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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
search space
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##### References:
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