Some operations and transductions that preserve rationality

When a language theorist encounters a new operation on languages, his first impulse is to know whether this operation preserves rational languages. If the answer appears to be positive, he proceeds immediately to the construction of a more or less complicated automaton to solve the problem. However there are many operations on languages, many language theorists (see the references) and many different constructions to study these many operations. The aim of this paper is to show that almost all these constructions are a particular case of a general and simple approach. It is fair to say immediately that a few operations are overlooked, such as the star operation, complementation and reversal. However, the scope of our method is quite broad, broader indeed than one would expect, and goes from “classical” operations such as union, intersection, concatenation, quotient, shuffle, inverse and direct morphisms, etc., to less classical ones such as infiltration, Dyck reduction, longest common prefix, Straubing’s counting, etc. It includes also questions that are not expressed directly as operations on languages, as, for example, Reutenauer’s theorem on TOL-systems. The interest of the method in not only to give a unified framework for all these results. Statements of the form “such an operation preserves rational languages” can be readily be refined into “such an an operation preserves star-free languages” or even more generally “such an operation preserves such a variety of rational languages”. The key idea of our construction is to consider an operation φ : A1 × · · · × An → A∗ as the inverse of a transduction τ : A∗ → A1 × · · · × An (whenever it is possible). Then, given monoids M1, . . . , Mn, recognizing the languages L1, . . . , Ln, of A ∗ 1, . . . , A ∗ n, respectively, we are able to construct a monoid M recognizing (L1, . . . , Ln)φ as soon as the transduction τ admits Université Paris VI et CNRS, LITP, 4 Place Jussieu, 75230 Paris Cedex 05, France.