Observations on Measure
and
P
Lowness for 2
Jack H. Lutz
Department of Computer Science
Iowa State University
Ames, IA 50011
Abstract
Assuming that k 2 and Pk does not have p-measure 0, it is
shown that BP Pk = Pk . This implies that the following conditions
hold if P2 does not have p-measure 0.
(i) AM \ co-AM is low for P2 . (Thus BPP and the graph isomorphism problem are low for P2 .)
(ii) If P2 6= PH, then NP does not have polynomial-size circuits.
This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International, Microware Systems Corporation, and Amoco Foundation.
1 Introduction
Many widely believed conjectures in computational complexity are \strong"
in the sense that they are known to imply that P 6= NP, but are not known
to follow from the P 6= NP hypothesis. Recent investigations have shown
that a number of these conjectures do follow from the (apparently) stronger
hypothesis that NP does not have p-measure 0. (This hypothesis, written
(NP) 6= 0, is dened in terms of resource-bounded measure, a theory developed in 18] and discussed briey in section 2 below. Intuitively, (NP) 6= 0
holds if NP contains a non-negligible subset of the exponential time class
E2 = DTIME(2polynomial) { the smallest deterministic time complexity class
known to contain NP.) For example, if (NP) 6= 0, it is now known that
NP contains P-bi-immune languages 25] there is an NP search problem that
is not eciently reducible to the corresponding decision problem 3, 23] every Pn ;tt-complete problem for NP ( < 1) is exponentially dense 22]
every Pm-complete problem for NP has an exponentially dense exponential
complexity core 6] and there are problems that are PT-complete, but not
Pm-complete, for NP 23]. These conclusions, which are not known to follow
from P 6= NP or other \traditional" complexity-theoretic hypotheses (e.g.,
the separation of the polynomial-time hierarchy), suggest that (NP) 6= 0
is a plausible scientic hypothesis with substantial explanatory power. (See
22, 6, 20] for further discussion of this hypothesis.)
This paper shows that the hypothesis (NP) 6= 0 also has consequences
involving the complexity classes BP Pk (k 2) and lowness for P2. In fact,
these consequences all follow from the hypothesis that the class P2 does not
have p-measure 0. Since NP P2, the hypothesis (P2) 6= 0 follows from,
and is thus at least as plausible as, the hypothesis (NP) 6= 0.
Section 3 contains the main observation of this paper, which concerns
the eect of the BP-operator on the classes Pk (k 2) of the polynomialtime hierarchy. The BP-operator, introduced by Schoning 31] and discussed
in section 2 below, assigns to each complexity class C a complexity class
BP C , which can be regarded as a \feasibly randomized version" of C . Two
important special-case values of this operator are the bounded-error probabilistic polynomial-time class BPP = BP P and the Arthur-Merlin class
AM = BP NP. Generalizing the proofs by Lautemann 15] and Sipser and
Gacs 32] that BPP P2 \ P2, Schoning 31] showed that, for all k 1,
BP Pk Pk+1. This result, in combination with more elementary facts,
established the inclusion structure depicted in Figure 1.
p
p
p
p
p
p
p
1
Figure 2: Inclusion structure if
(P2) 6= 0
Figure 1: Known inclusion structure
p
2
The hypothesis (P2) 6= 0 simplies this inclusion structure. Specifically, Theorem 3.3 below shows that, if some class Pj does not have pmeasure 0, then the classes Pj Pj+1 : : : are all xed points of the BPoperator. That is, if (Pj ) 6= 0, then for all k j , BP Pk = Pk . In
particular, if (P2) 6= 0, then the situation depicted in Figure 2 holds. Intuitively, Theorem 3.3 says that, if P2 does not have p-measure 0, then it
contains a language that is suciently random to simmulate a BP-operator.
The proof makes essential use of the construction by Nisan and Wigderson 27] of secure pseudorandom generators from languages that are hard to
approximate by circuits.
The remaining observations of this paper, presented in sections 4 and 5,
involve lowness for P2 and follow easily from Theorem 3.3 and recent results
in computational complexity.
The concept of lowness originated in recursion theory and was introduced
to complexity theory by Schoning 29]. A language A f0 1g is low for a
relativizable complexity class C if C (A) = C , i.e., if oracle access to A does
not increase the computational power of C . A class L of languages is then
low for C if C (L) = C , i.e., if every element of L is low for C . Kobler 12] has
recently provided a useful survey of lowness results in complexity theory.
Section 4 concerns the lowness of probabilistic complexity classes. Zachos
and Heller 36] proved that BPP is low for P2. Schoning 30] improved this
by showing that NP \ co-AM is low for P2, whence the graph isomorphism
problem is low for P2 . Klapper 9] strengthed this by establishing that all of
AM \ co-AM is low for P2. More recently, Kobler, Schoning, and Toran 13]
showed that AM \ co-AM is, in fact, low for AM. Theorem 4.2 below notes
that, under the hypothesis (P2) 6= 0, AM \ co-AM is also low for P2.
