A full separation theorem for the derivable rules of intuitionistic linear logic without bounds, 0 and exponentials is proved. Several structural consequences of this theorem for subreducts of (commutative) residuated lattices are obtained. The theorem is then extended to the logic LR+ and its proof is extended to obtain the finite embeddability property for the class of square increasing residuated lattices.

We apply the theory of partial algebras, following the approach developed by Van Alten, 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. (shrink)

In the original publication of the article, the authors name were abbreviated as “D. Shkatov” and “C. J. Van Alten”. However it should be “Dmitry Shkatov” and “Clint J. Van Alten”. The original article has been corrected.