Resolution-based decision procedures for the universal theory of some classes of distributive lattices with operators

In this paper we establish a link between satisfiability of universal sentences with respect to classes of distributive lattices with operators and satisfiability with respect to certain classes of relational structures. This justifies a method for structure-preserving translation to clause form of universal (Horn) sentences in such classes of algebras. We show that refinements of resolution yield decision procedures for the universal (Horn) theory of some such classes. In particular, we obtain exponential decision procedures for the universal Horn theory of (i) the class of all bounded distributive lattices with operators, (ii) for the class of all bounded distributive lattices with operators satisfying a set of (generalized) residuation conditions, and a doubly-exponential decision procedure for the universal Horn theory of the class of all Heyting algebras