We study the reachability problem for continuous one-counter automata, COCA for short. In such automata, transitions are guarded by upper and lower bound tests against the counter value. Additionally, the counter updates associated with taking transitions can be (non-deterministically) scaled down by a nonzero factor between zero and one. Our three main results are as follows: (1) We prove that the reachability problem for COCA with global upper and lower bound tests is in NC2; (2) that, in general, the problem is decidable in polynomial time; and (3) that it is decidable in the polynomial hierarchy for COCA with parametric counter updates and bound tests.

中文翻译：

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连续一计数器自动机

我们研究连续一计数器自动机（COCA）的可达性问题。在这种自动机中，通过上限和下限测试针对计数器值来保护过渡。另外，与转换相关联的计数器更新可以（不确定）按零和一之间的非零因子缩小。我们的三个主要结果如下：（1）证明了使用全局上限和下限测试的COCA的可达性问题在NC2中；（2）一般而言，问题可以在多项式时间内确定；（3）在带有参数计数器更新和绑定测试的COCA的多项式层次结构中是可确定的。