On Composing Finite Forests with Modal Logics

Abstract

We study the expressivity and complexity of two modal logics interpreted on finite forests and equipped with standard modalities to reason on submodels. The logic \(\mathsf {ML} (\,\operatorname{\raisebox {-2pt}{\rule {1.2pt}{2.1ex}}}\,) \) extends the modal logic K with the composition operator \(\operatorname{\raisebox {-2pt}{\rule {1.2pt}{2.1ex}}} \) from ambient logic, whereas \(\mathsf {ML} (\mathbin {\ast }) \) features the separating conjunction \(\mathbin {\ast } \) from separation logic. Both operators are second-order in nature. We show that \(\mathsf {ML} (\,\operatorname{\raisebox {-2pt}{\rule {1.2pt}{2.1ex}}}\,) \) is as expressive as the graded modal logic \(\mathsf {GML} \) (on trees) whereas \(\mathsf {ML} (\mathbin {\ast }) \) is strictly less expressive than \(\mathsf {GML} \). Moreover, we establish that the satisfiability problem is Tower-complete for \(\mathsf {ML} (\mathbin {\ast }) \), whereas it is (only) AExp\(_{\text{\textsc {Pol}}} \)-complete for \(\mathsf {ML} (\,\operatorname{\raisebox {-2pt}{\rule {1.2pt}{2.1ex}}}\,) \), a result which is surprising given their relative expressivity. As by-products, we solve open problems related to sister logics such as static ambient logic and modal separation logic.