# Partitioning a Symmetric Rational Relation into Two Asymmetric Rational Relations

## TL;DR

This paper addresses the problem of partitioning symmetric rational relations into two asymmetric rational relations, providing solutions for zero-avoiding transducer realizations and highlighting open problems and limitations.

## Contribution

It introduces a method to partition symmetric rational relations into asymmetric ones when realized by zero-avoiding transducers, expanding understanding of rational relation decompositions.

## Key findings

- Partitioning is possible for zero-avoiding transducer relations.
- Includes all recognizable, left synchronous, and right synchronous relations.
- Identifies open problems for non-zero-avoiding transducer relations.

## Abstract

We consider the problem of partitioning effectively a given symmetric (and irreflexive) rational relation R into two asymmetric rational relations. This problem is motivated by a recent method of embedding an R-independent language into one that is maximal R-independent, where the method requires to use an asymmetric partition of R. We solve the problem when R is realized by a zero-avoiding transducer (with some bound k): if the absolute value of the input-output length discrepancy of a computation exceeds k then the length discrepancy of the computation cannot become zero. This class of relations properly contains all recognizable, all left synchronous, and all right synchronous relations. We leave the asymmetric partition problem open when R is not realized by a zero-avoiding transducer. We also show examples of total wordorderings for which there is a relation R that cannot be partitioned into two asymmetric rational relations such that one of them is decreasing with respect to the given word-ordering.