Timely Common Knowledge

Coordinating activities at different sites of a multi-agentsystem typically imposes epistemic constraints on the par-ticipants. Specifying explicit bounds on the relative timesat which actions are performed induces combined tempo-ral and epistemic constraints on when agents can performtheir actions. This paper characterises the interactive epis-temic state that arises when actions must meet particulartemporal constraints. The new state, called timely commonknowledge, generalizes common knowledge, as well as othervariants of common knowledge. While known variants ofcommon knowledge are defined in terms of a fixed point ofan epistemic formula, timely common knowledge is definedin terms of a vectorial fixed point of temporal-epistemic for-mulae. A general class of coordination tasks with timingconstraints is defined, and timely common knowledge is usedto characterise both solvability and optimal solutions of suchtasks. Moreover, it is shown that under natural conditions,timely common knowledge is equivalent to an infinite con-junction of temporal-epistemic formulae, in analogy to thepopular definition of common knowledge.