$p\text{-MC}(\Sigma_1^+) \le^{\mathrm{fpt}} p\text{-MC}(\Sigma_1^+[2])$, where $\Sigma_1^+[2]$ is the class of positive $\Sigma_1$-formulas whose relation symbols have arity at most $2$ [FG06, Lemma 6.13]. This is the second of the three reductions showing Clique A[1]-hard ($WH_D06_CliqueA1Complete$).

Construction. The structure is replaced by its incidence structure [FG06, Definition 6.12]: its universe consists of the elements of $\mathcal A$ and one new element $b_{R,\bar a}$ for every tuple $\bar a$ of every relation $R$; a unary relation $P_R$ holds the new elements of $R$; and binary relations $E_1, \dots, E_r$ connect $a_i$ to $b_{R,\bar a}$ when $a_i$ is the $i$-th entry of $\bar a$, where $r$ is the largest arity of an atom of $\varphi$. Every atom $R x_1 \dots x_r$ is replaced by $\exists y\,(P_R\, y \wedge E_1 x_1 y \wedge \dots \wedge E_r x_r y)$, and the new quantifiers are moved to the front. The atoms $E_i\,x\,y$ force $x$ to be an element of $\mathcal A$, and a positive formula stays true when its remaining variables are moved to elements of $\mathcal A$, so the old elements need no marking relation.

Complexity. The reduction is computable in polynomial time, and the new formula has size $O(|\varphi|)$.