Knowledge and the Logic of Local ( Extended abstract ) Propositions

An agent's limited view of the state of a distributed system may render globally different situations indistinguishable. A proposition is local for this agent whenever his view suffices to decide this proposition. Motivated by a framework for the development of distributed programs from knowledge-based specifications, we introduce a modal logic of local propositions, in which it is possible to quantify over such propositions. We show that this logic is able to represent a rich set of epistemic notions. Under the usual strong semantics, this logic is not recursively axiomatizable, however. We show that by weakening the semantics of quantification, it is possible to obtain a logic that is axiomatizable and is still able to express interesting epistemic notions. 1 I n t r o d u c t i o n Yoram Moses 9 Alexander Yanal Tel Aviv 62382 Israel [email protected] Formal approaches to modelling knowledge have received considerable attention in the second half of the Twentieth century. In the philosophical literature, numerous models and axiomatizations were suggested and attacked, mainly in the 1960's and 1970's [vW51, Hin62, Get63, LenT8]. The modal logic $5, widely accepted as an appropriate characterization of necessity, has been attacked in many ways and is no longer considered by philosophers to be a good candidate for capturing knowledge. Since the 1970's formal treatments of knowledge and belief have been pursued in other fields. In economics [Aum76] and in the study of distributed computing systems [HM90, FHMV95], explicit concrete definitions for knowledge have been given and used in a variety of applications. Interestingly, the definitions in both cases are essentially equivalent, and define a notion of knowledge that satisfies $5. Very roughly, the essence of the definition used in economics and distributed systems can be described as follows. In every world, each agent is associated with a local state, a specific piece of information that completely determines the agent's knowledge: In world w, the world v is possible for agent i exactly if i has the same local state in both. The underlying structure immediately yields for each agent an equivalence relation (or partition) over the set of all worlds, and the properties of $5 follow. The properties of $5 are not the goal of the definition in this case, however. Rather, the definition is independently motivated, and a notion satisfying $5 is the outcome.