Disjunctive Datalog

Disjunctive Datalog is an extension of the logic programming language Datalog that allows disjunctions in the heads of rules. This extension enables disjunctive Datalog to express several NP-hard problems that are not known to be expressable in plain Datalog. Disjunctive Datalog has been applied in the context of reasoning about ontologies in the semantic web.^[1] DLV is an implementation of disjunctive Datalog.

Syntax

A disjunctive Datalog program is a collection of rules. A rule is a clause of the form:^[2]

${\displaystyle a_{1}\vee \dots \vee a_{n}\leftarrow b_{1}\wedge \dots \wedge b_{m}\quad 1\leq n,0\leq m}$

where ${\displaystyle b_{1}}$, ..., ${\displaystyle b_{m}}$ may be negated, and may include (in)equality constraints.

Semantics

There are at least three ways to define the semantics of disjunctive Datalog:^[3]

• Minimal model semantics
• Perfect model semantics
• Disjunctive stable model semantics, which generalizes the stable model semantics

Expressivity

Disjunctive Datalog can express several NP-complete and NP-hard problems, including the travelling salesman problem, graph coloring, maximum clique problem, and minimal vertex cover.^[3] These problems are only expressible in Datalog if the polynomial hierarchy collapses.

Implementations

The DLV (DataLog with Disjunction, where the logical disjunction symbol V is used) system implements the disjunctive stable model semantics.^[4]

See also

• Syntax and semantics of logic programming

Sources

Notes

1. Kaminski, Mark; Nenov, Yavor; Grau, Bernardo Cuenca (2014-06-21). "Datalog Rewritability of Disjunctive Datalog Programs and its Applications to Ontology Reasoning". Proceedings of the AAAI Conference on Artificial Intelligence. 28 (1). arXiv:1404.3141. doi:10.1609/aaai.v28i1.8854. ISSN 2374-3468. S2CID 17098158.
2. Eiter, Gottlob & Mannila 1997, p. 370.
3. Eiter, Gottlob & Mannila 1997.
4. Alviano, Mario; Faber, Wolfgang; Leone, Nicola; Perri, Simona; Pfeifer, Gerald; Terracina, Giorgio (2011), "The Disjunctive Datalog System DLV", Datalog Reloaded, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 282–301, doi:10.1007/978-3-642-24206-9_17, ISBN 978-3-642-24205-2, retrieved 2023-08-04

References

• Eiter, Thomas; Gottlob, Georg; Mannila, Heikki (1997-09-01). "Disjunctive datalog". ACM Transactions on Database Systems. 22 (3): 364–418. doi:10.1145/261124.261126. ISSN 0362-5915. S2CID 8755376.