We characterize the languages in the individual levels of the quantifier alternation hierarchy of first-order logic with two variables by identities. This implies decidability of the individual levels. More generally we show that two-sided semidirect products with J as the right-hand factor preserve decidability.
@InProceedings{krebs_et_al:LIPIcs.FSTTCS.2012.86, author = {Krebs, Andreas and Straubing, Howard}, title = {{An effective characterization of the alternation hierarchy in two-variable logic}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {86--98}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.86}, URN = {urn:nbn:de:0030-drops-38501}, doi = {10.4230/LIPIcs.FSTTCS.2012.86}, annote = {Keywords: FO\underline2, Quantifier Alternation, J, Pseudovarities, Identities} }
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