Computational complexity and practice of verified and efficient algorithms for dynamical systems

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Large progress has been made on the formalization of exact real computation in type theory and its implementation in the Coq proof assistant. Previous work on formalizing nondeterministic computation on real and complex numbers has been extended and unified. In particular, progress has been made on formalizing nondeterminism occuring in exact real computation and a sound notion of a multivalued limit operator was devised. The theory was implemented in the Coq proof assistant and using Coq's program extraction mechanism efficient Haskell programs for exact real computation can be extracted. As a non-trivial application we extract a nondeterministic program computing the complex square root.As Covid-19 related border restrictions loosened during this year, international joint research could be resumed. During a visit by Michal Konecny (Aston University, UK), we extended the aforementioned formalization to spaces of subsets and functions, which has important applications to differential equations, in particular regarding reachability questions.Several representations for dealing with subsets have been compared in terms of efficiency and certfied programs for generating drawings of subsets have been extracted and shown to behave well in terms of running time. Examples include fractals in two dimensional Euclidean space.Progress has also been made on integrating the logic based proof system IFP into our type theoretical work. A system for IFP style proofs in Coq and a new program extraction to untyped lambda calculus similar to IFP was developed based on the meta-coq framework.