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($Γ$) can be viewed as the problem of deciding the primitive positive theory of $Γ$, and pp-definability captures gadget reductions between CSPs. An important class of tractable constraint languages $Γ$ is characterized by having few subpowers, that is, the number of $n$-ary relations pp-definable from $Γ$ 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 $Γ$ 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：一个一阶公式若仅允许使用存在量词和合取，则被称为原始正定（pp）公式。Pp公式是（固定模板）约束满足问题中的核心概念，因为CSP($Γ$)可视为判定$Γ$的原始正定理论的问题，且pp可定义性刻画了CSP之间的构件归约。一类重要的易处理约束语言$Γ$具有少量次幂的特征，即从$Γ$可pp定义的$n$元关系数量受$2^{p(n)}$限制，其中$p(n)$为某个多项式。本文研究该性质的一个限制条件：每个pp可定义关系均可由多项式长度的pp公式定义。我们猜想此类简短定义的存在性实际上等价于$Γ$具有少量次幂，并针对一个包含所有三元素域上约束语言的大子类验证了该猜想。此外，我们讨论了该猜想如何对具有少量次幂的代数的次幂成员问题施加co-NP的上界复杂度约束。