Subset Space Logics of Knowledge and Time
In the present paper we study several logics of knowledge and effort, which originate from the modal system for subset spaces due to Moss and Parikh [Moss and Parikh 1992], [Dabrowski et al. 1996]. Our crucial means is the functional interpretation of the modal operator varying over subsets. In this way a "topological" single step-operator appears. Because of its closeness to the nexttime-operator in common temporal logic it will be named in this way subsequently. First we treat the case of a single knower. We obtain generalizations of certain fragments of propositional temporal logic to the subset space logic of knowledge. Both linear and branching time structures are considered. Afterwards the methods and results are extended to the multi-agent case. We are mainly interested in the basic topics concerning logics in computer science: completeness, decidability and complexity. In the multi-agent case, however, we examine also notions which only occur if more than one agent is present such as common knowledge.
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