Constraint satisfaction problems (CSPs) can be modelled in
terms of existence of homomorphism between structures.
Feder and Vardi have introduced a fragment of Monadic Second Order
logic, MMSNP, and proved that the class of problems captured by MMSNP
is computationally equivalent to the class of CSPs. However, this
computational equivalence involve a change of signature and a highly
non-trivial derandomisation due to Kun. In fact, in terms of
Descriptive Complexity, the picture is quite different: there are problems in
MMSNP that are not CSPs (over the same signature). Moreover, in general,
Bodirsky proved that problems captured by MMSNP are in fact (finite
union of) well behaved \emph{infinite} Constraint Satisfaction
Problems. This yields the following question: \emph{Given a sentence of
MMSNP, can we decide whether it captures a finite or an infinite
CSP?} This question, when restricted to the first-order fragment of
MMSNP is related to the notion of \emph{homomorphism duality} studied
in Structural Combinatorics. Another popular concept in this field is
that of \emph{restricted homomorphism duality}, which corresponds in
our setting to the following question: \emph{Given a sentence of
MMSNP, can we decide whether it captures a finite or an infinite
CSP, when restricted to a class of input $\mathcal{K}$?}
We will present an overview of results concerning MMSNP, CSP and
duality.