Reachability Analysis Using Message Passing over Tree Decompositions

In this paper, we study efficient approaches to reachability analysis for discrete-time nonlinear dynamical systems when the dependencies among the variables of the system have low treewidth. Reachability analysis over nonlinear dynamical systems asks if a given set of target states can be reached, starting from an initial set of states. This is solved by computing conservative over approximations of the reachable set using abstract domains to represent these approximations. However, most approaches must tradeoff the level of conservatism against the cost of performing analysis, especially when the number of system variables increases. This makes reachability analysis challenging for nonlinear systems with a large number of state variables. Our approach works by constructing a dependency graph among the variables of the system. The tree decomposition of this graph builds a tree wherein each node of the tree is labeled with subsets of the state variables of the system. Furthermore, the tree decomposition satisfies important structural properties. Using the tree decomposition, our approach abstracts a set of states of the high dimensional system into a tree of sets of lower dimensional projections of this state. We derive various properties of this abstract domain, including conditions under which the original high dimensional set can be fully recovered from its low dimensional projections. Next, we use ideas from message passing developed originally for belief propagation over Bayesian networks to perform reachability analysis over the full state space in an efficient manner. We illustrate our approach on some interesting nonlinear systems with low treewidth to demonstrate the advantages of our approach.