Adding Circumscription to Decidable Fragments of First-Order Logic: A Complexity Rollercoaster
<|reference_start|>Adding Circumscription to Decidable Fragments of First-Order Logic: A Complexity Rollercoaster: We study extensions of expressive decidable fragments of first-order logic with circumscription, in particular the two-variable fragment FO$^2$, its extension C$^2$ with counting quantifiers, and the guarded fragment GF. We prove that if only unary predicates are minimized (or fixed) during circumscription, then decidability of logical consequence is preserved. For FO$^2$ the complexity increases from $\textrm{coNexp}$ to $\textrm{coNExp}^\textrm{NP}$-complete, for GF it (remarkably!) increases from $\textrm{2Exp}$ to $\textrm{Tower}$-complete, and for C$^2$ the complexity remains open. We also consider querying circumscribed knowledge bases whose ontology is a GF sentence, showing that the problem is decidable for unions of conjunctive queries, $\textrm{Tower}$-complete in combined complexity, and elementary in data complexity. Already for atomic queries and ontologies that are sets of guarded existential rules, however, for every $k \geq 0$ there is an ontology and query that are $k$-$\textrm{Exp}$-hard in data complexity.<|reference_end|>
@article{lutz2024adding,
    title={Adding Circumscription to Decidable Fragments of First-Order Logic: A
  Complexity Rollercoaster},
    author={Carsten Lutz and Quentin Mani`ere},
    journal={arXiv preprint arXiv:2407.20822},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.20822},
    primaryClass={cs.AI cs.LO}
}