On the computational complexity of small descriptions

For a set L that is polynomial time reducible to some sparse set, the authors investigate the computational complexity of such sparse sets relative to L. They construct sets A and B such that both of them are polynomial time reducible to some sparse set, but A (resp., B) is polynomial time reducible to no sparse set in P^A (resp., NP^B ∩ co-NP^B); that is, the complexity of sparse sets to which A (resp., B) is reducible is more than P^A (resp., NP^B ∩ co-NP^B). From these results and/or application of their proof technique the authors obtain: (1) lower bounds for the relative complexity of finding polynomial size circuits for some sets in P/poly, and (2) separations of the equivalence classes of sparse sets under various reducibilities