  Minimal trap spaces (MTSs) capture subspaces in which the Boolean dynamics is
trapped, whatever the update mode. They correspond to the attractors of the
most permissive mode. Due to their versatility, the computation of MTSs has
recently gained traction, essentially by focusing on their enumeration. In this
paper, we address the logical reasoning on universal properties of MTSs in the
scope of two problems: the reprogramming of Boolean networks for identifying
the permanent freeze of Boolean variables that enforce a given property on all
the MTSs, and the synthesis of Boolean networks from universal properties on
their MTSs. Both problems reduce to solving the satisfiability of quantified
propositional logic formula with 3 levels of quantifiers
($\exists\forall\exists$). In this paper, we introduce a Counter-Example Guided
Refinement Abstraction (CEGAR) to efficiently solve these problems by coupling
the resolution of two simpler formulas. We provide a prototype relying on
Answer-Set Programming for each formula and show its tractability on a wide
range of Boolean models of biological networks.
