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Bounded-depth circuits cannot sample good codes
, 2010
"... We study a variant of the classical circuit-lower-bound problems: proving lower bounds for sampling distributions given random bits. We prove a lower bound of 1 − 1/n Ω(1) on the statistical distance between (i) the output distribution of any small constant-depth (a.k.a. AC 0) circuit f: {0, 1} poly ..."
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We study a variant of the classical circuit-lower-bound problems: proving lower bounds for sampling distributions given random bits. We prove a lower bound of 1 − 1/n Ω(1) on the statistical distance between (i) the output distribution of any small constant-depth (a.k.a. AC 0) circuit f: {0, 1} poly(n) → {0, 1} n, and (ii) the uniform distribution over any code C ⊆ {0, 1} n that is “good”, i.e. has relative distance and rate both Ω(1). This seems to be the first lower bound of this kind. We give two simple applications of this result: (1) any data structure for storing codewords of a good code C ⊆ {0, 1} n requires redundancy Ω(log n), if each bit of the codeword can be retrieved by a small AC 0 circuit; (2) for some choice of the underlying combinatorial designs, the output distribution of Nisan’s pseudorandom generator against AC 0 circuits of depth d cannot be sampled by small AC 0 circuits of depth less than d. 1
S.: Randomness condensers for efficiently samplable, seeddependent sources, full version of this paper. Available from authors’ websites
"... We initiate a study of randomness condensers for sources that are efficiently samplable but may depend on the seed of the condenser. That is, we seek functions Cond: {0, 1} n × {0, 1} d → {0, 1} m such that if we choose a random seed S ← {0, 1} d, and a source X = A(S) is generated by a randomized c ..."
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We initiate a study of randomness condensers for sources that are efficiently samplable but may depend on the seed of the condenser. That is, we seek functions Cond: {0, 1} n × {0, 1} d → {0, 1} m such that if we choose a random seed S ← {0, 1} d, and a source X = A(S) is generated by a randomized circuit A of size t such that X has min-entropy at least k given S, then Cond(X; S) should have min-entropy at least some k ′ given S. The distinction from the standard notion of randomness condensers is that the source X may be correlated with the seed S (but is restricted to be efficiently samplable). Randomness extractors of this type (corresponding to the special case where k ′ = m) have been implicitly studied in the past (by Trevisan and Vadhan, FOCS ‘00). We show that: • Unlike extractors, we can have randomness condensers for samplable, seed-dependent sources whose computational complexity is smaller than the size t of the adversarial sampling algorithm A. Indeed, we show that sufficiently strong collision-resistant hash functions are seed-dependent condensers that produce outputs with min-entropy k ′ = m − O(log t), i.e. logarithmic entropy deficiency.
Large deviation bounds for decision trees and sampling lower bounds for AC0-circuits
- In Proceedings of the 53rd IEEE Symposium on Foundations of Computer Science
, 2012
"... There has been considerable interest lately in the complexity of distributions. Re-cently, Lovett and Viola (CCC 2011) showed that the statistical distance between a uniform distribution over a good code, and any distribution which can be efficiently sampled by a small bounded-depth AC0 circuit, is ..."
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There has been considerable interest lately in the complexity of distributions. Re-cently, Lovett and Viola (CCC 2011) showed that the statistical distance between a uniform distribution over a good code, and any distribution which can be efficiently sampled by a small bounded-depth AC0 circuit, is inverse-polynomially close to one. That is, such distributions are very far from each other. We strengthen their result, and show that the distance is in fact exponentially close to one. This allows us to strengthen the parameters in their application for data structure lower bounds for succinct data structures for codes. From a technical point of view, we develop new large deviation bounds for functions computed by small depth decision trees, which we then apply to obtain bounds for AC0 circuits via the switching lemma. We show that if such functions are Lipschitz on average in a certain sense, then they are in fact Lipschitz almost everywhere. This type of result falls into the extensive line of research which studies large deviation bounds for the sum of random variables, where while not independent, exhibit large deviation bounds similar to these obtained by independent random variables. 1
The Complexity of Estimating Min-Entropy
, 2012
"... Goldreich, Sahai, and Vadhan (CRYPTO 1999) proved that the promise problem for estimating the Shannon entropy of a distribution sampled by a given circuit is NISZK-complete. We consider the analogous problem for estimating the min-entropy and prove that it is SBPcomplete, even when restricted to 3-l ..."
