This paper introduces Fossil 2.0, an advanced software tool designed for synthesizing certificates such as Lyapunov and barrier functions for dynamical systems represented by ordinary differential equations and difference equations. Fossil 2.0 features a range of significant enhancements, including improved user interfaces, an expanded library of certificates, controller synthesis capabilities, and an extensible architecture. These advancements are detailed as part of this paper. The core of Fossil is a counterexample-guided inductive synthesis (CEGIS) framework that ensures soundness. The tool employs neural networks as templates to generate candidate functions, which are rigorously validated using a satisfiability modulo theories (SMT) solver. Key improvements over the previous release include support for a broader class of certificates, integration of control law synthesis, and compatibility with discrete-time models.

An emerging branch of control theory specialises in certificate learning, concerning the specification of a desired (possibly complex) system behaviour for an autonomous or control model, which is then analytically verified by means of a function-based proof. However, the synthesis of controllers abiding by these complex requirements is in general a non-trivial task and may elude the most expert control engineers. This results in a need for automatic techniques that are able to design controllers and to analyse a wide range of elaborate specifications. In this paper, we provide a general framework to encode system specifications and define corresponding certificates, and we present an automated approach to formally synthesise controllers and certificates. Our approach contributes to the broad field of safe learning for control, exploiting the flexibility of neural networks to provide candidate control and certificate functions, whilst using SAT-modulo-theory (SMT)-solvers to offer a formal guarantee of correctness. We test our framework by developing a prototype software tool, and assess its efficacy at verification via control and certificate synthesis over a large and varied suite of benchmarks.

This paper presents Fossil 2.0, a new major release of a software tool for the synthesis of certificates (e.g., Lyapunov and barrier functions) for dynamical systems modelled as ordinary differential and difference equations. Fossil 2.0 is much improved from its original release, including new interfaces, a significantly expanded certificate portfolio, controller synthesis and enhanced extensibility. We present these new features as part of this tool paper. Fossil implements a counterexample-guided inductive synthesis (CEGIS) loop ensuring the soundness of the method. Our tool uses neural networks as templates to generate candidate functions, which are then formally proven by an satisfiability modulo theories solver acting as an assertion verifier. Improvements with respect to the first release include a wider range of certificates, synthesis of control laws, and support for discrete-time models.