  Knowledge Compilation (KC) studies compilation of boolean functions f into
some formalism F, which allows to answer all queries of a certain kind in
polynomial time. Due to its relevance for SAT solving, we concentrate on the
query type "clausal entailment" (CE), i.e., whether a clause C follows from f
or not, and we consider subclasses of CNF, i.e., clause-sets F with special
properties. In this report we do not allow auxiliary variables (except of the
Outlook), and thus F needs to be equivalent to f.
  We consider the hierarchies UC_k <= WC_k, which were introduced by the
authors in 2012. Each level allows CE queries. The first two levels are
well-known classes for KC. Namely UC_0 = WC_0 is the same as PI as studied in
KC, that is, f is represented by the set of all prime implicates, while UC_1 =
WC_1 is the same as UC, the class of unit-refutation complete clause-sets
introduced by del Val 1994. We show that for each k there are (sequences of)
boolean functions with polysize representations in UC_{k+1}, but with an
exponential lower bound on representations in WC_k. Such a separation was
previously only know for k=0. We also consider PC < UC, the class of
propagation-complete clause-sets. We show that there are (sequences of) boolean
functions with polysize representations in UC, while there is an exponential
lower bound for representations in PC. These separations are steps towards a
general conjecture determining the representation power of the hierarchies PC_k
< UC_k <= WC_k. The strong form of this conjecture also allows auxiliary
variables, as discussed in depth in the Outlook.
