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 # = {=

Regular matching problems for infinite trees

We study the matching problem of regular tree languages, that is, "$\exists \sigma:\sigma(L)\subseteq R$?" where $L,R$ are regular tree languages over the union of finite ranked alphabets $\Sigma$ and $\mathcal{X}$ where $\mathcal{X}$ is an alphabet of variables and $\sigma$ is a substitution such that $\sigma(x)$ is a set of trees in $T(\Sigma\cup H)\setminus H$ for all $x\in \mathcal{X}$. Here, $H$ denotes a set of "holes" which are used to define a "sorted" concatenation of trees. Conway studied this problem in the special case for languages of finite words in his classical textbook "Regular algebra and finite machines" published in 1971. He showed that if $L$ and $R$ are regular, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma(L)\subseteq R$?" is decidable. Moreover, there are only finitely many maximal solutions, the maximal solutions are regular substitutions, and they are effectively computable. We extend Conway's results when $L,R$ are regular languages of finite and infinite trees, and language substitution is applied inside-out, in the sense of Engelfriet and Schmidt (1977/78). More precisely, we show that if $L\subseteq T(\Sigma\cup\mathcal{X})$ and $R\subseteq T(\Sigma)$ are regular tree languages over finite or infinite trees, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma_{\mathrm{io}}(L)\subseteq R$?" is decidable. Here, the subscript "$\mathrm{io}$" in $\sigma_{\mathrm{io}}(L)$ refers to "inside-out". Moreover, there are only finitely many maximal solutions $\sigma$, the maximal solutions are regular substitutions and effectively computable. The corresponding question for the outside-in extension $\sigma_{\mathrm{oi}}$ remains open, even in the restricted setting of finite trees

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