LLM2Cert Uses LLMs to Synthesize Verifiable Stability and Safety Certificates

Introduction

For robotic controllers, autonomous systems, and other safety-critical applications, simulation alone cannot establish that a system is stable or will remain inside a safe region. Researchers instead use mathematical certificates such as Lyapunov functions and barrier certificates to turn those claims into proof obligations. The challenge is that constructing suitable certificates becomes difficult when the dynamics are nonlinear, high-dimensional, or subject to control constraints.

The paper LLM2Cert: LLM Synthesis of Certificates with SMT/Lean Verification explores a hybrid solution. Rather than treating a language model as the final authority, the framework uses it to propose candidate certificates and relies on formal tools to determine whether the resulting claims actually hold.

Key points

• A broader benchmark. The authors introduce DynaCertBench with 105 instances. It contains 80 non-redundant systems curated from the literature, along with high-dimensional variants, control-constrained variants, and newly introduced nonlinearly coupled systems. The tasks span stability, region of attraction, forward invariance, and safety.
• Generation is separated from proof. LLM2Cert asks an LLM to synthesize candidate Lyapunov functions or barrier certificates. The associated proof obligations are then routed to an SMT solver or directly to a Lean Proof-Repair agent.
• Verification feedback closes the loop. When a proposal fails, the information from the checker is used to guide subsequent revisions. This is different from one-shot generation, where an apparently plausible expression is accepted or rejected without an iterative repair process.
• The combined pipeline improves coverage. LLM2Cert verifies 81 of 105 benchmark instances, or 77.1%. This is a 17.1 percentage-point improvement over one-shot proposals without feedback. Among the 81 fixed certificates, 22 are cases that All-SMT leaves unresolved.

Why it matters

The central contribution is not simply asking an LLM to write mathematical formulas. It is a division of labor between search and proof: the model explores a large space of possible certificate structures, while SMT and Lean enforce logical and formal requirements. Such a workflow could reduce the amount of manual effort required to construct certificates and help address scaling limitations in numerical or solver-only approaches.

The benchmark design is also significant. If test suites contain mainly low-dimensional polynomial systems selected for solver convenience, reported progress may not reflect engineering difficulty. By adding higher-dimensional, coupled, and control-constrained cases, DynaCertBench creates a more demanding test of automated certificate synthesis.

The results should nevertheless be read as evidence for a promising workflow, not as a claim that certificate discovery has been solved. A substantial portion of the benchmark remains unresolved, and the cost of formal checking may still grow with system complexity. LLM2Cert’s main lesson is that iterative generation, verification, and repair can make language-model-assisted mathematical reasoning more dependable without replacing formal proof tools.

Source: Hugging Face Daily Papers