Locating P/poly Optimally in the Extended Low Hierarchy

The low hierarchy within NP and the extended low hierarchy h a ve turned out to be very useful in classifying many i n teresting language classes. We relocate P=poly from the third-level EL P 3 (Balcc azar et al., 1986) to the third-level EL P 3 of the extended low hierarchy. The location of P=poly in EL P 3 is optimal since, as shown by Allender and Hemachandra (1992), there exist sparse sets that are not contained in the next lower level EL P 2. A s a consequence of our result, all NP sets in P=poly are relocated from the third-level L P 3 (Ko and Schh oning, 1985) to the third-level L P 3 of the low hierarchy.