Zero-Avoiding Transducers, Length Separable Relations, and the Rational Asymmetric Partition Problem

We consider the problem of partitioning effectively a given irreflexive (and possibly symmetric) rational relation [Formula: see text] into two asymmetric rational relations. This problem is motivated by a recent method of embedding an [Formula: see text]-independent language into one that is maximal [Formula: see text]-independent, where the method requires to use an asymmetric partition of [Formula: see text]. We solve the problem when [Formula: see text] is length-separable, which means that the following two subsets of [Formula: see text] are rational: the subset of word pairs [Formula: see text] where [Formula: see text]; and the subset of word pairs [Formula: see text] where [Formula: see text]. This property is satisfied by all recognizable, all left synchronous, and all right synchronous relations. We leave it as an open problem when [Formula: see text] is not length-separable. We also define zero-avoiding transducers for length-separable relations, which makes our partitioning solution constructive.