Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S2(A)=S2. Building on this, we strengthen the Kaemper-AFK Theorem, namely, we prove that if NP subseteq (NP intersect coNP)/poly then the polynomial hierarchy collapses to S2(NP intersect coNP). We also strengthen Yap''s Theorem, namely, we prove that if NP subseteq coNP/poly then the polynomial hierarchy collapses to S2(NP). Under the same assumptions, the best previously known collapses were to ZPP(NP) and ZPP(NP(NP)) respectively ([KW98,BCK+94], building on [KL80,AFK89,Kaem91,Yap83]). It is known that S2 subseteq ZPP(NP) [Cai01]. That result and its relativized version show that our new collapses indeed improve the previously known results. Since the Kaemper-AFK Theorem and Yap''s Theorem are used in the literature as bridges in a variety of results---ranging from the study of unique solutions to issues of approximation---our results implicitly strengthen all those results.
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- Competing Provers Yield Improved Karp-Lipton Collapse Results
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