置换间关系的可测试性
Testability of relations between permutations
AI总结：
该研究首次探究置换间关系的属性测试问题，定义关联图编码可测试性信息，基于图扩张性质给出可测试/不可测试判定准则，还将置换稳定性概念纳入可测试性框架并梳理未来方向。
摘要：
我们首次开展与置换间关系相关的属性测试问题研究。这类问题的输入是定义在集合{1,…,n}上的一组置换(σ₁,…,σ_d)，研究目标是判定该组置换是否满足某个关系系统E，或是与所有满足E的置换组相距甚远。若该计算问题仅通过查询给定置换的少量元素即可求解，我们就称E是可测试的。例如，当d=2且E仅包含关系XY=YX时，这对应测试σ₁σ₂=σ₂σ₁是否成立，其中σ₁σ₂和σ₂σ₁表示置换的复合运算。\n 我们定义了一类与关系系统E自然关联的图，这类图编码了与E的可测试性相关的全部信息。随后我们证明了两个定理，分别基于这些图的扩张性质给出可测试性与不可测试性的判定准则。借助与群论的深层关联，这两个定理均可适用于广泛的关系系统类别。\n 此外，我们将已被充分研究的“置换稳定性”这一群论概念表述为上述可测试性概念的特例，将所有此前关于稳定性的研究解读为可测试性结果，从计算视角梳理了此前的稳定性研究成果，并提出了稳定性与可测试性领域的诸多未来研究方向。
英文摘要：
We initiate the study of property testing problems concerning relations between permutations. In such problems, the input is a tuple $(σ_1,\dotsc,σ_d)$ of permutations on $\{1,\dotsc,n\}$, and one wishes to determine whether this tuple satisfies a certain system of relations $E$, or is far from every tuple that satisfies $E$. 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 relation $\mathsf{XY=YX}$, this corresponds to testing whether $σ_1σ_2=σ_2σ_1$, where $σ_1σ_2$ and $σ_2σ_1$ denote composition of permutations. We define a collection of graphs, naturally associated with the system $E$, that encodes all the information relevant to the testability of $E$. We then prove two theorems that provide criteria for testability and non-testability in terms of expansion properties of these graphs. By virtue of a deep connection with group theory, both theorems are applicable to wide classes of systems of relations. In addition, we formulate the well-studied group-theoretic notion of stability in permutations as a special case of the testability notion above, interpret all previous works on stability as testability results, survey previous results on stability from a computational perspective, and describe many directions for future research on stability and testability.
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