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Average Case Lower Bounds for Monotone Switching Networks
Yuval Filmus∗∗ Toniann Pitassi∗† Robert Robere∗‡ Stephen A. Cook∗§
February 13, 2014
Abstract
An approximate computation of a function f:{0,1}n→ {0,1}by a circuit or switching network
Mis a computation in which Mcomputes fcorrectly on the majority of the inputs (rather than on all
inputs). Besides being interesting in their own right, lower bounds for approximate computation have
proved useful in many subareas of complexity theory, such as cryptography and derandomization.
Lower bounds for approximate computation are also known as correlation bounds or average case
hardness. In this paper, we obtain the first average case monotone depth lower bounds for a function
in monotone P. We tolerate errors that are asymptotically the best possible for monotone circuits.
Specifically, we prove average case exponential lower bounds on the size of monotone switching
networks for the GEN function. As a corollary, we establish that for every i, there are functions that
can be computed with no error in monotone NCi+1, but that cannot be computed without large error
by monotone circuits in NCi. Our proof extends and simplifies the Fourier analytic technique due
to Potechin [23], and further developed by Chan and Potechin [8]. As a corollary of our main lower
bound, we prove that the communication complexity approach for monotone depth lower bounds
does not naturally generalize to the average case setting.
∗Department of Computer Science, University of Toronto, yuvalf@cs.toronto.edu
†Department of Computer Science, University of Toronto, toni@cs.toronto.edu
‡Department of Computer Science, University of Toronto, robere@cs.toronto.edu
§Department of Computer Science, University of Toronto, sacook@cs.toronto.edu
1
1 Introduction
In this paper, we study the average case hardness of monotone circuits and monotone switching networks.
The first superpolynomial lower bounds on monotone circuit size is the celebrated result due to Razborov
[27], who showed that the clique function requires exponential-size monotone circuits, and thus also
requires large monotone depth. His result was improved and generalized by many authors to obtain other
exponential lower bounds for monotone circuits (for example [1,2, 4, 13, 15]). All of these bounds are
average case lower bounds for functions that lie outside of monotone P. The best known average case
lower bound for an explicit function is for Andreev’s polynomial problem, which is (1/2−1/n1/6)-hard
for subexponential-size circuits (follows from [4]); that means that there is a subexponential function
f(n)such that every circuit of size at most f(n)differs from Andreev’s polynomial problem on at least
a(1/2−1/n1/6)-fraction of the inputs. For an excellent survey on the applications of lower bounds on
approximate computations, see [5].
Beginning in 1990, lower bound research was aimed at proving monotone size/depth tradeoffs for
efficient functions that lie inside monotone P. The first such result, due to Karchmer and Wigderson [18],
established a beautiful equivalence between (monotone) circuit depth and the (monotone) communica-
tion complexity of a related communication game. They used this framework to prove that the NL-
complete directed connectivity problem requires Ω(log2n)monotone circuit depth, thus proving that
monotone NL (and thus also monotone NC2) is not contained in monotone NC1. Subsequently Grigni
and Sipser [12] used the communication complexity framework to separate monotone logarithmic depth
from monotone logarithmic space. Raz and McKenzie [25] generalized and improved these lower bounds
by defining an important problem called the GEN problem, and proved tight lower bounds on the mono-
tone circuit depth of this problem. As corollaries, they separated monotone NCifrom monotone NCi+1
for all i, and also proved that monotone NC is a strict subset of monotone P. Unlike the earlier results (for
functions lying outside of P), the communication-complexity-based method developed in these papers
seems to work only for exact computations.
Departing from the communication game methodology from the 1990’s, Potechin [23] recently in-
troduced a new Fourier-analytic framework for proving lower bounds for monotone switching networks.
Potechin was able to prove using his framework a nΩ(log n)size lower bound for monotone switch-
ing networks for the directed connectivity problem. (A lower bound of 2Ω(t)on the size of monotone
switching networks implies a lower bound of Ω(t)on the depth of monotone circuits, and thus the result
on monotone switching networks is stronger.) Recently, Chan and Potechin [8] improved on [23] by
establishing a nΩ(h)size lower bound for monotone switching networks for the GEN function, and also
for the clique function. Thus, they generalized most of the lower bounds due to Raz and McKenzie to
monotone switching networks. However, again their lower bounds apply only for monotone switching
networks that compute the function correctly on every input.
In this paper we obtain the first average case lower bounds on the size of monotone switching net-
works (and thus also the first such lower bounds on the depth of monotone circuits) for functions inside
monotone P. We prove our lower bounds by generalizing the Fourier-analytic technique due to Chan and
Potechin. In the process we first give a new presentation of the original method, which is simplified and
more intuitive. Then we show how to generalize the method in the presence of errors, which involves
handling several nontrivial obstacles.
We show that GEN is (1/2−1/n1/3−)-hard for subexponential-size circuits (under a specific dis-
tribution), and that directed connectivity is (1/2−1/n1/2−)-hard for nO(log n)-size circuits (also under
a specific distribution). In comparison, it is known that under the uniform distribution, no monotone
function is (1/2−log n/√n)-hard even for O(nlog n)-size circuits [22]. A related result shows that for
every there is a (non-explicit) function that is (1/2−1/n1/2−)-hard for subexponential-size circuits
2
under the uniform distribution [17].
As a corollary to the above theorem, we separate the levels of the NC hierarchy, as well as monotone
NC from monotone P, in the average case setting. That is, we prove that for all i, there are monotone
functions that can be computed exactly in monotone NCi+1 but such that any NCicircuit computing the
same function must have large error. And similarly, there are functions in monotone Pbut such that any
monotone NC circuit must have large error.
This leaves open the question of whether or not the original communication-complexity-based ap-
proach due to Karchmer and Wigderson can also be generalized to handle errors. This is a very interest-
ing question, since if this is the case, then average case monotone depth lower bounds would translate
to probabilistic communication complexity lower bounds for search problems. Developing communi-
cation complexity lower bound techniques for search problems is an important open problem because
such lower bounds have applications in proof complexity, and imply integrality gaps for matrix-cut algo-
rithms (See [3,16].) We show as a corollary of our main lower bound that the communication complexity
approach does not generalize to the case of circuits that make mistakes.
The outline for the rest of the paper is as follows. In Section 2 we give background information on
switching networks, the GEN function, and Fourier analysis. Section 3 is a summary of our main lower
bound, and corollaries, including a strong separation of the monotone NC hierarchy and of monotone NC
from monotone P. Section 4 gives an overview of the proof of our lower bound. The lower bound for
exact computations is presented in Section 5, and Section 6 extends it to average case lower bounds. In
Section 7 we discuss implications for randomized counterparts of the Karchmer-Wigderson construction.
Part of the construction involved in the lower bounds appears in Appendix A.
Section 4, Section 5 and Appendix A together form an exposition of the work of Chan and Potechin [8].
2 Preliminaries
In this section we give definitions that will be useful throughout the entire course of the paper. If nis
a positive integer then we define the set [n] = {1,2, . . . , n}, and for integers i, j with i < j we define
the set [i, j] = {i, i + 1, . . . , j }If Sis a subset of some universe U, we define S=U\Sto be the
complement of S. If Nand mare positive integers, we denote by [N]
mthe collection of all subsets of
[N]of size m. The notation 1denotes the vector all of whose entries are 1(the length of the vector will
always be clear from the context). For an input x∈ {0,1}n, we denote the ith index of xby xi. For a
pair of inputs x, y ∈ {0,1}n, we write x≤yif xi≤yifor all i. A boolean function fis monotone if
f(x)≤f(y)whenever x≤y.
Assume f:{0,1}n→ {0,1}is a monotone boolean function. An input x∈ {0,1}nis called a
maxterm of fif f(x)=0and f(x0)=1for the input x0obtained by flipping any 0in xto a 1. Similarly,
an input x∈ {0,1}nis called a minterm if f(x) = 1 and f(x0) = 0 for the input x0obtained by flipping
any 1in xto a 0. If fis a boolean function and f(x)=1we will call xan accepting instance of f,
otherwise it is a rejecting instance.
If P(x)is a boolean condition depending on an input x, then we write [P(x)] to denote the boolean
function associated with that condition. For example, if Vis a fixed set, we denote by [U⊆V]the
function P(U)which is 1if U⊆Vand 0otherwise.
Monotone circuits are circuits in which only AND gates and OR gates are allowed (unrestricted
circuits can also use NOT gates). Such circuits always compute monotone boolean functions. Monotone
Pis the class of languages computed by uniform polynomial size monotone circuits. Monotone NC
is the class of languages computed by uniform polynomial size monotone circuits of polylogarithmic
depth. Monotone NCiis the class of languages computed by uniform polynomial size monotone circuits
3
of depth O(login).
2.1 The GEN problem
We will prove lower bounds for the GEN function, originally defined by Raz and McKenzie [25].
Definition 2.1. Let N∈N, and let L⊆[N]3be a collection of triples on [N]called variables. For a
subset S⊆[N], the set of points generated from Sby Lis defined recursively as follows: every point in
Sis generated from S, and if i, j are generated from Sand (i, j, k)∈L, then kis also generated from S.
(If Lwere a collection of pairs instead of a collection of triples, then we could interpret Las a directed
graph, and then the set of points generated from Sis simply the set of points reachable from S.)
The GEN problem is as follows: given a collection of variables Land two distinguished points
s, t ∈[N], is tgenerated from {s}?
Formally, an instance of GEN is given by a number N, two numbers s, t ∈[N], and N3boolean
values coding the set L⊆[N]3. For definiteness, in the remainder of the paper we fix (arbitrarily) s= 1
and t=N.
We assume that every instance of GEN throughout the rest of the paper is defined on the set [N], and
we use sand tto denote the start and target points of the instance. Sometimes we want to distinguish a
particular variable in instances of GEN, so, if `is a variable appearing in an instance Ithen we call (I, `)
apointed instance.
We can naturally associate GEN instances with some graphs. If Gis a DAG with a unique source s,
a unique sink t, and in-degree at most 2, then we can form an instance of GEN from Gby identifying
the start and target points appropriately, and adding a variable (x, y, z)to the GEN instance if the edges
(x, z),(y, z)are in G. If zis a vertex of in-degree 1, say (x, z)∈G, then we add the variable (x, x, z)
instead.
If Gdoes not have a unique source or a unique sink then we can simply add one and connect it to all
of the sources or sinks, which is illustrated in Figure 1.
s
t
Figure 1: A pyramid graph and the corresponding GEN instance
The problem GEN is monotone: if we have an instance of GEN given by a set of variables L, and L
is an accepting input for GEN, then adding any variable l6∈ Lto Lwill not make La rejecting input.
Moreover, it can be computed in monotone P(we leave the proof as an easy exercise).
Theorem 2.2. GEN is in monotone P.
Let (C, C)be a partition of [N]\{s, t}into two sets. We call such a set Cacut in the point set [N].
We think of the cut Cas always containing sand never containing t, and so we define Cs=C∪ {s}.
4
Then we can define an instance I(C)of GEN, called a cut instance, as
I(C) = [N]3\ {(x, y, z)∈N3|x, y ∈Cs, z ∈Cs}.
The set of points generated in I(C)from {s}is precisely Cs. If I(C)is a cut instance and `= (x, y, z)
is a variable with x, y ∈Csand z∈Csthen we say that `crosses C.
It might seem more natural to define Cas a subset of [N]containing sand not containing t. However,
from the point of view of Fourier analysis, it is more convenient to remove both “constant” vertices s, t
from the equations.
The set of cut instances is exactly the set of maxterms of GEN.
Proposition 2.3. Let Lbe an instance of GEN. Then Lis rejecting if and only if there exists a cut
C⊆[N]such that L⊆I(C).
Let Cbe the collection of subsets of [N]\{s, t}, and note that every set C∈ C can be identified with
a cut (and therefore a cut instance). We also note that |C| = 2N−2.
