一句话：研究 MMSNP 与三个超类之间的守卫扩展，证明 GMSNP 的二分可归约到单关系签名，并引入仍具二分性的守卫不等式 MMSNP。
Feder 与 Vardi 证明 MMSNP 的 P/NP 二分等价于有限域 CSP 的二分，而去掉单调性、一元性或无不等式任一条件所得三个超类均无二分。本文研究 MMSNP 与这些超类之间二分性未知的缺口。对到无不等式单调 SNP 的缺口，研究 GMSNP，证明其二分性等价于单关系符号签名上的 GMSNP 二分性。对到带不等式 MMSNP 的缺口，引入带守卫不等式的 MMSNP，它严格强于 MMSNP 但仍具二分性。对到无不等式一元 SNP 的缺口，引入一种扩展 Matrix Partitions 的逻辑并提出二分性开放问题。最后详细重证 Feder–Vardi 关于 NP 嵌入带不等式 MMSNP 的定理，说明三个缺口右侧类均无二分。
Feder and Vardi showed that the class Monotone Monadic SNP without inequality (MMSNP) has a P vs NP-complete dichotomy if and only if such a dichotomy holds for finite-domain Constraint Satisfaction Problems (CSPs). Moreover, they showed that none of the three classes obtained by removing one of the defining properties of MMSNP (monotonicity, monadicity, no inequality) has a dichotomy. The overall objective of this paper is to study the gaps between MMSNP and each of these three superclasses, where the existence of a dichotomy remains unknown. For the gap between MMSNP and Monotone SNP without inequality, we study the class Guarded Monotone SNP without inequality (GMSNP) introduced by Bienvenu, ten Cate, Lutz, and Wolter, and prove that GMSNP has a dichotomy if and only if a dichotomy holds for GMSNP problems over signatures consisting of a unique relation symbol. For the gap between MMSNP and MMSNP with inequality, we introduce a new class MMSNP with guarded inequality, that lies between MMSNP and MMSNP with inequality and that is strictly more expressive than the former and still has a dichotomy. For the gap between MMSNP and Monadic SNP without inequality, we introduce a logic that extends the class of Matrix Partitions in a similar way how MMSNP extends finite-domain CSP, and pose an open question about the existence of a dichotomy for this class. Finally, we revisit the theorem of Feder and Vardi, which claims that the class NP embeds into MMSNP with inequality. We give a detailed proof of this theorem as it ensures no dichotomy for the right-hand side class of each of the three gaps.