Tractable Combinations of Theories via Sampling

For a first-order theory $T$, the Constraint Satisfaction Problem of $T$ is the computational problem of deciding whether a given conjunction of atomic formulas is satisfiable in some model of $T$. In this article we develop sufficient conditions for polynomial-time tractability of the constraint satisfaction problem for the union of two theories with disjoint relational signatures. To this end, we introduce the concept of sampling for theories and show that samplings can be applied to examples which are not covered by the seminal result of Nelson and Oppen.