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In search of an easy witness: Exponential time vs. probabilistic polynomial time. (English) Zbl 1059.68047
Summary: Restricting the search space \(\{0,1\}^n\) to the set of truth tables of “easy” Boolean functions on \(\log n\) variables, as well as using some known hardness-randomness tradeoffs, we establish a number of results relating the complexity of exponential-time and probabilistic polynomial-time complexity classes. In particular, we show that NEXP \(\subset\) P/poly \(\Leftrightarrow\) NEXT = MA; this can be interpreted as saying that no derandomization of MA (and, hence, of promise-BBP) is possible unless NEXT contains a hard Boolean function. We also prove several downward closure results for ZPP, RP, BPP, and MA; e.g., we show \(\text{EXP}=\text{BPP} \Leftrightarrow\text{EE}=\text{BPE}\), where EE is the double-exponential time class and BPE is the exponential-time analogue of BPP.

MSC:
68Q25 Analysis of algorithms and problem complexity
Keywords:
search space
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