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Complexity of the Universal Theory of Residuated Ordered Groupoids

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Abstract

We study the computational complexity of the universal theory of residuated ordered groupoids, which are algebraic structures corresponding to Nonassociative Lambek Calculus. We prove that the universal theory is co\(\textsf {NP}\)-complete which, as we observe, is the lowest possible complexity for a universal theory of a non-trivial class of structures. The universal theories of the classes of unital and integral residuated ordered groupoids are also shown to be co\(\textsf {NP}\)-complete. We also prove the co\(\textsf {NP}\)-completeness of the universal theory of classes of residuated algebras, algebraic structures corresponding to Generalized Lambek Calculus.

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Notes

  1. In alternative terminology, residuated groupoids (Buszkowski, 2005).

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Acknowledgements

We are grateful to the anonymous reviewers for their comments, which have helped to improve the paper.

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  1. School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa

    Dmitry Shkatov & C. J. Van Alten

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  1. Dmitry Shkatov
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Shkatov, D., Van Alten, C.J. Complexity of the Universal Theory of Residuated Ordered Groupoids. J of Log Lang and Inf 32, 489–510 (2023). https://doi.org/10.1007/s10849-022-09392-9

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