Deciding FO-rewritability of regular languages and ontology-mediated queries in Linear Temporal Logic

Our concern is the problem of determining the data complexity of answering an ontology-mediated query (OMQ) formulated in linear temporal logic LTL over (Z,<) and deciding whether it is rewritable to an FO(<)-query, possibly with some extra predicates. First, we observe that, in line with the circuit complexity and FO-definability of regular languages, OMQ answering in AC0, ACC0 and NC1 coincides with FO(<,equiv)-rewritability using unary predicates x equiv 0 (mod n), FO(<,MOD)-rewritability, and FO(RPR)-rewritability using relational primitive recursion, respectively. We prove that, similarly to known PSPACE-completeness of recognising FO(<)-definability of regular languages, deciding FO(<,equiv)- and FO(<,MOD)-definability is also PSPACE-complete (unless ACC0 = NC1). We then use this result to show that deciding FO(<)-, FO(<,equiv)- and FO(<,MOD)-rewritability of LTL OMQs is EXPSPACE-complete, and that these problems become PSPACE-complete for OMQs with a linear Horn ontology and an atomic query, and also a positive query in the cases of FO(<)- and FO(<,equiv)-rewritability. Further, we consider FO(<)-rewritability of OMQs with a binary-clause ontology and identify OMQ classes, for which deciding it is PSPACE-, Pi2p- and coNP-complete.