A generic characterization of Pol(C)

TL;DR

This paper explores the polynomial closure operation on regular language classes, establishing an algebraic link between separation and membership problems, enabling effective algorithms for complex classes based on simpler separation procedures.

Contribution

It provides a general algebraic characterization of Pol(C) for any pseudovariety C, connecting separation and membership problems and enabling reductions between them.

Findings

Algebraic characterization of Pol(C) for arbitrary C

Reduction from Pol(C)-membership to C-separation

Framework for designing membership algorithms from separation algorithms

Abstract

We investigate the polynomial closure operation (C -> 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). This connection is formulated as an algebraic characterization of Pol(C) which holds when C is an arbitrary \pvari of regular languages and whose statement is parameterized by C-separation. 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).