An errorless circuit for a boolean function is one that outputs the correct answer or “don’t know ” on each input (and never outputs the wrong answer). The goal of errorless hardness amplification is to show that if f has no size s errorless circuit that outputs “don’t know ” on at most a δ fraction of inputs, then some f ′ related to f has no size s ′ errorless circuit that outputs “don’t know ” on at most a 1 − ǫ fraction of inputs. Thus the hardness is “amplified” from δ to 1 −ǫ. Unfortunately, this amplification comes at the cost of a loss in circuit size. This is because such results are proven by reductions which show that any size s ′ errorless circuit for f ′ that outputs “don’t know ” on at most a 1 − ǫ fraction of inputs could be used to construct a size s errorless circuit for f that outputs “don’t know ” on at most a δ fraction of inputs. If the reduction makes q queries to the hypothesized errorless circuit for f ′, then plugging in a size s ′ circuit yields a circuit of size ≥ qs ′, and thus we must have s ′ ≤ s/q. Hence it is desirable to keep the query complexity to a minimum. The first results on errorless hardness amplification were obtained by Bogdanov and Safra. They achieved query complexity O ( ( 1 1 δ log ǫ)2 ·
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