具有折扣和长期平均目标的随机对策的放大镜抽象
Magnifying Lens Abstraction for Stochastic Games with Discounted and Long-run Average Objectives
摘要：
回合制随机博弈及其重要子类马尔可夫决策过程（MDP）为具有概率和非确定性行为的系统提供了模型。我们考虑两个经典的定量目标：折扣和长期平均目标的回合制随机博弈。博弈模型和定量目标在概率验证、规划、最优库存控制、网络协议和性能分析等方面有着广泛的应用。模拟现实系统的游戏和MDP通常具有非常大的状态空间，并且概率抽象技术是处理状态空间爆炸所必需的。通常使用的全抽象技术不产生节省空间的系统，有许多状态具有类似的值，但不一定有类似的过渡结构。提出了一种半抽象技术，即放大镜抽象（MLA），聚类状态的基础上的价值，无视其过渡关系的差异，提出了定性目标（可达性和安全性目标）。在本文中，我们扩展的MLA技术，以解决随机游戏的折扣和长期平均目标。我们提出了基于MLA技术的随机博弈和折扣和目标MDP的抽象-精化算法。对于长期平均目标，我们的解决方案适用于所有MDP和一类随机游戏，其中每个状态具有相同的值。
英文摘要：
Turn-based stochastic games and its important subclass Markov decision processes (MDPs) provide models for systems with both probabilistic and nondeterministic behaviors. We consider turn-based stochastic games with two classical quantitative objectives: discounted-sum and long-run average objectives. The game models and the quantitative objectives are widely used in probabilistic verification, planning, optimal inventory control, network protocol and performance analysis. Games and MDPs that model realistic systems often have very large state spaces, and probabilistic abstraction techniques are necessary to handle the state-space explosion. The commonly used full-abstraction techniques do not yield space-savings for systems that have many states with similar value, but does not necessarily have similar transition structure. A semi-abstraction technique, namely Magnifying-lens abstractions (MLA), that clusters states based on value only, disregarding differences in their transition relation was proposed for qualitative objectives (reachability and safety objectives). In this paper we extend the MLA technique to solve stochastic games with discounted-sum and long-run average objectives. We present the MLA technique based abstraction-refinement algorithm for stochastic games and MDPs with discounted-sum objectives. For long-run average objectives, our solution works for all MDPs and a sub-class of stochastic games where every state has the same value.
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