Chapter 7: Property testing, PCPPs, and CSPs

In this chapter we study several closely intertwined topics: property testing, probabilistically checkable proofs of proximity (PCPPs), and constraint satisfaction problems (CSPs). All of our work will be centred around the task of testing whether an unknown boolean function is a dictator. We begin by extending the BLR Test to give a $3$-query property testing algorithm for the class of dictator functions. This in turn allows us to give a $3$-query testing algorithm for any property, so long as the right “proof” is provided. We then introduce CSPs, which are in fact identical to string testing algorithms. Finally, we explain how dictator tests can be translated into computational complexity results for CSPs and we sketch the proofs of some of Håstad’s optimal inapproximability results.

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