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CS221: Computational Complexity Prof. Salil Vadhan
Lecture 22: AC
11/15 Scribe: Arthur Rudolph
Contents
1 Announcements 1
2 Recap of NC 1
3 AC 2
4 Computing Parity 2
1 Announcements
- Pick up survey and solution set 3.
- Section monday as 6pm.
- No office hours for Salil monday (11/18)
2 Recap of NC
Recall that we defined
NCk = uniform circuits of depth O(logk^ n) and poly size
Last time we showed:
Theorem 1
1. NC 1 ⊆ L
2. NL ⊆ NC 2
Corollary 2 ∪kSPACE(logk^ n) = ∪kDEPTH(logk^ n),
where DEPTH(logk^ n) denotes languages decided by uniform circuits of depth O(logk^ n). (This is not the same as NCk for k > 1 because there is no polynomial bound on the size of the circuits.)
Proof of Corollary: From 1 above and the fact that size ≤ 2 depth, we have
DEPTH(log n) = NC 1 ⊆ SPACE(log n)
Via padding, we conclude DEPTH(logk^ n) ⊆ SPACE(logk^ n)
Note additionally this shows us that NC 2 ⊆ L^2 , so that we have NL ⊆ NC 2 ⊆ L^2 , a strengthening of Savitch’s Theorem.
From 2 above we have that:
(N)SPACE(log n) ⊆ NC 2 ⊆ DEPTH(log^2 n), so SPACE(logk^ n) ⊆ DEPTH(log^2 k^ n)
In conclusions, circuit depth, space, and formula size are all closely related complexity measures.
3 AC
Given the depth restrictions we have looked at, it is natural to ask about circuits restricted to constant depth. Because we have restricted the fan-in (i.e. the number of input to each gate) to a constant in all of our circuits, this also means the circuits have constant size, and thus the output can depend on only a constant number of the input bits. Because of this, it is not a very useful class, as typically we want to compute functions that depend on all of their input.
Instead we consider circuits with unbounded fan-in ∧ and ∨ gates (in addition to ¬ gates of fan-in 1). We define
ACk = uniform, unbounded fan-in circuits of poly-size and depth O(logk^ n)
The “AC” stands for “Alternating circuit.” The reason for this name is that we can think of ∧ nodes as analogous to universal configurations in an alternating TM and ∨ nodes with existential configurations. In this way AC can be seen as the polynomial hierarchy, scaled down by an exponential.
Proposition 3 NCk ⊆ ACk ⊆ NCk+ 1 (In particular, AC = NC.)
To see this, realize that because of the polynomial size restriction, the fan-in to any node is equal to some m ≤ poly(n). We can expand this node to a binary tree of depth log m = O(log n).
Going back to the question of constant-depth circuits, constant-depth circuits with unbounded fan-in are interesting; in particular, they can compute every function (albeit with exponential size), because we can use the CNF representation and use our unbounded fan-in to use depth two, with just one “and” node at the top collecting all the clauses.
4 Computing Parity
The Parity function is Par : { 0 , 1 }∗^ → { 0 , 1 } defined by Par(x) = ⊕ixi. It turns out this simple function is very hard for constant-depth (unbounded fan-in) circuits. The smallest constant-depth circuits for this function are essentially given by the following.
The first step of the approximation method for our problem is given by the following:
Lemma 7 If f : { 0 , 1 }n^ → { 0 , 1 } is computed by a depth d, size s, unbounded fan-in circuit, then ∃ a polynomial g : Zn 3 → Z 3 of degree (log s)O(d)^ which 99%-approximates f.
Consider following special case of the function AND(x 1 ,... , xn). An error-less polynomial for this is the monomial x 1 x 2 · · · xn. This has degree n, so we need some way to decrease our degree by lessening our accuracy in some controlled way.
