Why is Bo olean Complexity Theory Dicult?
L.G. Valiant
Aiken Computation Lab oratory
Harvard University
Cambridge, MA 02138
1 Intro duction
In the last decade substantial progress has b een made in our understanding of re-
stricted classes of Bo olean circuits, in particular those restricted to have constant
depth (Furst, Sipser, Saxe [FSS81], Ajtai [Ajt83], Yao [Yao85], Hastad [Has86 ],
Razb orov [Raz87], Smolensky [Smo87]) or to b e monotone (Razb orov [Raz85], An-
dreev [And85], Alon and Boppana [AB87], Tardos [Tar88 ], Karchmer and Wigderson
[KW88]). The question arises, p erhaps more urgently than b efore, as to what ap-
proaches could b e pursued that mightcontribute to progress on the unrestricted
mo del.
In this note we rst argue that if P 6= NP then any circuit-theoretic pro of of this
would have to b e preceded by analogous results for the more constrained arithmetic
mo del. This is b ecause, as we shall observe, there are proven implications showing
that if, for example, the Hamiltonian cycle problem (HC) requires exp onential circuit
size, then so do es the analogous problem on arithmetic circuits. Since the set of valid
algebraic identities in the latter mo del form a prop er subset of those in the former, a
lower b ound pro of for it should b e strictly easier.
In spite of the ab ove relationship the algebraic mo del is often regarded as an
alternative, rather than a restriction of the Bo olean mo del. One reason for this is
that sp eci c computations are usually understandable in one of these mo dels, and
not in b oth. In particular, the main p ower of the algebraic mo del derives from the
p ossibility of cancellations, and it is usually dicult to express explicitly how these
help in computing combinatorial problems. Our second aim in this note is to give
an example of an algorithm, namely the Samuelson-Berkowitz metho d for computing
Research supp orted by the National Science Foundation NSF-CCR-89-02500, the Oce for Naval
Research ONR-N0014-85-K-0445, the Center for IntelligentControl ARODAAL 03-86-K-0171 and
byDARPAAFOSR 89-0506. 1
the determinant, where the intermediate terms that are computed but ultimately
cancelled by the arithmetic circuit can b e exhibited explicitly in combinatorial terms.
The ease of computing the determinant can b e attributed to the existence of such
an auxiliary set of monomials with certain computational prop erties. A pro of of an
exp onential lower b ound on the complexity of a p olynomial that is b elieved to b e
hard, such as the p ermanent or the Hamiltonian circuit p olynomial, would involve
establishing the nonexistence of such an auxiliary set. It is dicult to imagine how
such a pro of might go.
Finally,we observethatinlow-level complexity, the arguments giving precedence
to studying the algebraic mo del no longer hold. A ma jor op en problem area is that
of proving for some explicit problem that it cannot b e computed by unrestricted
Bo olean circuits simultaneously in size O (n) and depth O (log n). We describ e, via
some conjectures, one candidate approachtowards proving suchalower b ound for
problems such as sorting. Analogous conjectures exist for the algebraic mo del, but
resolution of those would not imply the same for the Bo olean case.
2 Algebraic Structures
Let x ; ;x b e a set of indeterminates and S a set of constants. Let and be
1 n
two binary op erators. We de ne a circuit or a straight line program syntactically as
a nite sequence of instructions of the form.
f := g op h i 2f1; ;Cg
i i i i
where, for each i; op 2f ; g,andg ;h 2fx ; ;x g[ S [ff ; ;f g. In
i i i 1 n 1 i 1
other words a circuit is a sequence of C binary instructions, where each argumentof
each one is either an indeterminate, a constant, or the result of the execution of an
instruction earlier in the sequence. The complexity of a circuit is C , the number of
instructions.
Such a circuit can b e interpreted in several ways. In this pap er we shall assume
that S is a commutative ring with identity. Then each instruction f canbeidenti ed
i
with the p olynomial that is computed at f ,if and are interpreted as the ring
i
op erations in the p olynomial ring S [x ; ;x ].
1 n
Among natural multivariate p olynomials whose complexity in this mo del is of
interest are Hamiltonian circuits (HC), the p ermanent (PERM) and the determinant
(DET). These are de ned over a matrix X of indeterminates fx ; ;x g where x
11 nn ij
can b e thought of as representing edge (i; j ) of the complete directed graph G on
n
no des f1; ;ng. They are de ned as follows: 2
n
X Y
PERM(X ) = x ;
i; (i)
" i=1
n
X Y
HC(X ) = x
i; (i);
i=1
"
has 1 cycle
n
Y X
#( )
x DE T (X ) = ( 1)
i; (i);
i=1 "
where is the set of p ermutations : f1; ;ng!f1; ;ng and #( )isthe
number of even length cycles in .
