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. 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 have two contributions. 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. Apart from that, we give a detailed proof that every problem in NP is polynomial-time equivalent to a problem in MMSNP with inequality, which implies the absence of a dichotomy for the latter. 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 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.

翻译：Feder和Vardi证明，当且仅当有限域约束满足问题具有P与NP完全的二象性时，无不等式的单调一元SNP类（MMSNP）才具有该二象性。此外，他们指出，移除MMSNP三个定义属性（单调性、一元性、无不等式）中任意一个所得到的三个类均不存在二象性。本文的核心目标是研究MMSNP与这三个超类之间的间隙——在这些间隙中二象性的存在性仍属未知。针对MMSNP与无不等式的单调SNP之间的间隙，我们研究了由Bienvenu、ten Cate、Lutz和Wolter提出的无不等式受保护单调SNP类（GMSNP），并证明GMSNP具有二象性当且仅当在仅包含单一关系符号的签名上的GMSNP问题存在二象性。对于MMSNP与带不等式的MMSNP之间的间隙，我们做出两项贡献：首先引入带受保护不等式的MMSNP新类，该类介于MMSNP与带不等式的MMSNP之间，其表达能力严格强于前者且仍保持二象性；其次，我们详细证明了NP中的每个问题都与带不等式的MMSNP中的某个问题具有多项式时间等价性，这直接导致后者不存在二象性。关于MMSNP与无不等式的单子SNP之间的间隙，我们提出一种逻辑，该逻辑以类似MMSNP扩展CSP的方式扩展了矩阵划分类，并就该类二象性的存在性提出开放性问题。最后，我们重新审视Feder和Vardi关于NP类可嵌入带不等式的MMSNP的定理，通过详细证明该定理，确认了三个间隙右侧类均不存在二象性。