Complexity of Combinations of Qualitative Constraint Satisfaction Problems
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The CSP of a first-order theory T is the problem of deciding for a given finite set S of atomic formulas whether T ∪ S is satisfiable. Let T_1 and T_2 be two theories with countably infinite models and disjoint signatures. Nelson and Oppen presented conditions that imply decidability (or polynomial-time decidability) of CSP(T_1 ∪ T_2) under the assumption that CSP(T_1) and CSP(T_2) are decidable (or polynomial-time decidable). We show that for a large class of ω-categorical theories T_1, T_2 the Nelson-Oppen conditions are not only sufficient, but also necessary for polynomial-time tractability of CSP(T_1 ∪ T_2) (unless P=NP).
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