论文标题
与状态的矢量增加系统的多维长期平均问题
Multi-dimensional Long-Run Average Problems for Vector Addition Systems with States
论文作者
论文摘要
带有状态（VAS）的矢量增加系统由一组有限的状态和计数器组成。一个过渡将当前状态更改为下一个状态，每个计数器要么是增加，要么减少或保持不变。每个计数器的状态和价值是一种配置。计算是一个无限的配置序列，在连续配置之间具有过渡。一个概率的VAS由一个VASS组成，以及每个状态过渡的概率分布。已广泛研究了诸如状态和配置可达性之类的定性属性。在这项工作中，我们考虑了VASS和概率VASS的多维长期平均目标。对于计数器，配置的成本是计数器的值；计算机计算的长期平均值是计算中配置成本的长期平均值。给定VASS和每个计数器的阈值值的多维长期平均问题，询问是否有一个计算，使每个计数器的长期平均值的柜台的长期平均值不会超过相应的阈值。对于概率vass，而不是计算的存在，我们考虑每个计数器的预期长期平均值是否不超过相应的阈值。我们的主要结果如下：我们表明，多维长期平均问题（a）对于整数价值vass是NP的完整问题； （b）对于自然价值的vass（即非负计数器）是不可决定的； （c）可以在多项式时间内解决概率整数值的vass，以及当所有计算都不终止时，概率的天然价值vass。
A vector addition system with states (VASS) consists of a finite set of states and counters. A transition changes the current state to the next state, and every counter is either incremented, or decremented, or left unchanged. A state and value for each counter is a configuration; and a computation is an infinite sequence of configurations with transitions between successive configurations. A probabilistic VASS consists of a VASS along with a probability distribution over the transitions for each state. Qualitative properties such as state and configuration reachability have been widely studied for VASS. In this work we consider multi-dimensional long-run average objectives for VASS and probabilistic VASS. For a counter, the cost of a configuration is the value of the counter; and the long-run average value of a computation for the counter is the long-run average of the costs of the configurations in the computation. The multi-dimensional long-run average problem given a VASS and a threshold value for each counter, asks whether there is a computation such that for each counter the long-run average value for the counter does not exceed the respective threshold. For probabilistic VASS, instead of the existence of a computation, we consider whether the expected long-run average value for each counter does not exceed the respective threshold. Our main results are as follows: we show that the multi-dimensional long-run average problem (a) is NP-complete for integer-valued VASS; (b) is undecidable for natural-valued VASS (i.e., nonnegative counters); and (c) can be solved in polynomial time for probabilistic integer-valued VASS, and probabilistic natural-valued VASS when all computations are non-terminating.
