Final Report Summary - PROPERTY TESTING (Property testing and sublinear algorithms for languages and combinatorial properties) Suppose we need to verify whether a function f:D→{0,1} satisfies a given property, but we might not even be able to read the entire function before making a decision. Instead, we are allowed to read a limited number of values from f, in the most extreme case a constant number independent of the size of D. While an exact answer is not possible under such circumstances, we may still be able to give an approximate one using a probabilistic “intelligent sampling” algorithm. Such algorithms are called property testing algorithms.A particular area in property testing is that of massively parametrized testing. Here there are two parts to a problem, a “setting” part which is known in advance (i.e. the corresponding function is read fully by the algorithm) and with respect to which all answers must be exact, and a function which is not read in full and for which approximations are allowed.This project was concerned with several topics within the field of property testing and related questions (such as that of examining an unknown distribution, where we only have access to samples drawn according to it). This project resolved some questions and opened up some research avenues with new questions, such as the following.Formula satisfaction testing deals with the following problem: A Boolean formula over n variables is given in advance (this is the “setting” part in a massively parametrized model), and we would like to test an assignment to these variables for the property of evaluating to “1” under the formula (the assignment can be thought of as a function f:{1,...,n}→{0,1}). A result of this project is that all formulas with bounded fan-in gates (even general ones) admit a test where the number of reads is a constant that depends only on the approximation parameter.On the other hand, non-Boolean formulas (e.g. formulas over {0,1,2,3} instead of {0,1}) do not necessarily admit such a test.Function isomorphism is the property of one function f:{0,1}n→{0,1} being identical to another function g:{0,1}n→{0,1} after a reordering of the variables. For example, the function f(x,y)=x∧¬y is isomorphic to g(x,y)=y∧¬x. Given a function g that is fully known in advance, one would like to know whether a function f can be tested for isomorphism with g. The stillstanding conjecture is that this is possible exactly for the functions g which can be written as a combination of fully symmetric functions and a only constant number of the values of the coordinates of {0,1}n.Conditional property testing is a new model (developed in this project and independently in a work by Canonne, Ron and Servedio) that lies between traditional function testing and traditional distribution testing. In the traditional distribution testing we can only obtain samples independently drawn according to the distribution. In the conditional model, we are also allowed to ask for samples drawn according to the distribution conditioned over subsets specified by the algorithm. This brings “algorithmic” aspects to distribution testing. The results obtained here provide both upper and lower bounds with regards to natural distribution testing tasks under this model.Some possible applications were also investigated. The main result here deals with the possibility to speed up database sorting operations in cases where the original database file satisfies a commonly occurring property. The property testing aspect comes in when trying to assess whether a given database file indeed has the property that allows for a faster sort.
