Bratman’s Belief-Desire-Intention (BDI) theory is seminal in the literature on BDI agents. His BDI theory is taken into account to extend Shoham’s database perspective on beliefs and intentions. In the extended framework, an intentions is considered as a high-level action, which cannot be executed directly, with a duration. They have to be progressively refined until executable basic actions are obtained. Higher- and lower-level actions are linked by the means-end relation, alias instrumentality relation. In this paper, we investigate the complexity of the decision problems for satisfiability, consequence, refinement and instrumentality in the database. Moreover, we translate these problems into the satisfiability and validity problems in propositional linear temporal logic (\(\mathsf {PLTL}\)). With such translations, we can utilize the efficient automated theorem provers for \(\mathsf {PLTL}\) to solve the problem of deciding the refinement relation between an intention and an intention set, as well as the instrumentality relation.