发表机构
Boston University（波士顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对结构化括号语言（如Dyck语言）的成员资格测试，证明了测试Dyck语言需Ω(n^{2/5})查询下界，并发现一类常数可测的excursion语言，同时构造了需要多项式查询的Hidden String语言，揭示了查询复杂度的边界。
AI 中文摘要
我们研究了结构化字符串语言中成员资格测试的查询复杂度，重点关注Dyck语言及其自然推广。测试者可以查询单词的某些位置，并且必须区分有效输入与在汉明距离上相距甚远的单词，同时仅检查次线性的位置数量。我们的结果界定了常数查询可测试性与多项式查询复杂度之间的边界。首先，我们证明了对于具有任意固定数量m≥2种括号类型的Dyck语言D_m，测试其需要Ω(n^{2/5})的下界，改进了Fischer、Magniez和Starikovskaya (SODA '18)先前Ω(n^{1/5})的下界，并几乎匹配了他们的O(n^{2/5+o(1)})上界。此外，我们表明针对这些问题的所有非自适应算法都需要Ω(n^{1/2})次查询。我们的下界使用Pólya-urn过程来构造困难分布；其次，我们确定了一类广泛的加权括号语言，称为“excursion languages”，它们仍然保持常数查询可测试性。这些语言编码了保持非负并返回零的有界步长游走。对于每个固定的excursion语言，我们给出一个查询复杂度为O(1/ε^2)的非自适应测试器，并证明这种对ε的依赖是最优的，即使对于自适应算法也是如此。作为特例，我们获得了测试D_1的紧致Θ(1/ε^2)查询复杂度，改进了先前的O(log(1/ε)/ε^2)上界，并给出了第一个匹配的双侧错误下界。第三，我们构造了一个简单的困难语言Hidden String，它由确定性线性文法生成，但测试它需要Ω(n^{2/5})次自适应查询和Ω(n^{1/2})次非自适应查询。这表明即使对于高度受限的字符串语言，也会出现多项式查询复杂度。
英文摘要
We study the query complexity of testing membership in structured string languages, focusing on Dyck languages and natural generalizations. A tester receives query access to a word and must distinguish valid inputs from words that are far in Hamming distance, while inspecting only a sublinear number of positions. Our results sharpen the boundary between constant-query testability and polynomial query complexity. First, we prove an $Ω(n^{2/5})$ lower bound for testing Dyck languages $D_m$ with any fixed number $m\ge2$ of parenthesis types, improving the previous $Ω(n^{1/5})$ lower bound of Fischer, Magniez, and Starikovskaya (SODA `18) and nearly matching their upper bound of $O(n^{2/5+o(1)})$. Furthermore, we show that all nonadaptive algorithms for these problems require $Ω(n^{1/2})$ queries. Our lower bounds use a Pólya-urn process to construct the hard distribution; Second, we identify a broad class of weighted-parenthesis languages, which we call {\em excursion languages,} that remain constant-query testable. These languages encode bounded-step walks that stay nonnegative and return to zero. For every fixed excursion language, we give a nonadaptive tester with query complexity $O(1/\varepsilon^2)$, and we prove this dependence on $\varepsilon$ is optimal, even for adaptive algorithms. As a special case, we obtain the tight $Θ(1/\varepsilon^2)$ query complexity of testing $D_1$, improving the previous $O(\log(1/\varepsilon)/\varepsilon^2)$ upper bound and giving the first matching two-sided-error lower bound. Third, we construct a simple hard language, Hidden String, that is generated by a deterministic linear grammar but nevertheless requires $Ω(n^{2/5})$ adaptive queries and $Ω(n^{1/2})$ nonadaptive queries to test. This shows that polynomial query complexity appears even for highly restricted string languages.
