Abstract
We study the computational problem of checking whether a quantified conjunctive query (a first-order sentence built using only conjunction as Boolean connective) is true in a finite poset (a reflexive, antisymmetric, and transitive directed graph). We prove that the problem is already \(\mathrm {NP}\)-hard on a certain fixed poset, and investigate structural properties of posets yielding fixed-parameter tractability when the problem is parameterized by the query. Our main algorithmic result is that model checking quantified conjunctive queries on posets of bounded width is fixed-parameter tractable (the width of a poset is the maximum size of a subset of pairwise incomparable elements). We complement our algorithmic result by complexity results with respect to classes of finite posets in a hierarchy of natural poset invariants, establishing its tightness in this sense.
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Notes
- 1.
Existential and universal logic are maximal syntactic fragments properly contained in first-order logic.
- 2.
Conjunctive positive logic and existential (respectively, universal) logic are incomparable syntactic fragments of first-order logic.
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Acknowledgments
This research was supported by the European Research Council (Complex Reason, 239962) and the FWF Austrian Science Fund (Parameterized Compilation, P26200 and X-TRACT, P26696).
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Bova, S., Ganian, R., Szeider, S. (2014). Quantified Conjunctive Queries on Partially Ordered Sets. In: Cygan, M., Heggernes, P. (eds) Parameterized and Exact Computation. IPEC 2014. Lecture Notes in Computer Science(), vol 8894. Springer, Cham. https://doi.org/10.1007/978-3-319-13524-3_11
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