On deciding the data complexity of answering Linear Monadic Datalog Queries with LTL Operators

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 ○/○- (at the next/previous moment) is either in AC⁰, or in ACC⁰\AC⁰, or NC¹-complete, or L-hard and in NL. Then we show that the problem of deciding L-hardness of answering such queries is PSpace-complete, while checking membership in the classes AC⁰ and ACC⁰ as well as NC¹-completeness can be done in ExpSpace. Finally, we prove that membership in AC⁰ or in ACC⁰, NC¹-completeness, and L-hardness are undecidable for queries with operators ◇/◇- (sometime in the future/past) provided that NC¹ ≠ NL and L ≠ NL.