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.

翻译：一阶公式被称为原始正（primitive positive, pp）公式，若它仅允许使用存在量词和合取。Pp-公式是（固定模板）约束满足问题的核心概念，因为CSP(Γ)可视为判定Γ的原始正理论的问题，而pp-可定义性捕捉了CSP之间的构件归约。一类重要的可处理约束语言Γ的特征在于具有少量子幂，即从Γ中pp-可定义的n元关系的数量以2^{p(n)}为界，其中p(n)为某个多项式。本文研究了该性质的一个限制：每个pp-可定义的关系均可通过一个长度多项式的pp-公式来定义。我们猜想这种短定义的存在实际上等价于Γ具有少量子幂，并针对一个大的子类（特别包括所有三元素域上的约束语言）验证了这一猜想。此外，我们讨论了该猜想如何对具有少量子幂的代数中的子幂成员问题施加co-NP的上界复杂度。