A Temporal Logic of Normative Systems

Abstract

We study Normative Temporal Logic (ntl), a formalism intended for reasoning about the temporal properties of normative systems. ntl is a generalisation of the well-known branching-time temporal logic ctl, in which the path quantifiers ${\mathsf{A}}$ (“on all paths…”) and ${\mathsf {E}}$ (“on some path…”) are replaced by the indexed deontic operators ${\mathsf{O}}_{\eta}$ (“it is obligatory in the context of the normative system η that …”) and ${\mathsf{P}}_{\eta}$ (“it is permissible in the context of the normative system η that…”). After introducing the logic, we give a sound and complete axiomatisation. We then present a symbolic representation language for normative systems, and we identify four different model checking problems, corresponding to whether or not a model is represented symbolically or explicitly, and whether or not we are given a concrete interpretation for the normative systems named in formulae to be model checked. We show that the complexity of model checking varies from p-complete in the simplest case (explicit state model checking where we are given a specific interpretation for all normative systems in the formula) up to exptime-hard in the worst case (symbolic model checking, no interpretation given). We present examples to illustrate the use of ntl, and conclude with discussions of related work (in particular, the relationship of ntl to other deontic logics), and some issues for future work.