 A similar analysis applies to infinite trees as well as to epistemic forests (cf.  Section 3.6).  More generally, any imperfect information structure can arise from information updates, provided players satisfy perfect recall and no miracles, and moves in the game have logically definable preconditions governing their availability. Precise formulations and proofs can be found in van Benthem, Gerbrandy, Hoshi, and Pacuit (2009). A generalization to game play without assumption of synchronicity is provided in Dégremont, Löwe, and Witzel (2011).