TLA + is a specification language based on standard set theory and temporal logic that has constructs for hierarchical proofs. We describe how to write TLA + proofs and check them with TLAPS, the TLA + Proof System. We use Peterson's mutual exclusion algorithm as a simple example to describe the features of TLAPS and show how it and the Toolbox (an IDE for TLA + ) help users to manage large, complex proofs.1 The TLA + module containing the specification and proof is accessible at the TLAPS Web page [20].2 The figure shows the pretty-printed version of PlusCal code and TLA + formulas. As an example of how they are typed, here is the ascii version of the variables declaration: variables flag = [i \in {0, 1} |-> FALSE], turn = 0;

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Abstract. Many puzzling social behaviors, such as avoiding eye contact, using innuendos, and insignificant events that trigger revolutions, seem to relate to common knowledge and coordination, but the exact relationship has yet to be formalized. Herein, we present such a formalization. We state necessary and sufficient conditions for what we call state-dependent equilibria -equilibria where players play different strategies in different states of the world. In particular, if everybody behaves a certain way (e.g. does not revolt) in the usual state of the world, then in order for players to be able to behave a different way (e.g. revolt) in another state of the world, it is both necessary and sufficient for it to be common p-believed that it is not the usual state of the world, where common p-belief is a relaxation of common knowledge introduced by Monderer and Samet [16]. Our framework applies to many player r-coordination games -a generalization of coordination games that we introduce -and common (r, p)-beliefs -a generalization of common p-beliefs that we introduce. We then apply these theorems to two particular signaling structures to obtain novel results.

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