Abstract
We consider the decomposability problem, i.e., the problem to decide whether a logical theory \({\mathcal{T}}\) is equivalent to a union of two (or several) components in signatures, which correspond to a partition of the signature of \({\mathcal{T}}\) ‘‘modulo’’ a given shared subset of symbols. We introduce several tools for proving that the computational complexity of this problem coincides with the complexity of entailment. As an application of these tools we derive tight bounds for the complexity of decomposability of theories in signature fragments of first-order logic, i.e., those fragments, which are obtained by restricting signature.
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REFERENCES
F. van Harmelen, V. Lifschitz, and B. Porter, Handbook of Knowledge Representation, Vol. 3: Foundations of Artificial Intelligence (Elsevier, Amsterdam, 2008).
S. Staab and R. Studer, Handbook on Ontologies, 2nd ed., Vol. 2 of International Handbooks on Information Systems (Springer, 2009).
C. Lutz and F. Wolter, ‘‘Mathematical logic for life science ontologies,’’ in Proceedings of the 16th International Workshop on Logic, Language, Information, and Computation, WoLLIC 2009, Tokyo, Japan (2009).
E. Börger, E. Grädel, and Y. Gurevich, The Classical Decision Problem, Perspectives in Mathematical Logic (Springer, 1997).
D. Ponomaryov, ‘‘On decomposability in logical calculi,’’ Bull. Novosib. Comput. Center 28, 111–120 (2008).
P. Emelyanov and D. Ponomaryov, ‘‘The complexity of AND-decomposition of Boolean functions,’’ Discrete Appl. Math. 280, 113–132 (2020).
P. Emelyanov, ‘‘On two kinds of dataset decomposition,’’ in Proceedings of the 18th International Conference on Computational Science ICCS 2018, Wuxi, China (Springer, 2018), pp. 171–183.
B. Konev, C. Lutz, D. Ponomaryov, and F. Wolter, ‘‘Decomposing description logic ontologies,’’ in Proceedings of the 12th International Conference on the Principles of Knowledge Representation and Reasoning KR 2010, Toronto, Canada (AAAI Press, 2010).
D. Ponomaryov and M. Soutchanski, ‘‘Progression of decomposed local-effect action theories,’’ ACM Trans. Comput. Logic 18, 16.1–16.41 (2017). https://doi.org/10.1145/3091119
A. Morozov and D. Ponomaryov, ‘‘On decidability of the decomposability problem for finite theories,’’ Sib. Math. J. 51, 667–674 (2010).
J. D. Monk, Mathematical Logic (Springer, Berlin, 1976).
L. Bachmair, H. Ganzinger, and U. Waldmann, ‘‘Set constraints are the monadic class,’’ in Proceedings of the 8th Annual IEEE Symposium on Logic in Computer Science, Montreal, Canada (IEEE Comput. Soc. Press, 1993), pp. 75–83.
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The work was supported by the Mathematical Center in Akademgorodok under Agreement no. 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation.
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(Submitted by I. Sh. Kalimullin)
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Ponomaryov, D. On the Relationship Between the Complexity of Decidability and Decomposability of First-Order Theories. Lobachevskii J Math 42, 2905–2912 (2021). https://doi.org/10.1134/S199508022112026X
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DOI: https://doi.org/10.1134/S199508022112026X