Competing Provers Yield Improved Karp-Lipton Collapse Results

Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S2 = S2. Building on this, we strengthen the Kämper– AFK Theorem, namely, we prove that if NP ⊆ (NP ∩ coNP)/poly then the polynomial hierarchy collapses to SNP∩coNP 2 . We also strengthen Yap’s Theorem, namely, we prove that if NP ⊆ coNP/poly then the polynomial hierarchy collapses to SNP 2 . Under the same assumptions, the best previously known collapses were to ZPP and ZPP NP respectively ([KW98, BCK+94], building on [KL80, AFK89, Käm91, Yap83]). It is known that S2 ⊆ ZPP NP [Cai01]. That result and its relativized version show that our new collapses indeed improve the previously known results. Since the Kämper– 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.