2.2 Vectors and Vector Spaces
In this section we recall some definitions from linear algebra. Consider the set of all cuts Con the point
set [N]of GEN. A cut vector is a function f:C → R, which we think of as a real-valued vector indexed
by cuts C∈ C. We define an inner product on the space of cut vectors by
hf, gi=1
|C| X
C∈C
f(C)g(C)
for any two cut vectors f, g. Two cut vectors f, g are orthogonal if hf, gi= 0, and a set of vectors Vis
orthogonal if every pair of vectors in Vis orthogonal. Using this inner product, we define the magnitude
of a cut vector fto be ||f|| =phf, f i.
We will also need some tools from Fourier analysis. Given a cut U∈ C, define the cut vector
χU:C → Rby
χU(C)=(−1)|U∩C|.
We have the following proposition regarding these vectors:
Proposition 2.4. The collection of vectors {χU}U∈C is orthonormal.
This particular collection of vectors forms a basis for the vector space of cut vectors known as the
Fourier basis. It follows that we can write any cut vector f:C → Ras f=PC∈Chf, χCiχC, where
hf, χCiis called the Fourier coefficient at C. Following convention, we will denote hf, χCiby ˆ
f(C).
We need some useful properties of the Fourier transform. First recall Parseval’s Theorem: let fbe
any cut vector, then
hf, f i=X
C∈C
ˆ
f(C)2,(1)
Parseval’s theorem also holds in a more general setting: If Bis an orthonormal set of cut vectors then
hf, f i ≥ X
φ∈B|hf, φi|2.(2)
5
2.3 Switching Networks
In this section we introduce switching networks, which are a computation model used to capture space-
bounded computation.
Definition 2.5. Let X={x1, . . . , xn}be a set of input variables. A monotone switching network Mon
the variables Xis specified as follows. There is an underlying graph undirected G= (V, E)whose nodes
are called states and whose edges are called wires, with a distinguished start state sand a distinguished
target state t. The wires of Mare labelled with variables from X.
Given an input x:X→ {0,1}(or an assignment of the variables), the switching network responds
as follows. Let ebe a wire in the switching network, and let xibe the variable labelling e. The edge eis
alive if xi= 1, and it is dead if xi= 0.
We say that Maccepts an input xif there exists a path from sto tusing only wires which are
alive under x. If no such path exists, then Mrejects x. The boolean function computed by Mis
f(x) = [Maccepts x].
Throughout the paper we follow the convention used in the previous definition and use bold face to
denote objects in switching networks.
We briefly mention some computation models extending monotone switching networks. If the un-
derlying graph Gis directed, then instead of a switching network we have a switching-and-rectifier
network. A non-monotone switching network can have negated variables on the wires. All switching
networks considered in this paper are monotone and undirected.
Before Chan and Potechin [8] there are very few results explicitly lower bounding the sizes of mono-
tone switching networks. Recall that the threshold function T hk,n :{0,1}n→ {0,1}takes on value 1iff
the input xhas at least k1s. Markov [20] determined that the size of an optimal monotone switching and
rectifier network computing the threshold function T hk,n is exactly k(n−k+ 1). For monotone switch-
ing networks computing threshold functions there are a series of results. Halld´
orsson, Radhakrishnan,
and Subrahmanyam [14] proved a lower bound of Ω(nlog log log n)for T hn−1,n. When the threshold k
is “small” there is a result due to Kricheviskii [19], which gives a lower bound of Ω(nlog n)for T h2,n.
This was later generalized by Radhakrishnan [24] to a lower bound of Ω(bk/2cnlog(n/(k−1))) for
T hk,n for all kwith 2≤k≤n/2. Each of these results were proved using combinatorial arguments.
Monotone switching networks and monotone circuits are connected by the following result essen-
tially proved by Borodin [6]: a monotone circuit of depth dcan be simulated by a monotone switching
network of size 2d, and a monotone switching network of size scan be simulated by a monotone circuit
of depth O(log2s). (The result holds also for non-monotone switching networks and non-monotone
circuits.) See Appendix B for a proof sketch.
Aswitching network for GEN is a switching network whose input is an instance of GEN. Such
a switching network is complete if it accepts all yes instances of GEN. It is sound if it rejects all no
instances of GEN. A switching network which is both complete and sound computes the GEN function.
Let Mbe a switching network for GEN. We can naturally identify each state uin the switching
network Mwith a reachability vector Ru:C → {0,1}for cut instances I(C)defined by
Ru(C) := (1if uis reachable on input I(C),
0otherwise.
Here are some basic properties of the reachability vectors.
Theorem 2.6. Let Mbe a switching network with start state sand target state t.
6
1. Rs≡1.
2. If Mis sound then Rt≡0.
3. If uand vare two states connected by a wire labelled `and Cis a cut with `∈I(C)then
Ru(C) = Rv(C).
2.4 Reversible Pebbling for GEN
Next we discuss the reversible pebbling game on graphs, which is a space-efficient way to perform
reachability tests on graphs. The particular form of this test gives an algorithm for GEN by applying it to
the underlying graph of a GEN instance.
Definition 2.7. Let G= (V , E)be a directed acyclic graph (DAG) with a unique source sand a unique
sink t. For a node v∈V, let P(v) = {u∈V: (u, v)∈E}be the set of all incoming neighbors of v. We
define the reversible pebbling game as follows. A pebble configuration is a subset S⊆Vof “pebbled”
vertices. For every x∈Vsuch that P(x)⊆S, a legal pebbling move consists of either pebbling or
unpebbling x, see Figure 2. Since sis a source, P(s) = ∅, and so we can always pebble or unpebble it.
u
v
u
v
xx
x
Figure 2: Legal pebbling moves involving x; the corresponding pebbling configurations are uand v
The goal of the reversible pebbling game is to place a pebble on t, using only legal pebbling moves,
starting with the empty configuration, while minimizing the total number of pebbles used simultaneously.
Formally, we want to find a sequence of pebbling configurations S0=∅, S1, . . . , Snsuch that t∈Sn,
and for each i∈ {0, . . . , n −1}, the configuration Si+1 is reachable from configuration Siby a legal
pebbling move. We call such a sequence a valid pebbling sequence for G. The pebbling cost of the
sequence is max(|S0|,...,|Sn|). The reversible pebbling number of a DAG Gis the minimal pebbling
cost of a valid reversible pebbling sequence for G.
Chan [7] defines the reversible pebbling number slightly differently, requiring a valid pebbling se-
quence to end in a configuration in which only tis pebbled. The reversible nature of the pebbling implies
that this increases the pebbling number by at most one: given a valid pebbling sequence in our sense,
which without loss of generality ends with a move pebbling tfor the first time, roll back all moves but
the very last one to obtain a configuration in which only tis pebbled.
The more common black pebbling game has the same rules for pebbling a node x∈V, but always
allows unpebbling a node. The black pebbling number of a dag Gis the minimal pebbling cost of a valid
black pebbling sequence for G. Every valid pebbling sequence is also a valid black pebbling sequence,
and so the black pebbling number is upper-bounded by the reversible pebbling number.
Dymond and Tompa [10] showed that the reversible pebbling number of any graph with nvertices
and in-degree 2is O(n/ log n). Gilbert and Tarjan [11] constructed matching graphs (see also Nord-
str¨
om’s excellent survey [21]).
7
Theorem 2.8. There is an explicit family of DAGs Gnwith in-degree 2such that Gnhas nvertices and
reversible pebbling number Ω(n/ log n).
In fact, the graphs Gnhave black-white pebbling number Ω(n/ log n)(see Nordstr¨
om [21] for the
definition of black-white pebbling). For more on reversible pebbling and its applications, see [7].
Pyramid graphs are graphs with O(h2)nodes and reversible pebbling number Θ(h)for each positive
integer h.
Definition 2.9. A directed graph P= (V, E)is a pyramid graph with hlevels if Vis partitioned into
hsubsets, V1, V2, . . . , Vh(called levels), where Vihas ivertices. Let Vi={vi1, vi2, . . . , vii}. For each
i∈[h−1], if vij and vi,j+1 are a pair of adjacent vertices in layer Vi, then there are edges (vij , vi+1,j−i−1)
and (vi,j+1, vi+1,j−i−1).
For example, the graph in Figure 2 is a pyramid with 3levels.
Cook [9] calculated the pebbling number of pyramids.
Theorem 2.10. Let Pbe any pyramid graph with hlevels. The reversible pebbling number of Pis Θ(h).
Proof. Cook [9] showed that the black pebbling number of Pis Θ(h), and so the reversible pebbling
number of Pis Ω(h). Conversely, induction on hshows that there is a valid pebbling sequence that
pebbles only the apex of a pyramid of height husing at most O(h)pebbles.
Another useful class of graphs are path graphs.
Definition 2.11. A directed graph P= (V, E)is a directed path of length nif V={v1, . . . , vn}and
E={(v1, v2),...,(vn−1, vn)}.
Potechin [23] computed the reversible pebbling number of directed paths. In hindsight, the same
computation appears in Raz and McKenzie [25]; they computed a different statistic of the graph, which
was shown to equal the reversible pebbling number by Chan [7].
Theorem 2.12. The reversible pebbling number of a directed path of length nis Θ(log n).
Every DAG with in-degree at most 2naturally defines a minterm of GEN which we call a graph
instance.
Definition 2.13. Let Gbe a DAG with in-degree at most 2and a single sink t, and suppose the vertex set
of Gis a subset of [N]not containing s. The GEN instance corresponding to Gcontains the following
triples: for each source x, the triple (s, s, x); for each vertex zwith inbound neighborhood {x}, the triple
(x, x, z); for each vertex zwith inbound neighborhood {x, y}, the triple (x, y, z). Such an instance is
called a graph instance isomorphic to G. The underlying vertex set consists of the vertex set of Gwith
the vertex tremoved.
For a graph G, the function G-GEN is the monotone function whose minterms are all graph instances
isomorphic to G.
Raz and McKenzie [25, Proposition 3.8] proved the following (easy) result.
Theorem 2.14. For each DAG Gthere is a polynomial size monotone circuit of depth O(hlog N)for
G-GEN, where his the height of G.
Their main contribution was to prove a lower bound of Ω(plog N)for G-GEN, where pis the re-
versible pebbling number of G, assuming m=NO(1). For pyramids, which satisfy p= Θ(h), this
lower bound is tight.
8
3 Statement of Results
In this paper we prove lower bounds for monotone switching networks computing GEN. Our first con-
tribution is a simplified proof of the following theorem [8] which gives an exponential lower bound for
monotone switching networks.
Theorem 3.1. Let N , m, h be positive integers satisfying m≥h≥3, and let Gbe a DAG on m
vertices with in-degree at most 2and reversible pebbling number at least h. Any sound monotone
switching network for GEN which accepts all graph instances isomorphic to Gmust have at least
Ω(hN/m3)(h−2)/3/O(m)states.
Corollary 3.2. For any > 0, any monotone switching network which computes GEN must have at least
2Ω(N1/2−)states.
Proof. Apply the theorem with m=N1/2−and a DAG Gwith reversible pebbling number h=
Θ(m/ log m), given by Theorem 2.8.
We also consider monotone switching networks which are allowed to make errors. Let Dbe any
distribution on instances of GEN. We say that a monotone switching network Mcomputes GEN with
error if the function computed by Mdiffers from GEN on an -fraction of inputs (with respect to D).
The distributions Dwe use are parametrized by DAGs. For any DAG Gwith in-degree at most 2
we define DGto be the distribution on instances of GEN which with probability 1/2chooses I(C)for
a uniformly random cut C∈ C, and with probability 1/2chooses a uniformly random graph instance
isomorphic to G.
Our major result in this paper is a strong extension of Theorem 3.1 for switching networks computing
GEN with error close to 1/2.
