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The expressivity of update logics

We prove two new results about logics involving updates and common knowledge. The first result is that the logic LAU∗ using Arrow Common Knowledge is more expressive than the logic LAR using Relativized Common Knowledge. The second result is that the logic LAUC using Arrow Updates and normal Common Knowledge is equally expressive as LAU∗ . Together with previously known results this fully determines the expressivity landscape of all logics involving any combination of normal Common Knowledge (C), Relativized Common Knowledge (R), Arrow Common Knowledge (U∗), Public Announcements (P) and Arrow Updates (U).