Reducts of finitely bounded homogeneous structures, and lifting tractability from finite-domain constraint satisfaction

Many natural decision problems can be formulated as constraint satisfaction problems for reducts of finitely bounded homogeneous structures. This class of problems is a large generalisation of the class of CSPs over finite domains. Our first result is a general polynomial-time reduction from such infinite-domain CSPs to finite-domain CSPs. We use this reduction to obtain new powerful polynomial-time tractability conditions that can be expressed in terms of topological polymorphism clones. Moreover, we study the subclass $\mathcal{C}$ of CSPs for structures that are first-order definable over equality with parameters. Also this class $\mathcal{C}$ properly extends the class of all finite-domain CSPs. We show that the tractability conjecture for reducts of finitely bounded homogeneous structures is for $\mathcal{C}$ equivalent to the finite-domain tractability conjecture.