On the Complexity of Reasoning About Dynamic Policies

Abstract

We study the complexity of satisfiability for DLP$_{\mathrm{dyn}}^+$, an expressive logic introduced by Demri that allows to reason about dynamic policies. DLP$_{\mathrm{dyn}}^+$ extends the logic DLP_dyn of Pucella and Weissman, which in turn extends van der Meyden’s Dynamic Logic of Permission (DLP). DLP$_{\mathrm{dyn}}^+$ generously enhances DLP and DLP_dyn by allowing to update the policy set by adding or removing policy transitions, which are defined as a direct product of two sets, each specified by a formula of the logic itself. It is proven that satisfiability for DLP$_{\mathrm{dyn}}^+$ is complete for deterministic exponential time. Our results close the complexity gap of satisfiability for DLP$_{\mathrm{dyn}}^+$ from 2EXP, and for DLP_dyn from NEXP, to EXP respectively, matching the EXP lower bound both inherit from Propositional Dynamic Logic (PDL). To prove the EXP upper bound for DLP$_{\mathrm{dyn}}^+$, we first proceed by accurately identifying a suitable generalization of PDL, which allows to use compressed programs and then find a satisfiability preserving translation from DLP$_{\mathrm{dyn}}^+$ to this extension of PDL. To finally show the EXP upper bound for DLP$_{\mathrm{dyn}}^+$, we prove that satisfiability of our extension of PDL lies in EXP!.