TC0

TC⁰ is a complexity class used in circuit complexity. It is the first class in the hierarchy of TC classes.

TC⁰ contains all languages which are decided by Boolean circuits with constant depth and polynomial size, containing only unbounded-fanin AND gates, OR gates, NOT gates, and majority gates. Equivalently, threshold gates can be used instead of majority gates.

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

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

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 ${\mbox{TC}}^{0}\subsetneq {\mbox{PP}}$ . (Allender 1996, as cited in Burtschick 1999).

Basis for uniform ${\mbox{TC}}^{0}$

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

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.

External links

Complexity Zoo: TC⁰

Important complexity classes (more)

Considered feasible	DLOGTIME AC⁰ ACC⁰ TC⁰ L SL RL NL NC SC CC P P-complete ZPP RP BPP BQP APX

Suspected infeasible	UP NP NP-complete NP-hard co-NP co-NP-complete AM QMA PH ⊕P PP #P #P-complete IP PSPACE PSPACE-complete

Considered infeasible	EXPTIME NEXPTIME EXPSPACE ELEMENTARY PR R RE ALL

Class hierarchies	Polynomial hierarchy Exponential hierarchy Grzegorczyk hierarchy Arithmetical hierarchy Boolean hierarchy

Families of classes	DTIME NTIME DSPACE NSPACE Probabilistically checkable proof Interactive proof system

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