AC (complexity)

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In circuit complexity, AC is a complexity class hierarchy. Each class, ACⁱ, consists of the languages recognized by Boolean circuits with depth $O(\log ^{i}n)$ and a polynomial number of unlimited fan-in AND and OR gates.

The name "AC" was chosen by analogy to NC, with the "A" in the name standing for "alternating" and referring both to the alternation between the AND and OR gates in the circuits and to alternating Turing machines.

The smallest AC class is AC⁰, consisting of constant-depth unlimited fan-in circuits.

The total hierarchy of AC classes is defined as

${\mbox{AC}}=\bigcup _{i\geq 0}{\mbox{AC}}^{i}$

Relation to NC

The AC classes are related to NC, ACC, and TC classes. For each i, we have

${\mathsf {NC}}^{i}\subseteq {\mathsf {AC}}^{i}\subseteq {\mathsf {ACC}}^{i}\subseteq {\mathsf {TC}}^{i}\subseteq {\mathsf {NC}}^{i+1}.$

As an immediate consequence of this, we have that NC = AC = ACC = TC.

We have ${\mathsf {NC}}^{0}\subsetneq {\mathsf {AC}}^{0}\subsetneq {\mathsf {ACC}}^{0}$. Specifically, PARITY is in ${\mathsf {ACC}}^{0}$ but not in ${\mathsf {AC}}^{0}$. And since NC requires bounded fan-in, any function of type $\{0,1\}^{n}\to \{0,1\}$ whose output depends on more than $O(1)$ inputs is beyond ${\mathsf {NC}}^{0}$. In particular, the unbounded fan-in OR is beyond ${\mathsf {NC}}^{0}$.

In detail, define $f_{n}:\{0,1\}^{n}\to \{0,1\}$ by $f_{n}(x_{1},\dots ,x_{n})=\sum _{i}x_{i}\mod 2$. Then it requires $\Omega (2^{\frac {n^{1/d}}{16}})$ gates to be computed by a ${\mathsf {AC}}^{0}$ circuit with depth $\leq d$.

Variations

The power of the AC classes can be affected by adding additional gates. If we add gates which calculate the modulo operation for some modulus m, we have the classes ACCⁱ[m].

References

• Arora, Sanjeev; Barak, Boaz (2009), Computational Complexity: A Modern Approach, Cambridge University Press, ISBN 978-0-521-42426-4, Zbl 1193.68112
• 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
• Pitassi, Toniann (Fall 2015), "Lecture #8" (PDF), CS 2401 – Introduction to Complexity Theory, University of Toronto
• Razborov, A. A. (April 1987), "Lower bounds on the size of bounded depth circuits over a complete basis with logical addition", Mathematical Notes of the Academy of Sciences of the USSR, 41 (4): 333–338, doi:10.1007/BF01137685, ISSN 0001-4346
• Regan, Kenneth W. (1999), "Complexity classes", Algorithms and Theory of Computation Handbook, CRC Press.
• Vollmer, Heribert (1998), Introduction to circuit complexity. A uniform approach, Texts in Theoretical Computer Science, Berlin: Springer-Verlag, ISBN 3-540-64310-9, Zbl 0931.68055