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In this paper we employ SMT solvers to soundly synthesise Lyapunov functions that assert the stability of a given dynamical model. The search for a Lyapunov function is framed as the satisfiability of a second-order logical formula, asking whether there exists a function satisfying a desired specification (stability) for all possible initial conditions of the model. We synthesise Lyapunov functions for linear, non-linear (polynomial), and for parametric models. For non-linear models, the algorithm also determines a region of validity for the Lyapunov function. We exploit an inductive framework to synthesise Lyapunov functions, starting from parametric templates. The inductive framework comprises two elements: a learner proposes a Lyapunov function, and a verifier checks its validity-its lack is expressed via a counterexample (a point over the state space), for further use by the learner. Whilst the verifier uses the SMT solver Z3, thus ensuring the overall soundness of the procedure, we examine two alternatives for the learner: a numerical approach based on the optimisation tool Gurobi, and a sound approach based again on Z3. The overall technique is evaluated over a broad set of benchmarks, which shows that this methodology not only scales to 10-dimensional models within reasonable computational time, but also offers a novel soundness proof for the generated Lyapunov functions and their domains of validity.

We introduce an automated, formal, counterexample-based approach to synthesise Barrier Certificates (BC) for the safety verification of continuous and hybrid dynamical models. The approach is underpinned by an inductive framework: this is structured as a sequential loop between a learner, which manipulates a candidate BC structured as a neural network, and a sound verifier, which either certifies the candidate’s validity or generates counter-examples to further guide the learner. We compare the approach against state-of-the-art techniques, over polynomial and non-polynomial dynamical models: the outcomes show that we can synthesise sound BCs up to two orders of magnitude faster, with in particular a stark speedup on the verification engine (up to three orders less), whilst needing a far smaller data set (up to three orders less) for the learning part. Beyond improvements over the state of the art, we further challenge the new approach on a hybrid dynamical model and on larger-dimensional models, and showcase the numerical robustness of our algorithms and codebase.

We introduce an automated, formal, counterexample-based approach to synthesise Barrier Certificates (BC) for the safety verification of continuous and hybrid dynamical models. The approach is underpinned by an inductive framework: this is structured as a sequential loop between a learner, which manipulates a candidate BC as a neural network, and a sound verifier, which either certifies through algorithmic proofs the candidate's validity or generates counter-examples to further guide the learner. We compare the approach against state-of-the-art techniques, over polynomial and non-polynomial dynamical models: the outcomes show that we can synthesise sound BCs up to two orders of magnitude faster, with in particular a stark speedup on the verification engine (up to five orders less), whilst needing a far smaller data set (up to three orders less) for the learning part. Beyond the state of the art, we further challenge the (verification side of the) approach on a hybrid dynamical model.