  A first-order formula is called primitive positive (pp) if it only admits the
use of existential quantifiers and conjunction. Pp-formulas are a central
concept in (fixed-template) constraint satisfaction since CSP($\Gamma$) can be
viewed as the problem of deciding the primitive positive theory of $\Gamma$,
and pp-definability captures gadget reductions between CSPs.
  An important class of tractable constraint languages $\Gamma$ is
characterized by having few subpowers, that is, the number of $n$-ary relations
pp-definable from $\Gamma$ is bounded by $2^{p(n)}$ for some polynomial $p(n)$.
In this paper we study a restriction of this property, stating that every
pp-definable relation is definable by a pp-formula of polynomial length. We
conjecture that the existence of such short definitions is actually equivalent
to $\Gamma$ having few subpowers, and verify this conjecture for a large
subclass that, in particular, includes all constraint languages on
three-element domains. We furthermore discuss how our conjecture imposes an
upper complexity bound of co-NP on the subpower membership problem of algebras
with few subpowers.
