发表机构
Weizmann Institute of Science（魏茨曼科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对剩余字符串相等性（ResStringEq）与Dyck括号语言的测试问题，证明了Ω(n^{2/5})的自适应查询下界、Ω(√n)的非自适应下界，给出了匹配的非自适应测试器，并优化了接近参数的依赖关系，几乎弥合了已有研究的查询复杂度差距。
AI 中文摘要
剩余字符串相等性（记为$\texttt{ResStringEq}$）是指所有由$\boldsymbol{\text{0,1,*}}$组成的字符串对，在删除所有`$*$`符号后彼此相等的性质。该性质由Fischer、Magniez和Starikovskaya（SODA 2018）首次提出，他们用其证明了$\texttt{Dyck}$语言（即$\texttt{Dyck}_m$，由$m$种括号类型组成的平衡括号序列构成的语言）的测试下界。他们证明，对长度为$n$的输入测试$\texttt{ResStringEq}$需要$\boldsymbol{\tilde{\boldsymbol{\text{Ω}}}(n^{1/5})}$次查询，并给出了从$\texttt{ResStringEq}$测试到$m\boldsymbol{\text{≥2}}$时$\texttt{Dyck}_m$测试的归约。此外他们还证明，对任意常数接近参数，$\texttt{Dyck}_m$可通过$\boldsymbol{\text{O}(n^{2/5+δ})}$次查询测试，其中$δ>0$是任意小的常数。\n在本研究中，我们几乎弥合了剩余的差距：证明测试$\texttt{ResStringEq}$（以及$m≥2$时的$\texttt{Dyck}_m$）需要$\boldsymbol{\text{Ω}(n^{2/5})}$次查询。我们还证明了非自适应查询测试器的更强下界$\boldsymbol{\text{Ω}(\boldsymbol{\text{√}}n)}$。通过提出一种用于$\texttt{ResStringEq}$的非自适应测试器（其查询复杂度为$\boldsymbol{\text{O}(n^{1/2+δ})}$，$δ>0$为任意小常数），我们证明该$\boldsymbol{\text{Ω}(\boldsymbol{\text{√}}n)}$界是近似紧的。此外，我们将该非自适应测试器扩展到$\texttt{Dyck}$语言，保持相同的查询复杂度。最后，我们改进了Fischer、Magniez和Starikovskaya的测试器中对接近参数$ε$的依赖，将其从$\boldsymbol{\text{O}(1/ε)^{\boldsymbol{\text{poly}}(1/δ)}}$降至$\boldsymbol{\text{O}(1/ε)^{\boldsymbol{\text{O}}(\boldsymbol{\text{log}}(1/δ))}}$。
英文摘要
Residual-String Equality, denoted $\texttt{ResStringEq}$, is the property consisting of all pairs of strings over $\{0,1,*\}$ that are equal after deleting all `$*$' symbols from them. This property was first introduced by Fischer, Magniez, and Starikovskaya (SODA 2018), who used it to show a lower bound on testing the $\texttt{Dyck}$ languages, where $\texttt{Dyck}_m$ is the language consisting of balanced sequences of parentheses over $m$ parenthesis types. They showed that testing $\texttt{ResStringEq}$ on inputs of length $n$ requires $Ω(n^{1/5})$ queries, and presented a reduction from testing $\texttt{ResStringEq}$ to testing $\texttt{Dyck}_m$ where $m \geq 2$. Furthermore, they showed that $\texttt{Dyck}_m$ can be tested with $O(n^{2/5+δ})$ queries for every constant proximity parameter, where $δ>0$ is an arbitrarily small constant. In this work, we nearly close the remaining gap, by showing that testing $\texttt{ResStringEq}$, and hence $\texttt{Dyck}_m$ where $m\geq 2$, requires $Ω(n^{2/5})$ queries. We also show a stronger lower bound of $Ω(\sqrt{n})$ for testers that make non-adaptive queries. We establish that the $Ω(\sqrt{n})$ bound is nearly tight, by presenting a non-adaptive tester for $\texttt{ResStringEq}$ that uses $O(n^{1/2+δ})$ queries for an arbitrarily small constant $δ>0$. Furthermore, we extend this non-adaptive tester to the $\texttt{Dyck}$ languages, with the same query complexity. Finally, we improve the dependence on the proximity parameter $ε$ in the tester of Fischer, Magniez, and Starikovskaya, reducing it from $O(1/ε)^{\mathrm{poly}(1/δ)}$ to $O(1/ε)^{O(\log(1/δ))}$.
