When dealing with unrealizable specifications in reactive synthesis, finding the weakest environment assumptions that ensure realizability is often considered a desirable property. However, little effort has been dedicated to defining or evaluating the notion of weakness of assumptions formally. The question of whether one assumption is weaker than another is commonly interpreted by considering the implication relationship between the two or, equivalently, their language inclusion. This interpretation fails to provide any insight into the weakness of the assumptions when implication (or language inclusion) does not hold. To our knowledge, the only measure that is capable of comparing two formulae in this case is entropy, but even it cannot distinguish the weakness of assumptions expressed as fairness properties. In this paper, we propose a refined measure of weakness based on combining entropy with Hausdorff dimension, a concept that captures the notion of size of the ω -language satisfying a linear temporal logic formula. We focus on a special subset of linear temporal logic formulae which is of particular interest in reactive synthesis, called GR(1). We identify the conditions under which this measure is guaranteed to distinguish between weaker and stronger GR(1) formulae, and propose a refined measure to cover cases when two formulae are strictly ordered by implication but have the same entropy and Hausdorff dimension. We prove the consistency between our weakness measure and logical implication, that is, if one formula implies another, the latter is weaker than the former according to our measure. We evaluate our proposed weakness measure in two contexts. The first is in computing GR(1) assumption refinements where our weakness measure is used as a heuristic to drive the refinement search towards weaker solutions. The second is in the context of quantitative model checking where it is used to measure the size of the language of a model violating a linear temporal logic formula.

中文翻译：

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GR(1) 公式的弱点度量

在处理反应式合成中无法实现的规范时，找到确保可实现性的最弱环境假设通常被认为是理想的属性。然而，很少有人致力于正式定义或评估假设弱点的概念。一个假设是否比另一个更弱的问题通常通过考虑两者之间的隐含关系或等效地考虑它们的语言包含来解释。当暗示（或语言包含）不成立时，这种解释无法提供对假设弱点的任何洞察。据我们所知，在这种情况下，唯一能够比较两个公式的度量是熵，但即使它也无法区分表示为公平属性的假设的弱点。在本文中， ω -满足线性时序逻辑公式的语言。我们专注于线性时序逻辑公式的一个特殊子集，它对反应合成特别感兴趣，称为 GR(1)。我们确定了保证该度量能够区分较弱和较强的 GR(1) 公式的条件，并提出了一种改进的度量来涵盖两个公式按隐含严格排序但具有相同熵和 Hausdorff 维数的情况。我们证明了我们的弱点度量和逻辑蕴涵之间的一致性，即如果一个公式暗示另一个，根据我们的度量，后者比前者弱。我们在两种情况下评估我们提出的弱点度量。第一个是在计算 GR(1) 假设细化时，我们的弱点度量被用作启发式方法来推动细化搜索向较弱的解决方案。第二个是在定量模型检查的背景下，它用于测量违反线性时间逻辑公式的模型语言的大小。