We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded systems and cyber-physical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multi-agent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic

We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important for modeling and understanding many applications, including embedded systems and cyber-physical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multi-agent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic dynamics with hybrid systems. We survey dynamic logics for specifying and verifying properties for each of those classes of dynamical systems. A dynamic logic is a first-order modal logic with a pair of parametrized modal operators for each dynamical system to express necessary or possible properties of their transition behavior. Due to their full basis of first-order modal logic operators, dynamic logics can express a rich variety of system properties, including safety, controllability, reactivity, liveness, and quantified parametrized properties, even about

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Biodiesel is an alternative fuel source that can be easily made by novices with an inexpensive home-made reactor using waste vegetable oil, but producing high quality fuel using a home-made reactor is difficult due to the complexity of the chemical interactions and the configurations of the proces-sor. A biodiesel processor is a complex system that can be modeled and simulated using formal modeling methods, but accurate modeling can require prohibitively expensive anal-ysis. In this work we present our model analysis approach designed to ‘crowdsource ’ simulations of a formal Stochastic Hybrid System model of biodiesel production. The approach uses an iPad application called Biodiesel Sim, which allows users to play a simulation game that calculates trajectories of a formal stochastic model of biodiesel production. The results of simulations run in the game are collected in a centralized database, which can be searched by users or researchers via a website to encourage users to improve their simulations. The framework used in this application can be used as a template for analysis of other formal models. 1.