Proof-theoretic strengths of weak theories for positive inductive definitions

In this paper the lightface ${\mathrm{\Pi }}_1^1$-Comprehension axiom is shown to be proof-theoretically strong even over $RC{A}_0^{\ast }$, and we calibrate the proof-theoretic ordinals of weak fragments of the theory $I{D}_1$ of positive inductive definitions over natural numbers. Conjunctions of negative and positive formulas in the transfinite induction axiom of $I{D}_1$ are shown to be weak, and disjunctions are strong. Thus we draw a boundary line between predicatively reducible and impredicative fragments of $I{D}_1$.