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.

中文翻译：

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模态代数普遍理论的复杂性

我们按照 Van Alten (Theor Comput Sci 501:82–92, 2013 ) 开发的方法，将偏代数理论应用于单调和正规模态代数通用理论的计算复杂性研究。我们展示了如何使用部分代数理论来分别获得单调和正规模态代数的通用理论的 co-NP 和 EXPTIME 上限。我们也得到了相应的下界，这意味着单调模态代数的普遍理论是co-NP-完全的，正规模态代数的普遍理论是EXPTIME-完全的。还可以推导出单调模态代数的拟方程理论是 co-NP 完全的。而正规模态代数的通用理论的 EXPTIME 上限可以通过更直接的方式获得，