This paper addresses the complexity of SAT-based invariant inference, a prominent approach to safety verification. We consider the problem of inferring an inductive invariant of polynomial length given a transition system and a safety property. We analyze the complexity of this problem in a black-box model, called the Hoare-query model, which is general enough to capture algorithms such as IC3/PDR and its variants. An algorithm in this model learns about the system's reachable states by querying the validity of Hoare triples. We show that in general an algorithm in the Hoare-query model requires an exponential number of queries. Our lower bound is information-theoretic and applies even to computationally unrestricted algorithms, showing that no choice of generalization from the partial information obtained in a polynomial number of Hoare queries can lead to an efficient invariant inference procedure in this class. We then show, for the first time, that by utilizing rich Hoare queries, as done in PDR, inference can be exponentially more efficient than approaches such as ICE learning, which only utilize inductiveness checks of candidates. We do so by constructing a class of transition systems for which a simple version of PDR with a single frame infers invariants in a polynomial number of queries, whereas every algorithm using only inductiveness checks and counterexamples requires an exponential number of queries. Our results also shed light on connections and differences with the classical theory of exact concept learning with queries, and imply that learning from counterexamples to induction is harder than classical exact learning from labeled examples. This demonstrates that the convergence rate of Counterexample-Guided Inductive Synthesis depends on the form of counterexamples.

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不变推理中的复杂性和信息

本文讨论了基于 SAT 的不变推理的复杂性，这是一种重要的安全验证方法。我们考虑在给定转换系统和安全属性的情况下推断多项式长度的归纳不变量的问题。我们在称为 Hoare-query 模型的黑盒模型中分析了这个问题的复杂性，该模型足够通用，可以捕获 IC3/PDR 及其变体等算法。该模型中的算法通过查询 Hoare 三元组的有效性来了解系统的可达状态。我们表明，通常 Hoare 查询模型中的算法需要指数数量的查询。我们的下界是信息论的，甚至适用于计算不受限制的算法，表明从多项式 Hoare 查询中获得的部分信息中没有选择泛化可以导致此类中有效的不变推理过程。然后，我们首次表明，通过使用丰富的 Hoare 查询，如在 PDR 中所做的那样，推理可以比 ICE 学习之类的方法更有效，后者仅利用候选者的归纳检查。我们通过构建一类转换系统来实现这一点，其中具有单帧的简单版本的 PDR 推断多项式查询中的不变量，而每个仅使用归纳检查和反例的算法都需要指数数量的查询。我们的结果还阐明了与带有查询的精确概念学习的经典理论的联系和差异，并暗示从反例学习到归纳比从标记示例中学习经典精确学习更难。这表明反例引导归纳综合的收敛速度取决于反例的形式。