We present a mathematical setting for attack-defense trees, a classic graphical model to specify attacks and countermeasures. We equip attack-defense trees with (trace) language semantics allowing to have an original dynamic interpretation of countermeasures. Interestingly, the expressiveness of attack-defense trees coincides with star-free languages, and the nested countermeasures impact the expressiveness of attack-defense trees. With an adequate notion of countermeasure-depth, we exhibit a strict hierarchy of the star-free languages that does not coincides with the classic one. Additionally, driven by the use of attack-defense trees in practice, we address the decision problems of trace membership and of non-emptiness, and study their computational complexities parameterized by the countermeasure-depth.

中文翻译：

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用于动态对抗的攻防树语义和新的无星语言层次结构

我们提出了攻击防御树的数学设置，这是一种指定攻击和对策的经典图形模型。我们为攻击防御树配备（跟踪）语言语义，从而可以对对策进行原始的动态解释。有趣的是，攻防树的表达能力与无星语言相吻合，嵌套的对策影响了攻防树的表达能力。有了足够的对抗深度概念，我们展示了一种与经典语言并不相符的严格的无星语言层次结构。此外，在实践中使用攻防树的驱动下，我们解决了踪迹成员资格和非空性的决策问题，并研究了它们由对策深度参数化的计算复杂度。