Let $\mathsf{KP\omega}_0$ be Kripke-Platek set theory with Infinity but Foundation (or $\in$-Induction) restricted to $\Delta_0$-formulas. $\mathsf{ZF}$ proves $\in$-Induction holds for arbitrary formulas, but it is because we have Full Separation which is unavailable over $\mathsf{KP\omega}_0$, so $\in$-Induction on more complex formulas become a non-trivial statement over $\mathsf{KP\omega}_0$.

I wonder whether the following is known:

Question. Working over $\mathsf{KP\omega}_0$,

1. Does $\Pi_1$-Foundation prove $\Sigma_1$-Foundation or vice versa?
2. What about their consistency strength? Does $\Pi_1$-Foundation have a higher consistency strength than that of $\Sigma_1$-Foundation?

Added. $\Gamma$-Foundation is the following schema: For a formula $\phi(x)$ in $\Gamma$, if $[\forall x\in a \phi(x)]\to \phi(a)$ for all $a$, then $\forall a\phi(a)$.

Let $\mathsf{KP_0\omega}$ be Kripke-Platek set theory with Infinity but Foundation (or $\in$-Induction) restricted to $\Delta_0$-formulas. $\mathsf{ZF}$ proves $\in$-Induction holds for arbitrary formulas, but it is because we have Full Separation which is unavailable over $\mathsf{KP_0\omega}$, so $\in$-Induction on more complex formulas become a non-trivial statement over $\mathsf{KP_0\omega}$.

I wonder whether the following is known:

Question. Working over $\mathsf{KP_0\omega}$,

1. Does $\Pi_1$-Foundation prove $\Sigma_1$-Foundation or vice versa?
2. What about their consistency strength? Does $\Pi_1$-Foundation have a higher consistency strength than that of $\Sigma_1$-Foundation?

Added. $\Gamma$-Foundation is the following schema: For a formula $\phi(x)$ in $\Gamma$, if $[\forall x\in a \phi(x)]\to \phi(a)$ for all $a$, then $\forall a\phi(a)$.

Let $\mathsf{KP\omega}_0$ be Kripke-Platek set theory with Infinity but Foundation (or $\in$-Induction) restricted to $\Delta_0$-formulas. $\mathsf{ZF}$ proves $\in$-Induction holds for arbitrary formulas, but it is because we have Full Separation which is unavailable over $\mathsf{KP\omega}_0$, so $\in$-Induction on more complex formulas become a non-trivial statement over $\mathsf{KP\omega}_0$.

I wonder whether the following is known:

Question. Working over $\mathsf{KP\omega}_0$,

1. Does $\Pi_1$-Foundation prove $\Sigma_1$-Foundation or vice versa?
2. What about their consistency strength? Does $\Pi_1$-Foundation have a higher consistency strength than that of $\Sigma_1$-Foundation?

Let $\mathsf{KP\omega}_0$ be Kripke-Platek set theory with Infinity but Foundation (or $\in$-Induction) restricted to $\Delta_0$-formulas. $\mathsf{ZF}$ proves $\in$-Induction holds for arbitrary formulas, but it is because we have Full Separation which is unavailable over $\mathsf{KP\omega}_0$, so $\in$-Induction on more complex formulas become a non-trivial statement over $\mathsf{KP\omega}_0$.

I wonder whether the following is known:

Question. Working over $\mathsf{KP\omega}_0$,

1. Does $\Pi_1$-Foundation prove $\Sigma_1$-Foundation or vice versa?
2. What about their consistency strength? Does $\Pi_1$-Foundation have a higher consistency strength than that of $\Sigma_1$-Foundation?

Added. $\Gamma$-Foundation is the following schema: For a formula $\phi(x)$ in $\Gamma$, if $[\forall x\in a \phi(x)]\to \phi(a)$ for all $a$, then $\forall a\phi(a)$.