  It is known that several variations of the axiom of determinacy play
important roles in the study of reverse mathematics, and the relation between
the hierarchy of determinacy and comprehension are revealed by Tanaka, Nemoto,
Montalb\'an, Shore, and others. We prove variations of a result by
Ko{\l}odziejczyk and Michalewski relating determinacy of arbitrary boolean
combinations of $\Sigma^0_2$ sets and reflection in second-order arithmetic.
Specifically, we prove that: over $\mathsf{ACA}_0$,
$\Pi^1_2$-$\mathsf{Ref}(\mathsf{ACA}_0)$ is equivalent to $\forall
n.(\Sigma^0_1)_n$-$\mathsf{Det}^*_0$;
$\Pi^1_3$-$\mathsf{Ref}(\Pi^1_1$-$\mathsf{CA}_0)$ is equivalent to $\forall
n.(\Sigma^0_1)_n$-$\mathsf{Det}$; and
$\Pi^1_3$-$\mathsf{Ref}(\Pi^1_2$-$\mathsf{CA}_0)$ is equivalent to $\forall
n.(\Sigma^0_2)_n$-$\mathsf{Det}$. We also restate results by Montalb\'an and
Shore to show that $\Pi^1_3$-$\mathsf{Ref}(\mathsf{Z}_2)$ is equivalent to
$\forall n.(\Sigma^0_3)_n$-$\mathsf{Det}$ over $\mathsf{ACA}_0$.
