For a new seminar of the proofs and algorithms pole of LIX, we are happy to welcome Georg Struth.

Link to the online conference

Abstract: Hybrid systems integrate continuous dynamics and discrete control. A prominent approach is differential dynamic logic (dL), a modal logic for reasoning about ordinary differential equations and their solutions within hybrid programs. Over the last decade, a domain-specific sequent calculus for dL has been developed together with an intricate explicit substitution calculus for it. It has been integrated into the KeYmaera X tool and proved its worth in numerous case studies.

This talk presents an algebraic reconstruction of this approach and the first verification and refinement components in an interactive proof assistant inspired by it. Our components are based on shallow embeddings in which the proof theory of dL is by and large replaced by semantic equational reasoning about orbits and trajectories of dynamical systems in combination with discrete state updates. We use algebras, in particular variants of Kleene algebras and predicate transformer algebras, for automatic verification condition generation. We currently support a component for reasoning with weakest liberal preconditions, a Hoare logic and a refinement calculus for hybrid programs.

These allow us to verify hybrid programs in various ways: starting from hybrid program specifications based on ordinary differential equations, we can certify solutions of locally Lipschitz-continuous vector fields using the Picard-Lindelöf theorem and then reason explicitly about them. Alternatively we can represent solutions of continuous vector fields implicitly by invariant sets. Finally, we can analyse hybrid program that specify trajectories or orbits of hybrid systems directly.

Apart from presenting the algebras used and their extension and instantiation to hybrid program semantics, I will present some examples, comment on the intricacies of formalising the approach within the Isabelle/HOL proof assistant, and outline some future directions for this line of work, time permitting.

This work is joint with Jonathan Julián Huerta y Munive, and partially funded by CONACYT.

The list of next seminars can be found at: https://smimram.gitlabpages.inria.fr/proofs-algorithms/seminar/

The calendar of seminars can be found at: https://smimram.gitlabpages.inria.fr/proofs-algorithms/seminar/calendar.ics