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On some topics around the Wadge rank $ω_2$

Kechris and Martin showed that the Wadge rank of the $ω$-th level of the decreasing difference hierarchy of coanalytic sets is $ω_2$ under the axiom of determinacy. In this article, we give an alternative proof of the Kechris-Martin theorem, by understanding the $ω$-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 $Π^1_1$- and $Σ^1_1$-least number principles, respectively. We also analyze Weihrauch degrees of related principles.