Untangled: A Complete Dynamic Topological Logic

TL;DR

This paper establishes a sound and complete dynamic topological logic for scattered spaces, providing the first such axiomatisation in the original trimodal language, with finite axiomatizability.

Contribution

It introduces the first sound and complete dynamic topological logic for scattered spaces in the original language, with finite axiomatisation and semantic insights.

Findings

First sound and complete logic for scattered spaces

Finite axiomatisation over the original language

Semantic connection to provability and fixed points

Abstract

Dynamic topological logic ($\mathbf{DTL}$) is a trimodal logic designed for reasoning about dynamic topological systems. It was shown by Fern\'andez-Duque that the natural set of axioms for $\mathbf{DTL}$ is incomplete, but he provided a complete axiomatisation in an extended language. In this paper, we consider dynamic topological logic over scattered spaces, which are topological spaces where every nonempty subspace has an isolated point. Scattered spaces appear in the context of computational logic as they provide semantics for provability and enjoy definable fixed points. We exhibit the first sound and complete dynamic topological logic in the original trimodal language. In particular, we show that the version of $\mathbf{DTL}$ based on the class of scattered spaces is finitely axiomatisable over the original language, and that the natural axiomatisation is sound and complete.