We observe that the W*-hierarchy, a variant (introduced by Downey, Fellows, and Taylor ) 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=W* and W=W*. 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 and W*.
"An analysis of the W*-hierarchy." J. Symbolic Logic 72 (2) 513 - 534, June 2007. https://doi.org/10.2178/jsl/1185803622