Algebraic recognizability of regular tree languages

We propose a new algebraic framework to discuss and classify recognizable tree languages, and to characterize interesting classes of such languages. Our algebraic tool, called preclones, encompasses the classical notion of syntactic Sigma-algebra or minimal tree automaton, but adds new expressivity to it. The main result in this paper is a variety theorem \`{a} la Eilenberg, but we also discuss important examples of logically defined classes of recognizable tree languages, whose characterization and decidability was established in recent papers (by Benedikt and S\'{e}goufin, and by Bojanczyk and Walukiewicz) and can be naturally formulated in terms of pseudovarieties of preclones. Finally, this paper constitutes the foundation for another paper by the same authors, where first-order definable tree languages receive an algebraic characterization

Eilenberg Theorems for Free

Eilenberg-type correspondences, relating varieties of languages (e.g. of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory. Numerous such correspondences are known in the literature. We demonstrate that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality. Our main contribution is a variety theorem that covers e.g. Wilke's and Pin's work on $\infty$-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two previous categorical approaches of Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new results, including an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words

Rational subsets of Baumslag-Solitar groups

We consider the rational subset membership problem for Baumslag-Solitar groups. These groups form a prominent class in the area of algorithmic group theory, and they were recently identified as an obstacle for understanding the rational subsets of $\text{GL}(2,\mathbb{Q})$. We show that rational subset membership for Baumslag-Solitar groups $\text{BS}(1,q)$ with $q\ge 2$ is decidable and PSPACE-complete. To this end, we introduce a word representation of the elements of $\text{BS}(1,q)$: their pointed expansion (PE), an annotated $q$-ary expansion. Seeing subsets of $\text{BS}(1,q)$ as word languages, this leads to a natural notion of PE-regular subsets of $\text{BS}(1, q)$: these are the subsets of $\text{BS}(1,q)$ whose sets of PE are regular languages. Our proof shows that every rational subset of $\text{BS}(1,q)$ is PE-regular. Since the class of PE-regular subsets of $\text{BS}(1,q)$ is well-equipped with closure properties, we obtain further applications of these results. Our results imply that (i) emptiness of Boolean combinations of rational subsets is decidable, (ii) membership to each fixed rational subset of $\text{BS}(1,q)$ is decidable in logarithmic space, and (iii) it is decidable whether a given rational subset is recognizable. In particular, it is decidable whether a given finitely generated subgroup of $\text{BS}(1,q)$ has finite index.Comment: Long version of paper with same title appearing in ICALP'2

Complementation of Rational Sets on Countable Scattered Linear Orderings

In a preceding paper (Bruyère and Carton, automata on linear orderings, MFCS'01), automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-infinite and even transfinite words studied by Büchi. Kleene's theorem has been generalized to these words. We prove that rational sets of words on countable scattered linear orderings are closed under complementation using an algebraic approach

Extended Temporal Logic on Finite Words and Wreath Product of Monoids with Distinguished Generators

We associate a modal operator with each language belonging to a given class of regular languages and use the (reverse) wreath product of monoids with distinguished generators to characterize the expressive power of the resulting logic

Macro tree transducers

Macro tree transducers are a combination of top-down tree transducers and macro grammars. They serve as a model for syntax-directed semantics in which context information can be handled. In this paper the formal model of macro tree transducers is studied by investigating typical automata theoretical topics like composition, decomposition, domains, and ranges of the induced translation classes. The extension with regular look-ahead is considered

Temporal Logic with Cyclic Counting and the Degree of Aperiodicity of Finite Automata

We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QA_M of finite automata whose degree of aperiodicity belongs to the division ideal generated by M is closed with respect to direct products, disjoint unions, subautomata, homomorphic images and renamings. These closure conditions define q-varieties of finite automata. We show that q-varieties are in a one-to-one correspondence with literal varieties of regular languages. We also characterize QA_M as the cascade product of a variety of counters with the variety of aperiodic (or counter-free) automata. We then use the notion of degree of aperiodicity to characterize the expressive power of first-order logic and temporal logic with cyclic counting with respect to any given set M of moduli. It follows that when M is finite, then it is decidable whether a regular language is definable in first-order or temporal logic with cyclic counting with respect to moduli in M

Temporal logic with cyclic counting and the degree of aperiodicity of finite automata

We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QAM of finite automata whose degree of aperiodicity belongs to the division ideal generated by M is closed with respect to direct products, disjoint unions, subautomata, homomorphic images and renamings. These closure conditions define q-varieties of finite automata. We show that q-varieties are in a one-to-one correspondence with literal varieties of regular languages. We also characterize QA M as the cascade product of a variety of counters with the variety of aperiodic (or counter-free) automata. We then use the notion of degree of aperiodicity to characterize the expressive power of first-order logic and temporal logic with cyclic counting with respect to any given set M of moduli. It follows that when M is finite, then it is decidable whether a regular language is definable in first-order or temporal logic with cyclic counting with respect to moduli in M