发表机构
University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对转移概率不确定的带标记马尔可夫链，提出普遍与存在扰动等价性，证明状态情形可多项式时间判定，分布情形分别达NL与NP完全，并实现鲁棒模型缩减算法。
AI 中文摘要
行为等价性，如语言等价性和概率互模拟等价性，是缩减概率模型规模的基本技术。然而，这些等价性对转移概率的精确值敏感，这使得它们在概率需要近似处理的应用中不适用。受带标记马尔可夫链的支持图已知但转移概率不确定的场景的启发，我们研究了这些等价性的鲁棒变体。我们引入了普遍（扰动）等价性，它捕捉了一种对转移概率的所有扰动都具有鲁棒性的等价性变体：如果两个状态或分布在所有与支持图一致的转移概率赋值下都保持等价，则它们是普遍等价的。我们还考虑了存在（扰动）等价性的对偶概念，只要存在一个产生等价性的转移概率赋值，该等价性就成立。我们证明了，对于状态而言，普遍语言等价性与普遍概率互模拟等价性是一致的，并发展了一种刻画方法，从而产生了一个多项式时间的分区细化算法。我们实现了该算法，并通过实验证明它作为一种鲁棒模型缩减技术是有效的。我们进一步证明了分布的普遍语言等价性与状态情形密切相关，并证明了判定状态和分布的普遍等价性的NL完全性。我们证明了，对于状态而言，存在语言等价性与存在概率互模拟等价性是一致的，且确定性见证转移函数总是足够的，这导致了一个NP完全性结果。相比之下，我们表明分布的存在语言等价性对于实数的存在性理论是完全的。
英文摘要
Behavioural equivalences, such as language equivalence and probabilistic bisimilarity, are fundamental techniques for reducing the size of probabilistic models. However, these equivalences are sensitive to the precise values of transition probabilities, making them unsuitable in applications where probabilities are subject to approximation. Motivated by settings in which the support graph of a labelled Markov chain is known but the transition probabilities are uncertain, we study robust variants of these equivalences. We introduce universal (perturbation) equivalence, which captures a variant of equivalence that is resilient to all perturbations of transition probabilities: two states or distributions are universally equivalent if they remain equivalent under every assignment of transition probabilities consistent with the support graph. We also consider the dual notion of existential (perturbation) equivalence, which holds whenever there exists an assignment of transition probabilities that yields equivalence. We establish that, for states, universal language equivalence coincides with universal probabilistic bisimilarity and develop a characterisation that yields a polynomial-time partition refinement algorithm. We implement the algorithm and demonstrate experimentally that it is effective as a technique for robust model reduction. We further show that universal language equivalence for distributions is closely related to the state case, and prove NL-completeness of deciding universal equivalence for both states and distributions. We prove that, for states, existential language equivalence coincides with existential probabilistic bisimilarity and deterministic witness transition functions always suffice, leading to an NP-completeness result. In contrast, we show that existential language equivalence for distributions is complete for the existential theory of the reals.
