Big brother logic: logical modeling and reasoning about agents equipped with surveillance cameras in the plane

We consider multi-agent scenarios where each agent controls a surveillance camera positioned in the plane, with fixed position and angle of view, but rotating freely. The agents can thus observe the surroundings and each other. They can also reason about each other’s observation abilities and knowledge derived from these observations. We introduce suitable logical languages for reasoning about such scenarios which involve atomic formulae stating what agents can see, multi-agent epistemic operators for individual, distributed and common knowledge, as well as dynamic operators reflecting the ability of cameras to turn around in order to reach positions satisfying formulae in the language. We introduce semantics of our basic logic BBL and its extensions on natural geometric models, as well as formal Kripke semantics for them in vision-based finite abstractions of the geometric models. We discuss the expressiveness of our logical languages and provide their translations in PDL style. Using these translations we develop algorithms and obtain complexity results for model checking and satisfiability testing for BBL and its extensions. Notably, we show that even for the extension with common knowledge, model checking remains in PSPACE. Finally, we discuss some further extensions: by adding obstacles, positioning the cameras in 3D or enabling them to change positions.