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Goldreich, Sahai, and Vadhan (CRYPTO 1999) proved that the promise problem for estimating the Shannon entropy of a distribution sampled by a given circuit is NISZK-complete. We consider the analogous problem for estimating the min-entropy and prove that it is SBPcomplete, even when restricted to 3-local samplers. For logarithmic-space samplers, we observe that this problem is NP-complete by a result of Lyngsø and Pedersen on hidden Markov models (JCSS 2002). 1
Extractors for Turing-machine sources
, 2012
"... We obtain the first deterministic randomness extractors for n-bit sources with minentropy ≥ n 1−α generated (or sampled) by single-tape Turing machines running in time n 2−16α, for all sufficiently small α> 0. We also show that such machines cannot sample a uniform n-bit input to the Inner Produc ..."
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We obtain the first deterministic randomness extractors for n-bit sources with minentropy ≥ n 1−α generated (or sampled) by single-tape Turing machines running in time n 2−16α, for all sufficiently small α> 0. We also show that such machines cannot sample a uniform n-bit input to the Inner Product function together with the output. The proofs combine a variant of the crossing-sequence technique by Hennie [SWCT 1965] with extractors for block sources, especially those by Chor and Goldreich [SICOMP 1988] and by Kamp, Rao, Vadhan, and Zuckerman [JCSS 2011].
Verifying Proofs in Constant Depth
, 2013
"... In this paper we initiate the study of proof systems where verification of proofs proceeds by NC 0 circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC 0 functions. Our result ..."
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In this paper we initiate the study of proof systems where verification of proofs proceeds by NC 0 circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC 0 functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC 0 proof systems for a variety of languages ranging from regular to NP complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC 0 proof systems. We also show that Majority does not admit NC 0 proof systems. Finally, we present a general construction of NC 0 proof systems for regular languages with strongly connected NFA’s.
Time Hierarchies for Sampling Distributions
, 2012
"... We prove that for every constant k ≥ 2, every polynomial time bound t, and every polynomially small ǫ, there exists a family of distributions on k elements that can be sampled exactly in polynomial time but cannot be sampled within statistical distance 1−1/k−ǫ in time t. Our proof involves reducing ..."
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We prove that for every constant k ≥ 2, every polynomial time bound t, and every polynomially small ǫ, there exists a family of distributions on k elements that can be sampled exactly in polynomial time but cannot be sampled within statistical distance 1−1/k−ǫ in time t. Our proof involves reducing the problem to a communication problem over a certain type of noisy channel. We solve the latter problem by giving a construction of a new type of list-decodable code, for a setting where there is no bound on the number of errors but each error gives more information than an erasure. 1
Randomness Condensers for Efficiently
"... (Article begins on next page) The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. ..."
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(Article begins on next page) The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters.
The Computational Complexity of Randomness
, 2013
"... This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to wh ..."
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This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to whether the output is random or the input is random. Part I concerns settings where the problem’s output is random. A sampling problem associates to each input x a probability distribution D(x), and the goal is to output a sample from D(x) (or at least get statistically close) when given x. Although sampling algorithms are fundamental tools in statistical physics, combinatorial optimization, and cryptography, and algorithms for a wide variety of sampling problems have been discovered, there has been comparatively little research viewing sampling throughthelens ofcomputational complexity. We contribute to the understanding of the power and limitations of efficient sampling by proving a time hierarchy theorem which shows, roughly, that “a little more time gives a lot more power to sampling algorithms.” Part II concerns settings where the algorithm’s output is random. Even when the specificationofacomputational problem involves no randomness, onecanstill consider randomized
SPECIAL ISSUE: APPROX-RANDOM 2012 Extractors for Polynomial Sources over Fields of Constant Order and Small Characteristic ∗
, 2012
"... Abstract: A polynomial source of randomness over Fn q is a random variable X = f (Z) where f is a polynomial map and Z is a random variable distributed uniformly over Fr q for some integer r. The three main parameters of interest associated with a polynomial source are the order q of the field, the ..."
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Abstract: A polynomial source of randomness over Fn q is a random variable X = f (Z) where f is a polynomial map and Z is a random variable distributed uniformly over Fr q for some integer r. The three main parameters of interest associated with a polynomial source are the order q of the field, the (total) degree D of the map f, and the base-q logarithm of the size of the range of f over inputs in Fr q, denoted by k. For simplicity we call X a (q,D,k)-source. Informally, an extractor for (q,D,k)-sources is a function E: Fn q → {0,1} m such that the distribution of the random variable E(X) is close to uniform over {0,1} m for any (q,D,k)source X. Generally speaking, the problem of constructing extractors for such sources becomes harder as q and k decrease and as D increases. A rather large number of recent ∗A conference version of this paper appeared in the Proceedings of RANDOM 2012 [1]. † Supported by funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 240258. ‡ Supported by funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 240258.
