Efficient enumeration of solutions produced by closure operations

Abstract: In this paper we address the problem of generating all elements obtained by the saturation of an initial set by some operations. More precisely, we prove that we can generate the closure by polymorphisms of a boolean relation with a polynomial delay. This implies for instance that we can compute with polynomial delay the closure of a family of sets by any set of "set operations" (e.g. by union, intersection, difference, symmetric difference$\dots$). To do so, we prove that for any set of operations $\mathcal{F}$, one can decide in polynomial time whether an elements belongs to the closure by $\mathcal{F}$ of a family of sets. When the relation is over a domain larger than two elements, our generic enumeration method fails for some cases, since the associated decision problem is $NP$-hard and we provide an alternative algorithm.