Property testing: theory and applications

Property testers are algorithms that distinguish inputs with a given property from those that are far from satisfying the property. Far means that many characters of the input must be changed before the property arises in it. Property testing was introduced by Rubinfeld and Sudan in the context of linearity testing and first studied in a variety of other contexts by Goldreich, Goldwasser and Ron. The query complexity of a property tester is the number of input characters it reads. This thesis is a detailed investigation of properties that are and are not testable with sublinear query complexity. We begin by characterizing properties of strings over the binary alphabet in terms of their formula complexity. Every such property can be represented by a CNF formula. We show that properties of n-bit strings defined by 2CNF formulas are testable with O( √ n) queries, whereas there are 3CNF formulas for which the corresponding properties require Ω(n) queries, even for adaptive tests. We show that testing properties defined by 2CNF formulas is equivalent, with respect to the number of required queries, to several other function and graph testing problems. These problems include: testing whether Boolean functions over general partial orders are close to monotone, testing whether a set of vertices is close to one that is a vertex cover of a specific graph, and testing whether a set of vertices is close to a clique. Testing properties that are defined in terms of monotonicity has been extensively investigated in the context of the monotonicity of a sequence of integers and the monotonicity of a function over the m-dimensional hypercube {1, . . . , a}m. We study the query complexity of monotonicity testing of both Boolean and integer functions over general partial orders. We show upper and lower bounds for the general problem and for specific partial orders. A few of our intermediate results are of independent interest. 1. If strings with a property form a vector space, adaptive 2-sided error tests for the property have no more power than non-adaptive 1-sided error tests. 2. Random LDPC codes with linear distance and constant rate are not locally testable. 3. There exist graphs with many edge-disjoint induced matchings of linear size. In the final part of the thesis, we initiate an investigation of property testing as applied to images. We study visual properties of discretized images represented by n× n matrices of binary pixel values. We obtain algorithms with the number of queries independent of n for several basic properties: being a half-plane, connectedness and convexity. Thesis Supervisor: Michael Sipser Title: Professor of Mathematics