Are all regular languages in TC?

Determine whether every regular language over a finite alphabet belongs to the nonuniform circuit complexity class TC (threshold circuits), i.e., whether membership in any regular language can be decided by constant-depth, polynomial-size Boolean circuits equipped with majority gates.

Background

The paper reviews standard connections between logic, automata, and circuit complexity to situate transformer expressivity results. It notes that NC contains all regular languages and that TC extends AC with majority gates, enabling arithmetic-like computations.

Despite these facts, it is a longstanding unresolved question whether TC already suffices to capture all regular languages. Resolving this would clarify the boundary between classes relevant to transformer models that align with TC-type computations and would inform how far Average Hard Attention Transformers (AHAT) might reach relative to regular languages.