Same as AC0[m], but now the constant-depth circuit can contain MOD m gates for any m.
Contained in TC0.
Indeed, can be simulated by depth-3 threshold circuits of quasipolynomial size [Yao90].
According to [All96], there is no good evidence for the existence of cryptographically secure functions in ACC0.
There is no non-uniform ACC0 circuits of polynomial size for NTIMES[2n] and no ACC0 circuit of size 2nO(1) for ENP (The class E with an NP oracle). These are the only two known nontrivial lower bounds against ACC0 and were recently discovered by [Wil11].
See NC for definition.
[KV94] give a family of functions that is computable in NC1, but not efficiently learnable unless there exists an efficient algorithm for factoring Blum integers.
Was shown to equal 5-PBP [Bar89]. On the other hand, width 5 is necessary unless NC1 = ACC0 [BT88].
As an application of this result, NC1 can be simulated on a quantum computer with three qubits, one initialized to a pure state and the remaining two in the maximally mixed state [ASV00]. Surprisingly, [AMP02] showed that only a single qubit is needed to simulate NC1 - i.e. that NC1 is contained in 2-EQBP. (Complex amplitudes are needed for this result.)
Contains TC0.
NC1 contains the integer division problem [BCH86], even if an L-uniformity condition is imposed [CDL01].
UE*-uniform NC1 is equal to ALOGTIME [RUZ81].
A branching program is a directed acyclic graph with a designated start vertex. Each (non-sink) vertex is labeled by the name of an input bit, and has two outgoing edges, one of which is followed if that input bit is 0, the other if the bit is 1. A sink vertex can be either an 'accept' or a 'reject' vertex.
The size of the branching program is the number of vertices. The branching program has width k if the vertices can be sorted into levels, each with at most k vertices, such that each edge goes from a level to the one immediately after it.
Then k-PBP is the class of decision problems solvable by a family of polynomial-size, width-k branching programs. (A uniformity condition may also be imposed.)
k-PBP equals (nonuniform) NC1 for constant k at least 5 [Bar89]. On the other hand, 4-PBP is in ACC0 [BT88].
Contained in k-EQBP, as well as PBP.
See also BPd(P).
Same as QAC0[m], except that Mod-m gates are allowed for any m.
Defined in [Moo99].
[GHP00] showed that QACC0 equals QAC0[p] for any prime p.
See TC for definition.
TC0 contains ACC0, and is contained in NC1.
TC0 circuits of depth 3 are strictly more powerful than TC0 circuits of depth 2 [HMP+93].
TC0 circuits of depth 3 and quasipolynomial size can simulate all of ACC0 [Yao90].
There is a function in AC0 (explicitly given in [She08]), whose computation with TC0 circuits of depth 2 requires an exponential number of gates.
[NR97] give a candidate pseudorandom function family computable in TC0, that is secure assuming a subexponential lower bound on the hardness of factoring. (See also [NRR01] for an improvement of this construction, as well as [Kha93].)
One implication is that, assuming such a bound, there is no natural proof in the sense of [RR97] separating TC0 from P/poly. (It is important for this that a function family, and not just a candidate pseudorandom generator, is computable in TC0.) Another implication is that functions in TC0 are likely to be difficult to learn.
The permanent of a 0-1 matrix cannot be computed in uniform TC0 [All99].
In a breakthrough result [Hes01] (building on [BCH86] and [CDL01]), integer division was shown to be in UD-uniform TC0. Indeed division is complete for this class under AC0 reductions.