The logic MMSNP is a well-studied fragment of Existential Second-Order logic that, from a computational perspective, captures finite-domain Constraint Satisfaction Problems (CSPs) modulo polynomial-time reductions. At the same time, MMSNP contains many problems that are expressible as $ω$-categorical CSPs but not as finite-domain ones. We initiate the study of Promise MMSNP (PMMSNP), a promise analogue of MMSNP. We show that every PMMSNP problem is poly-time equivalent to a (finite-domain) Promise CSP (PCSP), thereby extending the classical MMSNP-CSP correspondence to the promise setting. We then investigate the complexity of PMMSNPs arising from forbidding monochromatic cliques, a class encompassing promise graph colouring problems. For this class, we obtain a full complexity classification conditional on the Rich 2-to-1 Conjecture, a recently proposed perfect-completeness surrogate of the Unique Games Conjecture. As a key intermediate step which may be of independent interest, we prove that it is NP-hard, under the Rich 2-to-1 Conjecture, to properly colour a uniform hypergraph even if it is promised to admit a colouring satisfying a certain technical condition called reconfigurability. This proof is an extension of the recent work of Braverman, Khot, Lifshitz and Minzer (Adv. Math. 2025). To illustrate the broad applicability of this theorem, we show that it implies most of the linearly-ordered colouring conjecture of Barto, Battistelli, and Berg (STACS 2021).

翻译：逻辑MMSNP是存在性二阶逻辑的一个经过深入研究的片段，从计算角度来看，它包含了模多项式时间归约的有限域约束满足问题（CSPs）。同时，MMSNP包含许多可表达为ω-范畴CSP但非有限域CSP的问题。我们开启了承诺MMSNP（PMMSNP）的研究，它是MMSNP的承诺对应物。我们证明每个PMMSNP问题在多项式时间内等价于一个（有限域）承诺CSP（PCSP），从而将经典的MMSNP-CSP对应关系扩展到承诺设置。然后，我们研究了由禁止单色团产生的PMMSNP的复杂度，这类问题涵盖了承诺图着色问题。针对这一类别，我们在Rich 2-to-1猜想（近期提出的唯一博弈猜想的完美完备性替代）成立的条件下，获得了完整的复杂度分类。作为一个可能具有独立意义的关键中间步骤，我们证明在Rich 2-to-1猜想下，即使保证一个均匀超图存在满足称为可重构性的特定技术条件的着色，对其进行正确着色也是NP难的。这一证明推广了Braverman、Khot、Lifshitz和Minzer（Adv. Math. 2025）的最新工作。为展示该定理的广泛适用性，我们证明它蕴含了Barto、Battistelli和Berg（STACS 2021）的线性序着色猜想的大部分内容。