Equality of TC0 and PTIME

Determine whether \(\mathsf{TC}^0 = \mathsf{PTIME}\).

Background

The paper states that constant-depth threshold circuits are contained in polynomial-time computation, but the relationship between these classes is not known. Resolving whether TC0\mathsf{TC}^0 equals PTIME\mathsf{PTIME} would determine whether polynomial-time computation can be captured by constant-depth, unbounded-fan-in circuits with majority gates.

References

In particular, it is open whether $\mathsf{TC}0 = PTIME$.

— On the Expressive Power of Transformers  (2608.12671 - Kolaitis et al., 13 Aug 2026) in Section 3.2, subsection “Definitions”