# Tractable Combinations of Temporal CSPs

The constraint satisfaction problem (CSP) of a first-order theory T is the computational problem of deciding whether a given conjunction of atomic formulas is satisfiable in some model of T. We study the computational complexity of CSP(T_1 ∪ T_2) where T_1 and T_2 are theories with disjoint finite relational signatures. We prove that if T_1 and T_2 are the theories of temporal structures, i.e., structures where all relations have a first-order definition in (ℚ;<), then CSP(T_1 ∪ T_2) is in P or NP-complete. To this end we prove a purely algebraic statement about the structure of the lattice of locally closed clones over the domain ℚ that contain Aut(ℚ;<).

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