A generic characterization of Pol(C)

We investigate the polynomial closure operation (C 7→ Pol (C)) defined on classes of regular languages. We present an interesting and useful connection relating the separation problem for the class C and the membership problem for it polynomial closure Pol (C). It was first discovered in [6]. This connection is formulated as an algebraic characterization of Pol (C) which holds when C is an arbitrary quotienting lattice of regular languages and whose statement is parameterized by Cseparation. Its main application is an effective reduction from Pol (C)-membership to C-separation. Thus, as soon as one designs a C-separation algorithm, this yields “for free” a membership algorithm for the more complex class Pol (C). Additionally, we present a second transfer theorem which applies to a smaller class than Pol (C): the intersection class Pol(C) ∩ co-Pol(C). This is the class containing all languages L such that both L and its complement belong to Pol (C). This second transfer theorem is a simple corollary of the first one ans was originally formulated in [1]. However it is also stronger: it yields a reduction from Pol (C) ∩ co-Pol(C)-membership to C-membership.