Sapo: Reachability Computation and Parameter Synthesis of Polynomial Dynamical Systems

Sapo is a C++ tool for the formal analysis of polynomial dynamical systems. Its main features are: 1) Reachability computation, i.e., the calculation of the set of states reachable from a set of initial conditions, and 2) Parameter synthesis, i.e., the refinement of a set of parameters so that the system satisfies a given specification. Sapo can represent reachable sets as unions of boxes, parallelotopes, or parallelotope bundles (symbolic representation of polytopes). Sets of parameters are represented with polytopes while specifications are formalized as Signal Temporal Logic (STL) formulas

Reachability computation for polynomial dynamical systems

This paper is concerned with the problem of computing the bounded time reachable set of a polynomial discrete-time dynamical system. The problem is well-known for being difficult when nonlinear systems are considered. In this regard, we propose three reachability methods that differ in the set representation. The proposed algorithms adopt boxes, parallelotopes, and parallelotope bundles to construct flowpipes that contain the actual reachable sets. The latter is a new data structure for the symbolic representation of polytopes. Our methods exploit the Bernstein expansion of polynomials to bound the images of sets. The scalability and precision of the presented methods are analyzed on a number of dynamical systems, in comparison with other existing approaches

Automatic Dynamic Parallelotope Bundles for Reachability of Nonlinear Dynamical Systems

Reachable set computation is an important technique for the verification of safety properties of dynamical systems. In this thesis, we investigate reachable set computation for discrete non-linear systems based on parallelotope bundles. The crux of the reachability algorithm relies on computing an upper and lower bound on the supremum and infimum respectively of a non-linear function over a rectangular domain. Bernstein Expansion of a polynomial function has been explored as a traditional method for computing these bounds efficiently. In light of this, we aim to improve the traditional parallelotope-based reachability method by removing the manual step of parallelotope template selection in order to make the procedure fully automatic. Furthermore, we show that adding templates dynamically during computations can improve accuracy. To this end, we investigate two techniques for generating template directions. The first technique approximates the dynamics as a linear transformation and generates templates using this transformation. The second technique uses Principal Component Analysis (PCA) of sample trajectories for generating templates. We have implemented our approach in a Python-based tool called Kaa, which uses two types of global optimization solvers, the first using Bernstein polynomials and the second usingthe Kodiak library. We demonstrate the improved accuracy of our approach on several standard nonlinear benchmark systems, including a high-dimensional COVID19 model. Finally, we explore a potential application of the Bernstein expansion technique to real-time reachability. We present evidence of several hurdles and barriers against effectively utilizing our Bernstein coefficient pruning method.Master of Scienc

LNCS

Template polyhedra generalize intervals and octagons to polyhedra whose facets are orthogonal to a given set of arbitrary directions. They have been employed in the abstract interpretation of programs and, with particular success, in the reachability analysis of hybrid automata. While previously, the choice of directions has been left to the user or a heuristic, we present a method for the automatic discovery of directions that generalize and eliminate spurious counterexamples. We show that for the class of convex hybrid automata, i.e., hybrid automata with (possibly nonlinear) convex constraints on derivatives, such directions always exist and can be found using convex optimization. We embed our method inside a CEGAR loop, thus enabling the time-unbounded reachability analysis of an important and richer class of hybrid automata than was previously possible. We evaluate our method on several benchmarks, demonstrating also its superior efficiency for the special case of linear hybrid automata

A Classification-based Approach for Approximate Reachability

Hamilton-Jacobi (HJ) reachability analysis has been developed over the past decades into a widely-applicable tool for determining goal satisfaction and safety verification in nonlinear systems. While HJ reachability can be formulated very generally, computational complexity can be a serious impediment for many systems of practical interest. Much prior work has been devoted to computing approximate solutions to large reachability problems, yet many of these methods may only apply to very restrictive problem classes, do not generate controllers, and/or can be extremely conservative. In this paper, we present a new method for approximating the optimal controller of the HJ reachability problem for control-affine systems. While also a specific problem class, many dynamical systems of interest are, or can be well approximated, by control-affine models. We explicitly avoid storing a representation of the reachability value function, and instead learn a controller as a sequence of simple binary classifiers. We compare our approach to existing grid-based methodologies in HJ reachability and demonstrate its utility on several examples, including a physical quadrotor navigation task

ReachNN: Reachability Analysis of Neural-Network Controlled Systems

Applying neural networks as controllers in dynamical systems has shown great promises. However, it is critical yet challenging to verify the safety of such control systems with neural-network controllers in the loop. Previous methods for verifying neural network controlled systems are limited to a few specific activation functions. In this work, we propose a new reachability analysis approach based on Bernstein polynomials that can verify neural-network controlled systems with a more general form of activation functions, i.e., as long as they ensure that the neural networks are Lipschitz continuous. Specifically, we consider abstracting feedforward neural networks with Bernstein polynomials for a small subset of inputs. To quantify the error introduced by abstraction, we provide both theoretical error bound estimation based on the theory of Bernstein polynomials and more practical sampling based error bound estimation, following a tight Lipschitz constant estimation approach based on forward reachability analysis. Compared with previous methods, our approach addresses a much broader set of neural networks, including heterogeneous neural networks that contain multiple types of activation functions. Experiment results on a variety of benchmarks show the effectiveness of our approach

POLAR-Express: Efficient and Precise Formal Reachability Analysis of Neural-Network Controlled Systems

Neural networks (NNs) playing the role of controllers have demonstrated impressive empirical performance on challenging control problems. However, the potential adoption of NN controllers in real-life applications has been significantly impeded by the growing concerns over the safety of these neural-network controlled systems (NNCSs). In this work, we present POLAR-Express, an efficient and precise formal reachability analysis tool for verifying the safety of NNCSs. POLAR-Express uses Taylor model arithmetic to propagate Taylor models (TMs) layer-by-layer across a neural network to compute an over-approximation of the neural network. It can be applied to analyze any feed-forward neural networks with continuous activation functions, such as ReLU, Sigmoid, and Tanh activation functions that cover the common benchmarks for NNCS reachability analysis. Compared with its earlier prototype POLAR, we develop a novel approach in POLAR-Express to propagate TMs more efficiently and precisely across ReLU activation functions, and provide parallel computation support for TM propagation, thus significantly improving the efficiency and scalability. Across the comparison with six other state-of-the-art tools on a diverse set of common benchmarks, POLAR-Express achieves the best verification efficiency and tightness in the reachable set analysis. POLAR-Express is publicly available at https://github.com/ChaoHuang2018/POLAR_Tool