Abstract
We show that if a complexity classC is closed downward under polynomial-time majority truth-table reductions (≤ pmtt ), then practically every other “polynomial” closure property it enjoys is inherited by the corresponding bounded two-sided error class BP[C]. For instance, the Arthur-Merlin game class AM [B1] enjoys practically every closure property of NP. Our main lemma shows that, for any relativizable classD which meets two fairly transparent technical conditions, we haveC BP[C] \( \subseteq \)BP[D C]. Among our applications, we simplify the proof by Toda [Tol], [To2] that the polynomial hierarchy PH is contained in BP[⊕P]. We also show that relative to a random oracleR, PHR is properly contained in ⊕PR.
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The first author was supported in part by NSF Grant CCR-9011248 and the second author was supported in part by NSF Grant CCR-89011154.
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Regan, K.W., Royer, J.S. On closure properties of bounded two-sided error complexity classes. Math. Systems Theory 28, 229–243 (1995). https://doi.org/10.1007/BF01303057
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DOI: https://doi.org/10.1007/BF01303057