Safety verification ensures that a system avoids undesired behaviour. Liveness complements safety, ensuring that the system also achieves its desired objectives. A complete specification of functional correctness must combine both safety and liveness. Proving with mathematical certainty that a system satisfies a safety property demands presenting an appropriate inductive invariant of the system, whereas proving liveness requires showing a measure of progress witnessed by a ranking function. Neural model checking has recently introduced a data-driven approach to the formal verification of reactive systems, albeit focusing on ranking functions and thus addressing liveness properties only. In this paper, we extend and generalise neural model checking to additionally encompass inductive invariants and thus safety properties as well. Given a system and a linear temporal logic specification of safety and liveness, our approach alternates a learning and a checking component towards the construction of a provably sound neural certificate. Our new method introduces a neural certificate architecture that jointly represents inductive invariants as proofs of safety, and ranking functions as proofs of liveness. Moreover, our new architecture is amenable to training using constraint solvers, accelerating prior neural model checking work otherwise based on gradient descent. We experimentally demonstrate that our method is orders of magnitude faster than the state-of-the-art model checkers on pure liveness and combined safety and liveness verification tasks written in SystemVerilog, while enabling the verification of richer properties than was previously possible for neural model checking.

We introduce a machine learning approach to model checking temporal logic, with application to formal hardware verification. Model checking answers the question of whether every execution of a given system satisfies a desired temporal logic specification. Unlike testing, model checking provides formal guarantees. Its application is expected standard in silicon design and the EDA industry has invested decades into the development of performant symbolic model checking algorithms. Our new approach combines machine learning and symbolic reasoning by using neural networks as formal proof certificates for linear temporal logic. We train our neural certificates from randomly generated executions of the system and we then symbolically check their validity using satisfiability solving which, upon the affirmative answer, establishes that the system provably satisfies the specification. We leverage the expressive power of neural networks to represent proof certificates as well as the fact that checking a certificate is much simpler than finding one. As a result, our machine learning procedure for model checking is entirely unsupervised, formally sound, and practically effective. We experimentally demonstrate that our method outperforms the state-of-the-art academic and commercial model checkers on a set of standard hardware designs written in SystemVerilog.