Definable isomorphism problem

We investigate the isomorphism problem in the setting of definable sets (equiv-alent to sets with atoms): given two definable relational structures, are they related by adefinable isomorphism? Under mild assumptions on the underlying structure of atoms, weprove decidability of the problem. The core result is parameter-elimination: existence ofan isomorphism definable with parameters implies existence of an isomorphism definablewithout parameters.