Boolean function monotonicity testing requires (almost) n1/2n^{1/2} non-adaptive queries

We prove a lower bound of $\Omega(n^{1/2 - c})$, for all $c>0$, on the query complexity of (two-sided error) non-adaptive algorithms for testing whether an $n$-variable Boolean function is monotone versus constant-far from monotone. This improves a $\tilde{\Omega}(n^{1/5})$ lower bound for the same problem that was recently given in [CST14] and is very close to $\Omega(n^{1/2})$, which we conjecture is the optimal lower bound for this model

Symmetries in algebraic Property Testing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 94-100).Modern computational tasks often involve large amounts of data, and efficiency is a very desirable feature of such algorithms. Local algorithms are especially attractive, since they can imply global properties by only inspecting a small window into the data. In Property Testing, a local algorithm should perform the task of distinguishing objects satisfying a given property from objects that require many modifications in order to satisfy the property. A special place in Property Testing is held by algebraic properties: they are some of the first properties to be tested, and have been heavily used in the PCP and LTC literature. We focus on conditions under which algebraic properties are testable, following the general goal of providing a more unified treatment of these properties. In particular, we explore the notion of symmetry in relation to testing, a direction initiated by Kaufman and Sudan. We investigate the interplay between local testing, symmetry and dual structure in linear codes, by showing both positive and negative results. On the negative side, we exhibit a counterexample to a conjecture proposed by Alon, Kaufman, Krivelevich, Litsyn, and Ron aimed at providing general sufficient conditions for testing. We show that a single codeword of small weight in the dual family together with the property of being invariant under a 2-transitive group of permutations do not necessarily imply testing. On the positive side, we exhibit a large class of codes whose duals possess a strong structural property ('the single orbit property'). Namely, they can be specified by a single codeword of small weight and the group of invariances of the code. Hence we show that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing. These findings also reveal a new characterization of the extensively studied BCH codes. As a by-product, we obtain a more explicit description of structured tests for the special family of BCH codes of design distance 5.by Elena Grigorescu.Ph.D

Information theory in property testing and monotonicity testing in higher dimension

Abstract. In general property testing, we are given oracle access to a function f, and we wish to randomly test if the function satisfies a given property P, or it is ε-far from having that property. In a more general setting, the domain on which the function is defined is equipped with a probability distribution, which assigns different weight to different elements in the distance function. This paper relates the complexity of testing the monotonicity of a function over the d-dimensional cube to the Shannon entropy of the underlying distribution. We provide an improved upper bound on the property tester query complexity and we finetune the exponential dependence on the dimension d.

Limit structures and property testing

In the thesis, we study properties of large combinatorial objects. We analyze these objects from two different points of view. The first aspect is analytic - we study properties of limit objects of combinatorial structures. We investigate when graphons (limits of graphs) and permutons (limits of permutations) are finitely forcible, i.e., when they are uniquely determined by finitely many densities of their substructures. We give examples of families of permutons that are finitely forcible but the associated graphons are not and we disprove a conjecture of Lovasz and Szegedy on the dimension of the space of typical vertices of a finitely forcible graphon. In particular, we show that there exists a finitely forcible graphon W such that the topological spaces T(W) and T(W) have infinite Lebesgue covering dimension. We also study the dependence between densities of substructures. We prove a permutation analogue of the classical theorem of Erdos, Lovasz and Spencer on the densities of connected subgraphs in large graphs. The second aspect of large combinatorial objects we concentrate on is algorithmic|we study property testing and parameter testing. We show that there exists a bounded testable permutation parameter that is not finitely forcible and that every hereditary permutation property is testable (in constant time) with respect to the Kendall's tau distance, resolving a conjecture of Kohayakawa