$n$-permutability and linear Datalog implies symmetric Datalog

In the last decade, algebraic methods have led to much progress in classifying the complexity of the non-uniform Constraint Satisfaction Problem (CSP). The programming language Datalog, whose origins lie in logic programming and database theory, has been playing an important role in classifying complexity of CSP since at least the classic paper of Feder and M. Vardi [10]. Feder and Vardi used Datalog to define CSPs of bounded width. Later, V. Dalmau [7] asked which CSPs can be solved using the weaker language of linear Datalog, and finally L. Egri, B. Larose and P. Tesson [9] introduced even weaker symmetric Datalog, both in an effort to describe the finer hierarchy of CSP complexity. We want to show that if CSP(A) can be solved by a linear Datalog program (alternatively, has bounded pathwidth duality) and A is n-permutable for some n, then CSP(A) can be solved by a symmetric Datalog program (and so lies in L). While both conditions are necessary and so we have an “if and only if” description of symmetric Datalog, it is an open problem to describe those structures A such that CSP(A) is solvable by linear Datalog. However, once CSPs for which linear Datalog works are classified, we will immediately get an equally good classification of symmetric Datalog CSPs. In particular, should it turns out that SD(∨) implies bounded pathwidth duality, we would have a neat characterization of problems solvable by symmetric Datalog: It is the class of problems whose algebras omit all tame congruence theory types except for the Boolean type (we go into greater detail about this in the Conclusions).