Forcing and classes of $\mathsf{HYP}$-dominating functions

This paper is aimed at showing separations between three subsets of $ω^ω$, namely $\mathsf{HYP}\text{-}\mathsf{SNE}$, $\mathsf{HYP}\text{-}\mathsf{SME}$, and $\mathsf{HYP}\text{-}\mathsf{DOM}$. These classes are natural computational analogues of cardinal characteristics from Cichon's diagram and are known to satisfy $\mathsf{HYP}\text{-}\mathsf{SNE} \subseteq \mathsf{HYP}\text{-}\mathsf{SME} \subseteq \mathsf{HYP}\text{-}\mathsf{DOM}$. To show that both of these inclusions are strict we introduce effectivizations of Laver and Hechler forcing, which we believe are of independent interest. Our techniques allow us to show similar results relative to any Turing ideal closed under $\leq_{\mathsf{HYP}}$.