Shuffle factorization is unique
Theoretical Computer Science
Shuffle Quotient and Decompositions
DLT '01 Revised Papers from the 5th International Conference on Developments in Language Theory
Prime decompositions of regular languages
DLT'06 Proceedings of the 10th international conference on Developments in Language Theory
Algorithmic decomposition of shuffle on words
Theoretical Computer Science
The poor man's proof assistant: using prolog to develop formal language theoretic proofs
Proceedings of the 2013 companion publication for conference on Systems, programming, & applications: software for humanity
The poor man's proof assistant: using prolog to develop formal language theoretic proofs
Proceedings of the 2013 companion publication for conference on Systems, programming, & applications: software for humanity
Theoretical Computer Science
Hi-index | 5.23 |
We investigate a special variant of the shuffle decomposition problem for regular languages; namely, when the given regular language is the shuffle of finite languages. The shuffle decomposition into finite languages is, in general, not unique. That is, there are L"1,L"2,L"3,L"4 with but {L"1,L"2}{L"3,L"4}. However, if all four languages are singletons (with at least two combined letters), it follows by a result of Berstel and Boasson [J. Berstel, L. Boasson, Shuffle factorization is unique, Theoretical Computer Science 273 (2002) 47-67] that the solution is unique; that is, {L"1,L"2}={L"3,L"4}. We further show that if L"1 and L"2 are arbitrary finite sets and L"3 and L"4 are singletons (with at least two letters in each), the solution is unique. Therefore, shuffle decomposition of words is unique not only over words, but over arbitrary sets. This is strong as we cannot let all four be arbitrary finite sets. Hopefully, the obtained results will help to better understand the very nature of the shuffle operation.