We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying structure of atoms, we prove decidability of the problem. The core result is parameter-elimination: existence of an isomorphism definable with parameters implies existence of an isomorphism definable without parameters.

@article{paperbot738,
  title = {Definable isomorphism problem},
  author = {Khadijeh Keshvardoost and Bartek Klin and Sławomir Lasota and Joanna Ochremiak and Szymon Toruńczyk},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 15, Issue 4},
  year = {2019},
  doi = {10.23638/lmcs-15(4:14)2019}
}