Dynamic Topological Logic (𝒟𝒯ℒ) is a multimodal system for reasoning about dynamical systems. It is defined semantically and, as such, most of the work done in the field has been model-theoretic. In particular, the problem of finding a complete axiomatization for the full language of 𝒟𝒯ℒ over the class of all dynamical systems has proven to be quite elusive.
Here we propose to enrich the language to include a polyadic topological modality, originally introduced by Dawar and Otto in a different context. We then provide a sound axiomatization for 𝒟𝒯ℒ over this extended language, and prove that it is complete. The polyadic modality is used in an essential way in our proof.
"A sound and complete axiomatization for Dynamic Topological Logic." J. Symbolic Logic 77 (3) 947 - 969, September 2012. https://doi.org/10.2178/jsl/1344862169