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Recently I read all kinds of work from logic scientist in which epistemic logic was the main topic. Where epistmic change refers to change in knowledge of some agent in a multi-agent system (in a non-changing world), ontic change refers to change in the factual world.

What I was wondering about is how one can add the dimension of time to an epistemic system (or possibly in a system in which the ontic changes included). For example to formalize things like:

  • After a certain point (in the future) A will know that $\varphi$ is the case (but up to then A does not know)
  • A knows that after a certain point in the future, $\varphi$ will be the case (and she knows that until then, $\varphi$ is false.
  • $B$ previously knew that $\psi$ was the case, but now (because, maybe the facts in the world changed) he does not know wheter $\psi$ is true.

where $A$, $B$, $\psi$, $\varphi$ are agents and propositional formulas

My main question is: what is the default paradigm/model/language/logical systems to express the sentences above? And how can one visualize it (for example by using (extended) Kripke-models?

Another question is how we can add the time dimension to dynamic epistemic logic ("after action $\alpha$ happened $A$ knows that there will be a time step in future (from) where $\tau$ is true" or anything like that).

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I think that both of your questions are the same. If we want to reason about how agent's knowledge evolve, we should employ dynamic epistemic logic (DEL). Since DEL is about how agent's awareness changes after epistemic updates, it implicitly contains temporal domain (i.e. executing an update shows what is the case after it, based on what is the case before the update (preconditions)). Ontic changes also belong to the domain of DEL (a thorough treatment is given in van Benthem, J. et al. Logics of Communication and Change). Adding temporal transition to DEL is a little bit tricky. Having in mind the "forward-looking" feature of update models, in Renne, B. et al. Logics of Temporal-Epistemic Actions only "backward-looking", or "yesterday", operator is added. This logic turns out to be translatable to epistemic temporal logic without dynamic updates. However, I do not know any works on adding ontic changes to this dynamic epistemic temporal logic (DETL). (If you know any, please let me know.)

As for more classical approaches, a nice survey is Goranko, V., Pacuit, E. Temporal Aspects of the Dynamics of Knowledge. They, the approaches, are, however, not as intriguing as DETL, since they possess properties of no learning and no forgetting. Moreover, it is yet necessary to show that it is possible to reason about any asynchronous update within DETL. Thus, a sound and unified approach to reasoning about epistemic, ontic and temporal changes is yet to come.

P.S.: it might be possible to model your examples as public or private announcements or assignments within the framework of ontic DETL.

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