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We consider the problem of partitioning effectively a given symmetric (andirreflexive) rational relation R into two asymmetric rational relations. Thisproblem is motivated by a recent method of embedding an R-independent languageinto one that is maximal R-independent, where the method requires to use anasymmetric partition of R. We solve the problem when R is realized by azero-avoiding transducer (with some bound k): if the absolute value of theinput-output length discrepancy of a computation exceeds k then the lengthdiscrepancy of the computation cannot become zero. This class of relationsproperly contains all recognizable, all left synchronous, and all rightsynchronous relations. We leave the asymmetric partition problem open when R isnot realized by a zero-avoiding transducer. We also show examples of totalwordorderings for which there is a relation R that cannot be partitioned intotwo asymmetric rational relations such that one of them is decreasing withrespect to the given word-ordering.

Partitioning a Symmetric Rational Relation into Two Asymmetric Rational Relations