A Spatio-Temporal View of Knowledge

In this paper, we extend the well-known multi-modal lan-guage for knowledge of agents ‘faithfully’. That is, the framework we propose preserves in essence the modal way of describing knowledge. Two desirable properties arise from our approach. First, extending the language gives us clearly more expressive power, which is exploited for the develop-ment of a spatio-temporal view of knowledge, in particu-lar. And second, doing so modally enables us to prove some good-natured meta-properties of the new system. For a start, we integrate two additional features into the com-monly used logic of knowledge. The first one is as simple as natural: distinguished states are named, in fact, by nominals from hybrid logic. The second one is a new modal operator associated with every agent. Such a modality ascribes to the agent ‘access ’ to states that are complementary to his or her own view of the world and is, therefore, called a ‘distinction operator ’ (contrasting the knowledge operator which goes to-gether with indistinguishability of states). It turns out that distinction operators and hybrid names perfectly co-operate. In fact, we obtain completeness and decidability of the ap-pearing logical system as well as a corresponding complexity result. Then we add a spatio-temporal component to the just intro-duced hybrid logic of knowledge and distinction. To this end, an ‘effort operator ’ is incorporated, and the set of names is structured according to the additional dimension. We argue in favour of a formal basis for spatio-temporal epistemic reason-ing, provide a suitable axiomatization, and get a completeness theorem for the system derived from that, too