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The Intersection Problem for Finite Monoids Fleischer, Lukas Kufleitner, Manfred intersection problem finite monoid recognizing morphism complexity We investigate the intersection problem for finite monoids, which asks for a given set of regular languages, represented by recognizing morphisms to finite monoids from a variety V, whether there exists a word contained in their intersection. Our main result is that the problem is PSPACE-complete if V is contained in DS and NP-complete if V is non-trivial and contained in DO. Our NP-algorithm for the case that V is contained in DO uses novel methods, based on compression techniques and combinatorial properties of DO. We also show that the problem is log-space reducible to the intersection problem for deterministic finite automata (DFA) and that a variant of the problem is log-space reducible to the membership problem for transformation monoids. In light of these reductions, our hardness results can be seen as a generalization of both a classical result by Kozen and a theorem by Beaudry, McKenzie and Thérien. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Lukas Fleischer and Manfred Kufleitner 2018 Is Part Of LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2018.30 urn:nbn:de:0030-drops-85079 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.30 eng https://creativecommons.org/licenses/by/3.0/legalcode