The closure of complexity classes is a delicate question and the answer varies depending on the type of reduction considered. The closure of most classes under many-to-one (Karp) reductions is clear, but the question becomes complicated when oracle (Cook) reductions are allowed, and even more so when the oracle reductions are allowed to be randomized.

We first prove that $BPP^{SZK} \subseteq AM \cap coAM$, strengthening the previously known facts that $SZK$ is closed under $NC^1$ truth-table reductions (Sahai and Vadhan, J. ACM '03) and that $P^{SZK} \subseteq AM \cap coAM$ (Vadhan, as cited in Goldreich, ECCC '05). Our proof relies on showing that a certain class of real-valued functions that we call $RTFAM$ can be computed using an $AM$ protocol.

Then we investigate the power of randomized oracle reductions with relation to the notion of program checking (Blum and Kannan, J. ACM '95). We observe that a theorem of Beigel implies that if any problem in $TFNP$ such as Nash equilibrium is $NP$-hard under randomized oracle reductions, then $SAT$ is checkable.

We also observe that Beigel's theorem can be extended to an average-case setting by relating checking to the notion of program \emph{testing} (Blum et al., JCSS '93). From this, we derive that if one-way functions can be based on $NP$-hardness via a randomized oracle reduction, then $SAT$ is checkable. By showing that $NP$ has a \emph{non-uniform} tester, we also show that if $SAT$ has a worst-case to average-case randomized oracle reduction, then $SAT$ is checkable with a non-uniform instance checker. These results hold even for adaptive randomized oracle reductions.

Acknowledgements added.

The closure of complexity classes is a delicate question and the answer varies depending on the type of reduction considered. The closure of most classes under many-to-one (Karp) reductions is clear, but the question becomes complicated when oracle (Cook) reductions are allowed, and even more so when the oracle reductions are allowed to be randomized.

We first prove that $BPP^{SZK} \subseteq AM \cap coAM$, strengthening the previously known facts that $SZK$ is closed under $NC^1$ truth-table reductions (Sahai and Vadhan, J. ACM '03) and that $P^{SZK} \subseteq AM \cap coAM$ (Vadhan, as cited in Goldreich, ECCC '05). Our proof relies on showing that a certain class of real-valued functions that we call $RTFAM$ can be computed using an $AM$ protocol.

Then we investigate the power of randomized oracle reductions with relation to the notion of program checking (Blum and Kannan, J. ACM '95). We observe that a theorem of Beigel implies that if any problem in $TFNP$ such as Nash equilibrium is $NP$-hard under randomized oracle reductions, then $SAT$ is checkable.

We also observe that Beigel's theorem can be extended to an average-case setting by relating checking to the notion of program \emph{testing} (Blum et al., JCSS '93). From this, we derive that if one-way functions can be based on $NP$-hardness via a randomized oracle reduction, then $SAT$ is checkable. By showing that $NP$ has a \emph{non-uniform} tester, we also show that if $SAT$ has a worst-case to average-case randomized oracle reduction, then $SAT$ is checkable with a non-uniform instance checker. These results hold even for adaptive randomized oracle reductions.

Typos fixed (in the abstract above and in the proof of Theorem 1.1)

The closure of complexity classes is a elicate question and the answer varies depending on the type of reduction considered. The closure of most classes under many-to-one (Karp) reductions is clear, but the question becomes complicated when oracle (Cook) reductions are allowed, and even more so when the oracle reductions are allowed to be randomized.

We first prove that $BPP^{SZK} \subseteq AM \cap coAM$, strengthening the previously known facts that $SZK$ is closed under $NC^1$ truth-table reductions (Sahai and Vadhan, J. ACM '03) and that $P^{SZK} \subseteq AM \cap coAM$ (Vadhan, as cited in Goldreich, ECCC '05). Our proof relies on showing that a certain class of real-valued functions that we call $RTFAM$ can be computed using an $AM$ protocol.

Then we investigate the power of randomized oracle reductions with relation to the notion of program checking (Blum and Kannan, J. ACM '95). We observe that a theorem of Beigel implies that if any problem in $TFNP$ such as Nash equilibrium is $NP$-hard under randomized oracle reductions, then $SAT$ is checkable.

We also observe that Beigel's theorem can be extended to an average-case setting by relating checking to the notion of program \emph{testing} (Blum et al., JCSS '93). From this, we derive that if one-way functions can be based on $NP$-hardness via a randomized oracle reduction, then $SAT$ is checkable. By showing that $NP$ has a \emph{non-uniform} tester, we also show that if $SAT$ has a worst-case to average-case randomized oracle reduction, then $SAT$ is checkable with a non-uniform instance checker. These results hold even for adaptive randomized oracle reductions.