Decision problems among the main subfamilies of rational relations
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 2, pp. 255-275.

We consider the four families of recognizable, synchronous, deterministic rational and rational subsets of a direct product of free monoids. They form a strict hierarchy and we investigate the following decision problem: given a relation in one of the families, does it belong to a smaller family? We settle the problem entirely when all monoids have a unique generator and fill some gaps in the general case. In particular, adapting a proof of Stearns, we show that it is recursively decidable whether or not a deterministic subset of an arbitrary number of free monoids is recognizable. Also we exhibit a single exponential algorithm for determining if a synchronous relation is recognizable.

DOI : https://doi.org/10.1051/ita:2006005
Classification : 3D05,  68Q45
Mots clés : multitape automata, Presburger arithmetics, decision problems
@article{ITA_2006__40_2_255_0,
     author = {Carton, Olivier and Choffrut, Christian and Grigorieff, Serge},
     title = {Decision problems among the main subfamilies of rational relations},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {255--275},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {2},
     year = {2006},
     doi = {10.1051/ita:2006005},
     zbl = {1112.03008},
     mrnumber = {2252638},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ita:2006005/}
}
Carton, Olivier; Choffrut, Christian; Grigorieff, Serge. Decision problems among the main subfamilies of rational relations. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 2, pp. 255-275. doi : 10.1051/ita:2006005. http://archive.numdam.org/articles/10.1051/ita:2006005/

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