Relative leftmost path principles and omega-model reflections of transfinite inductions

In this paper, we give characterizations of Towsner’s relative leftmost path principles in terms of omega-model reflections of transfinite inductions. In particular, we show that the omega-model reflection of ${\mathrm{\Pi }}_{n+1}^1$ transfinite induction is equivalent to the ${\mathrm{\Sigma }}_n^0$ relative leftmost path principle over ${\mathsf{R}\mathsf{C}\mathsf{A}}_0$ for $n>1$. As a consequence, we have that ${\mathrm{\Sigma }}_{n+1}^0\mathsf{L}\mathsf{P}\mathsf{P}$ is strictly stronger than ${\mathrm{\Sigma }}_n^0\mathsf{L}\mathsf{P}\mathsf{P}$.