We propose and evaluate a new technique for learning hybrid automata automatically by observing the runtime behavior of a dynamical system. Working from a sequence of continuous state values and predicates about the environment, CHARDA recovers the distinct dynamic modes, learns a model for each mode from a given set of templates, and postulates causal guard conditions which trigger transitions between modes. Our main contribution is the use of information-theoretic measures (1) as a cost function for data segmentation and model selection to penalize over-fitting and (2) to determine the likely causes of each transition. CHARDA is easily extended with different classes of model templates, fitting methods, or predicates. In our experiments on a complex videogame character, CHARDA successfully discovers a reasonable over-approximation of the character's true behaviors. Our results also compare favorably against recent work in automatically learning probabilistic timed automata in an aircraft domain: CHARDA exactly learns the modes of these simpler automata.
 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. A hybrid dynamical system is a mathematical model suitable for describing an extensive spectrum of multi-modal, time-series behaviors, ranging from bouncing balls to air traffic controllers. This paper describes multi-modal symbolic regression (MMSR): a learning algorithm to construct non-linear symbolic representations of discrete dynamical systems with continuous mappings from unlabeled, time-series data. MMSR consists of two subalgorithms--clustered symbolic regression, a method to simultaneously identify distinct behaviors while formulating their mathematical expressions, and transition modeling, an algorithm to infer symbolic inequalities that describe binary classification boundaries. These subalgorithms are combined to infer hybrid dynamical systems as a collection of apt, mathematical expressions. MMSR is evaluated on a collection of four synthetic data sets and outperforms other multi-modal machine learning approaches in both accuracy and interpretability, even in the presence of noise. Furthermore, the versatility of MMSR is demonstrated by identifying and inferring classical expressions of transistor modes from recorded measurements. This paper describes the modeling language CHARON for modular design of interacting hybrid systems. The language allows specification of architectural as well as behavioral hierarchy and discrete as well as continuous activities. The modular structure of the language is not merely syntactic, but is exploited by analysis tools and is supported by a formal semantics with an accompanying compositional theory of refinement. We illustrate the benefits of CHARON in the design of embedded control software using examples from automated highways concerning vehicle coordination.