Rational Reductions and Regular Languages of Constant Circuit Complexity

2026-09-16

AI Summary

研究通过布尔电路复杂度分析正则语言，引入有理真值表归约方法，解决了确定给定正则语言是否具有常数电路复杂度的问题。

Abstract

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.

Problem

• regular languages
• circuit complexity
• Boolean circuits
• PSPACE-complete
• rational truth-table reduction

Innovation

• circuit complexity
• rational truth-table reduction
• constant circuit complexity
• regular languages