Chaos and Derivative Logic in Topological
Dynamics
Yoàv Montacute

Abstract:
This thesis explores the connection between dynamical systems and logic. The relationship between the two subjects was first established by Artemov et al., who developed Tarski’s idea of linking topology and logic in order to reason about dynamic topological systems. Their framework was later extended by Kremer, Mints and Fernández-Duque.
Expanding their framework, we introduce a non-deterministic generalisation of dynamical systems and a characterisation of chaos in such systems. Moreover, we provide axiomatisation for a natural sub-class of all topologically transitive non-deterministic dynamical systems.
Topological semantics with the Cantor derivative operator gives rise to derivative logics, also referred to as d-logics. These logics have not previously been studied in the framework of dynamical systems. We show that the logics
wK4C and GLC both have the finite model property and are sound and complete with respect to the d-semantics in the deterministic setting. In particular, we prove that wK4C is the d-logic of all dynamic topological systems and GLC is the d-logic of all dynamic topological systems based on a scattered space.
The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological d-logics. Furthermore, such a result for GLC may constitute the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation - something known to be impossible over the class