Abstract
We show that minimal models of positive propositional theories can be decomposed based on the structure of the dependency graph of the theories. This observation can be useful for many applications involving computation with minimal models. As an example of such benefits, we introduce new algorithms for minimal model finding and checking that are based on model decomposition. The algorithms’ temporal worst-case complexity is exponential in the size s of the largest connected component of the dependency graph, but their actual cost depends on the size of the largest source actually encountered, which can be far smaller than s, and on the class of theories to which sources belong. Indeed, if all sources reduce to an HCF or HEF theory, the algorithms are polynomial in the size of the theory.
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Notes
- 1.
This is a short paper, all missing references can be found in [2].
- 2.
Note that the syntax of (1) is a bit unusual for a clause; usually, the equivalent notation \(\lnot a_1 \vee \lnot a_2 \vee ... \vee \lnot a_m \vee c_1 \vee c_2 \vee ...\vee c_n\) is employed.
- 3.
Clause nodes in the dependency graph are mandatory to achieve a graph which is linear in the size of the theory.
References
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Ben-Eliyahu-Zohary, R., Angiulli, F., Fassetti, F., Palopoli, L.: Modular construction of minimal models. A full version of this paper, can be obtained from the authors
Cadoli, M.: The complexity of model checking for circumscriptive formulae. Inf. Process. Lett. 44(3), 113–118 (1992)
Cadoli, M.: On the complexity of model finding for nonmonotonic propositional logics. In: Proceedings of the 4th Italian Conference on Theoretical Computer Science, pp. 125–139. World Scientific Publishing Co., October 1992
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Ben-Eliyahu-Zohary, R., Angiulli, F., Fassetti, F., Palopoli, L. (2017). Modular Construction of Minimal Models. In: Balduccini, M., Janhunen, T. (eds) Logic Programming and Nonmonotonic Reasoning. LPNMR 2017. Lecture Notes in Computer Science(), vol 10377. Springer, Cham. https://doi.org/10.1007/978-3-319-61660-5_5
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DOI: https://doi.org/10.1007/978-3-319-61660-5_5
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