Abstract

We introduce a restricted second-order logic $\textrm{SO}^{\textit{plog}}$ for finite structures where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. We demonstrate the relevance of this logic and complexity class by several problems in database theory. We then prove a Fagin’s style theorem showing that the Boolean queries which can be expressed in the existential fragment of $\textrm{SO}^{\textit{plog}}$ correspond exactly to the class of decision problems that can be computed by a non-deterministic Turing machine with random access to the input in time $O((\log n)^k)$ for some $k \ge 0$, i.e. to the class of problems computable in non-deterministic poly-logarithmic time. It should be noted that unlike Fagin’s theorem which proves that the existential fragment of second-order logic captures NP over arbitrary finite structures, our result only holds over ordered finite structures, since $\textrm{SO}^{\textit{plog}}$ is too weak as to define a total order of the domain. Nevertheless, $\textrm{SO}^{\textit{plog}}$ provides natural levels of expressibility within poly-logarithmic space in a way which is closely related to how second-order logic provides natural levels of expressibility within polynomial space. Indeed, we show an exact correspondence between the quantifier prefix classes of $\textrm{SO}^{\textit{plog}}$ and the levels of the non-deterministic poly-logarithmic time hierarchy, analogous to the correspondence between the quantifier prefix classes of second-order logic and the polynomial-time hierarchy. Our work closely relates to the constant depth quasipolynomial size AND/OR circuits and corresponding restricted second-order logic defined by David A. Mix Barrington in 1992. We explore this relationship in detail.

中文翻译：

---

会议活动

摘要

对于有限结构，我们引入了受限的二阶逻辑$ \ textrm {SO} ^ {\ textit {plog}} $，其中二阶量化的范围最大为结构大小上的多对数大小关系。通过数据库理论中的几个问题，我们证明了这种逻辑和复杂性类的相关性。然后，我们证明了一种Fagin风格定理，该定理表明可以在$ \ textrm {SO} ^ {\ textit {plog}} $的现有片段中表达的布尔查询恰好对应于可由a计算的决策问题类别。非确定性Turing机器，它对于$ k \ ge 0 $可以随机访问时间为$ O（（\ log n）^ k）$的输入，即在非确定性对数时间可计算的问题类别。应该注意的是，不同于Fagin定理证明二阶逻辑的存在性片段捕获任意有限结构上的NP，我们的结果仅保留有序有限结构，因为$ \ textrm {SO} ^ {\ textit {plog}} $太弱，无法定义域的总顺序。然而，$ \ textrm {SO} ^ {\ textit {plog}} $提供了多对数空间内自然的可表达性，其方式与二阶逻辑如何提供多项式空间内自然的可表达性密切相关。实际上，我们显示了$ \ textrm {SO} ^ {\ textit {plog}} $的量词前缀类别和不确定的多对数时间层次结构的级别之间的精确对应关系，类似于量词前缀之间的对应关系二阶逻辑和多项式时间层次结构。