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 NC¹ 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.