On guarded extensions of MMSNP

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.