Theorem 3.3. Let αbe a real number in the range 0< α < 1. Let m, h, N be positive integers satisfying
324m2≤Nαand 3≤h≤m. Let Gbe a DAG with mvertices, in-degree 2and reversible pebbling
number at least h.
Any monotone switching network which computes GEN on [N+ 2] with error ≤1/2−1/N 1−α
must have at least Ω(hN/m3)(h−2)/3/O(mN )states.
Corollary 3.4. For any αin the range 0< α < 1, any monotone switching network computing GEN
with error at most 1/2−1/N 1−αmust have at least 2Ω((1−α)Nα/2)states.
Proof. Apply the theorem with m=Nα/2/324 and a DAG Gwith reversible pebbling number h=
Θ(m/ log m), given by Theorem 2.8.
Using this theorem, we get the following corollary separating NCiand NCi+1 in the presence of
errors. Recall from Theorem 2.10 that the pyramid graph with height hhas reversible pebbling number
Θ(h)and m=h(h+ 1)/2nodes.
Theorem 3.5. Let 0< δ < 1/3be any real constant. For each positive integer ithere exists a language
Lwhich is computable in monotone NCi+1, but there is no sequence of circuits in monotone NCiwhich
computes Lon inputs of length kwith error ≤1/2−1/k1/3−δ.
Proof. For each N, let LN3⊆ {0,1}N3be G-GEN, where Gis a pyramid of height h= logiN, and
let L=SN>0LN3. Theorem 2.14 shows that for each Nthere exists a polynomial size circuit of depth
O(hlog N) = O(logi+1 N)computing LN3, and so Lis in monotone NCi+1.
9
On the other hand, Any monotone circuit Cwith bounded fan-in and depth dcan be simulated by a
monotone switching network with 2dstates. Therefore Theorem 3.3 (with α= 3δ) implies that for large
enough N, any monotone circuit computing LN3with ≤1/2−1/N1−3δ≤1/2−1/k1/3−δerror must
have depth Ω(hlog N) = Ω(logi+1 N).
Similarly, we can separate monotone NC from monotone P.
Theorem 3.6. Let 0< δ < 1/3be any real constant. There exists a language Lwhich is computable in
monotone P, but there is no sequence of circuits in monotone NC which computes Lon inputs of length
kwith error ≤1/2−1/k1/3−δ.
Proof. The proof is similar to the proof of the previous theorem. Instead of choosing h= logiN, choose
h=N1/100.
We also get the following result for directed connectivity which approaches the optimal result men-
tioned in the introduction (albeit with a different input distribution).
Theorem 3.7. Let 0< δ < 1/2be any real constant. For k=N2, let fbe the function whose input
is a directed graph on Nvertices, and f(G)=1if in the graph G, the vertex Nis reachable from the
vertex 1. There exists a distribution Don directed graphs such that any monotone switching network
computing fwith error < 1/2−1/k1/2−δ(with respect to D) contains at least NΩ(log N)states.
Proof. We can reduce fto GEN by replacing each edge (u, v)in the input graph by a triple (u, u, v).
The resulting instance is accepted by GEN iff t:= Nis reachable from s:= 1 in the input graph.
Given N, let GNbe the directed path of length N1/100. Theorem 2.12 shows that GNhas reversible
pebbling number Θ(log N). Applying Theorem 3.3 (with α= 2δ) yields a lower bound of NΘ(log N)on
the size of monotone switching networks computing fwith error < 1/2−1/N2δ= 1/2−1/kδwith
respect to the distribution DGN.
4 Overview of Proof
We focus on a set of minterms and maxterm over a ground set of size N. The minterms are height h,
size mpyramids1, where m=O(N1/3), and the maxterms are cuts, given by a subset Cof the vertices
containing sand not containing t. The instance I(C)corresponding to Ccontains all triples except for
(i, j, k)where iand jare in Cand kis not in C. Our minterms will consist of a special exponential-sized
family of pyramid instances, P, with the property that their pairwise intersection is at most h, and our
maxterms, C, will consist of all cuts. Given a monotone switching network (M,s,t)solving GEN over
[N], for each state vwe define its reachability function Rv:C → {0,1}given by Rv(C) = 1 if vis
reachable from sin the instance I(C), and otherwise Rv(C)=0.
At the highest level, the proof is a bottleneck counting argument. For each pyramid P∈ P, we will
construct a function gPfrom the set Cof all cuts to the reals. This function will satisfy three properties.
(1) For every P∈ P there is a “complex” state vPin the switching network such that hgP, RvPi=
Ω(1/|M|).
(2) gPonly depends on coordinates from P, and gPhas zero correlation with any function which
depends on at most hcoordinates.
1Our proof is presented more generally for any fixed graph of size roughly Nbut for simplicity of the proof overview, we
will restrict attention to pyramid yes instances.
10
(3) Finally, kgPkis upper-bounded by mO(h).
The first property tells us that for every pyramid P∈ P, there is a complex state vPin the network
that is specialized for P. The second property, together with the fact that the P’s in Pare pairwise
almost-disjoint (overlap in at most h−2points), will imply that the functions {gP:P∈ P} are
orthogonal, and thus we have
1≥ hRv, Rvi ≥ X
P∈P |hRv,gP
kgPki|2=1
mO(h)X
P∈PhgP, Rvi2.
By the third property, kgPkis small, and thus it follows that no state vcannot be complex for more than
|M|2mO(h)different P’s (since otherwise, the quantity on the right side of the above equation would
be greater than 1.) This together with the fact that |P| is very large, so that |P|/(|M|2mO(h))is still
exponentially large, imply our lower bound.
It remains to come up with the magical functions gP. For the rest of this overview, fix a particular
pyramid P. How can we show that some state in the switching network is highly specific to P? We
will use the original Karchmer-Wigderson intuition which tells us that in order for a monotone circuit
to compute GEN correctly (specifically, to output “1” on Pand “0” on all cuts), it must for every cut
Cproduce a witness variable l∈Pthat is not in I(C). And (intuitively) because different variables
must be used as witnesses for different cuts, this should imply that a non-trivial amount of information
must be remembered in order to output a good witness. This intuition also occurs in many information
complexity lower bounds.
The above discussion motivates studying progress (with respect to our fixed pyramid P, and all
cuts), by studying progress of the associated search problem. To this end, for each pyramid P∈ P
and variable `∈P, we will consider `-nice functions gP,l, where `-nice means that gP,`(C)=0
whenever I(C)(`)=1. We will think of an `-nice function gP,` as a “pseudo” probability distribution
over no instances (cuts) that puts zero mass on all cuts Csuch that I(C)(`) = 1. (gP,` is not an actual
distribution since it attains both positive and negative values.) For a state uin the switching network, the
inner product hRu, gP,`iwill be our measure of progress of the GEN function with respect to the pyramid
P, on no instances where the witness cannot be `. In order for gP,` to behave like a distribution, we will
require that hgP,`,1i= 1
Because Rsaccepts all cuts, it follows that hRs, gP,`i= 1, which we interpret as saying that at the
start state, we have made no progress on rejecting the pseudo-distribution defined by gP,`. Similarly,
because Rtrejects all cuts, it follows that hRt, gP,`i= 0, which we interpret as saying that at the
final state, we have made full progress since we have successfully rejected the entire pseudo-distribution
defined by gP,`.
For our yes instance Pand some variable `∈P, let p=s,u1,u2,...,uq,tbe an accepting compu-
tation path on P. If we trace the corresponding inner products along the path, hRs, gP,` i,hRu1, gP,`i, . . . ,
hRt, gP,`i, they will start with value 1and go down to 0. Now if uand vare adjacent states in the switch-
ing network connected by the variable `, then progress at uwith respect to gP,` is the same as progress
at vwith respect to gP,`. This is because the pseudo-distribution defined by gP,` ignores inputs where `
crosses the cut (they have zero “probability”), and all other cuts reach uiff they reach v.
This allows us to invoke a crucial lemma that we call the gap lemma, which states the following.
Fix an accepting path pfor the yes instance P. Then since for every `∈P,hgP,`, Rsi= 1 and
hgP,`, Rti= 0, and for every pair of adjacent states uiand ui+1 along the path, one of these inner
products doesn’t change, then there must exist some node von the path and two variables `1, `2∈P
such that |hgP,`1, Rvi−hgP,`2, Rvi| ≥ 1/|M|. Thus the gap lemma for Pimplies that this state v
behaves significantly differently on the two pseudo-distributions gP,`1and gP,`2of cuts, and therefore
11
this node can distinguish between these two pseudo-distributions. We will let gP=gP,`1−gP,`2be the
pseudo-distribution associated with P. In summary, the gap lemma implies that for every yes instance
P, we have a pseudo-distribution gPand a “complex state” in the switching network which is highly
specific to gP. Thus we have shown property (1) above.
In order to boost the “complex state” argument and get an exponential size lower bound, as explained
earlier, we still need to establish properties (2) and (3). The construction proceeds in two steps: first we
construct functions gP,` satisfying properties (2) but whose norm is too large, and then we fix the norm.
Our construction is the same as in earlier papers, and this is the essential place in the proof where the
pebbling number of the graph comes into play. (For pyramid graphs, the pebbling number is Θ(h),
where his the height of the pyramid.) The construction, while natural, is technical and thus we defer its
explanation to Appendix A.
Now we want to generalize the above argument to switching networks that are allowed to make
errors. Several important things go wrong in the above argument. First, the set Pof pyramids that we
start with above may not be accepted by the network, and in fact it might even be that none of them are
accepted by the network. Secondly, it is no longer true that hgP,`, Rti= 0 as required in order to apply
the gap lemma, because now there may be many cuts which are incorrectly accepted by the switching
network.
The first problem can be easily fixed since if a random pyramid is accepted by the network, then
we can still find a large design consisting of good pyramids, by taking a random permutation of a fixed
design. Solving the second problem is more difficult. In the worst case, it may be that hgP,`, Rti 6= 0 for
all gP,`, and so the gap lemma cannot be applied at all. To address this issue, we will say that a pyramid
is good for a network if it is both accepted by the network, and if hgP,`, Rtiis small (say less than 1/2)
for some `∈P. We are able to prove (by estimating the second moment) a good upper bound on the
probability that hgP,`, Rtiis large, and thus we show that with constant probability, a random pyramid is
good. Then we generalize our gap lemma to obtain an “approximate” gap lemma which essentially states
that as long as the inner products with Rtare not too close to 1, then we can still find a complex state
for Pin the network; in fact, it is enough that one inner product is not too close to 1. Using these two
new ingredients, we obtain average case exponential lower bounds for monotone switching networks for
GEN.
5 Lower Bounds for Exact Computation
In this section we give a simplified proof of the main theorem from [8], which we restate here2.
Theorem 3.1. Let N , m, h be positive integers satisfying m≥h≥3, and let Gbe a DAG on m
vertices with in-degree at most 2and reversible pebbling number at least h. Any sound monotone
switching network for GEN which accepts all graph instances isomorphic to Gmust have at least
Ω(hN/m3)(h−2)/3/O(m)states.
As discussed in the overview, the proof makes use of `-nice vectors, which we proceed to define.
Definition 5.1. Let g:C → Rbe a cut vector and `∈[N]3. Recall that for each cut Cin GEN we
associate a cut instance I(C).
We say that gis `-nice if hg, 1i= 1 and g(C)=0whenever `6∈ I(C).
2The result proved in [8] is stated only for pyramids, but as noted in [7], the proof works for any DAG with in-degree at
most 2once we replace hwith the reversible pebbling number of the graph.
12
In a sense, vectors which are `-nice are “ignorant” of cuts which are crossed by the variable `. This
has the following implication.
Lemma 5.2. Let `∈[N]3and let Mbe a monotone switching network for GEN. If a cut vector gis
`-nice then for any pair of states u,v∈Mconnected by a wire labelled `we have hRu, gi=hRv, gi.
