Pseudorandomness from Shrinkage

One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer a quantitative loss in parameters, and hence do not give nontrivial implications for models where we don’t know super-polynomial lower bounds but do know lower bounds of a fixed polynomial. We show that when such lower bounds are proved using random restrictions, we can construct PRGs which are essentially best possible without in turn improving the lower bounds. More specifically, say that a circuit family has shrinkage exponent Γ if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of pΓ+o(1). Our PRG uses a seed of length s1/(Γ+1)+o(1) to fool circuits in the family of size s. By using this generic construction, we get PRGs with polynomially small error for the following classes of circuits of size s and with the following seed lengths: 1. For de Morgan formulas, seed length s1/3+o(1); 2. For formulas over an arbitrary basis, seed length s1/2+o(1); 3. For read-once de Morgan formulas, seed length s.234...; 4. For branching programs of size s, seed length s1/2+o(1). The previous best PRGs known for these classes used seeds of length bigger than n/2 to output n bits, and worked only when the size s = O(n) [BPW11]

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 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-ϵ fraction of inputs. Thus, the hardness is “amplified” from δ to 1-ϵ. 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 (Formula presented.) when f′ is the XOR of several independent copies of f. We improve the query complexity (and hence the loss in circuit size) to (Formula presented.), 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 (Formula presented.) 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 δ and ϵ. We improve the query complexity to (Formula presented.), where t≥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 (Formula presented.) for balanced functions f, improving on the (Formula presented.) 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

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 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 ·