Modal Logics with Composition on Finite Forests: Expressivity and Complexity (Extra Material)

We investigate the expressivity and computational complexity of two modal logics on finite forests equipped with operators to reason on submodels. The logic ML(|) extends the basic modal logic ML with the composition operator | from static ambient logic, whereas ML() contains the separating conjunction * from separation logic. Though both operators are second-order in nature, we show that ML(|) is as expressive as the graded modal logic GML (on finite trees) whereas ML() lies strictly between ML and GML. Moreover, we establish that the satisfiability problem for ML(*) is Tower-complete, whereas for ML(|) is (only) AExpPol-complete. As a by-product, we solve several open problems related to sister logics, such as static ambient logic, modal separation logic, and second-order modal logic on finite trees.