We exploit the convexity property of the barrier function to formulate the optimal control synthesis problem asalinear program.

Reasoning with LLMs increasingly unfolds inside a broader verification loop. Internally, systems use cheap checks, such as self-consistency or proxy rewards, which we call weak verification. Externally, users inspect outputs and steer the model through feedback until results are trustworthy, which we call strong verification. These signals differ sharply in cost and reliability: strong verification can establish trust but is resource-intensive, while weak verification is fast and scalable but noisy and imperfect. We formalize this tension through weak--strong verification policies, which decide when to accept or reject based on weak verification and when to defer to strong verification. We introduce metrics capturing incorrect acceptance, incorrect rejection, and strong-verification frequency. Over population, we show that optimal policies admit a two-threshold structure and that calibration and sharpness govern the value of weak verifiers. Building on this, we develop an online algorithm that provably controls acceptance and rejection errors without assumptions on the query stream, the language model, or the weak verifier.

In CBF-based control, the desired safety properties of the system are mapped to nonnegativity of a CBF, and the control input is chosen to ensure that the CBF remains nonnegative for all time.

We collect 312 programs from various sources, including daily programs from college homework, the international competition (SV -COMP), benchmarks from previous papers (SLING), and programs from real-world software systems (Linux Kernel, GlibC, LiteOS, and Zephyr).

Code is at this URL.

Learning reliably safe autonomous control is one of the core problems in trustworthy autonomy.

However, many use cases of neural network verification require solving the inverse problem, or over-approximating the set of inputs that lead to certain outputs.

Neural Control Barrier Functions (NCBFs) have shown significant promise in enforcing safety constraints on nonlinear autonomous systems.