Separating Positive and Negative Data Examples by Concepts and Formulas: The Case of Restricted Signatures

TL;DR

This paper investigates the problem of separating positive and negative data examples using description logic concepts and formulas within decidable FO fragments, considering signature restrictions and different separability notions.

Contribution

It introduces signature-restricted separation problems, analyzes their decidability across various logic fragments, and provides complexity bounds for these problems.

Findings

Weak separability is decidable in ALCI but undecidable in GF, GNF, and ALCFIO.

Strong separability is decidable in ALCI, GF, and GNF.

Complexity bounds are established for the separation problems.

Abstract

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.