Untangled: A Complete Dynamic Topological Logic
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Dynamic topological logic (πππ) is a trimodal logic designed for reasoning about dynamic topological systems. It was shown by FernΓ‘ndez-Duque that the natural set of axioms for πππ 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 πππ based on the class of scattered spaces is finitely axiomatisable over the original language, and that the natural axiomatisation is sound and complete.
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