  In convex integer programming, various procedures have been developed to
strengthen convex relaxations of sets of integer points. On the one hand, there
exist several general-purpose methods that strengthen relaxations without
specific knowledge of the set $ S $, such as popular linear programming or
semi-definite programming hierarchies. On the other hand, various methods have
been designed for obtaining strengthened relaxations for very specific sets
that arise in combinatorial optimization. We propose a new efficient method
that interpolates between these two approaches.
  Our procedure strengthens any convex set $ Q \subseteq \mathbb{R}^n $
containing a set $ S \subseteq \{0,1\}^n $ by exploiting certain additional
information about $ S $. Namely, the required extra information will be in the
form of a Boolean formula $ \phi $ defining the target set $ S $.
  The aim of this work is to analyze various aspects regarding the strength of
our procedure. As one result, interpreting an iterated application of our
procedure as a hierarchy, our findings simplify, improve, and extend previous
results by Bienstock and Zuckerberg on covering problems.
