ALOGTIME is the class of languages decidable in logarithmic time by a random access alternating Turing machine.
Known to be equal to UE*-uniform NC1.
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].