On Composing Finite Forests with Modal Logics

Bartosz Bednarczyk, Stéphane Demri, Raul Fervari, Alessio Mansutti

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  \(\mathsf {ML} ({\color{black}{{\vert\!\!\vert\!\vert}}})\) extends the modal logic K with the composition operator  \({\color{black}{{\vert\!\!\vert\!\vert}}}\) 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  \(\mathsf {ML} ({\color{black}{{\vert\!\!\vert\!\vert}}})\) 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) AExpPol-complete for  \(\mathsf {ML} ({\color{black}{{\vert\!\!\vert\!\vert}}})\) , a result that 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.