  Kechris and Martin showed that the Wadge rank of the $\omega$-th level of the
decreasing difference hierarchy of coanalytic sets is $\omega_2$ under the
axiom of determinacy. In this article, we give an alternative proof of the
Kechris-Martin theorem, by understanding the $\omega$-th level of the
decreasing difference hierarchy of coanalytic sets as the (relative)
hyperarithmetical processes with finite mind-changes. Based on this viewpiont,
we also examine the gap between the increasing and decreasing difference
hierarchies of coanalytic sets by relating them to the $\Pi^1_1$- and
$\Sigma^1_1$-least number principles, respectively. We also analyze Weihrauch
degrees of related principles.
