## Pattern spectra, substring enumeration, and automatic sequences.(English)Zbl 0753.11012

Let $$P=p_ 1\ldots p_ h$$ be a binary pattern, i.e., a string of 0’s and 1’s such that $$p_ 1=1$$ and let $$a_ P$$ be the corresponding pattern sequence defined by $$a_ P(n)=(-1)^{e_ P(n)}$$ where $$e_ P(n)$$ counts the number of occurrences of $$P$$ in the binary expansion of an integer $$n\geq 0$$. P. Morton and W. J. Mourant [Proc. Lond. Math. Soc., III. Ser. 59, 253-293 (1989; Zbl 0694.10009)] showed that every sequence $$s$$ on $$\{+1,-1\}$$ can be uniquely expressed as the product $$s(n)=s(0)\prod_{P\in{\mathcal P}}a_ P(n)$$ where $${\mathcal P}$$ is a finite or infinite set of patterns. One of the main results of this paper (Theorem 2.1) says that the sequence $$s$$ is 2-automatic if and only if $${\mathcal P}$$ is a regular set of words on the alphabet $$\{0,1\}$$ that is to say $${\mathcal P}$$ is recognizable by a finite 2-automaton.
This theorem is introduced by some interesting examples and is extended to the following purely language-theoretic result: Let $$\Sigma$$ be a finite alphabet and let $$\Sigma^*$$ be the set of strings drawn from $$\Sigma$$. For strings $$r$$ and $$w$$ define $$\sigma_ r(w)$$ to count the number of occurrences of $$r$$ as a substring of $$w$$ (for $$r$$ equal to the empty string, $$\sigma_ r(w)$$ counts the length of $$w$$ plus 1) and for any language $$L\subset\Sigma^*$$ define $$\sigma_ L:\Sigma^*\to\mathbb{N}$$ by $$\sigma_ L(w)=\sum_{r\in L}\sigma_ r(w)$$. Now we consider, for any integer $$k\geq 2$$, the Nerode equivalence $$\sim_ k$$ on $$\Sigma^*$$ given by $w\sim_ kw'\Leftrightarrow\forall v\in\Sigma^*,\quad\sigma_ L(wv)=\sigma_ L(w'v)\bmod k.$ We shall say that $$\sim_ k$$ is finite if the set of classes $${\Sigma/\sim_ k}$$ is finite. Then (Theorem 4.1) the following properties are equivalent: (i) The language $$L$$ is regular; (ii) The equivalent relations $$\sim_ k$$ all are finite; (iii) At least one relation $$\sim_ k(k\geq 2)$$ is finite (in other words, following the terminology of the authors, the map $$w\mapsto\sigma_ L(w)\bmod k$$ is given by a finite $$\Sigma$$-automaton with outputs). In the definition of $$\sim_ k$$, without changing the result, the map $$\sigma_ L$$ can be replaced by $$\pi_ L:w\mapsto\pi_ L(w)$$ where $$\pi_ L(w)$$ counts the number of strings in $$L$$ which occur as a prefix of $$w$$.

### MSC:

 11B85 Automata sequences 68Q45 Formal languages and automata

Zbl 0694.10009
Full Text:

### References:

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