Quantifying over coalitions in epistemic logic

Some natural epistemic properties which may arise in applications can only be expressed in standard epistemic logic by formulae which are exponentially long in the number of agents in the system. An example is the property “at least m agents know that at most n agents know φ”. We present Epistemic Logic with Quantification over Coalitions (ELQC), where the standard common knowledge operator has been replaced allowing expressions of the form 〈P〉Cφ and [P ]Cφ where P is a coalition predicate, meaning that there is a coalition satisfying P which have common knowledge of φ and that all coalitions satisfying P have common knowledge of φ, respectively; and similarly for distributed knowledge and everybody-knows. While the language is no more expressive than standard epistemic logic, it is exponentially more succinct. We give a sound and complete axiomatisation for ELQC, and characterise the complexity of its model checking problem.