Membership Testing in Threshold One Transformation Monoids

AbstractThe membership problem in transformation monoids is the natural extension to the well-studied membership problem in permutation groups. We consider the restriction of the problem to the varieties of threshold one monoids, those monoids in which every element ƒ satisfies ƒp+1 = ƒ, for some integer p. We find that each of the complexity classes AC0, NC, and P can be associated with a variety of threshold one monoids which, within the hypothesis that NC ≠ P and P ≠ NP, is the unique largest variety of monoids where the membership problem can be done with this complexity. We extend our study to other cases of threshold one monoids, for which we obtain NP-completeness results. We also consider the problem which consists in deciding whether the transformation monoid of an automaton belongs to a specific variety: we show that we can do in AC0 the characterization of monoids in three of the varieties most significant to our study of the membership problem