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) AExp Pol -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.
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