On the Power of Randomized Reductions and the Checkability of SAT

We prove new results regarding the complexity of various complexity classes under randomized oracle reductions. We first prove that BPP
^PSZK
⊆ AM ∩ coAM, where PSZK is the class of promise problems having statistical zero knowledge proofs. This strengthens the previously known facts that PSZK is closed under NC
¹
truth-table reductions (Sahai and Vadhan, J. ACM '03) and that P
^PSZK
⊆ AM ∩ coAM (Vadhan, personal communication). Our proof relies on showing that a certain class of real-valued functions that we call ℝ-TUAM can be approximated using an AM protocol. Then we investigate the power of randomized oracle reductions with relation to the notion of instance 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 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 non-uniform tester, we also show that worst-case to average-case randomized oracle reduction for any relation (or language) R E NP implies that R has a nonuniform instance checker. These results hold even for adaptive randomized oracle reductions.