TC (complexity)

In theoretical computer science, and specifically computational complexity theory and circuit complexity, TC is a complexity class of decision problems that can be recognized by threshold circuits, which are Boolean circuits with AND, OR, and Majority gates. For each fixed i, the complexity class TCⁱ consists of all languages that can be recognized by a family of threshold circuits of depth ${\displaystyle O(\log ^{i}n)}$, polynomial size, and unbounded fan-in. The class TC is defined via

${\displaystyle {\mbox{TC}}=\bigcup _{i\geq 0}{\mbox{TC}}^{i}.}$

Relation to NC and AC

The relationship between the TC, NC and the AC hierarchy can be summarized as follows:

${\displaystyle {\mbox{NC}}^{i}\subseteq {\mbox{AC}}^{i}\subseteq {\mbox{TC}}^{i}\subseteq {\mbox{NC}}^{i+1}.}$

In particular, we know that

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

The first strict containment follows from the fact that NC⁰ cannot compute any function that depends on all the input bits. Thus choosing a problem that is trivially in AC⁰ and depends on all bits separates the two classes. (For example, consider the OR function.) The strict containment AC⁰ ⊊ TC⁰ follows because parity and majority (which are both in TC⁰) were shown to be not in AC⁰. ^{[1] [2]}

As an immediate consequence of the above containments, we have that NC = AC = TC.

References

1. Furst, Merrick; Saxe, James B.; Sipser, Michael (1984), "Parity, circuits, and the polynomial-time hierarchy", Mathematical Systems Theory, 17 (1): 13–27, doi:10.1007/BF01744431, MR   0738749 .
2. Håstad, Johan (1989), "Almost Optimal Lower Bounds for Small Depth Circuits", in Micali, Silvio (ed.), Randomness and Computation (PDF), Advances in Computing Research, 5, JAI Press, pp. 6–20, ISBN   0-89232-896-7, archived from the original (PDF) on 2012-02-22

• Vollmer, Heribert (1999). Introduction to Circuit Complexity. Berlin: Springer. ISBN   3-540-64310-9.