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The purpose of this paper is to provide efficient algorithms that decide membership for classes of several Boolean hierarchies for which efficiency (or even decidability) were previously not known. We develop new forbidden-chain characterizations for the single levels of these hierarchies and obtain the following results: • The classes of the Boolean hierarchy over level Σ1 of the dot-depth hierarchy are decidable in NL (previously only the decidability was known). The same remains true ifdoi:10.1016/j.tcs.2016.07.017 fatcat:tqu5qwscmbbbvebstf2lkexy2m