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\(BPP\) has subexponential time simulations unless \(EXPTIME\) has publishable proofs. (English) Zbl 0802.68054
Summary: We show that \(BPP\) can be simulated in subexponential time for infinitely many input lengths unless exponential time
\(\circ\) collapses to the second level of the polynomial-time hierarchy,
\(\circ\) has polynomial-size circuits, and
\(\circ\) has publishable proofs \((EXPTIME = MA)\).
We also show that \(BPP\) is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, we show \(BPP\) can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages in \(MA\)-\(P\).
The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random self-reducibility via low-degree polynomials. They exhibit an interplay between Boolean circuit simulation, interactive proofs and classical complexity classes. An important feature of this proof is that it does not relativize.
One of the ingredients of our proof is a lemma that states that if \(EXPTIME\) has polynomial size circuits then \(EXPTIME=MA\). This extends previous work by A. Meyer.

MSC:
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
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