### Algebraic Recognizability of Languages [chapter]

Pascal Weil
2004 Lecture Notes in Computer Science
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 notion extends to more complex models, such as those developed for modeling distributed or timed behaviors. In the beginning was the Word. . . Recognizable languages of finite words are part of every computer science cursus, and they are routinely described as a
more » ... cornerstone for applications and for theory. We would like to briefly explore why that is, and how this wordrelated notion extends to more complex models, such as those developed for modeling distributed or timed behaviors. The notion of recognizable languages is a familiar one, associated with classical theorems by Kleene, Myhill, Nerode, Elgot, Büchi, Schützenberger, etc. It can be approached from several angles: recognizability by automata, recognizability by finite monoids or finite-index congruences, rational expressions, monadic second order definability. These concepts are expressively equivalent, and this leads to a great many fundamental algorithms in the fields of compilation, text processing, software engineering, etc. . . Moreover, it surely indicates that the class of recognizable languages is central. These equivalence results use the specific structure of words (finite chains, labeled by the letters of the alphabet), and the monoid structure of the set of all words. Since the beginnings of language theory, there has been an interest for other models than words -especially for the purpose of modeling distributed or timed computation (trees, traces, pomsets, graphs, timed words, etc) -, and for extending to these models the tools that were developped for words. For many models, some of these tools may not be defined, and those who are defined, may not coincide. In this paper, we concentrate on the algebraic notion of recognizability: that which, for finite words, exploits the monoid structure of the set of words, and relies on the consideration of monoid morphisms into finite monoids, or equivalently, of finite-index monoid congruences. Our aim is to examine why this ⋆