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 the recognizable, the left synchronous, and the right synchronous relations. We leave the asymmetric partition problem open when R is not zero-avoiding. We also show examples of total word-orderings for which there is a relation R that cannot be partitioned into two asymmetric rational relations with respect to the given word-orderings.
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Notes
- 1.
In general, \(\varvec{t}\) has an input and an output alphabet, but here these are equal.
- 2.
This is well-defined: if \(\varvec{t}\) is zero-avoiding with bound k then it is also zero-avoiding with bound \(k'\) for all \(k'>k\).
- 3.
Further explanations of claims will be given in a journal version of this paper.
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Acknowledgement
We thank Jacques Sakarovitch for looking at this open problem and offering the opinion that it indeed appears to be non trivial.
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Konstantinidis, S., Mastnak, M., Šebej, J. (2019). Partitioning a Symmetric Rational Relation into Two Asymmetric Rational Relations. In: Hospodár, M., Jirásková, G. (eds) Implementation and Application of Automata. CIAA 2019. Lecture Notes in Computer Science(), vol 11601. Springer, Cham. https://doi.org/10.1007/978-3-030-23679-3_14
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