Resynchronizing Classes of Word Relations

A natural approach to define binary word relations over a finite alphabet A is through two-tape finite state automata that recognize regular languages over {1, 2} x A, where (i,a) is interpreted as reading letter a from tape i. Accordingly, a word w in L denotes the pair (u_1,u_2) in A^* x 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 (a.k.a. non-deterministic finite state transducers), 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 onto {1,2}. In this way, for each regular language C subseteq {1,2}^*, one obtains a class Rel({C}) of relations. Regular, Recognizable, and length-preserving rational relations are all examples of classes that can be defined in this way. We study the problem of containment for synchronized classes of relations: given C,D subseteq {1,2}^*, is Rel({C}) subseteq Rel({D})? We show a characterization in terms of C and D which gives a decidability procedure to test for class inclusion. This also yields a procedure to re-synchronize languages from {1, 2} x A preserving the denoted relation whenever the inclusion holds.