Abstract: Dynamic (topological) systems are mathematical models of change or movement in space which have applications in various branches of mathematics and the physical sciences. Formally, they consist of pairs (X,S), where X is a topological space and S: X->X is continuous. From a modal logic perspective this natually leads to a combination of spatial and temporal modal logics called Dynamic Topological Logic (DTL).
In this two-day course we will cover the main results and techniques leading up to the state of the art in the field of Dynamic Topological Logic. The proposed schecule is as follows.
DAY 1 (24/06/2020)
We begin with a brief introduction to topological dynamics, then introduce the logic S4C and summarize its main results. We then introduce the full DTL with "henceforth" and sketch the proof of undecidability for this full logic. Finally we discuss Kripke-incompleteness and how it can be, in a sense, avoided with the use of non-deterministic quasimodels.
DAY 2 (01/07/2020)
We intrdouce the logic DTL*, which extends the logic DTL with the "tangled closure" operator, a generalization of the standard topological closure to a polyadic version defined via a greatest fixed point. With this we give an axiomatization for DTL* and sketch its completeness proof. We then introduce Intuitionistic Temporal Logic (ITL), a variant of DTL which omits the interior modality and instead interprets implication in the Heyting algebra of open sets. We conclude by detailing how the techniques developed for DTL can be modified to obtain a decidability proof in this setting.
Please send an email to: r.zoghi@gmail.com to join this lecture.