## Section: New Results

### Fundamental results and algorithms: dynamic epistemic logic

Participants : Guillaume Aucher, François Schwarzentruber.

Dynamic Epistemic Logic (DEL) deals with the representation and the study of knowledge and belief change in a multi-agent setting. The core representative task of this logical framework can be split up in three parts: 1/ the inital global state of the distributed system, 2/ an event occurring in this system, 3/ a product update taking as argument these two representations and yielding a new representation of the new global state of the distributed system. Therefore, we can express uniformly within the DEL framework epistemic statements about:

- (i)
what is true about an initial state

- (ii)
what is true about an event occurring in this initial state

- (iii)
what is true about the resulting state after the event has occurred.

We axiomatized 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). Given three logical formulas $\phi $, ${\phi}^{\text{'}}$ and ${\phi}^{\text{'}\text{'}}$ describing respectively (i), (ii) and (iii), we also showed how to build three formulas that capture respectively all the information which can be inferred about (iii) from $\phi $ and ${\phi}^{\text{'}}$, all the information which can be inferred about (ii) from $\phi $ and ${\phi}^{\text{'}\text{'}}$, and all the information which can be inferred about (i) from ${\phi}^{\text{'}}$ and ${\phi}^{\text{'}\text{'}}$. We showed how our results extend to other modal logics than the minimal modal logic $\U0001d5aa$. These results are to appear in [9] and [10] . In [19] , we also provided a tableau method deciding whether such inferences are valid. We implemented it in LOTRECscheme and showed that this decision problem is NEXPTIME-complete. This work contributes to the proof theory and the study of the computational complexity of DEL which have rather been neglected so far.

Application to fault localization in IMS network (see the UNIVERSELF project) has started. The various agents involved in an IMS network (clients, assistance, administrators...) have a partial view of the network and so need to communicate their partial knowledge of the network to each other in order to localize the fault in the network (each communication having possibly a different cost). One of the main problems is to determine which communication should occur and which agent should be queried so that the fault is eventually localized. This problem can naturally be expressed in the DEL framework. We have shown how the initial state of an IMS network representing the knowledge of each agent can be represented by a particular kind of epistemic model (i) and how the desired state where the fault is localized can be expressed by a logical formula (iii). The problem amounts to determining which communication or sequence of communications should occur (ii) so that one passes from the initial epistemic model (i) to another epistemic model where the fault is localized (iii), and also to determine if such a communication or sequence of communications is possible. We have focused so far on the case of a single communication, but we plan to extend it to a sequence of communications. Further theoretical work still needs to be done to address the issue of sequential communication.

In parallel to this work, we also axiomatized different notions of knowledge and belief which are defined by means of a `sphere' semantics. This work is the result of an invited contribution and is to appear in [48] .