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

Some closure features of locally testable affine-invariant properties

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 31-32).We prove that the class of locally testable affine-invariant properties is closed under sums, intersections and "lifts". The sum and intersection are two natural operations on linear spaces of functions, where the sum of two properties is simply their sum as a vector space. The "lift" is a less well-studied property, which creates some interesting affine-invariant properties over large domains, from properties over smaller domains. Previously such results were known for "single-orbit characterized" affine-invariant properties, which are known to be a subclass of locally testable ones, and are potentially a strict subclass. The fact that the intersection of locally-testable affine-invariant properties are locally testable could have been derived from previously known general results on closure of property testing under set-theoretic operations, but was not explicitly observed before. The closure under sum and lifts is implied by an affirmative answer to a central question attempting to characterize locally testable affine-invariant properties, but the status of that question remains wide open. Affine-invariant properties are clean abstractions of commonly studied, and extensively used, algebraic properties such linearity and low-degree. Thus far it is not known what makes affine-invariant properties locally testable - no characterizations are known, and till this work it was not clear if they satisfied any closure properties. This work shows that the class of locally testable affine-invariant properties are closed under some very natural operations. Our techniques use ones previously developed for the study of "single-orbit characterized" properties, but manage to apply them to the potentially more general class of all locally testable ones via a simple connection that may be of broad interest in the study of affine-invariant properties.by Alan Xinyu Guo.S.M

Local Decoding and Testing of Polynomials over Grids

The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that n-variate polynomials of total degree at most d over grids, i.e. sets of the form A_1 times A_2 times cdots times A_n, form error-correcting codes (of distance at least 2^{-d} provided min_i{|A_i|}geq 2). In this work we explore their local decodability and local testability. While these aspects have been studied extensively when A_1 = cdots = A_n = F_q are the same finite field, the setting when A_i\u27s are not the full field does not seem to have been explored before. In this work we focus on the case A_i = {0,1} for every i. We show that for every field (finite or otherwise) there is a test whose query complexity depends only on the degree (and not on the number of variables). In contrast we show that decodability is possible over fields of positive characteristic (with query complexity growing with the degree of the polynomial and the characteristic), but not over the reals, where the query complexity must grow with $n$. As a consequence we get a natural example of a code (one with a transitive group of symmetries) that is locally testable but not locally decodable. Classical results on local decoding and testing of polynomials have relied on the 2-transitive symmetries of the space of low-degree polynomials (under affine transformations). Grids do not possess this symmetry: So we introduce some new techniques to overcome this handicap and in particular use the hypercontractivity of the (constant weight) noise operator on the Hamming cube

Formal Firewall Conformance Testing: An Application of Test and Proof Techniques

Firewalls are an important means to secure critical ICT infrastructures. As configurable off-the-shelf prod\-ucts, the effectiveness of a firewall crucially depends on both the correctness of the implementation itself as well as the correct configuration. While testing the implementation can be done once by the manufacturer, the configuration needs to be tested for each application individually. This is particularly challenging as the configuration, implementing a firewall policy, is inherently complex, hard to understand, administrated by different stakeholders and thus difficult to validate. This paper presents a formal model of both stateless and stateful firewalls (packet filters), including NAT, to which a specification-based conformance test case gen\-eration approach is applied. Furthermore, a verified optimisation technique for this approach is presented: starting from a formal model for stateless firewalls, a collection of semantics-preserving policy transformation rules and an algorithm that optimizes the specification with respect of the number of test cases required for path coverage of the model are derived. We extend an existing approach that integrates verification and testing, that is, tests and proofs to support conformance testing of network policies. The presented approach is supported by a test framework that allows to test actual firewalls using the test cases generated on the basis of the formal model. Finally, a report on several larger case studies is presented

A process theoretic triptych: two roads to the emergence of classicality, reconstructing quantum theory from diagrams, looking for post-quantum theories

This thesis asks what can be learnt about quantum theory by investigating it from the perspective of process theories. This is based on the diagrammatic compositional structure of Categorical Quantum Mechanics, leading to a very general framework to describe alternate theories of nature. In particular this framework is well suited to understanding the relationship between different theories. In the first part of the thesis we investigate the relationship between quantum and classical theory, showing how an abstract description of decoherence in terms of leaking information leads to emergent classicality. Moreover, this process theoretic notion of a `leak' allows us to capture the distinction between quantum and classical theory in a particularly simple way, highlighting how the quantum and classical worlds diverge. In the second part we look at how to reconstruct quantum theory from diagrammatic principles showing that i) the existence of a classical interface with the theory plus ii) standard notions of composition and iii) a time symmetric form of purification are sufficient to reconstruct the standard quantum formalism. Thereby demonstrating that the standard tools of Categorical Quantum Mechanics come very close to capturing the essence of quantum theory. In the third part we abstract the key features of this emergence of classicality to define a notion of `hyperdecoherence' whereby some post-quantum theory might appear quantum due to an uncontrolled interaction with an environment. We prove a no-go theorem which states that any operational post-quantum theory must violate the purification principle, and so must radically challenge our understanding of how information behaves. To summarise, we use the framework of process theories to gain a better understanding of quantum theory, its sub-theories, and its potential super-theories.Open Acces