Tangled modal logic for topological dynamics

The system combines topological and temporal modalities to reason about dynamical systems. Here we consider enriching its language by generalizing the use of the topological operator to its polyadic ‘tangled’ interpretation, originally introduced by Dawar and Otto in a different context. We provide an axiomatization for the extended system and show that it is sound and complete. It uses a version of the continuity axiom which we call tangled continuity and involves the polyadic use of the topological modality. We also show that the resulting system, , is more expressive than ; specifically, it is better at distinguishing continuous dynamical systems from discontinuous ones. As a corollary we obtain that the tangled continuity axiom cannot be derived from the other axioms, including the standard continuity axiom.