Weighted recognizability over infinite alphabets

We introduce weighted variable automata over infinite alphabets and commutative semirings. We prove that the class of their behaviors is closed under sum, and under scalar, Hadamard, Cauchy, and shuffle products, as well as star operation. Furthermore, we consider rational series over infinite alphabets and we state a Kleene-Schützenberger theorem. We introduce a weighted monadic second order logic and a weighted linear dynamic logic over infinite alphabets and investigate their relation to weighted variable automata. An application of our theory, to series over the Boolean semiring, concludes to new results for the class of languages accepted by variable automata

Weighted automata and multi-valued logics over arbitrary bounded lattices

AbstractWe show that L-weighted automata, L-rational series, and L-valued monadic second order logic have the same expressive power, for any bounded lattice L and for finite and infinite words. We also prove that aperiodicity, star-freeness, and L-valued first-order and LTL-definability coincide. This extends classical results of Kleene, Büchi–Elgot–Trakhtenbrot, and others to arbitrary bounded lattices, without any distributivity assumption that is fundamental in the theory of weighted automata over semirings. In fact, we obtain these results for large classes of strong bimonoids which properly contain all bounded lattices

The measure transfer for subshifts induced by morphisms of free monoids

Every non-erasing monoid morphism $\sigma: {\cal A}^* \to {\cal B}^*$ induces a measure transfer map $\sigma_X^{\cal M}: {\cal M}(X) \to {\cal M}(\sigma(X))$ between the measure cones ${\cal M}(X)$ and ${\cal M}(\sigma(X))$, associated to any subshift $X \subseteq {\cal A}^\mathbb Z$ and its image subshift $\sigma(X) \subseteq {\cal B}^\mathbb Z$ respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective. Furthermore, an efficient technique to compute the value of the transferred measure $\sigma_X^{\cal M}(\mu)$ on any cylinder $[w]$ (for $w \in {\cal B}^\mathbb Z$) is presented. Theorem: If a non-erasing morphism $\sigma: {\cal A}^* \to {\cal B}^*$ is recognizable in some subshift $X \subseteq {\cal A}^\mathbb Z$, then $\sigma^{\cal M}_X$ is bijective. The notion of a "recognizable" subshift is classical in symbolic dynamics, and due to its long history and various transformations and sharpening over time, it plays a central role in the theory. In order to prove the above theorem we show here: Proposition: A non-erasing morphism $\sigma: {\cal A}^* \to {\cal B}^*$ is recognizable in some subshift $X \subseteq {\cal A}^\mathbb Z$ if and only if (1) the induced map from shift-orbits of $X$ to shift-orbits of $\sigma(X)$ is injective, and (2) $\sigma$ preserves the shift-period of any periodic word in $X$

Pure and O-Substitution

The basic properties of distributivity and deletion of pure and o-substitution are investigated. The obtained results are applied to show preservation of recognizability in a number of surprising cases. It is proved that linear and recognizable tree series are closed under o-substitution provided that the underlying semiring is commutative, continuous, and additively idempotent. It is known that, in general, pure substitution does not preserve recognizability (not even for linear target tree series), but it is shown that recognizable linear probability distributions (represented as tree series) are closed under pure substitution

Measure transfer and SS-adic developments for subshifts

Based on previous work of the authors, to any $S$-adic development of a subshift $X$ a "directive sequence" of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given $S$-adic development. The issuing rich picture enables one to deduce results about $X$ with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result we also exhibit, for any integer $d \geq 2$, an $S$-adic development of a minimal, aperiodic, uniquely ergodic subshift $X$, where all level alphabets ${\cal A}_n$ have cardinality $d\,$, while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subset {\cal A}_n^\mathbb Z$