发表机构
School of Electrical Engineering and Computer Science, Universität Kassel; School of Mathematics and Computer Science, Indian Institute of Technology Goa（卡塞尔大学电气与计算机工程学院; 印度理工学院果阿分校数学与计算机学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文刻画了恒定电路复杂性的正则语言，引入有理真值表归约，证明判定该性质为PSPACE完全，并给出对数上界。
AI 中文摘要
我们研究了正则语言在无界扇入布尔电路族意义上的电路复杂性。我们利用带正则谓词的一阶逻辑单变量片段、印章伪簇 $\mathbf{QEJ}_\mathbf{1}$、合适的字同余以及正则表达式，刻画了恒定电路复杂性的正则语言。我们类似地刻画了中性字母正则语言的恒定电路复杂性。我们的下界结果蕴含：亚对数电路复杂性的正则语言类与恒定电路复杂性的正则语言类重合。此外，我们证明，判定以非确定性有限自动机给出的正则语言是否具有恒定电路复杂性是 $\mathbf{PSPACE}$-完全的。我们引入了一种强归约概念，称为有理真值表归约，它针对代数定义的语言类而定制。我们证明，对于一类我们称为温和的函数，有理真值表归约同时保持电路复杂性的上界和下界。我们证明，电路复杂性受温和函数界定的正则语言类实际上是一个长度倍增的语言簇。略微扩展恒定电路复杂性的正则语言类，我们类似地刻画了属于伪簇 $\mathbf{QEACom}$ 的正则语言类。对于这些语言，我们推导出对数电路复杂性的上界。
英文摘要
We study the circuit complexity of regular languages in terms of unbounded fan-in Boolean circuit families. We characterize the regular languages of constant circuit complexity in terms of the one-variable fragment of first-order logic with regular predicates, in terms of the pseudovariety of stamps $\mathbf{QEJ}_\mathbf{1}$, suitable word congruences and regular expressions. We analogously characterize the neutral letter regular languages of constant circuit complexity. Our lower bound result implies that the class of regular languages of sublogarithmic circuit complexity coincides with the one of constant circuit complexity. In addition we show that deciding whether a regular language, given as a nondeterministic finite automaton, has constant circuit complexity is $\mathbf{PSPACE}$-complete. We introduce a strong notion of reduction, called rational truth-table reduction, that is tailored towards algebraically defined classes of languages. We show that, for a class of functions we call mild, rational truth-table reductions preserve both upper and lower bounds on circuit complexity. We show that the class of regular languages, whose circuit complexity is bounded by a mild function, is in fact a length-multiplying variety of languages. Slightly extending the class of regular languages of constant circuit complexity, we analogously characterize the class of regular languages that are in the pseudovariety $\mathbf{QEACom}$. For these we derive logarithmic circuit complexity upper bounds.
