Stability and testability: equations in permutations
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We initiate the study of property testing problems concerning equations in permutations. In such problems, the input consists of permutations σ_1,…,σ_d∈Sym(n), and one wishes to determine whether they satisfy a certain system of equations E, or are far from doing so. If this computational problem can be solved by querying only a small number of entries of the given permutations, we say that E is testable. For example, when d=2 and E consists of the single equation 𝖷𝖸=𝖸𝖷, this corresponds to testing whether σ_1σ_2=σ_2σ_1. We formulate the well-studied group-theoretic notion of stability in permutations as a testability concept, and interpret all works on stability as testability results. Furthermore, we establish a close connection between testability and group theory, and harness the power of group-theoretic notions such as amenability and property (T) to produce a large family of testable equations, beyond those afforded by the study of stability, and a large family of non-testable equations. Finally, we provide a survey of results on stability from a computational perspective and describe many directions for future research.
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