Clearly the monomials of PERM and DET corresp ond to cycle covers in G ,while
n
these of HC corresp ond to Hamiltonian circuits. These p olynomials can b e interpreted
in anyringS . An imp ortantcaseis S = GF[2], the nite eld having two elements,
in which case PERM = DET since then 1=1.
Eachsuch problem is actually a p olynomial family indexed bythenumber of
indeterminates. When we talk ab out the p ermanentwe mean the family PERM
1
PERM PERM ; where the subscript denotes the number of variables. The com-
4 9
putational complexityofsuch a family is determined by the family of circuits, one
for each p olynomial, that are the smallest circuits for eachmember. Thus PERM is
p olynomial time computable if and only if there is a family of circuits, of size growing
p olynomially in n, that computes PERM.
Di erentchoices of S allow for di erent circuits.. For example PERM is p olyno-
mial time computable for S =GF[2] but is not known to b e so for any ring S whose
characteristic is not a p ower of 2 (see [Val79]).
We shall de ne three complexity classes of families of p olynomials in which all
co ecients are integral multiples of unity: ARP is the class computable by p olynomial
size circuits in all rings, GFP is that computable by p olynomial size circuits for S =
GF[2], and GFB is that computable by p olynomial size circuits for S=GF[2], where
2
the algebra is assumed to ob ey the extra axiom x = x. By de nition, clearly,
ARP GF P GF B :
Now GFB is equivalent to the class of Bo olean functions of p olynomial circuit
2
size, since the p olynomial ring over GF[2] with x = x gives Bo olean algebra over
f0; 1g and the op erations and in GF[2] give a complete Bo olean basis. Our
observations here are rst that some natural p olynomials in this algebra have natural
combinatorial interpretations, and, second, that their complexity can b e related to
NP-completeness. 3
In particular, a GFB circuit for HC computes nothing other than the parityofthe
numb er of Hamiltonian circuits in a graph. In other words if the indeterminates x of
ij
such a circuit are set to 1 or 0 according to whether edge (i; j ) is present in the graph
then the circuit will compute 1 or 0 according to whether the numb er of Hamiltonian
circuits is o dd or even. Furthermore, the following is implicit in [VV86] since the
randomized reductions used there can b e simulated by small circuits [Adl78]:
Theorem 1 HC 2 GF B ) NP has polynomial size circuits.
Denoting the class of Bo olean functions of p olynomial Bo olean circuit size bypC
[Val79], we conclude, therefore, that since NP j pC ) HC 2j GFB, any e orts at
proving the former should b e retargeted at rst proving the latter or indeed anyof
its even more restricted versions:
NP j pC ) HC 2j GF B ) HC 2j GF P ) HC 2j ARP
We note that NP j pC is equivalenttothe P 6= NP question but with Turing
uniformity removed from the de nitions. It is argued in [Val79] that the nonuniform
versions of such complexity questions as P 6= NP are at least as natural as the
uniform versions. For example, the nonuniform version of NP, there denoted by
pD (for p olynomial de nable,) has a completeness class that includes the original
Hamiltonian circuits problem, just as NP has. Our conclusion, therefore, is that
unless the explanation of the p ossible intractability of NP is to do with uniformity,
one should seek it rst in the more restricted structure of ARP or GFP.
3 Cancellations in the Samuelson-Berkowitz Al-
gorithm
Any Bo olean circuit can b e reexpressed eciently as a Bo olean circuit over and
in GF[2]. In almost any ecient such circuit the computation and cancellation of
unwanted terms by the op erations plays a central role. Unfortunately,thewayin
which ecient computations exploit this facility is little understo o d. It is dicult
even to nd interesting examples where the structure of the cancelled terms can b e
exhibited explicitly.
In this section we study the one example wehave found in a nontrivial computation
where the terms that are computed but ultimately cancelled have a natural charac-
terization. The algorithm is due to Samuelson [Sam42] and adapted byBerkowitz
[Ber84] (see also Eb erly [Eb e85 ]), and computes the determinantinARP. (Note that
other standard algorithms, such as Gaussian elimination, either use division or work
only in certain rings.) 4
The algorithm for computing det(X )isasfollows. WeletB (0 k n 1) b e
k
the principal (n k ) (n k ) minor of X .Thenwe de ne an (n k ) 1 matrix C
k
and a 1 (n k ) matrix D for each k (1 k n 1) as follows
k
!
