On Deciding the Data Complexity of Answering Linear Monadic Datalog Queries with LTL Operators(Extended Version)

Our concern is the data complexity of answering linear monadic datalog queries whose atoms in the rule bodies can be prefixed by operators of linear temporal logic LTL. We first observe that, for data complexity, answering any connected query with operators $\bigcirc/\bigcirc^-$ (at the next/previous moment) is either in AC0, or in $ACC0\!\setminus\!AC0$, or $NC^1$-complete, or LogSpace-hard and in NLogSpace. Then we show that the problem of deciding LogSpace-hardness of answering such queries is PSpace-complete, while checking membership in the classes AC0 and ACC0 as well as $NC^1$-completeness can be done in ExpSpace. Finally, we prove that membership in AC0 or in ACC0, $NC^1$-completeness, and LogSpace-hardness are undecidable for queries with operators $\Diamond_f/\Diamond_p$ (sometime in the future/past) provided that $NC^1 \ne NLogSpace$, and $LogSpace \ne NLogSpace$.