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

TL;DR

This paper investigates the complexity of determining whether ontology-mediated queries in linear temporal logic can be rewritten into first-order logic, revealing PSPACE- and EXPSPACE-complete results depending on the query and ontology types.

Contribution

It establishes the computational complexity of FO-rewritability decision problems for LTL-based ontology-mediated queries, extending known results to new logical fragments and ontology classes.

Findings

Deciding FO(<, e9)-rewritability is PSPACE-complete.

Deciding FO(<)-rewritability is EXPSPACE-complete.

Complexity varies with ontology and query types, including PSPACE-, Pi_2^p-, and coNP-complete cases.

Abstract

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 AC^0, ACC^0 and NC^1 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 ACC^0 = NC^1). 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-, Pi_2^p- and coNP-complete.