TC⁰

We can relate TC⁰ to other circuit classes, including AC⁰ and NC¹; Vollmer 1999 p. 126 states:

${\displaystyle {\mbox{AC}}^{0}\subsetneq {\mbox{AC}}^{0}[p]\subsetneq {\mbox{TC}}^{0}\subseteq {\mbox{NC}}^{1}.}$

Vollmer states that the question of whether the last inclusion above is strict is "one of the main open problems in circuit complexity" (ibid.).

We also have that uniform ${\displaystyle {\mbox{TC}}^{0}\subsetneq {\mbox{PP}}}$. (Allender 1996, as cited in Burtschick 1999).

The functional version of the uniform ${\displaystyle {\mbox{TC}}^{0}}$ coincides with the closure with respect to composition of the projections and one of the following function sets ${\displaystyle \{n+m,n\,{\stackrel {.}{-}}\,m,n\wedge m,\lfloor n/m\rfloor ,2^{\lfloor \log _{2}n\rfloor ^{2}}\}}$, ${\displaystyle \{n+m,n\,{\stackrel {.}{-}}\,m,n\wedge m,\lfloor n/m\rfloor ,n^{\lfloor \log _{2}m\rfloor }\}}$.[2] Here ${\displaystyle n\,{\stackrel {.}{-}}\,m=\max(0,n-m)}$, ${\displaystyle n\wedge m}$ is a bitwise AND of ${\displaystyle n}$ and ${\displaystyle m}$. By functional version one means the set of all functions ${\displaystyle f(x_{1},\ldots ,x_{n})}$ over non-negative integers that are bounded by functions of FP and ${\displaystyle (y{\text{-th bit of }}f(x_{1},\ldots ,x_{n}))}$ is in the uniform ${\displaystyle {\mbox{TC}}^{0}}$.

• Allender, E. (1996). "A note on uniform circuit lower bounds for the counting hierarchy". Proceedings 2nd International Computing and Combinatorics Conference (COCOON). Springer Lecture Notes in Computer Science. 1090. pp. 127–135.
• Clote, Peter; Kranakis, Evangelos (2002). Boolean functions and computation models. Texts in Theoretical Computer Science. An EATCS Series. Berlin: Springer-Verlag. ISBN 3-540-59436-1. Zbl 1016.94046.
• Vollmer, Heribert (1999). Introduction to Circuit Complexity. A uniform approach. Texts in Theoretical Computer Science. Berlin: Springer-Verlag. ISBN 3-540-64310-9. Zbl 0931.68055.
• Burtschick, Hans-Jörg; Vollmer, Heribert (1999). "Lindström Quantifiers and Leaf Language Definability". ECCC TR96-005. Cite journal requires |journal= (help)

• Complexity Zoo: TC⁰