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：

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非确定性多对数时间的受限二阶逻辑

我们引入了受限的二阶逻辑| $ \ textrm {SO} ^ {\ textit {plog}} $ | 对于二阶量化范围在大小关系上最多为多对数的大小关系的有限结构。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，我们证明一个Fagin风格定理，该定理表明可以在| $ \ textrm {SO} ^ {\ textit {plog}} $ |的存在片段中表达的布尔查询。确切地对应于由不确定性图灵机可以计算的决策问题的类别，该不确定性图灵机可以在时间| $ O（（\ log n）^ k）$ |中随机访问输入 对于| $ k \ ge 0 $ |⁠，即在不确定的对数时间内可计算的问题类别。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} } $ | 太弱而无法定义域的总顺序。不过，| $ \ textrm {SO} ^ {\ textit {plog}} $ | 在多对数空间内提供自然水平的可表达性，其方式与二阶逻辑在多项式空间内如何提供自然水平的可表达性密切相关。实际上，我们显示了| $ \ textrm {SO} ^ {\ textit {plog}} $ |的量词前缀类之间的确切对应关系。以及非确定性多对数时间层次结构的级别，类似于二阶逻辑的量词前缀类别与多项式时间层次结构之间的对应关系。我们的工作与恒定深度拟多项式大小的AND / OR电路以及由David A. Mix Barrington在1992年定义的相应的受限二阶逻辑密切相关。我们详细探讨了这种关系。