Abstract. We study the complexity of constructing pseudorandom generators (PRGs) from hard functions, focussing on constant-depth circuits. We show that, starting from a function f: {0, 1} l → {0, 1} computable in alternating time O(l) with O(1) alternations that is hard on average (i.e. there is a constant ɛ> 0 such that every circuit of size 2 ɛl fails to compute f on at least a 1/poly(l) fraction of inputs) we can construct a PRG: {0, 1} O(log n) → {0, 1} n computable by DLOGTIMEuniform constant-depth circuits of size polynomial in n. Such a PRG implies BP · AC 0 = AC 0 under DLOGTIME-uniformity. On the negative side, we prove that starting from a worst-case hard function f: {0, 1} l → {0, 1} (i.e. there is a constant ɛ> 0 such that every circuit of size 2 ɛl fails to compute f on some input) for every positive constant δ < 1 there is no black-box construction of a PRG: {0, 1} δn → {0, 1} n computable by constant-depth circuits of size polynomial in n. We also study worst-case hardness amplification, which is the related problem of producing an average-case hard function starting from a worst-case hard one. In particular, we deduce that there is no blackbox worst-case hardness amplification within the polynomial time hierarchy. These negative results are obtained by showing that polynomialsize constant-depth circuits cannot compute good extractors and listdecodable codes.

We give the first nontrivial model-independent time-space tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general random-access nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for log-space uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomial-time hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...

this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equivalent formulations of uniform circuit complexity classes in terms of descriptive complexity classes

We extend the lower bound techniques of [14], to the unbounded-error probabilistic model. A key step in the argument is a generalization of Nepomnjasci's theorem from the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspace-uniform TC 0 circuits for iterated multiplication [9]. Here is an

We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and alternating time, Σ O(1)Time (t). Our main results are the following: 1. We prove that 2 n0.1�-size depth-3 circuits for Approximate Majority on n bits have bottom fan-in Ω(log n). As a corollary we obtain that BPTime (t) �⊆ Σ2Time � o(t 2) � with respect to some oracle. This complements the result that BPTime (t) ⊆ Σ2Time � t 2 · poly log t � with respect to every oracle (Sipser and Gács, STOC ’83; Lautemann, IPL ’83). 2. We prove that Approximate Majority is computable by uniform polynomial-size circuits of depth 3. Prior to our work, the only known polynomial-size depth-3 circuits for Approximate Majority were non-uniform (Ajtai, Ann. Pure Appl. Logic ’83). We also prove that BPTime (t) ⊆ Σ3Time (t · poly log t). This complements our results in (1). 3. We prove new lower bounds for solving QSAT 3 ∈ Σ3Time (n · poly log n) on probabilistic computational models. In particular, we prove that solving QSAT 3 requires time n 1+Ω(1) on Turing machines with a random-access input tape and a sequential-access work tape that is initialized with random bits. No lower bound was previously known on this model (for a function computable in linear space). ∗ Author supported by NSF grant CCR-0324906. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the

We study the complexity of building pseudorandom generators (PRGs) with logarithmic seed length from hard functions. We show that, starting from a function f: {0, 1} l → {0, 1} that is mildly hard on average, i.e. every circuit of size 2 Ω(l) fails to compute f on at least a 1/poly(l) fraction of inputs, we can build a PRG: {0, 1} O(log n) → {0, 1} n computable in ATIME(O(1), log n) = alternating time O(log n) with O(1) alternations. Such a PRG implies BP · AC0 = AC0 under DLOGTIME-uniformity. On the negative side, we prove a tight lower bound on black-box PRG constructions that are based on worst-case hard functions. We also prove a tight lower bound on blackbox worst-case hardness amplification, which is the problem of producing an average-case hard function starting from a worst-case hard one. These lower bounds are obtained by showing that constant depth circuits cannot compute extractors and list-decodable codes.

For circuit classes R, the fundamental computational problem Min-R asks for the minimum R-size of a Boolean function presented as a truth table. Prominent examples of this problem include Min-DNF, which asks whether a given Boolean function presented as a truth table has a k-term DNF, and Min-Circuit (also called MCSP), which asks whether a Boolean function presented as a truth table has a size k Boolean circuit. We present a new reduction proving that Min-DNF is NP-complete. It is significantly simpler than the known reduction of Masek [30], which is from Circuit-SAT. We then give a more complex reduction, yielding the result that Min-DNF cannot be approximated to within a factor smaller than (logN) γ, for some constant γ> 0, assuming that NP is not contained in quasipolynomial time. The standard greedy algorithm for Set Cover is often used in practice to approximate Min-DNF. The question of whether Min-DNF can be approximated to within a factor of o(logN) remains open, but we construct an instance of Min-DNF on which the solution produced by the greedy algorithm is Ω(logN) larger than optimal. Finally, we turn to the question of approximating circuit size for slightly more general classes of circuits. DNF formulas are depth two circuits of AND and OR gates. Depth d circuits are denoted by AC0 d. We show that it is hard to approximate the size of AC0 d circuits (for large enough d) under cryptographic assumptions.

We prove several new results regarding the relationship between probabilistic time, BPTime(t), and alternating time, Σ O(1)Time(t). Our main results are the following: 1. We prove that BPTime(t) ⊆ Σ3Time(t · poly log t). Previous results show that BPTime (t) ⊆ Σ2Time � t 2 · log t � (Sipser and Gács, STOC ’83; Lautemann, IPL ’83) and BPTime(t) ⊆ ΣcTime(t) for a large constant c> 3 (Ajtai, Adv. in Comp. Complexity Theory ’93). 2. We prove that BPTime(t) � ⊆ Σ2Time � o(t 2) � with respect to some oracle. This complements our result (1), and shows that the running time of the Sipser-Gács-Lautemann simulation is optimal, up to a log t factor, for relativizing techniques. (All the results in (1) relativize.) This result is obtained as a corollary from a new circuit lower bound for approximate majority: poly(n)-size depth-3 circuits for approximate majority have bottom fan-in Ω(log n). 3. We prove that solving QSAT 3 ∈ Σ3Time(n · poly log n) requires time n 1+Ω(1) on probabilistic Turing machines using space n.9, with random access to input and work tapes, and two-way sequential access to the random-bit tape. This is the first lower bound of the form t = n 1+Ω(1) on a model with random access to the input and two-way access to the random bits. 4. We prove that solving QSAT 3 ∈ Σ3Time(n · poly log n) requires time n 1+Ω(1) on Turing machines with an input tape and a sequential work tape that is initialized with random bits. This is the first lower bound on a probabilistic extension of the off-line Turing machine model with one work tape.

In this work we discuss selected topics on small-depth computation, presenting a few unpublished proofs along the way. The four chapters contain: 1. A unified treatment of the challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over {0, 1}. 2. An unpublished proof that small bounded-depth circuits (AC 0) have exponentially small correlation with the parity function. The proof is due to Klivans and Vadhan; it builds upon and simplifies previous ones. 3. Valiant’s simulation of log-depth linear-size circuits of fan-in 2 by sub-exponential size circuits of depth 3 and unbounded fan-in. To our knowledge, a proof of this result has never appeared in full.

This article is a unified treatment of the state-of-the-art on the fundamental challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over {0, 1}. It discusses long-standing results and recent developments, related proof techniques, and connections with pseudorandom generators. It also suggests several research directions. 1