebra Quasi-equational theory 

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Notes

Acknowledgements

We are grateful to the anonymous referees for suggestions that improved the presentation of this article.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Computer Science and Applied MathematicsUniversity of the Witwatersrand, JohannesburgWitsSouth Africa

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