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On the Relationship Between the Complexity of Decidability and Decomposability of First-Order Theories

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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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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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Correspondence to D. Ponomaryov.

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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).

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