On the Trade-off Between Efficiency and Precision of Neural Abstraction

Neural abstractions have been recently introduced as formal approximations of complex, nonlinear dynamical models. They comprise a neural ODE and a certified upper bound on the error between the abstract neural network and the concrete dynamical model. So far neural abstractions have exclusively been obtained as neural networks consisting entirely of $ReLU$ activation functions, resulting in neural ODE models that have piecewise affine dynamics, and which can be equivalently interpreted as linear hybrid automata. In this work, we observe that the utility of an abstraction depends on its use: some scenarios might require coarse abstractions that are easier to analyse, whereas others might require more complex, refined abstractions. We therefore consider neural abstractions of alternative shapes, namely either piecewise constant or nonlinear non-polynomial (specifically, obtained via sigmoidal activations). We employ formal inductive synthesis procedures to generate neural abstractions that result in dynamical models with these semantics. Empirically, we demonstrate the trade-off that these different neural abstraction templates have vis-a-vis their precision and synthesis time, as well as the time required for their safety verification (done via reachability computation). We improve existing synthesis techniques to enable abstraction of higher-dimensional models, and additionally discuss the abstraction of complex neural ODEs to improve the efficiency of reachability analysis for these models.Comment: To appear at QEST 202

Reachability analysis of linear hybrid systems via block decomposition

Reachability analysis aims at identifying states reachable by a system within a given time horizon. This task is known to be computationally expensive for linear hybrid systems. Reachability analysis works by iteratively applying continuous and discrete post operators to compute states reachable according to continuous and discrete dynamics, respectively. In this paper, we enhance both of these operators and make sure that most of the involved computations are performed in low-dimensional state space. In particular, we improve the continuous-post operator by performing computations in high-dimensional state space only for time intervals relevant for the subsequent application of the discrete-post operator. Furthermore, the new discrete-post operator performs low-dimensional computations by leveraging the structure of the guard and assignment of a considered transition. We illustrate the potential of our approach on a number of challenging benchmarks.Comment: Accepted at EMSOFT 202

Tropical Fourier-Motzkin elimination, with an application to real-time verification

We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier-Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata).Comment: 29 pages, 8 figure

Provable Run Time Safety Assurance for a Non-Linear System

Systems that are modeled by non-linear continuous-time differential equations with uncertain parameters have proven to be exceptionally difficult to formally verify. The past few decades have produced a number of useful verification tools which can be applied to such systems but each is applicable to only a subset of possible verification scenarios. The Level Sets Toolbox (LST) is one such tool which is directly applicable to non-linear systems, however, it is limited to systems of relatively small continuous state space dimension. Other tools such as PHAVer and the SpaceEx invariant of the Le Guernic-Girard (LGG) support function algorithm are specifically designed for hybrid systems with linear dynamics and linear constraints but can accommodate hundreds of continuous states. The application of these linear reachability tools to non-linear models has been achieved by approximating non-linear systems as linear hybrid automata (LHA). Unfortunately, the practical applicability and limitations of this approach are not yet well documented. The purpose of this thesis is to evaluate the performance and dimensionality limitations of PHAVer and the LGG support function algorithm when applied to a LHA approximation of a particular non-linear system. A collision avoidance scenario with autonomous differential drive robots is used as a case study to demonstrate that an over-approximated reachable set boundary can be generated and implemented as a run time safety assurance mechanism