Adding Circumscription to Decidable Fragments of First-Order Logic: A Complexity Rollercoaster
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913476863066112 |
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| author | Lutz, Carsten Manière, Quentin |
| author_facet | Lutz, Carsten Manière, Quentin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20822 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Adding Circumscription to Decidable Fragments of First-Order Logic: A Complexity Rollercoaster Lutz, Carsten Manière, Quentin Artificial Intelligence Logic in Computer Science 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. |
| title | Adding Circumscription to Decidable Fragments of First-Order Logic: A Complexity Rollercoaster |
| topic | Artificial Intelligence Logic in Computer Science |
| url | https://arxiv.org/abs/2407.20822 |