Computational complexity of recognizing pre-principal and pre-ideal collections of subcubes

Determine the computational complexity of the following recognition problems for collections of subcubes of the Boolean cube {0,1}^n: (i) given a finite collection Q of subcubes, decide whether Q is pre-principal, that is, whether Q equals { Q(x) : x in {0,1}^n }, where Q(x) is the intersection of all subcubes in Q that contain x (with {0,1}^n used if none contain x); and (ii) given a finite collection J of subcubes, decide whether J is pre-ideal, that is, whether J contains {0,1}^n, is closed under intersection (whenever the intersection is nonempty), and is closed under arbitrary unions of its subcollections whenever the union is itself a subcube. Establish the complexity classes (e.g., polynomial time, NP-complete) of these decision problems.

Background

The paper characterizes exactly which collections of subcubes arise as principal trapspaces and as trapspaces of Boolean networks. Principal trapspace collections are shown to be precisely the pre-principal collections, defined by the condition Q = μ(Q), where μ(Q) collects the smallest subcube in Q containing each configuration; trapspace collections are precisely the pre-ideal collections, defined by closure under appropriate intersections and unions and containing the full cube.

Although these structural characterizations are complete, the algorithmic status of recognizing whether an arbitrary input collection of subcubes satisfies the pre-principal or pre-ideal conditions is not determined in the paper. Establishing the computational complexity of these recognition problems would clarify the feasibility of certifying whether a given family of subcubes can arise as the principal trapspaces or all trapspaces of some Boolean network.