## Abstract

This paper addresses complexity issues for important problems arising with disjunctive logic programming. In particular, the complexity of deciding whether a disjunctive logic program is consistent is investigated for a variety of well-known semantics, as well as the complexity of deciding whether a propositional formula is satisfied by all models according to a given semantics. We concentrate on finite propositional disjunctive programs with as well as without integrity constraints, i.e., clauses with empty heads; the problems are located in appropriate slots of the polynomial hierarchy. In particular, we show that the consistency check is Σ
^{p}_{2}
-complete for the disjunctive stable model semantics (in the total as well as partial version), the iterated closed world assumption, and the perfect model semantics, and we show that the inference problem for these semantics is Π
^{p}_{2}
-complete; analogous results are derived for the answer sets semantics of extended disjunctive logic programs. Besides, we generalize previously derived complexity results for the generalized closed world assumption and other more sophisticated variants of the closed world assumption. Furthermore, we use the close ties between the logic programming framework and other nonmonotonic formalisms to provide new complexity results for disjunctive default theories and disjunctive autoepistemic literal theories.

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Parts of the results in this paper appeared in form of an abstract in the Proceedings of the Twelfth ACM SIGACT SIGMOD-SIGART Symposium on Principles of Database Systems (PODS-93), pp. 158–167. Other parts appeared in shortened form in the Proceedings of the International Logic Programming Symposium, Vancouver, October 1993 (ILPS-93), pp. 266–278. MIT Press.

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Eiter, T., Gottlob, G. On the computational cost of disjunctive logic programming: Propositional case.
*Ann Math Artif Intell* **15, **289–323 (1995). https://doi.org/10.1007/BF01536399

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### Keywords

- Model Semantic
- Integrity Constraint
- Propositional Formula
- Default Theory
- Disjunctive Program