Dynamic Epistemic Logic (DEL) deals with the representation and the study in a multi-agent setting of knowledge and belief change. It can express in a uniform way epistemic statements about: (i) what is true about an initial situation (ii) what is true about an event occurring in this situation (iii) what is true about the resulting situation after the event has occurred. We axiomatize within the DEL framework what we can infer about (ii) given (i) and (iii) and what we can infer about (i) given (ii) and (iii). Given three formulas ϕ, ϕ′ and ϕ′′ describing respectively (i), (ii) and (iii), we also show how to build two formulas which capture all the information which can be inferred about (ii) from ϕ and ϕ′′, and all the information which can be inferred about (i) from ϕ′ and ϕ′′. We show how our results extend to other modal logics than K. Finally, we generalize the classical language of dynamic epistemic logic, where one can reason only with complete specifications of events, in order to account also for incomplete description of events. In the companion paper (Aucher, 2011), we axiomatize what we can infer about (iii) given (i) and (ii), and show how to build a formula which captures all the information which can be inferred about (iii) from ϕ and ϕ′.