Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, we introduce a dynamic logic for hybrid programs, which is a program notation for hybrid systems. As a verification technique that is suitable for automation, we introduce a free variable proof calculus with a novel combination of real-valued free variables and Skolemisation for lifting quantifier elimination for real arithmetic to dynamic logic. The calculus is compositional, i.e., it reduces properties of hybrid programs to properties of their parts. Our main result proves that this calculus axiomatises the transition behaviour of hybrid systems completely relative to differential equations. In a case study with cooperating traffic agents of the European Train Control System, we further show that our calculus is well-suited for verifying realistic hybrid systems with parametric system dynamics.

This article introduces a relatively complete proof calculus for differential dynamic logic (dL) that is entirely based on uniform substitution, a proof rule that substitutes a formula for a predicate symbol everywhere. Uniform substitutions make it possible to use axioms instead of axiom schemata, thereby substantially simplifying implementations. Instead of subtle schema variables and soundness-critical side conditions on the occurrence patterns of logical variables to restrict infinitely many axiom schema instances to sound ones, the resulting calculus adopts only a finite number of ordinary dL formulas as axioms, which uniform substitutions instantiate soundly. The static semantics of differential dynamic logic and the soundness-critical restrictions it imposes on proof steps is captured exclusively in uniform substitutions and variable renamings as opposed to being spread in delicate ways across the prover implementation. In addition to sound uniform substitutions, this article introduces differential forms for differential dynamic logic that make it possible to internalize differential invariants, differential substitutions, and derivatives as first-class axioms to reason about differential equations axiomatically. The resulting axiomatization of differential dynamic logic is proved to be sound and relatively complete.

Abstract. We generalise dynamic logic to a logic for differential-algebraic programs, i.e., discrete programs augmented with first-order differentialalgebraic formulas as continuous evolution constraints in addition to first-order discrete jump formulas. These programs characterise interacting discrete and continuous dynamics of hybrid systems elegantly and uniformly. For our logic, we introduce a calculus over real arithmetic with discrete induction and a new differential induction with which differential-algebraic programs can be verified by exploiting their differential constraints algebraically without having to solve them. We develop the theory of differential induction and differential refinement and analyse their deductive power. As a case study, we present parametric tangential roundabout maneuvers in air traffic control and prove collision avoidance in our calculus.

Hybrid systems are a fusion of continuous dynamical systems and discrete dynamical systems. They freely combine dynamical features from both worlds. For that reason, it has often been claimed that hybrid systems are more challenging than continuous dynamical systems and than discrete systems. We now show that, proof-theoretically, this is not the case. We present a complete proof-theoretical alignment that interreduces the discrete dynamics and continuous dynamics of hybrid systems. We give a sound and complete axiomatization of hybrid systems relative to continuous dynamical systems and a sound and complete axiomatization of hybrid systems relative to discrete dynamical systems. Thanks to our axiomatization, proving properties of hybrid systems is exactly the same as proving properties of continuous dynamical systems and again, exactly the same as proving properties of discrete dynamical systems. This fundamental cornerstone sheds light on the nature of hybridness and enables flexible and provably perfect combinations of discrete reasoning with continuous reasoning that lift to all aspects of hybrid systems and their fragments.

Cyber-physical systems (CPSs) are important whenever computer technology interfaces with the physical world as it does in selfdriving cars or aircraft control support systems. Due to their many subtleties, controllers for cyber-physical systems deserve to be held to the highest correctness standards. Their correct functioning is crucial, which explains the broad interest in safety analysis technology for their mathematical models, which are called hybrid systems because they combine discrete dynamics with continuous dynamics. Differential dynamic logic (dL) provides logical specification and rigorous reasoning techniques for hybrid systems. The logic dL is implemented in the theorem prover KeYmaera X, which has been instrumental in verifying ground robot controllers, railway systems, and the next-generation airborne collision avoidance system ACAS X. This article provides an informal overview of this logical approach to CPS safety that is detailed in a recent textbook on Logical Foundations of Cyber-Physical Systems. It also explains how safety guarantees obtained in the land of verified models reach the level of CPS execution unharmed. Keywords. Cyber-physical systems and Differential dynamic logic and Hybrid systems and Theorem proving and Formal verification arXiv:1910.11232v1 [cs.LO] 24 Oct 20191 The KeYmaera X prover is available at http://keymaeraX.org/ 2 Including supporting slides and video lectures are at http://lfcps.org/lfcps/