  We investigate logics and classes of problems below Fagin's existential
second-order logic (ESO) and above Feder and Vardi's logic for constraint
satisfaction problems (CSP), the so called monotone monadic SNP without
inequality (MMSNP). It is known that MMSNP has a dichotomy between P and
NP-complete but the removal of any of these three restrictions imposed on SNP
yields a logic that is Ptime equivalent to ESO: so by Ladner's theorem we have
three stronger sibling logics that are nondichotomic above MMSNP. In this
paper, we explore the area between these four logics, mostly by considering
guarded extensions of MMSNP, with the ultimate goal being to obtain logics
above MMSNP that exhibit such a dichotomy.
