🤖 AI Summary
This work systematically characterizes the boundary of Craig interpolation within decidable fragments of first-order logic. Focusing on key fragments—guarded fragment (GFO), two-variable fragment (FO₂), forward fragment (FO-forward), and fluted fragment—the study employs model-theoretic analysis, type construction, interpolation proofs, and reductions encoding decidability to precisely identify minimal extensions preserving interpolation: GNFO is the minimal decidable extension of GFO retaining interpolation; FO is the minimal extension preserving interpolation for both FO₂ and FO-forward; and all interpolation-preserving extensions of FO₂ and the fluted fragment are undecidable. The work establishes, for the first time within a unified framework, the fundamental tension between interpolation and decidability, and determines critical extension structures for these fragments.
📝 Abstract
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.