An analysis of the W*-hierarchy

Abstract

We observe that the W^*-hierarchy, a variant (introduced by Downey, Fellows, and Taylor [7]) 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 “Fagin-definable 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].