具有状态依赖折扣与奇偶目标的同时随机博弈：复杂性与算法
Concurrent Stochastic Games with Stateful-discounted and Parity Objectives: Complexity and Algorithms
AI总结：
本文研究双人零和同时随机博弈中状态依赖折扣目标极限值与奇偶目标的值近似问题，确立其TFNP[NP]复杂性，并提出算法将对动作数的指数依赖从线性降至对数，状态数为常数时可在多项式时间内运行。
摘要：
我们研究具有有限状态和动作空间、进行无限步的双人零和同时随机博弈。在每一步中，两个参与者同时且独立地选择一个动作。给定当前状态和所选动作，下一状态根据随机转移函数获得。目标函数是博弈路径（或无限轨迹）上的可测函数，目标的值是参与者能够对抗对手所保证的最大期望。我们考虑：（a）状态依赖折扣目标，这类目标类似于经典的折扣和目标，但状态关联不同的折扣因子而非单一折扣因子；以及（b）奇偶目标，它是ω-正则目标的规范表示。对于状态依赖折扣目标，给定折扣因子的一个排序，极限值是指当折扣因子按照给定顺序趋近于零时，状态依赖折扣目标值的极限。我们考虑的计算问题是值在任意加性误差内的近似。上述问题已知对于状态依赖折扣目标的极限值属于EXPSPACE，对于奇偶目标属于PSPACE。对于这两个问题，已知最优算法至少是指数时间的，并且对状态数和动作数呈指数依赖。我们针对状态依赖折扣目标极限值和奇偶目标的值近似问题的主要结果如下：（a）我们确立了TFNP[NP]复杂性；（b）我们提出的算法将指数中对动作数的依赖从线性改进为对数。特别地，如果状态数为常数，我们的算法在多项式时间内运行。
英文摘要：
We study two-player zero-sum concurrent stochastic games with finite state and action space played for an infinite number of steps. In every step, the two players simultaneously and independently choose an action. Given the current state and the chosen actions, the next state is obtained according to a stochastic transition function. An objective is a measurable function on plays (or infinite trajectories) of the game, and the value for an objective is the maximal expectation that the player can guarantee against the adversarial player. We consider: (a) stateful-discounted objectives, which are similar to the classical discounted-sum objectives, but states are associated with different discount factors rather than a single discount factor; and (b) parity objectives, which are a canonical representation for $ω$-regular objectives. For stateful-discounted objectives, given an ordering of the discount factors, the limit value is the limit of the value of the stateful-discounted objectives, as the discount factors approach zero according to the given order. The computational problem we consider is the approximation of the value within an arbitrary additive error. The above problem is known to be in EXPSPACE for the limit value of stateful-discounted objectives and in PSPACE for parity objectives. The best-known algorithms for both the above problems are at least exponential time, with an exponential dependence on the number of states and actions. Our main results for the value approximation problem for the limit value of stateful-discounted objectives and parity objectives are as follows: (a) we establish TFNP[NP] complexity; and (b) we present algorithms that improve the dependency on the number of actions in the exponent from linear to logarithmic. In particular, if the number of states is constant, our algorithms run in polynomial time.
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