ABSTRACT. In this talk we will try to explain the use of some importnat model-theoretic notions, focusing on the model-theory of finite rank groups and on the notion of orthogonality. Their use in applications to algebraic geometry will be gently illustrated by some examples. This talk is partly inspired by a series of recent joint papers with Franck Benoist (Paris-Sud) and Anand Pillay (Notre-Dame), giving new model theoretic proofs of the original results of Ehud Hrushovski on the Mordell-Lang Conjecture for function fields (1994).

ABSTRACT. Towards the development of more reliable computerized systems, expressive models are designed, targetting application to automatic verification (model-checking). As part of this effort, timed automata have been proposed in the early nineties [2] as a powerful and suitable model to reason about (the correctness of) real-time computerized systems. Timed automata extend finite-state automata with several clocks, which can be used to enforce timing constraints between various events in the system. They provide a convenient formalism and enjoy reasonably-efficient algorithms (e.g. reachability can be decided using polynomial space), which explains the enormous interest that they provoked in the community of formal methods. Timed games [4] extend timed automata with a way of modelling systems interacting with external, uncontrollable components: some transitions of the automaton cannot be forced or prevented to happen. The reachability problem then asks whether there is a strategy (or controller) to reach a given state, whatever the (uncontrollable) environment does. This problem can also be decided, in exponential time. Timed automata and games are not powerful enough for representing quantities like resources, prices, temperature, etc. The more general model of hybrid automata [14] allows for accurate modelling of such quantities using hybrid variables. The evolution of these variables follow differential equations, depending on the state of the system, and this unfortunately makes the reachability problem undecidable, even in the very restricted case of stopwatches (stopwatches are clocks that can be stopped, and hence, automata with stopwatches only are the simplest hybrid automata one can think of). Weighted (or priced) timed automata [3, 5] and games [15, 1, 9] have been proposed in the early 2000’s as an intermediary model for modelling resource consumption or allocation problems in real-time systems (e.g. optimal scheduling [6]). As opposed to (linear) hybrid systems, an execution in a weighted timed model is simply one in the underlying timed model: the extra quantitative information is just an observer of the system, and it does not modify the possible behaviours of the system. In this tutorial, we will present basic results concerning timed automata and games, and we will further investigate the models of weighted timed automata and games. We will present in particular the important optimal reachability problem: given a target location, we want to compute the optimal (i.e. smallest) cost for reaching a target location, and a corresponding strategy. We will survey the main results that have been obtained on that problem, from the primary results of [3, 5, 16, 13, 8, 17, 7] to the most recent developments [11, 10]. We will also mention our new tool TiAMo, which can be downloaded at https://git.lsv.fr/colange/tiamo. We will finally show that weighted timed automata and games have applications beyond that of model- checking [12].

ABSTRACT. In this presentation I focus on a logical-philosophical study of group beliefs and collective ``knowledge'', and their dynamics in communities of interconnected agents capable of reflection, communication, reasoning, argumentation etc. In particular, the aim is to study belief formation and belief diffusion (doxastic influence) in social networks, and to characterize a group's ``epistemic potential". This covers cases in which a group's ability to track the truth is higher than that of each of its members (the ``wisdom of the crowds": distributed knowledge, epistemic democracy and other beneficial forms of belief aggregation and deliberation), as well as situations in which the group's dynamics leads to informational distortions (the ``madness of the crowds": informational cascades, ``groupthink", the curse of the committee, pluralistic ignorance, group polarization etc). I look at several logical formalisms that make explicit various factors affecting the epistemic potential of a group: the agents' degree of interconnectedness, their degree of mutual trust, their different epistemic interests (their ``questions"), their different attitudes towards the available evidence and its sources etc. In this presentation I refer to a number of recent papers (1,2,3,4,5,6), that make use a variety of formal tools ranging from dynamic epistemic logics and probabilistic logics. I conclude with some philosophical reflections about the nature and meaning of group knowledge, as well as about the epistemic opportunities and dangers posed by informational interdependence.

REFERENCES:

{1} A. Achimescu, A. Baltag and J. Sack. The Probabilistic Logic of Communication and Change. The Journal of Logic and Computation, issue 07 January 2016.

{2} A. Baltag, Z. Christoff, J.U. Hansen and S. Smets, Logical Models of Informational Cascades. in J. van Benthem and F. Liu (Eds.): Logic across the University: Foundations and Applications, — Proceedings of the Tsinghua Logic Conference, Beijing, 14-16 October 2013, Studies in Logic, Volume 47, pp.405-432, College Publications, London, 2013.

{3} A. Baltag, Z. Christoff, R.K Rendsvig, S. Smets. Dynamic epistemic logics of diffusion and prediction in social networks (extended abstract). In Bonanno, G., van der Hoek, W., Perea, A., eds.: Proceedings of LOFT, 2016.

{4} A. Baltag, R. Boddy and S. Smets. Group Knowledge in Interrogative Epistemology. Forthcoming in the Outstanding Contributions to Logic Series, volume dedicated to J. Hintikka, Springer, 2017.

{5} F. Liu, J. Seligman, and P. Girard. Logical Dynamics of Belief Change in the Community, Synthese, Volume 191, Issue 11, pp 2403-2431, 2014.

{6} S. Smets, F.R. Velasquez-Quesada. How to make friends: A logical approach to social group creation. To appear in Baltag, A. and Seligman, J., eds.: Proceedings of the Sixth International Workshop LORI, 2017.