Query Complexity in Errorless Hardness Amplification

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 $${\delta}$$ fraction of inputs, then some f′ related to f has no size s′ errorless circuit that outputs “don’t know” on at most a $${1-\epsilon}$$ fraction of inputs. Thus, the hardness is “amplified” from $${\delta}$$ to $${1-\epsilon}$$ . Unfortunately, this amplification comes at the cost of a loss in circuit size. If the reduction makes q queries to the hypothesized errorless circuit for f′, then we obtain a result with 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 $${O\big((\frac{1}{\delta} \log\frac{1}{\epsilon})^2\cdot\frac{1}{\epsilon} \log\frac{1}{\delta} \big)}$$ when f′ is the XOR of several independent copies of f. We improve the query complexity (and hence the loss in circuit size) to $${O\big(\frac{1}{\epsilon} \log\frac{1}{\delta} \big)}$$ , which is optimal up to constant factors for non-adaptive black-box errorless hardness amplification. Bogdanov and Safra also proved a result that allows for errorless hardness amplification within NP. They achieved query complexity $${O\big(k^3\cdot\frac{1}{\epsilon^2} \log\frac{1}{\delta} \big)}$$ when f′ consists of any monotone function applied to the outputs of k independent copies of f, provided the monotone function satisfies a certain combinatorial property parameterized by $${\delta}$$ and $${\epsilon}$$ . We improve the query complexity to $${O\big(\frac{k}{t} \cdot\frac{1}{\epsilon} \log\frac{1}{\delta} \big)}$$ , where $${t\geq 1}$$ is a certain parameter of the monotone function. Using the best applicable monotone functions (which were constructed by Bogdanov and Safra), our result yields a query complexity of $${\widetilde{O} \big(\frac{1}{\epsilon^3} \cdot\frac{1}{\delta} \big)}$$ for balanced functions f, improving on the $${\widetilde{O} \big(\frac{1}{\epsilon^8} \cdot\frac{1}{\delta^6} \big)}$$ query complexity that follows from the Bogdanov–Safra result. As a side result, we prove a lower bound on the advice complexity of black-box reductions for errorless hardness amplification.