As large language models (LLMs) become increasingly deployed in safety-critical applications, the lack of systematic methods to assess their vulnerability to jailbreak attacks presents a critical security gap. We introduce the jailbreak oracle problem: given a model, prompt, and decoding strategy, determine whether a jailbreak response can be generated with likelihood exceeding a specified threshold. This formalization enables a principled study of jailbreak vulnerabilities. Answering the jailbreak oracle problem poses significant computational challenges, as the search space grows exponentially with response length. We present Boa, the first system designed for efficiently solving the jailbreak oracle problem. Boa employs a two-phase search strategy: (1) breadth-first sampling to identify easily accessible jailbreaks, followed by (2) depth-first priority search guided by fine-grained safety scores to systematically explore promising yet low-probability paths. Boa enables rigorous security assessments including systematic defense evaluation, standardized comparison of red team attacks, and model certification under extreme adversarial conditions. Code is available at https://github.com/shuyilinn/BOA/tree/mlsys2026ae