Résumé | A standard approach to define k-ary word relations over a finite alphabet A is through k-tape finite state automata that recognize regular languages L over {1, ... , k} x A, where (i,a) is interpreted as reading letter a from tape i. Accordingly, a word w in L denotes the tuple (u_1, ... , u_k) of words over A in which u_i is the projection of w onto i-labelled letters. While this formalism defines the well-studied class of Rational relations, enforcing restrictions on the reading regime from the tapes, which we call "synchronization", yields various sub-classes of relations. Such synchronization restrictions are imposed through regular properties on the projection of the language L onto {1, ... , k}. In this way, for each regular language C over the alphabet {1, ... , k}, one obtains a class Rel(C) of relations. Synchronous, Recognizable, and Length-preserving rational relations are all examples of classes that can be defined in this way.
We study basic properties of these classes of relations, in terms of closure under intersection, complement, concatenation, Kleene star and projection. In each case we characterize the classes with the property through a decidable property. |