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 study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The Blum-Shub-Smale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model ” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer N, decide whether N> 0. • In the Blum-Shub-Smale model, polynomial time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. • The Generic Task of Numerical Computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.

We prove that constant depth circuits of size n over the basis AND, OR, PARITY are no more powerful than circuits of this size with depth four. Similar techniques are used to obtain several other depth reduction theorems; in particular, we show every set in AC can be recognized by a family of depth three . The size bound n is optimal when considering depth reduction over AND, OR, and PARITY. Most of our results hold both for the uniform and the nonuniform case.

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.

Oracle constructions have long been used to provide evidence that certain questions in complexity theory cannot be resolved using the usual techniques of simulation and diagonalization. However, the existence of nonrelativizing proof techniques seems to call this practice into question. This paper reviews the status of nonrelativizing proof techniques, and argues that many oracle constructions still yield valuable information about problems in complexity theory.

We introduce a new method to separate counting classes of a special type by oracles. Among the classes, for which this method is applicable, are NP, coNP, US (also called 1-NP), \PhiP and all other MOD-classes, PP and C= P, classes of Boolean Hierarchies over the named classes, classes of finite acceptance type, and many more. As an important special case, we completely characterize all relativizable inclusions between classes NP(k) from the Boolean Hierarchy over NP and other classes defined by what we will call bounded counting. Supported by the Alexander von Humboldt foundation under a Feodor Lynen scholarship, while the author held a visiting position at the University of California at Santa Barbara. y Supported in part by DFG grant no. Wa 847/3-1 and NSF grant no. CCR-93-02057. 1 Introduction A probabilistic Turing machine is essentially a nondeterministic Turing machine M , where we view M 's nondeterministic choices as outcomes of a fair coin toss. Thus, every input is...

We investigate alternative notions of approximation for problems inside P (deter-ministic polynomial time), and show that even a slightly nontrivial information about a problem may be as hard to obtain as the solution itself. For example, we prove that if one could eliminate even a single possibility for the value of an arithmetic circuit on a given input, then this would imply that the class P has fast (polygarithmic time) parallel solutions. In other words, this would constitute a proof that there are no inherently sequential problems in P, which is quite unlikely. The result is robust with respect to eliminating procedures that are allowed to err (by excluding the correct value) with small probability. We also show that several fundamental linear algebra problems are hard in this sense. It turns out that it is as hard to substantially reduce the number of possible values for the determinant and rank as to compute them exactly. Finally, we show that (in some precise sense) randomness can be nontrivially substituted for nondeterminism in space. Although it is believed that randomness does not give more than a constant factor advantage in space over determinism, it is not even known whether it is no more powerful than nondeterminism. We will show that the latter is true for a restricted version of probabilistic logspace, where the error is potentially larger than what can be achieved by amplification.

We give two time- and space-efficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that every language solvable by a bounded-error quantum algorithm running in time t and space s is also solvable by an unbounded-error randomized algorithm running in time O(t · log t) and space O(s + log t), as well as by a bounded-error quantum algorithm restricted to use an arbitrary universal set and running in time O(t · polylog t) and space O(s + log t), provided the universal set is closed under adjoint. We also develop a quantum model that is particularly suitable for the study of general computations with simultaneous time and space bounds. As an application of our randomized simulation, we obtain the first nontrivial lower bound for general quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are the problems of determining the truth value of a given Boolean formula whose variables are fully quantified by one or two majority quantifiers, respectively. We prove that for every real d and every positive real δ there exists a real c> 1 such that either • MajMajSAT does not have a bounded-error quantum algorithm running in time O(n c), or • MajSAT does not have a bounded-error quantum algorithm running in time O(n d) and space O(n 1−δ). In particular, MajMajSAT does not have a bounded-error quantum algorithm running in time O(n 1+o(1) ) and space O(n 1−δ) for any δ> 0. Our lower bounds hold for any reasonable uniform model of quantum computation, in particular for the model we develop. 1

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In this thesis, we study small, yet important, circuit complexity classes within NC 1, such as ACC 0 and TC 0. We also investigate the power of a closely related problem called Iterated Matrix Multiplication and its implications in low levels of algebraic complexity theory. More concretely, • We show that extremely modest-sounding lower bounds for certain problems can lead to non-trivial derandomization results. – If the word problem over S5 requires constant-depth threshold circuits of size n1+ɛ for some ɛ> 0, then any language accepted by uniform polynomial-size probabilistic threshold circuits can be solved in subexponential time (and more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size.) – If there are no constant-depth arithmetic circuits of size n1+ɛ for the problem of multiplying a sequence of n 3-by-3 matrices, then for every constant d, black-box identity testing for depth-d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC circuits of subexponential size).