We consider the problem of partitioning effectively a given irreflexive (and possibly symmetric) 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 length-separable, which means that the following two subsets of R are rational: the subset of word pairs (u,v) where |u|≤|v|; and the subset of word pairs (u,v) where |u|≥|v|. This property is satisfied by all recognizable, all left synchronous, and all right synchronous relations. We leave it as an open problem when R is not length-separable. We also define zero-avoiding transducers for length-separable relations, which makes our partitioning solution constructive.

中文翻译：

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零避让传感器、长度可分关系和有理非对称分区问题

我们考虑有效划分给定的非自反（并且可能是对称的）有理关系的问题R成两个不对称的有理关系。这个问题是由最近一种嵌入R- 独立语言变成一种最大的语言R-独立的，该方法需要使用非对称分区R. 我们解决问题的时候R是长度可分的，这意味着以下两个子集R是有理的：词对的子集(你,v)在哪里|你|≤|v|; 和词对的子集(你,v)在哪里|你|≥|v|. 所有可识别、所有左同步和所有右同步关系都满足此属性。我们把它作为一个悬而未决的问题R不是长度可分的。我们还为长度可分离的关系定义了零回避传感器，这使得我们的分区解决方案具有建设性。