随机动力学的记忆依赖区间马尔可夫链抽象

中文摘要 AI 辅助

有限状态区间马尔可夫链（IMC）抽象通过将单元间转移概率和成本限制在区间内，为连续状态随机系统提供可靠的验证和性能界限。有限状态马尔可夫抽象通常是有损的，因为状态聚合往往会破坏马尔可夫性质。近年来，人们开发了随机系统的记忆依赖马尔可夫链（MC）抽象，通过记录最近访问的单元来缓解这一问题。在本文中，我们开发了IMC抽象的记忆依赖扩展。基于记忆缩小了当前单元内可能出现的状态分布族这一见解，我们证明了记忆能够收紧局部区间，但代价是抽象规模更大。我们还提出了无记忆IMC抽象，在不扩大状态空间的情况下收紧经典IMC抽象区间。为了表征相对于记忆的空间细化，我们推导了两个成本保证紧性上界。这两个上界分别依赖于单元大小，以及在对记忆过去进行过滤后最坏情况下的剩余分布模糊性。然后我们将结果专门应用于线性高斯系统，为此我们推导了该模糊性的闭式上界。数值示例表明，记忆依赖和无记忆构造可以产生比类似分区细化模型更紧的期望成本区间。

英文摘要

Finite-state interval Markov-chain (IMC) abstractions provide sound verification and performance bounds for continuous-state stochastic systems by enclosing cell-to-cell transition probabilities and costs in intervals. Finite-state Markovian abstractions are generally lossy, as state aggregation often destroys the Markov property. Recently, memory-dependent Markov-chain (MC) abstractions of stochastic systems have been developed to mitigate this by recording recently visited cells. In this paper, we develop the memory-dependent extension of IMC abstractions. Building on the insight that memory narrows the family of state distributions that could occur inside the current cell, we prove that memory tightens the local intervals, at the price of a larger abstraction. We also formulate no-memory IMC abstractions, tightening classic IMC abstraction intervals without enlarging the state space. To characterize spatial refinement relative to memory, we derive two cost-guarantee tightness upper bounds. The bounds depend, respectively, on cell size, and on the worst-case remaining distributional ambiguity after filtering over the remembered past. We then specialize our result to linear-Gaussian systems, for which we derive a closed-form upper bound on this ambiguity. Numerical examples show that the memory-dependent and no-memory constructions can produce tighter expected-cost intervals than comparable partition refined models.

发表机构

• Eindhoven University of Technology（埃因霍温理工大学）
• ETH Zurich（苏黎世联邦理工学院）
• The Italian Institute of Artificial Intelligence for Industry (AI4I)（意大利工业人工智能研究所）

机构由 AI 辅助整理，请以论文原文为准。