Proof. Suppose u,v∈Mare connected by a wire labelled `. Theorem 2.6 shows that Ru(C) = Rv(C)
whenever `∈I(C). Since g(C)=0whenever ` /∈I(C),
hRu, gi=1
|C| X
C∈C
`∈I(C)
Ru(C)g(C) = 1
|C| X
C∈C
`∈I(C)
Rv(C)g(C) = hRv, gi.
As discussed in the overview, for each graph instance Pof GEN we will come up with `-nice func-
tions gP,` for each `∈P. By tracing out the inner products of gP,` with reachability vectors along an
accepting path for P, we will be able to come up with a complex state specific to P. To find the complex
state, we make use of the following arithmetic lemma.
Lemma 5.3. Let `, m be integers, and let xt,i be real numbers, where 0≤t≤`and 1≤i≤m.
Suppose that for all t<`there exists isuch that xt,i =xt+1,i. Then
max
t,i,j |xt,i −xt,j| ≥ 1
2`max
i|x`,i −x0,i|.
Proof. Let i= argmaxi|x`,i −x0,i|and ∆ = |x`,i −x0,i|. Since
|x`,i −x0,i| ≤
`
X
t=1 |xt,i −xt−1,i|,
there exists t > 0such that |xt,i −xt−1,i| ≥ ∆/`. Let jbe an index such that xt,j =xt−1,j . Then
∆
`≤ |xt,i −xt−1,i|≤|xt,i −xt,j |+|xt−1,j −xt−1,i|,
and so for some s∈ {t−1, t},|xs,i −xs,j| ≥ ∆/2`.
Lemma 5.4 (Gap Lemma).Let Pbe any accepting instance of GEN, and let {g`}`∈Pbe a collection of
vectors indexed by variables in Psuch that for each `∈Pthe corresponding vector g`is `-nice. Let
Mbe any sound monotone switching network computing GEN with nstates, let {Ru}u∈Mbe the set of
reachability vectors for M, and let Wbe an sto tpath in Mwhich accepts P. Then there is a node u
on Wand two variables `1, `2∈Pfor which
|hRu, g`1−g`2i| ≥ 1
2n.
Proof. Denote the nodes on Wby {u1,u2,...,um}, where u1=sand um=t. Let xt,` =hRut, g`i.
Lemma 5.2 implies that for each t < m there exists `such that xt,` =xt+1,` , namely the variable `that
labels the edge connecting utand ut+1. Apply Lemma 5.3 to get a node uand two variables `1, `2such
that
|hRu, g`1i−hRu, g`2i| ≥ 1
2mmax
`|hRs, g`i−hRt, g`i|.
Since Rs≡1and Rt≡0,
|hRu, g`1−g`2i| ≥ 1
2mmax
`|h1, g`i−h0, g`i| =1
2m≥1
2n.
13
We defer the construction of the functions gP,` to Appendix A. The end result is the following
theorem.
Theorem 5.5. Let mand hbe positive integers. Let Pbe a graph instance of GEN with vertex set VP
isomorphic to a graph Gwith mvertices and reversible pebbling number at least h. There exist cut
vectors gP,` for each `∈Pwith the following properties:
1. For any `∈P,hgP,` ,1i= 1.
2. For any `∈P,gP,` is `-nice.
3. For any `∈P,gP,` depends only on vertices in VP.
4. For any `∈P,kgP,` k2≤(9m)h+1.
5. For any `1, `2∈Pand S∈ C of size |S| ≤ h−2,ˆgP,`1(S) = ˆgP,`2(S).
The third property implies that the function constructed by the gap lemma is specific to P. The final
property shows that all the small Fourier coefficients of gP,`1−gP,`2vanish. To take advantage of this
property, we employ a combinatorial design in which any two sets intersect in fewer than hpoints.
Lemma 5.6 (Trevisan [28]).For any positive integers q, m, h with h≤m, there exist qsets Q1, Q2, . . . , Qq⊆
[N], where N=m2e1+ln(q)/h/h, such that |Qi|=mfor each iand |Qi∩Qj| ≤ hfor each i6=j.
We are now ready to put the pieces together to prove an exponential lower bound on the size of
monotone switching networks for GEN that are correct on all inputs.
Proof of Theorem 3.1. Lemma 5.6 gives a design {Q1, . . . , Qq}of size q= ((h−2)N/e(m−1)2)h−2
in which |Qi|=m−1and |Qi∩Qj| ≤ h−2for all i6=j. For each Qi, choose some graph instance
Piisomorphic to Gwhose underlying vertex set is Qi. Apply Theorem 5.5 to each instance Pito get a
collection {gPi,`}`∈Piof cut vectors. Note that m≥h≥3implies that q≥(hN/4em2)h−2.
Let Mbe a sound monotone switching network for GEN of size nwhich accepts all graph instances
isomorphic to G. We apply the gap lemma (Lemma 5.4) to each collection of vectors, which gives a set of
vectors {gPi}q
i=1 and a collection of states {ui}q
i=1 such that hgPi,uii ≥ 1/2nand gPi=gPi,`1−gPi,`2
for some pair of variables `1, `2∈Pi.
The third property in Theorem 5.5 shows that gPidepends only on vertices in Qi, and the final
property guarantees that ˆgP,i(C) = 0 for all |C| ≤ h−2. It follows that for i6=j, the functions gPi, gPj
are orthogonal: if ˆgPi(C),ˆgPj(C)6= 0 then |C| ≥ h−1and C⊆Qi, Qj, contradicting the fact that
|Qi∩Qj| ≤ h−2. Finally, since kgPi,`k ≤ p(9(m−1))h+1 we get kgPik ≤ 2p(9m)h+1.
Since the set {gPi/kgPik: 1 ≤i≤q}is orthonormal, Parseval’s theorem implies that
n=X
u∈M
1≥X
u∈MkRuk2≥X
u∈M
q
X
i=1 hRu, gPii
kgPik2
≥q·1
4n2
1
4(9m)h+1 .
We deduce that
n3≥q
16 ·(9m)h+1 ≥hN
36em3h−2
(27m)−3.
6 Average Case Lower Bounds
In this section we extend the above exponential lower bound to deterministic monotone switching net-
works which are allowed to make errors on some distribution of inputs3. We begin by defining how we
3By Yao’s Minimax Theorem [29] this is equivalent to a lower bound for randomized monotone switching networks.
14
measure the error rates of switching networks.
If xis an instance of GEN, then we will write GEN(x) = 1 to denote that xis an accepting instance,
and GEN(x) = 0 to denote that xis a rejecting instance.
Definition 6.1. Let Dbe a distribution on GEN inputs, and let 0≤ < 1/2be a real number. We say that
a monotone switching network Mcomputes GEN with error on Dif Prx∼D[M(x)6=GEN(x)] = .
For the rest of the section fix a DAG Gon m+ 1 vertices with a unique sink t, in-degree at most 2,
and reversible pebbling number at least h+ 2. We will use the following distribution Dover instances of
GEN: with probability 1/2, choose a random graph instance isomorphic to G, and with probability 1/2
choose I(C)for a random cut C.
Our main result is Theorem 3.3, restated here.
Theorem 3.3. Let αbe a real number in the range 0< α < 1. Let m, h, N be positive integers satisfying
324m2≤Nαand 3≤h≤m. Let Gbe a DAG with mvertices, in-degree 2and reversible pebbling
number at least h.
Any monotone switching network which computes GEN on [N+ 2] with error ≤1/2−1/N 1−α
must have at least Ω(hN/m3)(h−2)/3/O(mN )states.
If Mis a monotone switching network computing GEN with errors on some distribution, say that
a instance Pis good for Mif Pis accepted by the network and hRt, gP,`i ≤ 1−1/N, where Rtis
reachability vector of the target state tin M,`∈Pis any variable, and gP,` is the `-nice vector given
by Theorem 5.5. We implement the techniques at the end of Section 4 by way of three technical results.
First, in Lemma 6.2, we generalize the gap lemma (cf. Lemma 5.4) to monotone switching networks
with errors. Next, Lemma 6.3 gives us a method to construct combinatorial designs (from Lemma 5.6)
which contain only good instances. Finally — and in the most technically involved work of this section
— we show in Section 6.2 how to bound the probability that hRt, gP,`iis bounded away from 1 over
random instances P. To aid understanding, the technique used in Section 6.2 is illustrated in Section 6.1
on a restricted form of the problem. Finally, in Section 6.3 we prove our main result.
We begin with a proof of the generalized gap lemma.
Lemma 6.2 (Generalized Gap Lemma).Let Pbe an accepting instance of GEN, and let {g`}`∈Pbe a
collection of vectors indexed by variables in Psuch that for each `∈Pthe corresponding vector g`
is `-nice. Let Mbe a monotone switching network for GEN with nstates. Let {Ru}u∈Mbe the set of
reachability vectors for M, and let Wbe an sto tpath in Mwhich accepts P. Suppose that for some
`∈Pwe have hRt, g`i ≤ β, where tis the target state of M. Then there is a node uon Wand two
variables `1, `2∈Pfor which
|hRu, g`1−g`2i| ≥ |1−β|
2n.
Proof. Once again, denote the nodes on Wby {u1,u2,...,um}, where u1=sand um=t. Apply
Lemma 5.3 with xt,` =hRut, g`ito obtain a node uand two variables `1, `2such that
|hRu, g`1−g`2i| =|hRu, g`1i−hRu, g`2i| ≥ 1
2mmax
`|hRs, g`i−hRt, g`i| ≥ |1−β|
2n.
The next lemma shows that if a uniformly random instance is good with high probability, then we
can find a block design with “many” good instances.
Lemma 6.3. Let N , m, h, q be positive integers. Suppose that there are qsets S1, S2,...Sq⊆[N]such
that the following holds:
15
1. |Si|=mfor all i∈[q], and
2. |Si∩Sj| ≤ hfor all i6=j.
Additionally, suppose that at most a fraction of the sets [N]
mhave some property P. Then there exists
a collection of q0= (1 −)qsets S0
1, S0
2, . . . , S0
q0⊆[N]for which both properties stated above hold, and
additionally none of the sets S0
ihas property P.
Proof. Let πbe a uniformly random permutation on [N], and for any set S⊆[N]let π(S) = {π(x) :
x∈S}. For each i∈[q], let Xibe the indicator random variable which is 1 if and only if the set π(Si)
has property P. Since πis chosen uniformly at random we have Pr[Xi= 1] = , and so by linearity of
expectation we get
E
π"q
X
i=1
Xi#=
q
X
i=1
E
π[Xi] =
q
X
i=1
Pr
π[Xi= 1] = q.
By the probabilistic method it follows that there must exist qsets U0
1, . . . , U 0
q, such that at most q of the
sets have property P. Therefore, there exist q0= (1 −)qsets S0
1, . . . , S0
q0such that both properties in
the statement of the lemma hold and none of the sets have property P.
The actual result we will need is slightly different, but the same proof will work.
Our goal is to upper-bound the probability that a uniformly random instance is bad, which we do
by upper-bounding the expectation of hRt, fP,`i2for a randomly chosen pointed instance (P, `)and
applying Markov’s inequality. (A pointed instance (P, `)is a graph instance Pisomorphic to Galong
with a variable `∈P.)
For the rest of this section, let Nbe an integer and let Mbe a monotone switching network of size
ncomputing GEN on [N+ 2] with error . For convenience in this section, we assume that s=N+ 1
and t=N+ 2. Recall that Rtis the reachability vector for the target state of M. For a pointed instance
(P, `),gP,` is the vector constructed using Theorem 5.5.
The error results from a combination of two errors: the error 1on yes instances, and the error 2
on no instances. According to the definition of D, we have = (1+2)/2. We record this observation.
Lemma 6.4. Let 1be the probability that Mrejects a random pointed instance, and let 2be the
probability that Maccepts a random cut instance (that is, an instance of the form I(C)for a random
cut C). Then = (1+2)/2.
Proof. Follows directly from the definition of D.
