Abstract:
In the late 80's Blum, Luby, Rubinfeld, Kannan et al. pioneered the theory of self-testing as an alternative way of dealing with the problem of software reliability. Over the last decade this theory played a crucial role in the construction of probabilistically checkable proofs and the derivation of hardness of approximation results. Applications in areas like computer vision, machine learning, and self-correcting programs were also established. In the self-testing problem one is interested in determining (maybe probabilistically) whether a function to which one has oracle access satisfies a given property. We consider the problem of testing algebraic functions and survey over a decade of research in the area. Special emphasis is given to illustrate the scenario where the problem takes place and to the main techniques used in the analysis of tests. A novel aspect of this work is the separation it advocates between the mathematical and algorithmic issues that arise in the theory of self-testing.