We consider a system consisting of a sequential composition of Mealy machines, called head and tail. We study two problems related to these systems. First, models of both head and tail components are available, and the aim is to obtain a replacement for the tail with the minimum number of states. We introduce a minimization method for this context which yields an exponential improvement over the state of the art. In the second problem, only the head is known and a desired model for the whole system is given. The objective is to construct a tail that causes the system to behave according to the given model. We give a polynomial time algorithm that computes whether such a tail exists, whereas methods developed for general unknown component equations take exponential time. However, we also show that there are instances of the problem where the size of the tail is bounded from bellow by an exponential function in the sizes of the head and the desired system. This shows that the complexity of the synthesis procedure is at least exponential, matching the upper bound in complexity provided by the existing methods.

中文翻译：

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小型机尾序的最小化与合成

我们考虑一个由连续组成的Mealy机器组成的系统，称为头和尾。我们研究了与这些系统有关的两个问题。首先，可以使用头部和尾部组件的模型，其目的是用最少的状态数获得尾部的替换。我们针对这种情况引入了一种最小化方法，该方法相对于现有技术产生了指数级的改进。在第二个问题中，只有杆头是已知的，并且给出了整个系统的期望模型。目的是构造一条导致系统根据给定模型运行的尾部。我们给出了多项式时间算法，该算法可计算是否存在这种尾部，而针对一般未知分量方程式开发的方法则需要花费指数时间。然而，我们还表明，在某些情况下，尾巴的大小受到波纹管的限制，其大小取决于头部和所需系统的大小的指数函数。这表明合成过程的复杂度至少是指数的，与现有方法提供的复杂度的上限匹配。