Our proofs crucially rely on the following upper bound on the Fourier coefficients of gP,`.
Lemma 6.5. Let (P, `)be a pointed instance with underlying vertex set D, and let C∈ C be any cut. If
C⊆Dthen ˆgP,` (C)2≤9|C|, otherwise ˆgP,`(C)=0.
Proof. Since gP,` depends only on the vertices D,ˆgP,`(C)=0unless C⊆D, so suppose that C⊆D.
The construction of ˆgP,` in Theorem 5.5 shows that either ˆgP,` = 0 or ˆgP,` =ˆ
fP,`, where fP,` is the
vector constructed in Lemma A.6. In the latter case, the upper bound follows from the lemma.
16
6.1 Warmup
In this section we introduce our technique by proving an upper bound on |E(P,`)hRt, gP,`i|. While this
result isn’t strong enough to prove strong lower bounds for monotone switching networks, it will serve
to introduce the ideas subsequently used to prove the actual lower bounds.
Instead of estimating E(P,`)hRt, gP,`idirectly, we will estimate a sum of many such terms all at
once. Using this device we are able to take advantage of the fact that the large Fourier coefficients of
gP,` appear on larger sets, and larger sets are shared by fewer pointed instances. The sum we are going
to consider includes a pointed instance (P, `)for each subset Qof [N]of size m(recall s, t /∈[N]in this
section).
Definition 6.6. Apointed instance function ℘associates with each set Q∈[N]
ma graph instance P
isomorphic to Gand a variable `∈P.
We will compute the expectation of
X
Q∈([N]
m)hRt, g℘(Q)i=hRt,X
Q∈([N]
m)
g℘(Q)i
over a random ℘(in fact, we will upper-bound this expression for every pointed instance function). This
will allow us to upper-bound E(P,`)hRt, gP,`iusing the following lemma.
Lemma 6.7. Let ℘be a pointed instance function chosen uniformly at random, and let (P, `)be a
random pointed instance. We have
E
℘[hRt,X
D∈([N]
m)
g℘(D)i] = N
mE
(P,`)[hRt, gP,`i].
Proof. Linearity of expectation shows that
E
℘hRt,X
D∈([N]
m)
g℘(D)i= E
℘X
D∈([N]
m)hRt, g℘(D)i=X
D∈([N]
m)
E
℘[hRt, g℘(D)i] = N
mE
S,℘[hRt, g℘(S)i],
where Sis chosen randomly from [N]
m. The lemma follows since ℘(S)is simply a random pointed
instance.
Here is the actual upper bound, which works for any pointed instance function ℘.
Lemma 6.8. Suppose N≥18m2. Let ℘be any pointed instance function. We have
hRt,X
D∈([N]
m)
g℘(D)i ≤ s21 + 18m2
NN
m.
Proof. By applying Parseval’s Theorem and then Cauchy-Schwarz, we get
hRt,X
D∈([N]
m)
g℘(D)i2≤ X
C∈C
ˆ
Rt(C)X
D∈([N]
m)
ˆg℘(D)(C)!2
≤ X
C∈C
ˆ
Rt(C)2! X
C∈CX
D∈([N]
m)
ˆg℘(D)(C)2!.
17
Recall that the network Mcomputes GEN with error 2on cut instances, and so Rt(C) = 1 for 2|C|
many cuts C. By Parseval’s theorem we have
hRt, Rti=kRtk2=X
C∈C
ˆ
Rt(C)2.
Using this, we have X
C∈C
ˆ
Rt(C)2=1
|C| X
C∈C
Rt(C)2=2|C|
|C| =2,
and substituting back yields
hRt,X
D∈([N]
m)
g℘(D)i2≤2X
C∈CX
D∈([N]
m)
ˆg℘(D)(C)2
.
Lemma 6.5 shows that ˆg℘(D)(C)=0unless C⊆D, in which case ˆg℘(D)(C)2≤9|C|. Additionally, we
have the useful estimates N
i≤Niand N−i
m−i≤N
mmi
Ni.
Continuing our bound of hRt,PD∈([N]
m)g℘(D)iand letting i=|C|for any set Cyields
2X
C∈CX
D∈([N]
m)
ˆg℘(D)(C)2
=2X
C∈CX
D∈([N]
m)
D⊇C
ˆg℘(D)(C)2
≤2
m
X
i=0 N
iN−i
m−i2
9i
≤2
m
X
i=0 N
m2Nim2i
N2i9i≤2N
m2m
X
i=0 9m2
Ni
.
We have 9m2/N ≤1/2by assumption, and so we can bound this sum by a geometric series like so:
m
X
i=0 9m2
Ni
≤1 + 9m2
N
∞
X
i=1 1
2i
= 1 + 18m2
N.
Taking square roots finally yields
hRt,X
D∈([N]
m)
g℘(D)i ≤ s21 + 18m2
NN
m.
Corollary 6.9. Suppose N≥18m2. For a random pointed instance (P, `),
E
(P,`)[hRt, gP,`i]≤s21 + 18m2
N.
Proof. Follows directly from Lemma 6.7.
In this way we can actually get a bound on |E(P,`)[hRt, gP,`i]|. However, this bound isn’t helpful for
showing that hRt, gP,`iis often bounded away from 1. It could be that most of the time, hRt, gP,`i= 1,
and rarely hRt, gP,`iattains large negative values. In the following section, we rule out this possibility
by obtaining a bound on E(P,`)[hRt, gP,`i2].
18
6.2 Tensor Square
We now repeat our calculations, this time bounding E(P,`)[hRt, gP,`i2]instead of E(P,`)[hRt, gP,` i].
Markov’s inequality will then imply that hRt, gP,` iis bounded away from 1most of the time.
At first glance it seems that our trick of taking a sum of several different pointed instances fails, since
h·,·i2isn’t linear. We fix that by taking the tensor square of all parties involved.
The tensor product of two cut vectors uand vis the vector u⊗v:C2→Rdefined by (u⊗v)(C, D) =
u(C)v(D). We recall the following useful lemma which connects tensor products to the squares of inner
products.
Lemma 6.10. Let uand vbe cut vectors. Then hu⊗u, v ⊗vi=hu, vi2, and [
f⊗g(C, D) = ˆ
f(C)ˆg(D).
Using tensor products, we are able to extend Lemma 6.7 to a result which is useful for bounding
E(P,`)[hRt, gP,`i2].
Lemma 6.11. Let ℘be a pointed instance function chosen uniformly at random, and let (P, `)be a
random pointed instance. We have
E
℘[hRt⊗Rt,X
D∈([N]
m)
g℘(D)⊗g℘(D)i] = N
mE
(P,`)[hRt, gP,`i2].
Proof. As in the proof of Lemma 6.7, we get
E
℘[hRt⊗Rt,X
D∈([N]
m)
g℘(D)⊗g℘(D)i] = N
mE
(P,`)[hRt⊗Rt, gP,` ⊗gP,` i].
The lemma now follows from Lemma 6.10.
We proceed with the analog of Lemma 6.8, whose proof is similar in spirit to the proof of Lemma 6.8.
Lemma 6.12. Suppose N≥162m2. Let ℘be any pointed instance function. We have
hRt⊗Rt,X
D∈([N]
m)
g℘(D)⊗g℘(D)i ≤ s21 + 324m2
NN
m.
Proof. Since the network Maccepts a random cut with probability 2we have hRt, Rti=2. We can
therefore apply Lemma 6.10 to get
hRt⊗Rt, Rt⊗Rti=hRt, Rti2=2
2.
Using this fact, applying Parseval’s identity, Cauchy-Schwarz, and Parseval again yields
hRt⊗Rt,X
D∈([N]
m)
g℘(D)⊗g℘(D)i2≤ X
C1∈C X
C2∈C
(\
Rt⊗Rt)(C1, C2)X
D∈([N]
m)
(\
g℘(D)⊗g℘(D))(C1, C2)!2
≤2
2X
C1∈C X
C2∈C X
D∈([N]
m)
\
(g℘(D)⊗g℘(D))(C1, C2)!2
.
19
Applying the second conclusion of Lemma 6.10 we get
hRt⊗Rt,X
D∈([N]
m)
g℘(D)⊗g℘(D)i2≤2
2X
C1∈C X
C2∈C X
D∈([N]
m)
ˆg℘(D)(C1)ˆg℘(D)(C2)2
.
Recall from Lemma 6.5 that |ˆg℘(D)(C)| ≤ 3|C|if C⊆D, and otherwise ˆg℘(D)(C) = 0. We can use this
to simplify the sum and get
X
C1∈C X
C2∈CX
D∈([N]
m)
ˆg℘(D)(C1)ˆg℘(D)(C2)2
≤X
C1∈C X
C2∈CX
D∈([N]
m)
D⊇C1∪C2
3|C1|+|C2|2
.
Now, for any two cuts C1, C2∈ C appearing in the above sum, let i=|C1∩C2|, j =|C1\C2|and
k=|C2\C1|. We can rewrite the sum as
X
C1∈C X
C2∈CX
D∈([N]
m)
D⊇C1∪C2
3|C1|+|C2|2
≤
m
X
i=0
m−i
X
j=0
m−i−j
X
k=0 N
i, j, kN−i−j−k
m−i−j−k2
92i+j+k,
where N
i,j,kis a multinomial coefficient. We can use the upper bound N
i,j,k≤Ni+j+kto get
m
X
i=0
m−i
X
j=0
m−i−j
X
k=0 N
i, j, kN−i−j−k
m−i−j−k2
92i+j+k
≤
m
X
i=0
m−i
X
j=0
m−i−j
X
k=0
Ni+j+kN
m2m2(i+j+k)
N2(i+j+k)92i+j+k
≤N
m2m
X
i=0
m−i
X
j=0
m−i−j
X
k=0 81m2
Ni+j+k
.
We can upper-bound this sum by factoring a larger sum:
N
m2m
X
i=0
m−i
X
j=0
m−i−j
X
k=0 81m2
Ni+j+k
≤N
m2m
X
i=0
m
X
j=0
m
X
k=0 81m2
Ni+j+k
≤N
m2 m
X
i=0
81im2i
Ni!
m
X
j=0
81jm2j
Nj
m
X
k=0
81km2k
Nk!.
Since N≥162m2by assumption we can upper-bound these using geometric series, as in the proof of
Lemma 6.8:
N
m2 m
X
i=0
81im2i
Ni!
m
X
j=0
81jm2j
Nj
m
X
k=0
81km2k
Nk!≤N
m21 + 162m2
N3
.
20
Taking square roots, we get
hRt⊗Rt,X
D∈([N]
m)
g℘(D)⊗g℘(D)i ≤ 21 + 162m2
N3/2N
m≤21 + 324m2
NN
m.
Corollary 6.13. Suppose N≥162m2. For a random pointed instance (P, `),
E
(P,`)[hRt, gP,`i2]≤21 + 324m2
N.
Proof. Follows directly from Lemma 6.11.
Applying Markov’s inequality, we immediately get the following corollary.
Corollary 6.14. Suppose N≥162m2. Let (P, `)be a random pointed instance. For any δ > 0,
Pr
(P,`)[hRt, gP,`i ≥ δ]≤δ−21 + 324m2
N2.
6.3 Exponential Lower Bounds
We are now ready to prove our main theorem (Theorem 3.3), which gives an exponential lower bound
on the size of monotone switching networks computing GEN with error close to 1/2.
Proof of Theorem 3.3. Lemma 5.6 gives a design {Q1, . . . , Qq}of size q= ((h−2)N/e(m−1)2)h−2
in which |Qi|=m−1and |Qi∩Qj| ≤ h−2for all i6=j. Since m≥h≥3we note that
q≥(hN/4em2)h−2. Let πbe a random permutation on [N], and let ℘be a random pointed instance
function. For each Qi,℘(π(Qi)) is a random pointed instance, and hence for any δin the range 0< δ <
1,
Pr
π,℘[hRt, g℘(π(Qi))i ≥ δ]≤δ−21 + 324(m−1)2
N2≤δ−21 + 324m2
N2.
