1.4 Computational Complexity
1.4.3 The Complexity Class AC 0
Figure 1.1: Unconditionally known results on BPP.
While it is not known whether BPP ⊆ NP, Sipser and Gács [Sip83] proved that BPP⊆Σp2∩Πp2, i.e., BPP is contained in the second level of the polynomial hierarchy.
A simpler proof of this fact was given by Lautemann [Lau83] in the same year, and we will rely on ideas from that proof in section 4.2.
Arguably the most important open question concerning BPP is whether it is equal to PTIME or not. Implagliazzo and Wigderson [IW97], building on a long line of work, proved that BPP = PTIME if there is a language decidable in deterministic time 2O(n) which requires circuits of size at least 2Ω(n). The proof works by constructing for everyc a pseudorandom generator which, givenO(logn) truely random bits, computes a string ofncpseudorandom bits that looks random to any circuit of size at mostnc. This result is generally seen as evidence towards the fact that BPP = PTIME. On the other hand, it is conditional under a very strong circuit lower bound, something far beyond current techniques.
1.4.3 The Complexity Class AC0
Instead of giving, as in the case of Turing machines, a single algorithm for inputs of arbitrary sizes, we may also specify a family (Cn)n≥1 of Boolean circuits such that each
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1.4 Computational Complexity circuitCnhasninputs and one output. Here, by a(Boolean) circuitwe mean a directed acyclic graph in which each node of in-degree >1 is labelled as and-node or as or-node, each node of in-degree 1 is labelled as negation node and all nodes of in-degree 0 are input nodes. Furthermore one node with out-degree 0 is labelled as output node. The size of a circuit C is the total number of nodes and edges and is denoted by|C|. Given an assignment a ∈ {0,1}n for a circuit C with n input nodes, we say that a satisfies C if the value computed by C on input a is 1. The depth of a circuit is the length of a longest path from its output to one of its input nodes. The fan-in of a circuit is the maximal in-degree among its nodes.
Definition 2. Given a circuit family (Cn)n≥1, the language accepted by it is the set {x∈ {0,1}∗|C|x|accepts x}.
The class AC0 is defined as the class of all languagesL⊂ {0,1}∗ for which there exists a circuit family (Cn)n≥1 which accepts it and such that there is a d > 1 such that all Cn have depth at mostd, and for which|Cn|=nO(1). Note that we do not assume any bound on the fan-in of theCn.
Although lower bounds on computational resources have been the core goal of re-search in computational complexity for several decades now, unconditional results are still very few. The class AC0 is a notable exeption, because Håstad’s Switching Lemma for bounded depth-circuits [Hå86] can be used to obtain exponential lower bounds for constant depth circuits. We say that a function f :{0,1}n→ {0,1}can be expressed as a k-DNF if it can be written as
f(x) :=_
i
λi,1∧. . .∧λi,k
for some i ≥ 1 and some choice of literals λi,j, each of which is either some xk or its negate¬xk. Expressibility as ak-CNF is defined similarly. Both for CNFs and for DNFs, tight lower bounds are easy to obtain by means of so called prime-implicants.
A randomp-restrictionρ is a tuple (ρ1, . . . , ρn) of independent random variables such that
P(ρi =∗) =p and P(ρi = 0) =P(ρi = 1) = 1−p 2 .
For any outcome ρ of this random variable, the restricted function f|ρ is a function on those variables xi for which ρi =∗, such that
f|ρ(x) =f(y), whereyi =
(xi ifρi=∗, ρi otherwise.
The use of random restrictions to obtain lower bounds for bounded-depth circuits has been pioneered by Furst, Saxe, and Sipser, who in [FSS81] used it to prove that the parity function has no AC0 circuits; the journal version appeared in [FSS84]. Building on the technique of random restrictions, subsequently Ajtai [Ajt83] and Yao [Yao85] obtained
1 Mathematical Preliminaries
size lower bounds of Ω(ncdlogn) and Ω(2n1/4d), respectively, for circuits of depthd. (Here, cd is a constant depending on d). In [Hå86], Johan Håstad obtained a lower bound of Ω(2cdn1/(d−1)) which is optimal in the sense that for some ˜cd, there are depth-d circuit families of sizeO(2c˜dn1/(d−1)) computing the parity function.
The common scheme in proving these lower bounds is the use of so called switching lemmaswhich state that, after applying a random restriction, with high probability each of the subcircuits in the two lowest levels of the circuit (those closest to the input gates) may be switched from ∧-∨-circuits to ∨-∧-circuits of similar size or vice versa. After merging two subsequent layers of gates of the same type, one obtains a circuit of depth one lower than that of the original circuit, until eventually one arrives at a CNF or DNF, for which known lower bounds apply. The switching lemma as proved by Håstad reads:
Theorem 3 (Håstad’s Switching Lemma). Let f : {0,1}n → {0,1} be expressible as
An overview of the switching lemma and its applications is given in [Bea94]. The most well-known consequences are the above-mentioned lower bound for the parity function and other functions with known lower bounds on DNFs or CNFs, in particular the majority function.
Another important consequence of Håstad’s Switching Lemma is the low average sen-sitivity of AC0 circuits:
Definition 4. Let f :{0,1}n → {0,1} be a Boolean function. Thesensitivity of f at
~x∈ {0,1}n is defined as
S(f;~x) :=|{1≤i≤n|f(~x)6=f(~x⊕~ei)}|,
where~x⊕~ei is the vector ~xwith thei-th bit flipped. The average sensitivity off is S(f) := 2−nX
~ x
S(f;~x).
The average sensitivity off may be interpreted as follows: Arrange the 2nelements of {0,1}ninto a hypercube, connecting two vertices~x, ~y∈ {0,1}nby an edge iff they differ in exactly one coordinate. Colour the vertices of this hypercube red and black according tof(~x), and call an edge coloured if it connects two vertices of different colour. Then S(f)/nis the probability that a randomly chosen edge in this hypercube is coloured. In the extreme case thatf is the parity function or its complement, all edges are coloured.
In this caseS(f) =S(f;~x) =nfor all ~x∈ {0,1}n.
In [LMN89] Linial, Mansour, and Nisan gave a bound on the average sensitivity of functions computable by AC0 circuit families, which was later strengthened by
Bop-16
1.4 Computational Complexity
pana [Bop97]:
Theorem 5. Let f : {0,1}∗ → {0,1} be a Boolean function computable by a family (Cn)n≥0 of Boolean circuits of depthdand size nO(1) consisting of negation gates and∨ and ∧-gates of unbounded fan-in. Then
S(fn)≤O(logd−1n).
Note that Boppana’s bound is optimal, as the parity of logd−1nmany input bits may be computed by polynomial-size circuits of depth d.
Linial et al. used their bound on the average sensitivity of AC0 functions to give an O(nlogO(1)n)-time algorithm for learning functions in AC0 [LMN89]. Another important application was found by Rossman [Ros08], who used Boppanas result to prove that bounded-depth circuits for detecting cliques of size k in graphs must have size at least
|V|2k/9, independent of the bound on their depth. Amano [Ama10] extended this result to arbitrary subgraphs. Another application to finite model theory is given in [Ros09], where Rossman used Boppana’s result to show that certain strategies in Ehrenfeucht-Fraïssé games on random structures are winning strategies with high probability.