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In a previous paper, we showed that profinite $L$-algebras (where $L$ is a variety of modal algebras generated by its finite members) are monadic over $\mathbf{Set}$. This monadicity result suggests that profinite $L$-algebras could be…
The theory of finite automata concerns itself with words in a free monoid together with concatenation and without further structure. There are, however, important applications which use alphabets which are structured in some sense. We…
In previous articles, we showed that the category of profinite $L$-algebras (where $L$ is a normal modal logic with the finite model property) is monadic over $\textbf{Set}$. Then, we developed sequent calculi for extensions of the language…
Indexed languages are a classical notion in formal language theory, which has attracted attention in recent decades due to its role in higher-order model checking: They are precisely the languages accepted by order-2 pushdown automata. The…
Inspired by distributed algorithms, we introduce a new class of finite graph automata that recognize precisely the graph languages definable in monadic second-order logic. For the cases of words and trees, it has been long known that the…
The aim of the paper is to build a connection between two approaches towards categorical language theory: the coalgebraic and algebraic language theory for monads. For a pair of monads modelling the branching and the linear type we defined…
This paper proposes a definition of recognizable transducers over monads and comonads, which bridges two important ongoing efforts in the current research on regularity. The first effort is the study of regular transductions, which extends…
Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.
We give an algebraic characterization of the tree languages that are defined by logical formulas using certain Lindstr\"om quantifiers. An important instance of our result concerns first-order definable tree languages. Our characterization…
The use of monoids in the study of word languages recognized by finite-state automata has been quite fruitful. In this work, we look at the same idea of "recognizability by finite monoids" for other monoids. In particular, we attempt to…
Monads in category theory are algebraic structures that can be used to model computational effects in programming languages. We show how the notion of "centre", and more generally "centrality", i.e. the property for an effect to commute…
In the classical theory of regular languages the concept of recognition by profinite monoids is an important tool. Beyond regularity, Boolean spaces with internal monoids (BiMs) were recently proposed as a generalization. On the other hand,…
Recognizable languages of finite words are part of every computer science cursus, and they are routinely described as a cornerstone for applications and for theory. We would like to briefly explore why that is, and how this word-related…
Profinite congruences on profinite algebras determining profinite quotients are difficult to describe. In particular, no constructive description is known of the least profinite congruence containing a given binary relation on the algebra.…
Profinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety $\mathcal V$ is standard if every Boolean topological algebra with the…
Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion…
We prove a topological completeness theorem for the modal logic GLP containing operators $\langle\lambda\rangle$ for $\lambda \in$ Ord intended to capture progressively stronger notions of consistency in mathematical theories. We show that,…
The clone of term operations of an algebraic structure consists of all operations that can be expressed by a term in the language of the structure. We consider bounds for the length and the height of the terms expressing these functions,…
This work studies the question of learning probabilistic deterministic automata from language models. For this purpose, it focuses on analyzing the relations defined on algebraic structures over strings by equivalences and similarities on…
Monads are of interest both in semantics and in higher dimensional algebra. It turns out that the idea behind usual notion finitary monads (whose values on all sets can be computed from their values on finite sets) extends to a more general…