  Answering Conjunctive Queries (CQs) and solving Constraint Satisfaction
Problems (CSPs) are arguably among the most fundamental tasks in Computer
Science. They are classical NP-complete problems. Consequently, the search for
tractable fragments of these problems has received a lot of research interest
over the decades. This research has traditionally progressed along three
orthogonal threads. a) Reformulating queries into simpler, equivalent, queries
(semantic optimization) b) Bounding answer sizes based on structural properties
of the query c) Decomposing the query in such a way that global consistency
follows from local consistency. Much progress has been made by various works
that connect two of these threads. Bounded answer sizes and decompositions have
been shown to be tightly connected through the important notions of fractional
hypertree width and, more recently, submodular width. recent papers by
Barcel\'o et al. study decompositions up to generalized hypertree width under
semantic optimization. In this work, we connect all three of these threads by
introducing a general notion of semantic width and investigating semantic
versions of fractional hypertree width, adaptive width, submodular width and
the fractional cover number.