B C
k k
B =
k 1
D X
k n k +1;n k +1
where X denotes the (i; i) elementofmatrix X .
ii
For each k (1 k n)we de ne T to b e the (n +2 k ) (n +1 k ) matrix
k
de ned as follows:
8
0 if i> j +1
>
>
>
<
1 if i = j +1
(T ) =
k
ij
>
X if i = j
n k +1;n k +1
>
>
:
j i 1
D B C if i< j
k k
k
It turns out that the co ecients of the characteristic p olynomial are given bythe
(n +1) 1 matrix
n
Y
T
k
k =1
where the (1,1) term gives the determinant.
The combinatorial interpretation of this computation of the determinant can b e
derived by noting that it is made up of the sum of terms of the form:
(T ) (T ) (T )i ;i (T ) (T ) ()
1 1;i 2 i ;i 3 2 3 n 1 i ;i n i ;1
1 1 2 n 2 n 1 n 1
Now the sequence of subscripts 1;i ;i ; i ; 1 imp oses constraints on the pro d-
1 2 n 1
ucts of matrix elements that o ccur in the term having this sequence of subscripts.
From the de nition of T it can b e seen that (T ) = X and (T ) =
k k ii n k +1;n k +1 k ij
j i 1
C if i< j.Interpreted as paths in the underlying complete graph G ; (T ) D B
k n k i;i k
k
is a self-lo op at no de n k + 1, while (T ) consists of closed walks of length j i +1
k ij
going through only no des 1; 2; ;n k + 1, and going through the last no de n k +1
exactly once. In b oth cases we can interpret (T ) as a set of closed walks of length
k ij
j i + 1 since a self lo op is of length one.
If s is the size of a closed walk generated by T ;s b eing zero if i = i +1,
k k k k 1 k
then clearly
s = i i +1:
k k k 1
Since 1 = i = i = 1 in (), it follows that
n 5
n n
X X
(i i )= (s 1) = 0:
k k 1 k
1 1
P
n
Hence s = n.Furthermore the sign of the term in ()isgiven by( 1) to
k
1
P
the p ower of p = (s 1) with summation over all k such that s 2. Hence
k k
s 1
k
eacheven walk contributes a factor ( 1) = 1 and eachoddwalk a factor of +1.
Therefore, the pro duct () contributes a factor
# even length walks
( 1)
which is exactly the same factor as in the determinant, if the only sets of walks
counted are the sets of disjoint cycles.
To summarize, we de ne an m-lo op in a graph with no des f1; ;ng to b e a closed
walk going through no de m exactly once, and through every r>mzero times. A
loop-cover of G is a set of m-lo ops, at most one for each m, such that the sum of the
lengths of the lo ops is n. Note that a lo op may rep eat no des and edges and that its
length is the numberofedgesoccurringinitallowing for multiplicity.
We claim that what the algorithm computes is the set of terms that corresp ond
to lo op covers of G , eachtermhaving sign + or according to whether the lo op
n
cover has even or o dd number of loops of even length.
~
Nowamulti-set of n edges E corresp onds to more than one lo op cover. For
example the set of six edges shown b elow
1
2
3
4
5
6
has six distinct lo op covers. As can b e veri ed, two of these corresp ond to single lo ops
(of size 6), three corresp ond to two lo ops (of sizes 1 and 2 resp ectively in all cases) and 6
one to three lo ops. Since all the lo ops involved haveeven length the algorithm will
assign p ositive sign to the three covers with an even numb er of them, and negative
sign to the three with an o dd numb er of them. In other words these lo op covers will
cancel out.
Now the terms in the determinant corresp ond to lo op covers that consist only of
cycles that are disjoint. Clearly such sets of edges have only one lo op cover, and the
algorithm will give it the correct sign.
From the correctness of the algorithm and our combinatorial interpretation of the
terms computed byitwe conclude that
~
Theorem 2 If E is any multiset of n edges in G then
n
~ ~
1. E is a cycle cover , E forms exactly one loop cover,
~ ~
2. E is not a cycle cover , E forms an even number of loop covers, exactly a half
of which have an odd number of even length loops.
We note in conclusion that the Samuelson-Berkowitz algorithm is not multilinear, in
the sense that it computes p owers of the variables higher than one. It is an op en
problem as to whether a multilinear ARP algorithm exists for the determinant.
