#
TC^{0}

**TC ^{0}** is a complexity class used in circuit complexity. It is the first class in the hierarchy of TC classes.

TC^{0} contains all languages which are decided by Boolean circuits with constant depth and polynomial size, containing only unbounded fan-in AND gates, OR gates, NOT gates, and majority gates. Equivalently, threshold gates can be used instead of majority gates.

TC^{0} contains several important problems, such as sorting *n* *n*-bit numbers, multiplying two *n*-bit numbers, integer division[1] or recognizing the Dyck language with two types of parentheses.

## Complexity class relations

We can relate TC^{0} to other circuit classes, including AC^{0} and NC^{1}; Vollmer 1999 p. 126 states:

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 . (Allender 1996, as cited in Burtschick 1999).

## Basis for uniform

The functional version of the uniform coincides with the closure with respect to composition of the projections and one of the following function sets , .[2] Here , is a bitwise AND of and . By functional version one means the set of all functions over non-negative integers that are bounded by functions of FP and is in the uniform .

## References

- Hesse, William; Allender, Eric; Mix Barrington, David (2002). "Uniform constant-depth threshold circuits for division and iterated multiplication" (PDF).
*Journal of Computer and System Sciences*.**65**: 695–716. doi:10.1016/S0022-0000(02)00025-9. - Volkov, Sergey. "Finite Bases with Respect to the Superposition in Classes of Elementary Recursive Functions, dissertation". arXiv:1611.04843.

- 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
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