用有理基数制再论正则数列
Revisiting regular sequences in light of rational base numeration systems
摘要：
正则序列推广了广泛研究的自动序列。设$S$是一个抽象的计数系统。当计数语言$L$是前缀封闭的和正则的时，如果由它的$S$ -内核生成的模块是递归生成的，则序列被称为$S$ -正则的。在本文中，我们给出了一个新的特征，这种序列的基础编号树$T(L)$，其节点是$L$的字。我们可以通过广度优先枚举后的感兴趣序列来装饰这些节点。对于前缀闭正则语言$L$，我们证明了一个序列是$S$ -正则的当且仅当由该序列修饰的树$T(L)$是线性的，即，节点的装饰线性地依赖于固定数目的祖先的装饰。接下来，我们介绍并研究了一个有理基计数系统中的正则序列，其计数语言是高度非正则的。我们的动机和评论我们的定义，一个序列是$\frac{p}{q}$ -正则的，如果底层的编号树装饰的序列是线性的。我们给出了这样的序列的前几个性质，我们提供了一些例子，我们提出了一种猜测$\frac{p}{q}$ -正则性的方法。然后讨论了$\frac{p}{q}$ -自动序列与$\frac{p}{q}$ -正则序列之间的关系。最后给出了$\frac{p}{q}$ -正则序列的图定向线性表示.我们的研究使我们能够突出的地方，规律性的计数语言起着主导作用。
英文摘要：
Regular sequences generalize the extensively studied automatic sequences. Let $S$ be an abstract numeration system. When the numeration language $L$ is prefix-closed and regular, a sequence is said to be $S$-regular if the module generated by its $S$-kernel is finitely generated. In this paper, we give a new characterization of such sequences in terms of the underlying numeration tree $T(L)$ whose nodes are words of $L$. We may decorate these nodes by the sequence of interest following a breadth-first enumeration. For a prefix-closed regular language $L$, we prove that a sequence is $S$-regular if and only if the tree $T(L)$ decorated by the sequence is linear, i.e., the decoration of a node depends linearly on the decorations of a fixed number of ancestors. Next, we introduce and study regular sequences in a rational base numeration system, whose numeration language is known to be highly non-regular. We motivate and comment our definition that a sequence is $\frac{p}{q}$-regular if the underlying numeration tree decorated by the sequence is linear. We give the first few properties of such sequences, we provide a few examples of them, and we propose a method for guessing $\frac{p}{q}$-regularity. Then we discuss the relationship between $\frac{p}{q}$-automatic sequences and $\frac{p}{q}$-regular sequences. We finally present a graph directed linear representation of a $\frac{p}{q}$-regular sequence. Our study permits us to highlight the places where the regularity of the numeration language plays a predominant role.
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