4 Simultaneous Lower Bounds on Size and Depth
Wehave argued that for NP-complete problems such as Hamiltonian circuits, sup er-
p olynomial lower b ounds on Bo olean circuits imply similar lower b ounds on more
restricted algebraic mo dels of computation. Hence one would exp ect that resolving
the former problem should b e mathematically more tractable then resolving the latter.
In low-level complexity,however, this formal relationship app ears to break down, and
there is fuller justi cation for working separately on b oth Bo olean and algebraic
problems. A case in question is that of simultaneous lower b ounds on size and depth.
For Bo olean circuits it a ma jor op en problem to nd an explicit family of sets of
Bo olean functions such that there is no family of Bo olean circuits over a complete
basis computing them that has size O (n) and depth O (log n)simultaneously. Here
n is the numb er of arguments plus functions. Belowwe shall givea combinatorial
conjecture the truth of whichwould imply suchlower b ounds for problems including
shifting and sorting. Analogous conjectures have b een prop osed in the algebraic
context [Val77] and some progress made on them ([Fri90]).
Consider a bipartite graph G with no de set X [ Y where X = fx ; ;x g and
1 n
Y = fy ; ;y g denote input variables and output functions resp ectively. Supp ose
1 n
the edges are de ned implicitly by a mapping where (y ) X is the set of input
i
no des that are adjacentto y in the graph.
i 7
Nowwede nem Bo olean functions f (x); ;f (x) and n further Bo olean func-
1 m
tions g ; ;g ; where g has m+ j (y ) j Bo olean arguments, such that, with some
1 n i i
); ;f (x); (y )): In other words wehave m func- abuse of notation, y = g (f (x
m i i i 1
tions of the inputs, and each output y can b e an arbitrary function of these m common
i
bits and of the inputs (y )towhich it is connected directly in G.
i
Wesay that G realizes permutation with m common bits if there exist
f ; ;f ;g ; ;g such that for all i,1 i n
1 m 1 n
x = g (f (x ); ;f (x); (y )):
i 1 m i
(i)
In other words, for the xed G and given one can nd the appropriate Bo olean
functions fg g; ff g such that the fy g realize the p ermutation of fx g, for all truth
i j i i
assignments to the fx g.
i
1 "
Now it app ears that if G is sparse, say with degree less than n for some "> 0,
and if m is small enough (e.g. m< n=2) then the common bits form an information
b ottleneck, and G cannot realize all p ermutations. In particular:
Conjecture 1 If G has degree 3 then thereisapermutation such that G does not
realize with n=2 common bits.
Conjecture 2 If G has degree 3 then there is a cyclic shift such that G does not
realize with n=2 common bits.
0 0 1+"
Conjecture 1 and 2 Conjectures 1 and 2 but for G with O (n ) edges for some
">0.
0
Now Conjecture 2 would imply that the following shifting problem has no circuit
of size O (n) and depth O (log n). The shifting problem is de ned with n + dlog ne
2
inputs x ; ;x ;s ; ;s where the value of s de nes in binary the amount
0 n 1 1
dlog ne
2
s to b e shifted. In other words output y equals x where s is the shift.
i i s mo d n
To prove the implication we app eal to Prop osition 6.2 in [Val77]which implies
that for any ">0, in any n-input n-output graph family of size O (n) and depth
O (log n) some n=2 no des and adjacent edges can b e removed so that fewer than
1+"
O (n ) input output pairs remain connected. Hence if such a computation graph
existed for shifting (i.e. if weidenti ed x ; ;x of the shifter as the inputs of the
1 n
graph) then we could identify the n=2 removed no des as the common bits. Each
setting of the shift bits fs g could b e seen as setting the functions ff g and fg g so
i i i
0
as to contradict Conjecture 2 .
0 0
Conjecture 1 is a weaker statement than Conjecture 2 , and might b e easier to
prove. Potentially it could yield a lower b ound on sorting, sayof n=(2 log n)numbers 8
each of size 2 log n. The conjecture as stated is not quite enough but the reader can
verify that it can b e adapted in various ways so as to imply sucha lower b ound.
As our conjectures imply,littleisknown ab out the p ermutations that can b e
realized even for low degrees such as three. The degree one case can b e analyzed
completely. The canonical case is the identity graph G( (y )= x )which, with n=2
i i
common bits, can realize such p ermutations as y = x ;y = x by letting
2i 2i 1 2i 1 2i
f = x , y = f x ;y = f x .
i 2i 1 2i i 2i 2i 1 i 2i 1
The degree two case has b een analysed partially by K. Kalorkoti.
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