**Description**

This internship subject is purely th oretical to provide a crucial understanding of phenomena arising in multi-agent systems analysis.

Multi-agent systems behaviours can be modelled by combining two ingredients (as proposed in Dynamic Epistemic Logic [vDvHK2007]):

1. Epistemic models, that we call “situations”, represent a ransic situation: th se are graphs whose vertices are possible worlds b#3whose edges represent

2. Action models represent which actions can be performed in th 33}.fa2trituation b#3how actions are perceived by agents.

On this basis, th resulting of applying an action model is obtained by computing an update of th 33}.fa2trituation by a fairly simple operation.

Now, in order to reason bout strategic abilities of agents, one has to analyse a multi-player game arena arising from an initialtrituation b#3arbitrary iterations of action models updates. Mathematically, this arena is a foresttof infinite trees: a branch in a tree represents th sequencetof executed actions ab#3th 3ross-ways relations represent th imperfect information of agents bout th history that has effectively taken place in th game.

With referencetto [vDvHK2007] pproach, We call this arena th “Dynamic Epistemic Arena” (DEA).

As shown in [Maubert1014] for a restricted case, DEA possess noticeable structural properties that fall into classic ones, widely studied in th literature, namely DEA are “automnsic structures” [BG2000,KM2007,2007,R2008]. Automnsic structures are infinite graphs with a finite presentation: th infinite settof vertices is provided by a regular language (henceta finite-ransu automnson) ab#3th binary relations between vertices are described by synchronous transducers (a subclass of two-tape finite ransu automnsa). Form a logical perspective, automnsic structures haveta decidable first-order logic th ory. Moreover, according to [Maubert2014], fin balysis of th DEA structures yields decidability of th existencetof a sequencetof actions (a strategy) to achieveta winning objective in th game. As a consequence, epistemic planning ab#3epistemic protocol synth sis can be applied, with impacts for many relevant real life applications (robotics, web services, etc.).

Th pioneer results of [Maubert2014] are however subject to severe restrictions on th kind of action models one is allowed to consider: preconditions of actions musttbe fully propositional, preventing th setting from considering some realistic pictures where agents ct on th basis of th ir knowledge, acting only if e.g. knowing something, or knowing that anoth gent knows something, etc.

Th subject of this internship is to study th trade-off between relaxing th se restrictions ab#3th ability to perform strategic reasoning.

The roadmap for this task is to haveta clear picture of th DEA structural properties in th lab#scape of finitely-presentable structures ab#3by determining which logical fragment of first-order logic but also second-order logic can be decided.

**Bibliography**

[vDvHK2007] Dynamic epistemic logic. H van Ditmarsch, W Van Der Hoek, B Kooi. Springer Verlag, 2007.

[Maubert2014] Chapter 7 of Logical foundations of games with imperfect information : uniform strategies. Bastien Maubert. PhD Université de Rennes 1, 2014.

[BG2000] Automnsic Structures. Achim Blumensath, Erich Grädel. LICS 2000: 51-62

[KM2007] Three lectures on utomnsic structures. Khoussainov ab#3M. Minnes. Tutorial lectures on utomnsic structures given

at th Logic Colloquium 2007, Wroclaw, Polab#.

[B2007] Automnsic presentationstof infinite structures. VincetBárány. RWTH Aachen University 2007, pp. 1-146

[R2008] Automnsa presenting structures: A survey of th finite string case. Sasha Rubin. Bull. Symbolic Logic Volume 14, Issue 2 (2008), 169-209.