Testability of relations between permutations

Abstract

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 whetherWe 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.

Testability in group theory

“…This paper is a journal counterpart to [5], which appeared in the proceedings of the 2021 IEEE Annual Symposium on Foundations of Computer Science (FOCS). In that paper we initiated a systematic study of testability of relations between permutations.…”

“…That paper was written from the point of view of property testing-a core subject in theoretical computer science. Here, we present the content of [5] (plus some supplements) from a very different perspective. While that paper was written in a mainly combinatorial language, the current paper is mostly group theoretic.…”

“…While that paper was written in a mainly combinatorial language, the current paper is mostly group theoretic. We will explain how the results of [5] can be viewed (and proved) as group theoretic statements, and through the language of invariant random subgroups. We hope that this presentation will attract the group theory community to join this line of research.…”

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This paper is a journal counterpart to [5], in which we initiate the study of property testing problems concerning a finite system of relations E between permutations, generalizing the study of stability in permutations. To every such system E, a group Γ = ΓE is associated and the testability of E depends only on Γ (just like in Galois theory, where the solvability of a polynomial is determined by the solvability of the associated group).

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Ultrametric analogues of Ulam stability of groups

“…Pointwise stability of equations in permutation was initially considered by Glebsky and Rivera [GR09], then by Arzhantseva and Pȃunescu [AP15] who proved that this can be translated to a group property, as in Definition 1.3. Since then this pointwise stability problem has been under intense investigation, as well as some variants thereof: flexible [BL20,LLM19], quantitative [BM18], uniform, probabilistic [BC20], and connections to computer science [BML20].…”