We consider Pareto analysis of multi-priced timed automata (MPTA) having multiple observers recording costs (to be minimised) and rewards (to be maximised) along a computation.

We study the Pareto Domination Problem, which asks whether it is possible to reach a target location such that the accumulated costs and rewards Pareto dominate a given vector. We show that this problem is undecidable in general, but decidable for MPTA with at most three observers. We show the problem to be PSPACE-complete for MPTA recording only costs or only rewards. We also consider an approximate Pareto Domination that is decidable in exponential time with no restrictions on types and number of observers.

We develop connections between MPTA and Diophantine equations. Undecidability of the Pareto Domination Problem is shown by reduction from Hilbert's 10th Problem, while decidability for three observers entails translation to a decidable fragment of arithmetic involving quadratic forms.

我们考虑多定价定时自动机 (MPTA) 的帕累托分析，该分析具有多个观察者记录成本（最小化）和奖励（最大化）的计算。

我们研究帕累​​托支配问题，该问题询问是否有可能到达目标位置，使得累积成本和奖励帕累托支配给定向量。我们表明这个问题一般是不可判定的，但对于最多三个观察者的 MPTA 是可判定的。我们表明，对于 MPTA 仅记录成本或仅记录奖励，该问题是 PSPACE 完备的。我们还考虑了一种近似的帕累托支配，它可以在指数时间内决定，对观察者的类型和数量没有限制。

我们在 MPTA 和丢番图方程之间建立联系。帕累托支配问题的不可判定性由希尔伯特第 10 个问题的简化表示，而三个观察者的可判定性需要转换为涉及二次形式的算术的可判定片段。