[Paper Review] A General Framework for Verification and Control of Dynamical Models via Certificate Synthesis

Alec Edwards, Andrea Peruffo+1 more|arXiv (Cornell University)|Sep 12, 2023

Adversarial Robustness in Machine LearningComputer Science3 citations

TL;DR

This paper presents a general framework for formally verifying and synthesizing controllers for dynamical systems using certificate synthesis via SMT-solving. It combines neural networks for candidate certificate and controller functions with SMT solvers to automatically generate provably correct certificates for complex specifications like safety, reachability, and their combinations, achieving formal correctness guarantees across diverse benchmarks.

ABSTRACT

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 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.

Motivation & Objective

• To address the challenge of automatically designing controllers and formally verifying complex system behaviors in nonlinear dynamical systems.
• To unify diverse specifications—such as safety, reachability, and their combinations—under a single certificate-based formal verification framework.
• To enable provably correct controller and certificate synthesis using neural networks as function templates and SMT solvers for formal verification.
• To extend beyond polynomial templates and convexity assumptions by leveraging SMT-solving for non-polynomial and complex system dynamics.
• To provide a scalable, automated, and formally sound approach for safety-critical control applications.

Proposed method

• Formulate system specifications (safety, reachability, reach-avoid, etc.) as logical constraints over state trajectories.
• Use neural networks with ReLU activation functions as parametric templates for candidate Lyapunov or barrier certificates and controllers.
• Encode the certificate synthesis problem as a logical formula in SMT-solver-relevant theories (e.g., nonlinear real arithmetic, polynomial constraints).
• Leverage SMT solvers (e.g., Z3, dReal, CVC5) to verify the correctness of candidate certificates and find valid parameterizations.
• Integrate counterexample-guided refinement loops to iteratively improve neural network-based certificates when initial candidates fail.
• Support modular composition of specifications using temporal logic (LTL) and finite automata, enabling synthesis for complex properties like weak until and release operators.

Experimental results

Research questions

• RQ1 Can a unified framework be developed to synthesize controllers and certificates for a broad class of dynamical system specifications?
• RQ2 How can neural networks be effectively used as expressive function templates for certificates while ensuring formal correctness?
• RQ3 To what extent can SMT solvers be leveraged to verify and synthesize certificates for non-polynomial and hybrid system dynamics?
• RQ4 Can complex specifications such as reach-while-avoid and reach-and-stay-while-stay be systematically encoded and verified using this framework?
• RQ5 How does the approach scale across diverse benchmarks with varying dynamical complexity and specification types?

Key findings

• The framework successfully synthesizes provably correct certificates and controllers for a wide range of specifications, including safety, reachability, and their combinations, across a large and diverse benchmark suite.
• The integration of neural networks with SMT solvers enables flexible, high-capacity function approximation while maintaining formal correctness guarantees.
• The approach outperforms traditional template-based methods by handling non-polynomial dynamics and complex specifications beyond sum-of-squares or linear programming constraints.
• The use of SMT solvers allows for sound verification even in the presence of non-convex and nonlinear system dynamics, ensuring robustness in safety-critical applications.
• The framework supports modular specification composition via LTL and finite automata, enabling synthesis for complex temporal properties such as weak until and release operators.
• Empirical evaluation demonstrates the efficacy and scalability of the prototype tool across benchmarks involving nonlinear ODEs and hybrid system behaviors.