This report presents the results of a friendly competition for formal verification and policy synthesis of stochastic models. The friendly competition took place as part of the workshop Applied Verification for Continuous and Hybrid Systems (ARCH) in 2018. In this first edition, we present five benchmarks with different levels of complexities and stochastic flavours. We make use of six different tools and frameworks (in alphabetical order): Barrier Certificates, FAUST2, FIRM-GDTL, Modest, SDCPN modelling & MC simulation and SReachTools; and attempt to solve instances of the five different benchmark problems. Through these benchmarks, we capture a snapshot on the current state-of the art tools and frameworks within the stochastic modelling domain. We also present the challenges encountered within this domain and highlight future plans which will push forward the development of more tools and methodologies for performing formal verification and optimal policy synthesis of stochastic processes.

We examine Lagrangian techniques for computing underapproximations of finite-time horizon, stochastic reachavoid level-sets for discrete-time, nonlinear systems. We use the concept of reachability of a target tube in the control literature to define robust reach-avoid sets which are parameterized by the target set, safe set, and the set in which the disturbance is drawn from. We unify two existing Lagrangian approaches to compute these sets and establish that there exists an optimal control policy of the robust reach-avoid sets which is a Markov policy. Based on these results, we characterize the subset of the disturbance space whose corresponding robust reachavoid set for the given target and safe set is a guaranteed underapproximation of the stochastic reach-avoid level-set of interest. The proposed approach dramatically improves the computational efficiency for obtaining an underapproximation of stochastic reach-avoid level-sets when compared to the traditional approaches based on gridding. Our method, while conservative, does not rely on a grid, implying scalability as permitted by the known computational geometry constraints. We demonstrate the method on two examples: a simple twodimensional integrator, and a space vehicle rendezvous-docking problem.

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This report presents the results of a friendly competition for formal verification and policy synthesis of stochastic models. It also introduces new benchmarks within this category, and recommends next steps for this category towards next year's edition of the competition. The friendly competition took place as part of the workshop Applied Verification for Continuous and Hybrid Systems (ARCH) in Spring 2019.

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Abstract-We present a scalable underapproximation of the terminal hitting time stochastic reach-avoid probability at a given initial condition, for verification of high-dimensional stochastic LTI systems. While several approximation techniques have been proposed to alleviate the curse of dimensionality associated with dynamic programming, these techniques are limited and cannot handle larger, more realistic systems. We present a scalable method that uses Fourier transforms to compute an underapproximation of the reach-avoid probability for systems with disturbances with arbitrary probability densities. We characterize sufficient conditions for Borel-measurability of the value functions. We exploit fixed control sequences parameterized by the initial condition (an open-loop control policy) to generate the underapproximation. For Gaussian disturbances, the underapproximation can be obtained using existing efficient algorithms by solving a convex optimization problem. Our approach produces non-trivial lower bounds and is demonstrated on a chain of integrators with 40 states.

We propose a method to efficiently compute the forward stochastic reach (FSR) set and its probability measure for nonlinear systems with an affine disturbance input, that is stochastic and bounded. This method is applicable to systems with an a priori known controller, or to uncontrolled systems, and often arises in problems in obstacle avoidance in mobile robotics. When used as a constraint in finite horizon controller synthesis, the FSR set and its probability measure facilitate probabilistic collision avoidance, in contrast to methods which presume the obstacles act in a worst-case fashion, and generate hard constraints that cannot be violated. We tailor our approach to accommodate rigid body constraints, and show convexity is assured so long as the rigid body shape of each obstacle is also convex. We extend methods for multi-obstacle avoidance through mixed integer linear programming (with linear robot and obstacle dynamics) to accommodate chance constraints that represent the FSR set probability measure. We demonstrate our method on a rigid-body obstacle avoidance scenario, in which a receding horizon controller is designed to avoid several stochastically moving obstacles while reaching a desired goal. Our approach can provide solutions when approaches that presume a worst-case action from the obstacle fail.