Geodesic
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

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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 : 10.1051/ita:2006005
Classification : 3D05, 68Q45
Keywords: multitape automata, Presburger arithmetics, decision problems 
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     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},
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     year = {2006},
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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

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