We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group over rational numbers. Furthermore, we prove two decidability results for the Half-Space Reachability Problem. Namely, we show that this problem is decidable for sub-semigroups of GL(2,Z) and of the Heisenberg group over rational numbers.
@InProceedings{colcombet_et_al:LIPIcs.ICALP.2019.44, author = {Colcombet, Thomas and Ouaknine, Jo\"{e}l and Semukhin, Pavel and Worrell, James}, title = {{On Reachability Problems for Low-Dimensional Matrix Semigroups}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.44}, URN = {urn:nbn:de:0030-drops-106209}, doi = {10.4230/LIPIcs.ICALP.2019.44}, annote = {Keywords: membership problem, half-space reachability problem, matrix semigroups, Heisenberg group, general linear group} }
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