DEL-sequents for progression, regression and epistemic planning

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 (iii) given (i) and (ii), what we can infer about (ii) given (i) and (iii), and what we can infer about (i) given (ii) and (iii). These three inference problems are related to classical problems addressed under different guises in artificial intelligence and theoretical computer science, which we call respectively progression, epistemic planning and regression. Given three formulas F(i), F(ii) and F(iii) describing respectively (i), (ii) and (iii), we also show how to build three formulas which capture respectively all the information which can be inferred about (iii) from F(i) and F(ii), all the information which can be inferred about (ii) from F(i) and F(iii), and all the information which can be inferred about (i) from F(ii) and F(iii). We show how our results extend to other modal logics than K.