We show that the guarded-negation fragment is, in a precise sense, the smallest extension of the guarded fragment with Craig interpolation. In contrast, we show that full first-order logic is the smallest extension of both the two-variable fragment and the forward fragment with Craig interpolation. Similarly, we also show that all extensions of the two-variable fragment and of the fluted fragment with Craig interpolation are undecidable.

@article{paperbot3381,
  title = {Craig Interpolation for Decidable First-Order Fragments},
  author = {Balder ten Cate and Jesse Comer},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 21, Issue 3},
  year = {2025},
  doi = {10.46298/lmcs-21(3:22)2025}
}