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We show that if $\mathbb A$ is a core relational structure such thatCSP($\mathbb A$) can be solved by a linear Datalog program, and $\mathbb A$ is$n$-permutable for some $n$, then CSP($\mathbb A$) can be solved by a symmetricDatalog program (and thus CSP($\mathbb A$) lies in deterministic logspace). Atthe moment, it is not known for which structures $\mathbb A$ will CSP($\mathbbA$) be solvable by a linear Datalog program. However, once somebody obtains acharacterization of linear Datalog, our result immediately gives acharacterization of symmetric Datalog.

$n$-permutability and linear Datalog implies symmetric Datalog