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

We consider the decomposability problem, i.e., the problem to decide whether a logical theory T is equivalent to a union of two (or several) components in signatures, which correspond to a partition of the signature of 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.