Approximation of hyperarithmetic analysis by $\omega$-model reflection

This paper presents two types of results related to hyperarithmetic analysis. First, we introduce new variants of the dependent choice axiom, namely $\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}{\mathrm{\Pi }}_0^1(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{.}{\mathrm{\Sigma }}_1^1)\text{-}{\mathsf{D}\mathsf{C}}_0$ and $\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}{\mathrm{\Pi }}_0^1(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{.}{\mathrm{\Sigma }}_1^1)\text{-}{\mathsf{D}\mathsf{C}}_0$. These variants imply ${\mathsf{A}\mathsf{C}\mathsf{A}}_0^{+}$ but do not imply ${\mathrm{\Sigma }}_1^1\mathrm{I}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$. We also demonstrate that these variants belong to hyperarithmetic analysis and explore their implications with well-known theories in hyperarithmetic analysis. Second, we show that ${\mathsf{R}\mathsf{F}\mathsf{N}}^{-1}({\mathsf{A}\mathsf{T}\mathsf{R}}_0)$, a class of theories defined using the $\omega$-model reflection axiom, approximates to some extent hyperarithmetic analysis, and investigate the similarities between this class and hyperarithmetic analysis.