[论文解读] Oracle Pushdown Automata, Nondeterministic Reducibilities, and the CFL Hierarchy over the Family of Context-Free Languages

Tomoyuki Yamakami|arXiv (Cornell University)|Mar 7, 2013

semigroups and automata theory参考文献 31被引用 4

一句话总结

本文引入了CFL层级——一种基于Oracle下推自动机和诸如单映射、有界真值表及图灵CFL归约等非确定性归约的上下文无关语言（CFL）新型结构复杂性框架。该研究证明：除非多项式层次坍塌，否则该层级是无限的；除非NP = NC²，否则第二层不在NC²中；并为每一层级提供了三种等价刻画，将其与布尔层级及电路类联系起来。

ABSTRACT

To expand a fundamental theory of context-free languages, we equip nondeterministic one-way pushdown automata with additional oracle mechanisms, which naturally induce various nondeterministic reducibilities among formal languages. As a natural restriction of NP-reducibility, we introduce a notion of many-one CFL reducibility and conduct a ground work to formulate a coherent framework for further expositions. Two more powerful reducibilities--bounded truth-table and Turing CFL-reducibilities--are also discussed in comparison. The Turing CFL-reducibility, in particular, helps us introduce an exquisite hierarchy, called the CFL hierarchy, built over the family CFL of context-free languages. For each level of this hierarchy, its basic structural properties are proven and three alternative characterizations are presented. The second level is not included in NC(2) unless NP= NC(2). The first and second levels of the hierarchy are different. The rest of the hierarchy (more strongly, the Boolean hierarchy built over each level of the hierarchy) is also infinite unless the polynomial hierarchy over NP collapses. This follows from a characterization of the Boolean hierarchy over the k-th level of the polynomial hierarchy in terms of the Boolean hierarchy over the k+1st level of the CFL hierarchy using log-space many-one reductions. Similarly, the complexity class Theta(k) is related to the closure of the k-th level of the CFL hierarchy under log-space truth-table reductions. We also argue that the CFL hierarchy coincides with a hierarchy over CFL built by application of many-one CFL-reductions. We show that BPCFL--a bounded-error probabilistic version of CFL--is not included in CFL even in the presence of advice. Employing a known circuit lower bound and a switching lemma, we exhibit a relativized world where BPCFL is not located within the second level of the CFL hierarchy.

研究动机与目标

• 通过将归约概念扩展至非确定性模型，为形式语言的结构复杂性理论（特别是上下文无关语言，CFL）建立一个连贯的框架。
• 使用图灵CFL归约形式化CFL上的层级结构，类比Meyer-Stockmeyer多项式层次。
• 研究CFL层级与其他复杂性类（包括多项式层次、NC²以及BPCFL等概率类）之间的关系。
• 探索相对化世界与Oracle构造，以更好地理解CFL层级内部的分离与包含关系。
• 识别CFL及其扩展在结构复杂性领域的开放问题与未来研究方向。

提出的方法

• 为非确定性单向下推自动机（npda）赋予对CFL中语言的Oracle访问能力，从而定义新的归约类型，如单映射、有界真值表及图灵CFL归约。
• 使用图灵CFL归约定义CFL层级，其中每一层Σ^CFL_k由通过Oracle访问可归约至层级第k级的语言构成。
• 使用对数空间单映射归约，将CFL层级第k级之上的布尔层级与多项式层次的第(k+1)级联系起来。
• 应用电路下界与开关引理，构造一个相对化世界，其中BPCFL不包含于CFL层级的第二层。
• 以Dyck语言及其性质作为CFL的典型完全语言，作为归约与层级构造的锚点。
• 为CFL层级的每一层级建立三种等价刻画，包括对数真值表归约下的封闭性以及与布尔层级的关联。

实验结果

研究问题

• RQ1 CFL层级是否无限？其坍塌的条件是什么？
• RQ2 CFL层级的第二层是否属于NC²？若属于，其含义为何？
• RQ3 CFL层级各层级之上的布尔层级与多项式层次之间有何关系？
• RQ4 BPCFL能否被包含于CFL层级中？此类包含在相对化世界中的含义为何？
• RQ5 CFL层级与AC¹、NC²、L等著名复杂性类之间的精确关系为何？

主要发现

• 除非多项式层次坍塌，否则CFL层级是无限的，该结论通过将CFL层级第k级之上的布尔层级与多项式层次的第(k+1)级相联系的刻画得以证明。
• 除非NP = NC²，否则CFL层级的第二层不包含于NC²，表明其与低深度电路类存在强分离。
• CFL层级的第一层与第二层互不相同，证实了在初始层级即存在非坍塌结构。
• 在CFL层级每一层级上构建的布尔层级是无限的，除非多项式层次坍塌，该结果将无限性结果扩展至更广泛的闭包结构。
• 即使有建议，BPCFL也不包含于CFL中，且存在一个相对化世界，其中BPCFL不在CFL层级的第二层。
• 类Θ^P_k（O(log n)-查询有界）与CFL层级第k级在对数真值表归约下的闭包性质相关，建立了查询有界与基于归约的层级之间的紧密联系。