Moreover, the probability that ℘(π(Qi)) is rejected by Mis 1. The probability that either of these bad
events happen is at most
τ:= 1+δ−21 + 324m2
N2≤δ−21 + 324m2
N2,
using Lemma 6.4. The technique of Lemma 6.3 shows that for some πand ℘, the number of Qifor
which either hRt, g℘(π(Qi))i ≥ δor ℘(π(Qi)) is rejected by Mis at most τ q. In other words, there exists
a set (P1, `1),...,(Pq∗, `q∗)of q∗:= (1 −τ)qpointed instances with underlying sets Q∗
1, . . . , Q∗
q∗such
that |Q∗
i∩Q∗
j| ≤ h−2for all i6=j.
The rest of the proof directly follows the proof of Theorem 3.1. We can apply Theorem 5.5 to
each instance Piand get a collection of vectors {gPi,`}`∈Pi, and then apply the generalized gap lemma
(Lemma 6.2) to each such collection of vectors, which yields a set of vectors {gPi}q∗
i=1 and a set of states
{ui}q∗
i=1 in Msatisfying hgPi, Ruii ≥ (1 −δ)/2n(recall nis the size of M). This collection of vectors
is orthogonal, and each vector satisfies the upper bound kgPik2≤4(9(m−1))h+1 ≤4(9m)h+1. Since
the set {gPi/kgPik: 1 ≤i≤q}is orthonormal, Parseval’s theorem implies that
n=X
u∈M
1≥X
u∈MkRuk2≥X
u∈M
q∗
X
i=1 hRu, gPii
kgPik2
≥q∗·(1 −δ)2
4n2
1
4(9m)h+1 .
21
We deduce that
n3≥q∗(1 −δ)2
16 ·(9m)h+1 ≥(1 −τ)(1 −δ)2hN
36em3h−2
(27m)−3.
(Compare what we got in Theorem 5.5.) An auspicious choice for δis δ= 1 −1/N. Since ≤
1/2−1/N1−αand 324m2/N ≤Nα−1,
τ≤1 + 324m2
N2≤1 + 1
N1−α1−2
N1−α≤1−1
N1−α.
Therefore
n3≥1
N1−α
1
N2hN
36em3h−2
(27m)−3≥hN
36em3h−2
(27mN)−3.
7 On Randomized Karchmer-Wigderson
For a boolean function f:{0,1}n→ {0,1}let Rf⊆ {0,1}2n×[n]be the following relation associated
with f. If α∈f−1(1) and β∈f−1(0), then Rf(α, β , i)holds if and only if α(xi)6=β(xi). Intuitively,
if αis a 1-input of the function and βis a 0-input of the function, then iis an index where αand β
differ. Similarly, if fis a monotone boolean function, then for α∈f−1(1) and β∈f−1(0) we define
Rm
f(α, β, i)if and only if α(xi) = 1 and β(xi)=0.
Karchmer and Wigderson [18] proved that for any boolean function, the minimum depth of any
circuit computing fis equivalent to the communication complexity of Rf, and similarly for any mono-
tone boolean function, the minimum monotone circuit depth for fis equivalent to the communication
complexity of Rm
f.
We are interested in whether there is an analog of the Karchmer-Wigderson result in the case of
circuits and communication complexity protocols that make errors. That is, is it true that for any boolean
function f, the minimum depth of any circuit that computes fcorrectly on most inputs over a distribution
D, is related to the communication complexity for Rfwith respect to D? And similarly is monotone
circuit depth related to communication complexity in the average case setting?
Such a connection would be very nice for several reasons. First, a communication complexity ap-
proach to proving average case circuit lower bounds is appealing. Secondly, proving lower bounds on the
average case communication complexity of relations is an important problem as it is related to proving
lower bounds in proof complexity. In particular, average case communication complexity lower bounds
for relations associated with unsatisfiable formulas imply lower bounds for several proof systems, includ-
ing the cutting planes and Lov´
asz-Schrijver systems [3,16]. Thus an analog of Karchmer-Wigderson in
the average case setting, together with our new lower bounds for monotone circuits, would imply a new
technique for obtaining average case communication complexity lower bounds for search problems.
In the forward direction, it is not hard to see that if Cis a circuit that computes fcorrectly on all
but an fraction of inputs, then there is a communication complexity protocol for Rf(or for Rm
fin the
case where Cis a monotone circuit) with low error. If the circuit Con inputs αand βyields the correct
answer, then the Karchmer-Wigderson protocol will be correct on the pair α, β.
However, the reverse direction is not so clear. Raz and Wigderson [26] showed that in the non-
monotone case, the Karchmer-Wigderson connection is false. To see this, we note that given inputs α, β,
there is an O(log2n)randomized protocol that finds a bit isuch that αi6=βi, or that determines that
α=β. (Recursively apply the randomized equality testing protocol on the left/right halves of the inputs
in order to find a bit where the strings are different, if such an index exists; a more efficient protocol along
22
similar lines has complexity O(log n).) Using this protocol, it follows that for every boolean function,
Rfhas an efficient randomized protocol. Now by Yao’s theorem, this implies that for every function f
and every distribution D, there is an efficient protocol for Rfwith low error with respect to D. On the
other hand, by a counting argument, almost all boolean functions require exponential-size circuits even
to approximate.
The intuitive reason that the Karchmer-Wigderson reduction works only in the direction circuits to
protocols is that protocols are cooperative while circuits are, in some sense, competitive. We make this
argument more precise in Appendix C.
In the monotone case, it is not as easy to rule out an equivalence between randomized monotone
circuit depth for fand randomized communication complexity for Rm
f. The above argument for the
non-monotone case breaks down here because the communication problem of finding an input isuch
that αi> βi, given the promise that such an iexists, is equivalent to the promise set disjointness problem
where Alice and Bob are given two sets with the promise that they are not disjoint, and they should output
an element that is in both sets. Promise set disjointness is as hard as set disjointness by the following
reduction. Given an efficient protocol for promise set disjointness, and given an instance α, β of Rm
f
where |α|=|β|=n, the players create strings α0,β0of length n+ 1 by selecting an index i∈[n+ 1]
randomly, and inserting a 1in position iinto both αand β. If αand βwere disjoint, running a protocol
for promise disjointness on α0and β0is guaranteed to return the index of the planted 1. Otherwise, the
promise disjointness protocol should return a different index (not the planted one) with probability at
least 1/2. Repeating logarithmically many times yields a protocol that solves set disjointness with low
error.
We will now prove that the Karchmer-Wigderson equivalence is also false in the monotone case.
Theorem 7.1. The monotone Karchmer-Wigderson reduction does not hold in the average-case setting.
That is, there is a monotone function f:{0,1}n→ {0,1}, a distribution Dand an satisfying 0< <
1/2, such that there is an efficient protocol for Rm
fwith error at most with respect to Dbut such that
any subexponential-size monotone circuit for fof depth nhas error greater than with respect to D.
Proof. We will consider the GEN problem on a universe of size N. Our distribution will consist of a
distribution of minterms and maxterms. The minterms will be pyramids of size n, and the maxterms will
be all cuts. We give an efficient randomized protocol for solving Rm
fon the uniform distribution over
minterms and maxterms. A random pyramid Pis generated by choosing mpoints uniformly at random
from [N]and constructing the pyramid triples arbitrarily. (For example, pick a permutation of the m
points, and let this define the pyramid triple.) Add the triples (s, s, vi)for all 1≤i≤h, where vi,
i∈[h]are the elements at the bottom of the pyramid, and add the triple (u, u, t), where uis the element
at the top of the pyramid. A random cut Cis generated by choosing each i∈[N]to be on the s-side with
probability 1/2, and otherwise on the t-side. The instance corresponding to Cis obtained by adding all
triples except for those triples that “cross the cut” – that is, the triples (i, j, k)where i, j ∈Cand k∈C.
Now consider the following protocol for solving Rm
f. On input Pfor Alice, Alice deterministically
selects a set of log mdisjoint triples (i, j, k)in Pand sends them to Bob. Then Bob checks whether
or not any of these triples crosses his cut, and if so he outputs one of these triples, and otherwise the
protocol fails, and Bob outputs an arbitrary triple.
We will call a cut Cbad for Pif the above algorithm fails. For each P, we will upper bound the
fraction of cuts that are bad for P. If (i, j, k)is a triple appearing in P, then the probability over all cuts
Cthat (i, j, k)does not cross Cis at most 1/8(since it is the probability that i, j lie on the s-side, and
klies on the t-side). Since we have log mdisjoint triples, the probability that none of them cross Cis
at most 1/m. Overall, our protocol has communication complexity O(log2m)and solves Rm
fover our
distribution of instances with probability at least 1−1/m.
23
On the other hand, by our main lower bound, any small monotone circuit for solving GEN errs with
constant probability over this same distribution of instances.
We note that the hard examples that were previously known to be hard for monotone circuits with
errors, such as clique/coclique, are not good candidates for the above separation. To see this note that
if yes instances are k-cliques and no instances are k−1colorings, then the clique player needs to send
about √kvertices in order to have a good chance of finding a repeated color, and thus the protocol is not
efficient in the regime where clique/coclique is exponentially hard (for k=n).
We also note that our result implies that the proof technique of [25] does not generalize to prove aver-
age case lower bounds for monotone circuits. This leaves the interesting open question of establishing a
new connection between randomized circuit depth and communication complexity. One possibility is to
consider randomized protocols for Rfor Rm
fwhich succeed with high probability on all pairs of inputs
(the randomized counterpart of the protocol in the proof of Theorem 7.1 fails in some extreme cases),
and connect them to some “distribution-less” notion of randomized circuits.
A Construction of Nice Vectors
In this section we construct the cut vectors which underlie the lower bounds proved in Section 5 and
Section 6, proving Theorem 5.5.
Theorem 5.5. Let mand hbe positive integers. Let Pbe a graph instance of GEN with vertex set VP
isomorphic to a graph Gwith mvertices and reversible pebbling number at least h. There exist cut
vectors gP,` for each `∈Pwith the following properties:
1. For any `∈P,hgP,` ,1i= 1.
2. For any `∈P,gP,` is `-nice.
3. For any `∈P,gP,` depends only on vertices in VP.
4. For any `∈P,kgP,` k2≤(9m)h+1.
5. For any `1, `2∈Pand S∈ C of size |S| ≤ h−2,ˆgP,`1(S) = ˆgP,`2(S).
As explained in the proof outline, the construction proceeds in two steps. In the first step, taken in
Section A.3, we construct vectors fP,` satisfying all the required properties other than the bound on the
norm. In the second step we “trim” the vectors fP,` (by intelligently applying a low-pass filter) to vectors
gP,` which satisfy all the required properties.
The difficult part of the construction is to control the Fourier spectrum of gP,` while guaranteeing
that gP,` is `-nice. In order to control the Fourier spectrum of gP,`, we will construct gP,` as a linear
combination of vectors SCwith known Fourier spectrum. In order to guarantee that gP,` is `-nice, we
will find an equivalent condition which involves inner products with a different set of vectors KD. The
two sets of vectors SC, KDare related by the identity
hSC, KDi= [C=D].
(In linear algebra parlance, {SC}and {KD}are dual bases.) Since gP,` is a linear combination of the
vectors SC, we will be able to verify that gP,` is `-nice using the equivalent condition.
24
For the rest of this section, we fix a graph instance Pwhose underlying vertex set is VP. All cut
vectors we define will depend only on vertices in VP, and so we think of them as real functions on the
set CPwhich consists of all subsets of VP(recall that s, t /∈VP). A vector fdefined on CPcorresponds
to the cut vector f0given by
f0(C) = f(C∩VP).
Henceforth, cut vector will simply mean a vector defined on CP. We endow these cut vectors with the
inner product
hf, gi=1
|CP|X
C∈CP
f(C)g(C).
