Regular matching problems for infinite trees

We investigate regular matching problems. The classical reference is Conway's textbook "Regular algebra and finite machines". Some of his results can be stated as follows. Let $L\subseteq(\Sigma\cup X)^*$ and $R\subseteq\Sigma^*$ be regular languages where $\Sigma$ is a set of constants and $X$ is a set of variables. Substituting every $x\in X$ by a regular subset $\sigma(x)$ of $\Sigma^*$ yields a regular set $\sigma(L)\subseteq\Sigma^*$. A substitution $\sigma$ solves a matching problem "$L\subseteq R$?" if $\sigma(L)\subseteq R$. There are finitely many maximal solutions $\sigma$; they are effectively computable and $\sigma(x)$ is regular for all $x\in X$; and every solution is included in a maximal one. Also, in the case of words "$\exists\sigma:\sigma(L)=R$?" is decidable. Apart from the last property, we generalize these results to infinite trees. We define a notion of choice function $\gamma$ which for any tree $s$ over $\Sigma\cup X$ and position $u$ of a variable $x$ selects at most one tree $\gamma(u)\in\sigma(x)$; next, we define $\gamma_\infty(s)$ as the limit of a Cauchy sequence; and the union over all $\gamma_\infty(s)$ yields $\sigma(s)$. Since our definition coincides with that for IO substitutions, we write $\sigma_{io}(L)$ instead of $\sigma(L)$. Our main result is the decidability of "$\exists\sigma:\sigma_{io}(L)\subseteq R$?" if $R$ is regular and $L$ belongs to a class of tree languages closed under intersection with regular sets. Such a special case pops up if $L$ is context-free. Note that "$\exists\sigma:\sigma_{io}(L)=R$?" is undecidable, in general in that case. However, the decidability of "$\exists\sigma:\sigma_{io}(L)=R$?" if both $L$ and $R$ are regular remains open because, in contrast to word languages, the homomorphic image of a regular tree language is not always regular if $\sigma(x)$ is regular for all $x\in X$.Comment: 18 pages. This replacement eliminates a false claim from the previous arXiv version of this paper: Item 4 of Theorem 1 did not hold for # = {=

On polynomial recursive sequences

International audienceWe study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where (non)expressiveness results translate to class separations. A typical example of a polynomial recursive sequence is $b_n=n!$. Our main result is that the sequence $u_n=n^n$ is not polynomial recursive