This book introduces a temporal type theory, the first of its kind as far as we know. It is based on a standard core, and as such it can be formalized in a proof assistant such as Coq or Lean by adding a number of axioms. Well-known temporal logics---such as Linear and Metric Temporal Logic (LTL and MTL)---embed within the logic of temporal type theory. The types in this theory represent "behavior types". The language is rich enough to allow one to define arbitrary hybrid dynamical systems, which are mixtures of continuous dynamics---e.g. as described by a differential equation---and discrete jumps. In particular, the derivative of a continuous real-valued function is internally defined. We construct a semantics for the temporal type theory in the topos of sheaves on a translation-invariant quotient of the standard interval domain. In fact, domain theory plays a recurring role in both the semantics and the type theory.
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This morning at ACT2018, David Spivak gave a VERY cool talk on using topos theory to model how airplanes can maintain a safe distance from each other in flight. You can watch the talk here! https://t.co/pUXZhj6SXA Also check out “Temporal Type Theory” at https://t.co/6LWNOQWtqw pic.twitter.com/7x9yjBwVIA
— Tai-Danae Bradley (@math3ma) May 2, 2018