Separating Positive and Negative Data Examples by Concepts and Formulas: The Case of Restricted Signatures
This work addresses a foundational problem in knowledge representation and reasoning for AI, focusing on decidability and complexity in logical separability with practical constraints.
The paper tackles the problem of separating positive and negative data examples using description logic concepts and formulas under a restricted signature, showing that weak separability is decidable in ALCI but undecidable in GF, GNF, and ALCFIO, while strong separability is decidable in ALCI, GF, and GNF, with tight complexity bounds provided.
We study the separation of positive and negative data examples in terms of description logic (DL) concepts and formulas of decidable FO fragments, in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols from the data and ontology that can be used for separation. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated. Our main results are that (a projective form of) the weak version is decidable in $\mathcal{ALCI}$ while it is undecidable in the guarded fragment GF, the guarded negation fragment GNF, and the DL $\mathcal{ALCFIO}$, and that strong separability is decidable in $\mathcal{ALCI}$, GF, and GNF. We also provide (mostly tight) complexity bounds.