The CSP of a first-order theory $T$ is the problem of deciding for a givenfinite set $S$ of atomic formulas whether $T cup S$ is satisfiable. Let $T_1$and $T_2$ be two theories with countably infinite models and disjointsignatures. Nelson and Oppen presented conditions that imply decidability (orpolynomial-time decidability) of $mathrm{CSP}(T_1 cup T_2)$ under theassumption that $mathrm{CSP}(T_1)$ and $mathrm{CSP}(T_2)$ are decidable (orpolynomial-time decidable). We show that for a large class of$omega$-categorical theories $T_1, T_2$ the Nelson-Oppen conditions are notonly sufficient, but also necessary for polynomial-time tractability of$mathrm{CSP}(T_1 cup T_2)$ (unless P=NP).
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机译：一阶理论的CSP $ T $是决定特定菲涅石的问题，无论是$ T CUP S $是否满足的原子公式。 让$ t_1 $和$ t_2 $是两个有关无限模型和脱节的理论。 纳尔逊和州呈现的条件意味着暗示$ mathrm {csp}（t_1 cup t_2）$下的可解锁性（Orpolynomial-time可译种性）在$ mathrm {csp}（t_1）$和$ mathrm {csp}下（t_2 ）$是可判定的（Orpolynomial-time Deacinable）。 我们表明，对于大量的$ omega $ -categorical理论$ t_1，t_2 $ t_2 $ t_2 $ t_2 $ inden-oppen条件是不够的，而且还需要$ mathrm {csp}的多项式易易行动（t_1 cup t_2）所必需的 $（除非p = np）。
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