  In this paper we study possibilities of interpolation and symbol elimination
in extensions of a theory $\mathcal{T}_0$ with additional function symbols
whose properties are axiomatised using a set of clauses. We analyze situations
in which we can perform such tasks in a hierarchical way, relying on existing
mechanisms for symbol elimination in $\mathcal{T}_0$. This is for instance
possible if the base theory allows quantifier elimination. We analyze
possibilities of extending such methods to situations in which the base theory
does not allow quantifier elimination but has a model completion which does. We
illustrate the method on various examples.
