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. 2020 Mar 13;12078:191–208. doi: 10.1007/978-3-030-45190-5_11

Relational Differential Dynamic Logic

Juraj Kolčák 10, Jérémy Dubut 11,12,✉, Ichiro Hasuo 11,13, Shin-ya Katsumata 11, David Sprunger 11, Akihisa Yamada 11
Editors: Armin Biere8, David Parker9
PMCID: PMC7439732

Abstract

In the field of quality assurance of hybrid systems, Platzer’s differential dynamic logic (dL) is widely recognized as a deductive verification method with solid mathematical foundations and sophisticated tool support. Motivated by case studies provided by our industry partner, we study a relational extension of dL, aiming to formally prove statements such as “an earlier engagement of the emergency brake yields a smaller collision speed.” A main technical challenge is to combine two dynamics, so that the powerful inference rules of dL (such as the differential invariant rules) can be applied to such relational reasoning, yet in such a way that we relate two different time points. Our contributions are a semantical theory of time stretching, and the resulting synchronization rule that expresses time stretching by the syntactic operation of Lie derivative. We implemented this rule as an extension of KeYmaera X, by which we successfully verified relational properties of a few models taken from the automotive domain.

Keywords: hybrid system, cyber-physical system, formal verification, theorem proving, dynamic logic

Footnotes

Thanks are due to Stefan Mitsch, André Platzer, and Yong Kiam Tan for useful tips on the KeYmaera X source code; and to Kenji Kamijo, Yoshiyuki Shinya, and Takamasa Suetomi from Mazda Motor Corporation for helpful discussions. The authors are supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST. I.H. is supported by Grant-in-Aid No. 15KT0012, JSPS. J.D. is supported by Grant-in-aid No. 19K20215, JSPS. The work was done during J.K.’s internship at the National Institute of Informatics, Tokyo, Japan.

Contributor Information

Armin Biere, Email: biere@jku.at.

David Parker, Email: d.a.parker@cs.bham.ac.uk.

Juraj Kolčák, Email: kolcak@lsv.fr.

Jérémy Dubut, Email: dubut@nii.ac.jp.

Ichiro Hasuo, Email: hasuo@nii.ac.jp.

Shin-ya Katsumata, Email: s-katsumata@nii.ac.jp.

David Sprunger, Email: sprunger@nii.ac.jp.

Akihisa Yamada, Email: akihisayamada@nii.ac.jp.

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