Logical Foundations of Cyber-Physical Systems

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About this Textbook

Cyber-physical systems (CPSs) combine cyber capabilities, such as computation or communication, with physical capabilities, such as motion or other physical processes. Cars, aircraft, and robots are prime examples, because they move physically in space in a way that is determined by discrete computerized control algorithms. Designing these algorithms is challenging due to their tight coupling with physical behavior, while it is vital that these algorithms be correct because we rely on them for safety-critical tasks.

This textbook teaches undergraduate students the core principles behind CPSs. It shows them how to develop models and controls; identify safety specifications and critical properties; reason rigorously about CPS models; leverage multi-dynamical systems compositionality to tame CPS complexity; verify CPS models of appropriate scale in logic; and develop an intuition for operational effects.

The book is supported with homework exercises, lecture videos, and slides.

About the Author

The author is an Associate Professor in the Computer Science Department at Carnegie Mellon University. He develops the logical foundations of cyber-physical systems to characterize their fundamental principles and to show how we can design computers that are guaranteed to interact correctly with the physical world. He has a Ph.D. from the University of Oldenburg, Germany. He received an ACM Doctoral Dissertation Honorable Mention Award and NSF Career Award, and he was named one of the Brilliant 10 Young Scientists by the Popular Science magazine and one of AI's 10 to Watch by the IEEE Intelligent Systems Magazine.


"This excellent textbook marries design and analysis of cyber-physical systems with a logical and computational way of thinking. The presentation is exemplary for finding the right balance between rigorous mathematical formalization and illustrative case studies rooted in practical problems in system design."

(Rajeev Alur, University of Pennsylvania)

Book Review
Form. Asp. Comp.

"This book provides a wonderful introduction to cyber-physical systems, covering fundamental concepts from computer science and control theory from the perspective of formal logic. The theory is brought to life through many didactic examples, illustrations, and exercises. A wealth of background material is provided in the text and in an appendix for each chapter, which makes the book self-contained and accessible to university students of all levels."

(Goran Frehse, Université Grenoble Alpes)

"[The author] has developed major important tools for the design and control of those cyber-physical systems that increasingly shape our lives. This book is a 'must' for computer scientists, engineers, and mathematicians designing cyber-physical systems."

(Anil Nerode, Cornell University)

"As computing interfaces increasingly with our physical world, resulting in so-called cyber-physical systems, our foundations of computing need to be enriched with suitable physical models. This book strikes a wonderful balance between rigorous foundations for this next era of computing with illustrative examples and applications that drive the developed methods and tools. A must read book for anyone interested in the development of a modern and computational system science for cyber-physical systems."

(George J. Pappas, University of Pennsylvania)

"This definitive textbook on cyber-physical systems lays the formal foundations of their behavior in terms of a single logical framework. Platzer's logic stands out among all other approaches because it provides a uniform treatment of both the discrete and continuous nature of cyber-physical systems, and does not shy away from their complex behavior due to stochasticity, uncertainty, and adversarial agents in the environment. His computational thinking approach makes this work accessible to practicing engineers who need to specify and verify that cyber-physical systems are safe."

(Jeannette M. Wing, Columbia University)

Book Reviews:
Knapp, A. and Roggenbach, M. André Platzer: Logical foundations of cyber-physical systems. Form. Asp. Comp. 32, 149-151, 2020.

Table of Contents

(S) = slides      (V) = videos
  1. Foreword by Goran Frehse
  2. Cyber-Physical Systems: Overview (S) (V)
Part I - Elementary Cyber-Physical Systems (V)
  1. Differential Equations & Domains (S) (V)
  2. Choice & Control (S) (V)
  3. Safety & Contracts (S) (V)
  4. Dynamical Systems & Dynamic Axioms (S) (V)
  5. Truth & Proof (S) (V)
  6. Control Loops & Invariants (S) (V)
  7. Events & Responses (S) (V)
  8. Reactions & Delays (S) (V)
Part II - Differential Equations Analysis (V)
  1. Differential Equations & Differential Invariants    (S) (V)
  2. Differential Equations & Proofs (S) (V)
  3. Ghosts & Differential Ghosts (S) (V)
  4. Differential Invariants & Proof Theory (S) (V)
Part III - Adversarial Cyber-Physical Systems (V)
  1. Hybrid Systems & Games (S) (V)
  2. Winning Strategies & Regions (S) (V)
  3. Winning & Proving Hybrid Games (S) (V)
  4. Game Proofs & Separations (S) (V)
Part IV - Comprehensive CPS Correctness (V)
  1. Axioms & Uniform Substitutions (S) (V)
  2. Verified Models & Verified Runtime Validation (S) (V)
  3. Virtual Substitution & Real Equations (S) (V)
  4. Virtual Substitution & Real Arithmetic (S) (V)
  5. Operators and Axioms
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Videos (V) or
Slides (S)

Supporting Slides are indicated with (S). Supporting Lecture Videos for the chapters are indicated with (V). Both the slides and the videos are supplementary material and, despite their useful overview, of course, cannot cover the entire contents of the textbook in full.

Sincere thanks to Jessica Packer for her wonderful video editing, without whom the videos would never have been completed.

Brief Introduction to Logical Foundations of Cyber-Physical Systems (10min)

[All Videos]

Online Material

There is a verification tool implementing the approach described in this book, called KeYmaera X.

Operators and
Slides and

Where to Get This Book

The book has appeared with Springer under ISBN 978-3-319-63587-3. Find

Bibliographic Information

This textbook just appeared with Springer.
  1. André Platzer.
    Logical Foundations of Cyber-Physical Systems.
    Springer, Cham, 2018. 659 pages. ISBN 978-3-319-63587-3.
    [bib | | doi | slides | video | book | web | errata | abstract]

Subject Index:
Artificial Intelligence (incl. Robotics)
Mathematical Logic and Formal Languages
Mathematical Logic and Foundations
Control, Robotics, Mechatronics
Quality Control, Reliability, Safety and Risk
Logical Foundations of Cyber-Physical Systems.
Springer, 2018. 659 pages, ISBN 978-3-319-63587-3. Published 2018-07-30. DOI 10.1007/978-3-319-63588-0.