Strictness of the TC0–LOGSPACE and LOGSPACE–PTIME Inclusions

Determine whether the inclusions [0m\(\mathsf{TC}^0 \subseteq \mathsf{LOGSPACE}\) and \(\mathsf{LOGSPACE} \subseteq \mathsf{PTIME}\) are strict.

Background

The paper introduces the circuit-complexity hierarchy $$\mathsf{AC}^0 \subsetneq \mathsf{TC}^0 \subseteq \mathsf{LOGSPACE} \subseteq \mathsf{PTIME}$$. While the first inclusion is known to be strict, the strictness of the subsequent inclusions remains unresolved. Establishing strictness would separate important classes of constant-depth threshold circuits, logarithmic-space computation, and polynomial-time computation.