Notions of transition invariants and closure certificates have seen recent use in the formal verification of controlled dynamical systems against \omega-regular properties. Unfortunately, existing approaches face limitations in two directions. First, they require a closed-form mathematical expression representing the model of the system. Such an expression may be difficult to find, too complex to be of any use, or unavailable due to security or privacy constraints. Second, finding such invariants typically rely on optimization techniques such as sum-of-squares (SOS) or satisfiability modulo theory (SMT) solvers. This restricts the classes of systems that need to be formally verified. To address these drawbacks, we introduce a notion of neural closure certificates. We present a data-driven algorithm that trains a neural network to represent a closure certificate. Our approach is formally correct under some mild assumptions, i.e., one is able to formally show that the unknown system satisfies the \omega-regular property of interest if a neural closure certificate can be computed. Finally, we demonstrate the efficacy of our approach with relevant case studies.

A barrier certificate, defined over the states of a dynamical system, is a real-valued function whose zero level set characterizes an inductively verifiable state invariant separating reachable states from unsafe ones. When combined with powerful decision procedures—such as sum-of-squares programming (SOS) or satisfiability-modulo-theory solvers (SMT)—barrier certificates enable an automated deductive verification approach to safety. The barrier certificate approach has been extended to refute LTL and ω -regular specifications by separating consecutive transitions of corresponding ω -automata in the hope of denying all accepting runs. Unsurprisingly, such tactics are bound to be conservative as refutation of recurrence properties requires reasoning about the well-foundedness of the transitive closure of the transition relation. This paper introduces the notion of closure certificates as a natural extension of barrier certificates from state invariants to transition invariants. We augment these definitions with SOS and SMT based characterization for automating the search of closure certificates and demonstrate their effectiveness over some case studies.

With the focus in the realm of automotive testing, process thermostats play a vital role in providing the required operating environment. These process thermostats, with the operating medium of an ethylene glycol-water ratio, play a crucial role in terms of controlling their thermal properties. With an emphasis on identifying the best preprocessing method for classifying these ratios, this study utilises a detailed comparison of statistical methods and the wrapper method-based Genetic Algorithm as a search method for the most relevant feature selections. Given the huge number of existing sensor parameters in the system, thereby emphasising the importance of feature selection criteria for effective analysis and model training. Furthermore, a random forest-based classifier is used with these parameters to predict the accurate ethylene glycol to water ratio.

Barrier certificates are inductive invariants that provide guarantees on the safety and reachability behaviors of continuous dynamical systems. For stochastic dynamical systems, barrier certificates take the form of inductive “expectation” invariants. In this context, a barrier certificate is a non-negative real-valued function over the state space of the system satisfying a strong supermartingale condition: it decreases in expectation as the system evolves The existence of barrier certificates, then, provides lower bounds on the probability of satisfaction of safety or reachability specifications over unbounded-time horizons. Unfortunately, establishing supermartingale conditions on barrier certificates can often be restrictive. In practice, we strive to overcome this challenge by utilizing a weaker condition called c-martingale that permits a bounded increment in expectation at every time step; unfortunately this only guarantees the property of interest for a bounded time horizon.