It is not difficult to check that kf0k=kfk,hf0, g0i=hf, giand that for C⊆VP,ˆ
f0(C) = ˆ
f(C). (All
other Fourier coefficients of f0vanish since f0depends only on VP.)
A.1 Universal Pebbling Networks
We start by describing the cut vectors KDand the condition involving them which is equivalent to being
`-nice. The cut vectors KDare defined by
KD(C) = [D⊆C].
We think of Das a pebbling configuration on the graph G, and of KDas a kind of reachability vector. To
make this identification more precise, we define a monotone switching network related to the reversible
pebbling game on G.
Definition A.1. The universal pebbling network for Pis a monotone switching network MPdefined as
follows. With every subset of vertices D⊆VP∪ {t}we associate a state Din MPand we think of the
vertices in D∪ {s}as the “pebbled vertices at D”. The start state of the network is s=∅, and we add a
dummy target state t.
Given two states D1,D2in MPand a variable `= (x, y, z), we connect D1and D2with an
`-labelled wire if the corresponding pebble configurations D1∪ {s}and D2∪ {s}differ by a single
legal pebbling move which either pebbles or unpebbles the vertex z. Additionally, for any pebbling
configuration Dwhich contains the target point t, we connect the corresponding state Dto the target
state tby a blank wire (i.e. one that is always alive).
Figure 3 shows two adjacent pebble configurations on the pyramid and the corresponding connec-
tions in the universal pebbling network. The pebbling configurations uand von the pyramid have
corresponding states in the network, and since they are reachable from one another using the variable x
we connect the corresponding states in the universal pebbling network.
The vectors KDare not quite reachability vectors4, but they satisfy the following crucial property
which is enjoyed by reachability vectors.
Lemma A.2. Suppose two states D1,D2of the universal switching network for Pare connected by a
wire labelled `. For every cut Csuch that `∈I(C),KD1(C) = KD2(C).
Proof. Let `= (x, y, z), and suppose that D2=D1∪ {z}. Suppose Cis a cut such that `∈I(C).
If KD2(C)=1then clearly KD1(C)=1, since D2⊇D1. If KD1(C)=1then x, y ∈Cs. Since
`∈I(C), this implies that z∈Cs, and so KD2(C)=1.
4For example, let Pbe a pyramid with vertices v31, v32, v31 , v21, v22 , v11, and let C={v31, v32 , v33, v11}. The graph
I(C)doesn’t contain the variables (v31, v32, v21 )and (v32, v33, v22 ), and so the pebbling configuration Cisn’t reachable. Yet
KC(C) = 1.
25
u
v
u
v
xx
x
Figure 3: Adjacent pebbling configurations and universal switching network states
We are now ready to state the property involving KCwhich is equivalent to being `-nice.
Lemma A.3. Let `∈Pbe any variable and let gbe any cut vector. If hKD1, gi=hKD2, gifor any two
states D1,D2connected by a wire labelled `then gis `-nice, and the converse also holds.
Before proving the lemma, we comment that up to the fact that gDis not quite the reachability
vector corresponding to D, the property given by the lemma is the same as being `-nice for the universal
switching network.
Proof. We start by proving the converse. Let `∈P, and suppose that gis `-nice. Let D1,D2be two
states connected by a wire labelled `. Lemma A.2 implies that
hKD1, gi=X
C∈CP
`∈I(C)
KD1(C)g(C) = X
C∈CP
`∈I(C)
KD2(C)g(C) = hKD2, gi.
The other direction is more involved. Let `= (x, y, z), and let Dbe any cut such that ` /∈I(D),
that is x, y ∈Dsand z /∈Ds. The two states Dand D∪ {z}are connected by a wire labelled `, and so
Lemma A.3 implies that
0 = |CP|hKD−KD∪{z}, gi=X
C
([D⊆C]−[D∪ {z} ⊆ C])g(C) = X
C⊇D
z /∈C
g(C).(3)
We use (3) along with reverse induction on |D|to show that g(D) = 0 whenever ` /∈I(D). Given
D, suppose that g(C) = 0 for all C)Dsatisfying ` /∈I(C), or equivalently, z /∈C. Equation (3)
implies that
0 = X
C⊇D
z /∈C
g(C) = g(D) + X
C)D
z /∈C
g(C) = g(D).
26
A.2 Dual Basis
In view of Lemma A.3, we will be interested in the value of hgP,`, KCifor the function gP,` which we
will construct. If K={KC:C∈ CP}were an orthonormal basis like the Fourier basis, then hgP,`, KCi
would be the coefficient of KCin the unique representation of gP,` in the basis K. However, the vectors
KCare not orthogonal. We therefore construct another basis S={SC:C∈ CP}with the following
property:
hKC, SDi= [C=D].
The basis Sis known as the dual basis to K. This property implies that hgP,` , KCiequals the coefficient
of SCin the unique representation of gP,` in the basis S: if
gP,` =X
C∈CP
αCSC
then hgP,`, KCi=αC. The dual basis is given by the following formula:
SD(C)=[C⊆D]· |CP|(−1)|D\C|.
We first prove that the two bases are in fact dual.
Lemma A.4. For any C, D ∈ CP,hKC, SDi= [C=D].
Proof. We have
hKC, SDi=X
E∈CP
[C⊆E][E⊆D](−1)|D\E|=X
C⊆E⊆D
(−1)|D\E|.
If Cis not a subset of Dthen clearly hKC, SDi= 0. If C⊆Dthen
hKC, SDi=X
E⊆D\C
(−1)|D|−|C|−|E|= [C=D].
The other crucial property of the dual basis is its Fourier expansion.
Lemma A.5. For any C, D ∈ CP,ˆ
SC(D)=[C⊆D](−2)|C|.
Proof. We have
ˆ
SC(D) = hSC, χDi=X
E∈CP
[E⊆C](−1)|C\E|(−1)|D∩E|=X
E⊆C
(−1)|C\E|+|D∩E|.
Let C={c1, . . . , ck}. Breaking up the sum over C, we have
ˆ
SC(D) = X
E1⊆{c1}
(−1)[c1/∈E1]+[c1∈E1∩D]·· · X
Ek⊆{ck}
(−1)[ck/∈Ek]+[ck∈Ek∩D].
By considering each sum separately, we see that ˆ
SC(D)=0unless C⊆D. When C⊆D,
ˆ
SC(D) = X
E1⊆{c1}
(−1)[c1/∈E1]+[c1∈E1]·· · X
Ek⊆{ck}
(−1)[ck/∈Ek]+[ck∈Ek]
=X
E1⊆{c1}
(−1) ··· X
Ek⊆{ck}
(−1) = (−2)|C|.
27
A.3 Nice Vector Construction
Let `= (x, y, z)∈P. Our plan is to construct gP,` as a linear combination of the form
gP,` =X
C∈CP
αC,`SC.
Since K∅=1, the property hgP,`,1i= 1 implies that α∅,` = 1. Lemma A.3 shows that for gP,` to be
`-nice, we must have αD,` =αD∪{z},` whenever x, y ∈Ds(recall Ds=D∪{s}). Finally, Lemma A.5
shows that for D∈ CPand any two `1, `2∈P,
ˆgP,`1(D)−ˆgP,`2(D) = X
C∈CP
(αC,`1−αC,`2)[C⊆D]2|C|.
Since we want all small Fourier coefficients to be the same for different `, we need αC,`1=αC,`2for all
small cuts C.
We can satisfy all the required constraints by setting all αC,` to binary values: for each `we will
construct a set A`⊆ CPcontaining ∅and define αC,` = [C∈A`], or equivalently
gP,` =X
C∈A`
SC.
Lemma A.3 shows that for gP,` to be `-nice, the set A`, as a set of pebbling configurations, must be
closed under legal pebbling of z. Lemma A.5 shows that for D∈ CPand any two `1, `2∈P,
ˆgP,`1(D)−ˆgP,`2(D) = X
C∈CP
([C∈A`1]−[C∈A`2])[C⊆D]2|C|.
Since we want all small Fourier coefficients to be the same for different `, we need A`14A`2to contain
only large cuts.
When z=t, the set At:= A`must be closed under legal pebbling of t. Since no cuts in CP
contain t, this implies that in all pebbling configurations in At, the vertex tcannot be pebbled. We will
enforce this by requiring that each C∈Atbe reachable from ∅by using at most h−2pebbles. Since
Ghas reversible pebbling number h, that ensures that tcannot be pebbled. This leads to the following
construction, which constructs the preliminary version fP,` of gP,`.
Lemma A.6. Let Atbe the set of pebbling configurations reachable from ∅using at most h−2pebbles.
We think of Atas a set of states in the universal switching network. For each variable `= (x, y, z ), let A`
be the closure of Atunder wires labelled `(that is, A`contains Atas well as any pebbling configuration
reachable from Atby a wire labelled `). Since the reversible pebbling number of Gis h, no pebbling
configuration in A`pebbles t, and so A`⊆ CP.
Define
fP,` =X
C∈A`
SC.
The functions fP,` satisfy the following properties:
1. For any `∈P,hfP,` ,1i= 1.
2. For any `∈P,fP,` is `-nice.
3. For any `1, `2∈Pand D∈ CPof size |D| ≤ h−2,ˆ
fP,`1(D) = ˆ
fP,`2(D).
28
4. For any `∈Pand D∈ CP,ˆ
fP,`(D)2≤9|D|.
Proof. Since K∅=1,
hfP,`,1i=hfP,`, K∅i= [{s} ∈ A`]=1.
To show that fP,` is `-nice, let D1,D2be two states of the reversible pebbling network connected
by a wire labelled `. We have
hfP,`, KD1i= [D1∈A`],hfP,`, KD2i= [D2∈A`].
Since A`is closed under wires labelled `,D1∈A`if and only if D2∈A`, and so hfP,` , KD1i=
hfP,`, KD2i. Lemma A.3 implies that fP,` is `-nice.
For the third property, let `1, `2∈P. We have
ˆ
fP,`1(D)−ˆ
fP,`2(D) = X
C∈CP
([C∈A`1]−[C∈A`2])[C⊆D]2|C|.
All pebbling configurations in A`14A`2must contain exactly h−1pebbles, and so [C∈A`1]6=
[C∈A`2]implies |C|=h−1. If |D| ≤ h−2then no subset of Dsatisfies this condition, and so
ˆ
fP,`1(D) = ˆ
fP,`2(D).
Finally, let D∈ CP. Lemma A.5 implies that
ˆ
fP,`(D) = X
C∈A`
[C⊆D](−2)|C|≤
|D|
X
k=0 m
k2k= 3|D|.
Similarly we get ˆ
fP,`(D)≥ −3k, implying ˆ
fP,`(D)2≤9|D|.
There are at most roughly mhpebbling configurations in each A`, and the norm of each SCis
|CP|2|C|/2≤2m+h/2. Therefore we expect the norm of each fP,` to be roughly 2m(√2m)h, which is
too high. In the next section, we fix the situation by applying a low-pass filter to fP,`.
A.4 Trimming the Nice Vectors
The vectors fP,` satisfy all the required properties other than having a small norm. In this section we
rectify this situation by relating the Fourier expansion of fP,` to the property of being `-nice. We will
show that a function is `-nice if its Fourier expansion satisfies certain homogeneous linear equations. If
`= (x, y, z)then each equation involves Fourier coefficients C∪Xfor X⊆ {x, y, z}. If we remove the
high Fourier coefficients of fP,` in a way which either preserves or removes all coefficients of the form
C∪X, then we preserve the property of being `-nice while significantly reducing the norm. We need to
be careful to maintain the property that the small Fourier coefficients are the same for all fP,`.
We start by expressing the property of being `-nice in terms of the Fourier coefficients of a cut vector
f. For a variable `, define the cut vector
I`(C) = [` /∈I(C)] = [x∈Cs][y∈Cs][z /∈Cs].
