最小化的标量的扩展
Extending the scalars of minimizations
摘要：
在形式语言的经典理论中，有限状态自动机允许识别$Σ^*$的有理子集的单词，其中$Σ$是一组符号（或字母表）。现在，给定一个半环$(\K,+,.)$，人们可以构造$\K$ -在Eilenberg意义下的$Σ^*$的子集，它们也被称为非交换形式幂级数，为它们构造了一个非常类似于语言理论的框架特定的非交换形式幂级数，被称为有理级数，是一族加权自动机（或$\K$ -自动机）的行为。为了得到一个有效的编码，指出其中一个状态数最少的可能是有趣的。$\K$ -自动机的最小化过程对于$\K$已经存在：{ \bf a）}是一个域，{ \bf b）}是一个非交换域，{ \bf c）}是一个PID。当$\K$是布尔半环时，这样的极小化过程（具有极小对象的同构）在确定性自动机的范畴内是已知的。证明了极小自动机在{ \bf（a）}和{ \bf（b）}情形下是同构的。但对（b）给出的证明不是构造性的。事实上，它是建立在$\K^n$的一个子模块的基础上的。这里我们给出一个独立的算法来重新证明这一事实，并给出一对非同构极小自动机的例子。此外，我们还研究了扩展{ \bf（c）}的可能性。为此，我们提供了一个{ \em有效最小化过程}（或{ \em EMP}），它可以用于更一般的系数集。
英文摘要：
In the classical theory of formal languages, finite state automata allow to recognize the words of a rational subset of $Σ^*$ where $Σ$ is a set of symbols (or the alphabet). Now, given a semiring $(\K,+,.)$, one can construct $\K$-subsets of $Σ^*$ in the sense of Eilenberg, that are alternatively called noncommutative formal power series for which a framework very similar to language theory has been constructed Particular noncommutative formal power series, which are called rational series, are the behaviour of a family of weighted automata (or $\K$-automata). In order to get an efficient encoding, it may be interesting to point out one of them with the smallest number of states. Minimization processes of $\K$-automata already exist for $\K$ being: {\bf a)} a field, {\bf b)} a noncommutative field, {\bf c)} a PID . When $\K$ is the bolean semiring, such a minimization process (with isomorphisms of minimal objects) is known within the category of deterministic automata. Minimal automata have been proved to be isomorphic in cases {\bf (a)} and {\bf (b)}. But the proof given for (b) is not constructive. In fact, it lays on the existence of a basis for a submodule of $\K^n$. Here we give an independent algorithm which reproves this fact and an example of a pair of nonisomorphic minimal automata. Moreover, we examine the possibility of extending {\bf (c)}. To this end, we provide an {\em Effective Minimization Process} (or {\em EMP}) which can be used for more general sets of coefficients.
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