Complexity of Model Checking Second-Order Hyperproperties on Finite Structures

Abstract: We study the model checking problem of Hyper2LTL over finite structures. Hyper2LTL is a second-order hyperlogic, that extends the well-studied logic HyperLTL by adding quantification over sets of traces, to express complex hyperproperties such as epistemic and asynchronous hyperproperties. While Hyper2LTL is very expressive, its expressiveness comes with a price, and its general model checking problem is undecidable. This motivates us to study the model checking problem for Hyper2LTL over finite structures -- tree-shaped or acyclic graphs, which are particularly useful for monitoring purposes. We show that Hyper2LTL model checking is decidable on finite structures. It is in PSPACE (in the size of the model) on tree-shaped models and in EXPSPACE on acyclic models. Additionally, we show that for an expressive fragment of Hyper2LTL, namely the Fixpoint Hyper2LTLfp fragment, the model checking problem is much simpler and is P-complete on tree-shaped models and EXP-complete on acyclic models. Last, we present some preliminary results that take into account not only the size of the model, but also the formula size.