  Symmetric Datalog, a fragment of the logic programming language Datalog, is
conjectured to capture all constraint satisfaction problems (CSP) in L.
Therefore developing tools that help us understand whether or not a CSP can be
defined in symmetric Datalog is an important task. It is widely known that a
CSP is definable in Datalog and linear Datalog if and only if that CSP has
bounded treewidth and bounded pathwidth duality, respectively. In the case of
symmetric Datalog, Bulatov, Krokhin and Larose ask for such a duality (2008).
We provide two such dualities, and give applications. In particular, we give a
short and simple new proof of the result of Dalmau and Larose that "Maltsev +
Datalog -> symmetric Datalog" (2008).
  In the second part of the paper, we provide some evidence for the conjecture
of Dalmau (2002) that every CSP in NL is definable in linear Datalog. Our
results also show that a wide class of CSPs-CSPs which do not have bounded
pathwidth duality (e.g., the P-complete Horn-3Sat problem)-cannot be defined by
any polynomial size family of monotone read-once nondeterministic branching
programs.
