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.
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}, mrnumber = {2252638}, zbl = {1112.03008}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ita:2006005/} }
TY - JOUR AU - Carton, Olivier AU - Choffrut, Christian AU - Grigorieff, Serge TI - Decision problems among the main subfamilies of rational relations JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2006 SP - 255 EP - 275 VL - 40 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ita:2006005/ DO - 10.1051/ita:2006005 LA - en ID - ITA_2006__40_2_255_0 ER -
%0 Journal Article %A Carton, Olivier %A Choffrut, Christian %A Grigorieff, Serge %T Decision problems among the main subfamilies of rational relations %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2006 %P 255-275 %V 40 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ita:2006005/ %R 10.1051/ita:2006005 %G en %F ITA_2006__40_2_255_0
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://www.numdam.org/articles/10.1051/ita:2006005/
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