An analysis of the W*-hierarchy

We observe that the W∗ -hierarchy, a variant (introduced by Downey, Fellows, and Taylor [8]) of the better known W-hierarchy, coincides with the W-hierarchy, though not level wise, but just as a whole hierarchy. More precisely, we prove that W[t] ⊆ W∗ [t] ⊆ W[2t − 2] for each t ≥ 2. It was known before that W[1] = W∗ [1] and W[2] = W∗ [2]. Our second main result is a new logical characterization of the W∗ -hierarchy in terms of “Fagindefinable problems.” As a by-product, we also obtain an improvement of our earlier characterization of the hierarchy in terms of model-checking problems. Furthermore, we obtain new complete problems for the classes W[3] and W∗ [3].

• Type: Journal
• Year: 2007
• Where: JSYML
• Authors: Yijia Chen, Jörg Flum, Martin Grohe
• Keywords: By-product, Fagindefinable Problems, JSYML 2007, Level Wise