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 polynomialtime 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).

No abstract

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$(T_1 \cup T_2)$ 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$(T_1 \cup T_2)$ 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$(Q;<)$.

No abstract