Closure Properties and Witness Reduction

Witness reduction has played a crucial role in several recent results in complexity theory. These include Toda's result that PH BP P, the \collapsing" of PH into P with a high probability; Toda and Ogiwara's results which \collapses" PH into various counting classes with a high probability; and hard functions for various function classes studied by Ogiwara and Hemachandra. Ogiwara and Hemachandra's results establish a connection between functions being hard for #P and functions interacting with the class to e ect witness reduction. In fact, we believe that the ability to achieve some form of witness reduction is what makes a function hard for a class of functions. To support our thesis we de ne new function classes and obtain results analogous to those of Ogiwara and Hemachandra. We also introduce the notion of randomly hard functions and obtain similar results.