A function fis `-nice if for every C, either `∈I(C)or f(C) = 0. In other words, either I`(C) = 0
or f(C)=0, that is to say I`f≡0. In order to express this condition as a condition on the Fourier
coefficients of f, we use the well-known convolution property of the Fourier transform.
29
Lemma A.7. Let f , g be cut vectors. Define another cut vector f∗gby
(f∗g)(C) = X
D⊆VP
f(D)g(D4C).
Then [
f∗g(C) = |CP|ˆ
f(C)ˆg(C)and c
fg(C) = ˆ
f(C)∗ˆg(C).
Since the Fourier expansion of I`is supported on coefficients which are a subset of {x, y, z}, the
convolution ˆ
I`∗ˆ
fhas a particularly simple form.
Lemma A.8. Let `= (x, y, z)∈P, and let A={x, y, z} \ {s, t}. For a cut vector f, the property of
being `-nice is equivalent to a set of homogeneous equations of the form
X
X⊆A
αi,X ˆ
f(Ci∪X)=0 (4)
for various cuts Ci⊆A. (The cuts Cican be repeated, and the coefficients αi,X can be different in
different equations.)
Proof. Let I`(C) = [x∈C][y∈C][z /∈C]. As we remarked above, fis `-nice if and only if I`f≡0,
which is true if and only if for all C∈ CP,0 = ˆ
I`f(C) = (ˆ
I`∗ˆ
f)(C)(using the convolution property).
Since all cuts C, the set Cscontains sand does not contain t,I`depends only on A, and so the equation
for Creads
0=(ˆ
I`∗ˆ
f)(C) = X
X⊆A
ˆ
I`(X)ˆ
f(X4C).
Write C=D14D2, where D1=C\Aand D2=C∩A. Then the equation for Cis equivalent to
0 = X
X⊆A
ˆ
I`(X4D2)ˆ
f((D14D2)4(X4D2)) = X
X⊆A
ˆ
I`(X4D2)ˆ
f(D1∪X),
which is of the advertised form.
Because of the particular form of I`, one can show that the equations we get really depend only on
D1, but we do not need this fact in the proof.
Lemma A.8 points the way toward the construction of the cut vectors gP,`.
Theorem 5.5. Let mand hbe positive integers. Let Pbe a graph instance of GEN with vertex set VP
isomorphic to a graph Gwith mvertices and reversible pebbling number at least h. There exist cut
vectors gP,` for each `∈Pwith the following properties:
1. For any `∈P,hgP,` ,1i= 1.
2. For any `∈P,gP,` is `-nice.
3. For any `∈P,gP,` depends only on vertices in VP.
4. For any `∈P,kgP,` k2≤(9m)h+1.
5. For any `1, `2∈Pand S∈ C of size |S| ≤ h−2,ˆgP,`1(S) = ˆgP,`2(S).
30
Proof. Let `={x, y, z}and A={x, y, z} \ {s, t}. We define the cut vector gP,` (as a function on CP)
by
gP,` =X
C∈CP
|C\A|≤h−2
ˆ
fP,`(C)χC.
We proceed to verify the properties of gP,` one by one. First,
hgP,`,1i= ˆgP,`(∅) = ˆ
fP,`(∅) = hfP,`,1i= 1.
Second, for each C⊆A, either all or none of the Fourier coefficients {ˆ
fP,`(C∪X) : X⊆A}
appear in gP,`, and in both cases the condition given by Lemma A.8 is maintained, showing that gP,` is
`-nice since fP,` is, by Lemma A.6.
Third, gP,` depends only on VPby construction.
Fourth, Parseval’s identity in conjunction with Lemma A.6 shows that
kgP,`k2=X
C∈CP
ˆgP,` (C)2≤X
C∈CP
|C|≤h+1
ˆ
fP,`(C)2≤X
C∈CP
|C|≤h+1
9|C|≤(9m)h+1.
Fifth, for S∈ CPof size |S| ≤ h−2and `1, `2∈P,
ˆgP,`1(S) = ˆ
fP,`1(S) = ˆ
fP,`2(S) = ˆgP,`2(S),
using Lemma A.6 once again.
B Relating Monotone Switching Network Size to Monotone Circuit Depth
In this appendix we show that a monotone circuit of depth dcan be simulated by a monotone switching
network of size 2d, and a monotone switching network of size scan be simulated by a monotone circuit of
depth O(log2s). The corresponding results for the nonmonotone case (with switching networks replaced
by branching programs) were proved by Borodin [6].
The second result is easy: Given a monotone switching network of size stogether with its input bits
x1, . . . , xn, we construct an s×sboolean matrix A, where Aij = 1 iff i=jor there is a wire between
state iand state jwith a label xk= 1. Now the circuit computes Asby squaring the matrix log stimes.
The network accepts the input iff Aab = 1, where ais the initial state and bis the accepting state.
Now we show how a monotone switching network can evaluate a monotone circuit Cof depth d. We
may assume that Cis a balanced binary tree where the root is the output gate (at level 1) and the nodes
at levels 1 to dare gates, each labelled either ∧or ∨, and each leaf at level d+ 1 is labelled with one of
the input variables xi. The idea is to simulate an algorithm which uses a depth first search of the circuit
to evaluate its gates.
We assume that the circuit is laid out on the plane with the output gate at the bottom and the input
nodes at the top, ordered from left to right. A full path is a path in the circuit from the output gate to
some input node.
The states of the network consist of a start state s, a target state t, and a state upfor each full path p
in the circuit (so there are 2dsuch states).
Definition B.1. We say that a path pfrom a gate gin Cto some input node is initial if pleaves every
∧-gate gon the path via the left input to g. We say that pis final if pleaves every ∧-gate via the right
input to g. If pis a full path, then the state upis initial if pis initial, and upis final if pis final.
31
We say that a state upis sound (for a given setting of the input bits ~x) if for each ∧-gate gon the
path p, if the path leaves gvia its right input, then the left input to ghas value 1.
We will construct our network so that following holds.
Claim B.1.For each input ~x, a state upis reachable iff upis sound.
Now we define the wires and their labels in the network.
The start state sis connected via a wire labelled 1(that is, this wire is always alive) to every initial
state up. Note that every initial state is (vacuously) sound, as required by the claim.
For every nonfinal full path pto some input xi,upis connected by a wire labelled xito every state
up0such that p0is a full path which follows pup to the last ∧-gate gsuch that pleaves gvia its left input,
and then p0leaves gvia its right input and continues along any initial path to a circuit input.
The following is easy to verify:
Claim B.2.It p, p0and xiare as above, and xi= 1, then upis sound iff up0is sound.
Claim B.1 follows from Claim B.2 and the facts that all initial states are sound, and every noninitial
sound state uq0is connected by a wire labelled 1to a sound state uqwhere the path qis to the left of
path q0.
For every final full path pto some input xi, the state upis connected to the target tby a wire labelled
xi. Note that if upis reachable and sound, and xi= 1, then every gate along phas value 1, including
the output gate. This and Claim B.1 shows that the network is sound (the circuit output is 1if the target
tin the network is reachable).
Conversely, if the circuit outputs 1, then there is a sound final full path pwhich witnesses it.
Claim B.1 shows that upis reachable, and so tis reachable. We conclude that the network is com-
plete.
C More on Randomized Karchmer-Wigderson
Our work in Section 7 implies that the Karchmer-Wigderson reduction fails in the randomized setting.
Here is a concrete example. Let f4be the function whose input is an undirected graph Gon nvertices,
and f4(G) = 1 if Gcontains some triangle. The corresponding communication problem is: Alice gets a
graph Awith a triangle, Bob gets a graph Bwithout a triangle, and they have to come up with an edge e
such that e∈Aand e /∈B. We consider the following probability distributions over yes and no inputs:
Alice gets a random triangle, and Bob gets a complete bipartite graph generated by a random partition of
{1, . . . , n}. Here is a deterministic protocol for Rm
f4with success probability 3/4:
1. Alice sends Bob the vertices i, j, k of her input triangle (we assume i < j < k).
2. If (i, j)/∈Bthen Bob outputs (i, j), else Bob outputs (i, k).
The Karchmer-Wigderson transformation produces the following formula:
φ=_
i<j<k
xi,j ∧xj,k,
where xs,t is the input variable corresponding to the edge (s, t). The formula φis true with high proba-
bility over a random no input. Intuitively, Alice can cheat: instead of choosing a bona fide triangle i, j, k,
she identifies two edges (i, j),(j, k)in the input graph, and pretends as if her input triangle were i, j, k.
In order to make this intuition more precise, we digress into the game semantics of monotone circuits.
32
Let Cbe a monotone circuit with input x. We define a combinatorial game a(C)between two
players, Alice and Bob, whose value depends on the input x. The game starts at the root node of the
circuit, and progresses toward the leaves. Alice’s goal is to guide the game toward a leaf corresponding
to a 1input bit, and Bob’s goal is to guide the game toward a leaf corresponding to a 0input bit. At a node
v=v1∨···∨vk, Alice decides which of the nodes v1, . . . , vkto go next to. At a node v=v1∧···∧vk,
Bob decides which of the nodes v1, . . . , vkto go next to. At a leaf f=xi, the input bit xiis revealed:
if xi= 1 then Alice wins, if xi= 0 then Bob wins. Under optimal play, one of the two players always
wins (this is a property of combinatorial games with perfect information). In fact, if C(x)=1then
Alice wins under optimal play, and if C(x) = 0 then Bob wins under optimal play. This can be proved
by induction on the structure of the circuit.
Having identified monotone circuits with games of the form a(C), we describe the Karchmer-
Wigderson reduction as a game. Given a function fand a (correct) protocol Pfor Rm
f, the game a(P)
proceeds as follows. The states of the game are partial transcripts of the protocol P. We only consider
partial transcripts which actually occur for some yes input of Alice and some no input of Bob. The initial
state is the empty transcript. At a partial transcript τ, if it’s Alice’s turn to speak in the protocol, then she
chooses whether to proceed to τ0or to τ1. If one of these, say τ0, never occurs as a partial transcript
of the protocol, then she is forced to choose the other, in this case τ1. When it’s Bob’s turn to speak, he
gets to decide which way to proceed. At a completed transcript τwith output i, the game terminates by
revealing the input bit xi.
The game a(P)has the feature that if f(x)=1then Alice wins, while if f(x)=0then Bob wins.
Since the situation is symmetric, we concentrate on the case that f(x) = 1. We show that Alice has a
winning strategy for a(P). Her winning strategy is simple: at a partial transcript τwhen it’s her turn
to speak, Alice acts according to the protocol P, assuming her input is x. The game terminates at a
transcript τwhich corresponds to some input yfor Bob. Since Pis correct, it outputs a bit isuch that
xi= 1 and yi= 0. Since xi= 1, Alice wins a(P).
There is an important difference between the communication protocol Pand the corresponding game
a(P): in the communication protocol P, both players are cooperating to find the bit iwhich monotoni-
cally distinguishes their inputs. In contrast, the game a(P)is competitive. If f(x) = 1 then Alice wins
since she has a winning strategy, in which she truthfully follows the protocol P. Whatever Bob does is
consistent with some input yfor him, and since Pis correct, at the end the protocol outputs a bit isuch
that xi= 1.
At this point, we come back to our earlier protocol for Rm
F4. The formula φcorresponds to the
following game:
1. Alice chooses vertices i, j, k.
2. Bob chooses whether to reveal xi,j or xj,k.
This game is heavily skewed toward Alice. Suppose xis a bipartite graph which contains two edges
(i, j),(j, k). Alice acts as if her input was the triangle {i, j, k}, and wins the game. Even though for
roughly 3/4of the triangles, the protocol would succeed (the exact probability depends on x), in the
corresponding game Alice always wins, since she gets to choose which triangle to run the protocol
against, and she chooses one for which the protocol fails.
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