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Volume 11 (2015) Article 1 pp. 1-34
The Complexity of Deciding Statistical Properties of Samplable Distributions
Received: December 10, 2013
Revised: February 13, 2015
Published: March 12, 2015
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Keywords: complexity theory, algorithms, complexity classes, completeness, samplable distributions
ACM Classification: F.1.3
AMS Classification: 68Q17, 68Q15

Abstract: [Plain Text Version]

$ \newcommand{\BCeP}{\mathsf{BC}_=\mathsf{P}} \newcommand{\SZK}{\mathsf{SZK}} \newcommand{\CeP}{\mathsf{C}_=\mathsf{P}} $

We consider the problems of deciding whether the joint distribution sampled by a given circuit has certain statistical properties such as being i.i.d., being exchangeable, being pairwise independent, having two coordinates with identical marginals, having two uncorrelated coordinates, and many other variants. We give a proof that simultaneously shows all these problems are $\CeP$-complete, by showing that the following promise problem (which is a restriction of all the above problems) is $\CeP$-complete: Given a circuit, distinguish the case where the output distribution is uniform and the case where every pair of coordinates is neither uncorrelated nor identically distributed. This completeness result holds even for samplers that are depth-$3$ circuits.

We also consider circuits that are $d$-local, in the sense that each output bit depends on at most $d$ input bits. We give linear-time algorithms for deciding whether a $2$-local sampler's joint distribution is fully independent, and whether it is exchangeable.

We also show that for general circuits, certain approximation versions of the problems of deciding full independence and exchangeability are $\SZK$-complete.

We also introduce a bounded-error version of $\CeP$, which we call $\BCeP$, and we investigate its structural properties.

An extended abstract of this paper appeared in the Proceedings of the 31st International Symposium on Theoretical Aspects of Computer Science (STACS), 2014.