Priced timed games (PTGs) are two-player zero-sum games played on the infinite graph of configurations of priced timed automata where two players take turns to choose transitions in order to optimize cost to reach target states. Bouyer et al. and Alur, Bernadsky, and Madhusudan independently proposed algorithms to solve PTGs with nonnegative prices under certain divergence restriction over prices. Brihaye, Bruyere, and Raskin later provided a justification for such a restriction by showing the undecidability of the optimal strategy synthesis problem in the absence of this divergence restriction. This problem for PTGs with one clock has long been conjectured to be in polynomial time, however the current best known algorithm, by Hansen, Ibsen-Jensen, and Miltersen, is exponential. We extend this picture by studying PTGs with both negative and positive prices. We refine the undecidability results for optimal strategy synthesis problem, and show undecidability for several variants of optimal reachability cost objectives including reachability cost, time-bounded reachability cost, and repeated reachability cost objectives. We also identify a subclass with bi-valued price-rates and give a pseudo-polynomial algorithm to partially answer the conjecture on the complexity of one-clock PTGs.

中文翻译：

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向定价定时游戏添加负价格

定价定时游戏 (PTG) 是在定价定时自动机的无限配置图上进行的两人零和游戏，其中两个玩家轮流选择转换以优化达到目标状态的成本。布耶等人。Alur、Bernadsky 和 ​​Madhusudan 分别提出了在一定的价格背离限制下解决非负价格 PTG 的算法。Brihaye、Bruyere 和 Raskin 后来通过展示在没有这种发散限制的情况下最优策略综合问题的不可判定性，为这种限制提供了理由。对于具有一个时钟的 PTG 的这个问题长期以来一直被推测为多项式时间，然而目前最知名的算法是由 Hansen、Ibsen-Jensen 和 Miltersen 提出的，是指数级的。我们通过研究具有负价格和正价格的 PTG 来扩展此图。我们改进了最优策略合成问题的不可判定性结果，并展示了最优可达性成本目标的几种变体的不可判定性，包括可达性成本、限时可达性成本和重复可达性成本目标。我们还确定了一个具有双值价格率的子类，并给出了一个伪多项式算法来部分回答关于单时钟 PTG 复杂性的猜想。