DEL-sequents for progression

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: 1. what is true about an initial situation 2. what is true about an event occurring in this situation 3. 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). Given three formulas φ,φ' and φ" describing respectively (i), (ii) and (iii), we also show how to build a formula φ × φ' which captures all the information which can be inferred about (iii) from φ and φ'. We show how our results extend to other modal logics than K. In our proofs and definitions, we resort to a large extent to the normal form formulas for modal logic originally introduced by Kit Fine. In a companion paper (Aucher, 2012), we axiomatize what we can infer about (ii) given (i) and (iii), and what we can infer about (i) given (ii) and (iii), and show how to build two formulas φ ⦸ φ" and φ' ⊘ φ" which capture respectively 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".