%0 Conference Paper %A Fleischer, Lukas %A Kufleitner, Manfred %D 2018 %T The intersection problem for finite monoids %U https://repository.lboro.ac.uk/articles/conference_contribution/The_intersection_problem_for_finite_monoids/9404153 %2 https://repository.lboro.ac.uk/ndownloader/files/17020838 %K Intersection problem %K Finite monoid %K Recognizing morphism %K Complexity %K Information and Computing Sciences not elsewhere classified %X 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. %I Loughborough University