 Rasmussen 2015 and Bjerring and Skipper (forthcoming) present a dynamic impossible worlds solution to the logical omniscience problem. Agents’ beliefs evolve over time due to epistemic actions, on this approach (see the entry on  dynamic epistemic logic  for background). Bjerring and Skipper focus on deductive actions. Agents count as competent insofar as they unfold the consequences of their beliefs, up to a certain depth of reasoning. Their operator “\(\langle n\rangle KA\)”, says: “After some \(n\)-step chain of logical reasoning, the agent comes to know that \(A\)”. The agent can update its epistemic state by kicking out choices of impossible worlds which were epistemic possibilities before the deduction took place. One can show that if a formula \(A\) follows from formulas \(A_1, \ldots, A_n\) in \(n\) steps of reasoning, then \(KA_1, \ldots, KA_n\) together entail \(\langle n\rangle KA\).