Thus if (P2) 6= 0 and the polynomial-time hierarchy does not collapse to
P2, then the graph isomorphism problem is not Pm-complete, PT-complete,
or SNP
T -complete for NP.
Section 5 concerns the lowness of self-reducible languages with polynomialsize circuits. Karp and Lipton 8] used self-reducibility to show that, if
the polynomial hierarchy does not collapse to P2, then NP 6 P=Poly,
i.e., NP does not have polynomial-size circuits. Ko and Schoning 11] rened this by showing that every language in NP that has polynomial-size
circuits is low for P2. Very recently, Kobler and Watanabe 14] have signicantly improved upon these results by showing that every self-reducible
language with polynomial-size circuits { in fact, every self-reducible language
in (NP \ co-NP)/Poly { is low for ZPP(NP). Thus, if the polynomial-time
p
P
p
p
p
3
hierarchy does not collapse to ZPP(NP), then NP does not have polynomialsize circuits. In Theorem 5.3 (which follows immediately from Theorem 3.3
and the result of Kobler and Watanabe), it is noted that, if (P2) 6= 0, then
every self-reducible language in (NP \ co-NP)/Poly is low for P2. Thus, if
(P2) 6= 0 and polynomial-time hierarchy does not collapse to P2, then NP
does not have polynomial-size circuits.
p
p
2 Preliminaries
The reader is referred to any of the texts 2, 4, 13, 28] for basic material on
complexity classes, relativized complexity classes, the polynomial-time hierarchy, feasible reductions, self-reducibility, polynomial advice, and (Boolean)
circuits. Oracle circuits are described in 34, 24]. For each k 1, QBFk is
the well-known k-quantied Boolean formula problem. Stockmeyer 33] and
Wrathall 35] have shown that QBFk is Pm-complete for pk , and it is clear
that QBFk 2 E, where E = DTIME(2linear). Other specic terminology and
notation used here include the following.
For languages A B f0 1g, the symmetric dierence of A and B is
A 4 B = (A ; B ) (B ; A)
and the tagged union of A and B is
n
o
n
o
A B = x0 x 2 A x1 x 2 B :
Using the standard enumeration
s0 = s1 = 0 s2 = 1 s3 = 00 s4 = 01 : : :
of f0 1g, each language A f0 1g is identied with its characteristic sequence A 2 f0 1g1 , whose ith bit (i 0) is A i] = if si 2 A then 1 else 0.
The cylinder generated by a string w 2 f0 1g is the set
n
Cw = A f0 1g A0::jwj ; 1] = w
o
where A0::l ; 1] is the string consisting of the rst l bits of A.
Resource-bounded measure was introduced in 18]. Introductions to this
subject may be found in the papers 16, 22, 7, 6, 23, 21, 20], and in the theses
4
26, 5]. For the purpose of this note, it suces to indicate the intuition and
cite a result that is used in the proof of Lemma 3.2.
Intuitively, a set X of languages has p-measure 0 (polynomial-time measure 0) if the following two conditions hold.
(i) If a language A f0 1g is chosen probabilistically according to a
random experiment in which an independent toss of a fair coin is used
to decide membership of each string in A, then the probability is 0 that
A 2 X.
(ii) Condition (i) holds in a manner that can be computationally veried
in polynomial time.
If a set X has p-measure 0, then X \ E is, in a precise sense, a negligibly
small subset of E 18].
More formally, a supermartingale is a function d : f0 1g ! 0 1) such
that, for all w 2 f0 1g ,
d(w) d(w0) +2 d(w1) :
If d is a supermartingale, then the success set of d is
S 1d] =
A lim sup d(A0::l ; 1]) = 1 l!1
and the unitary success set of d is
S 1d] =
d(w)1
n
o
Cw = A (9l 2 N)d(A 0::l ; 1]) 1 :
For any i 0, a real-valued function f : Ni f0 1g ! R is p-computable
if there is a function f^ : Ni+1 f0 1g ! Q such that f^(r k1 ki w)
is computable in time polynomial in r + k1 + + ki + jwj and, for all
r k1 ki 2 N and w 2 f0 1g ,
f^(r k1 ki w) ; f (k1 ki w) 2;r :
De nition. A set X of languages has p-measure 0 if there is a p-computable
supermartingale d such that X S 1 d].
5
The expression p(X ) = 0 means that X has p-measure 0. The expression
p (X ) 6= 0 means that X does not have p-measure 0. (This does not imply
that \ p(X )"Phas some nonzero value.)
A series 1
reals is p-convergent if there is a polyk=0 ak of nonnegative
P
nomial q such that, for all r 2 N 1k=q(r) ak 2;r . The proof of Lemma 3.2
uses the following polynomial-time version of the classical rst Borel-Cantelli
lemma.
Theorem 2.1 (Lutz 18]). Assume that d : N f0 1g ! 0 1) is a function
with the following properties.
(i) d is p-computable.
(ii) For each k 2 N, the function dk , dened by dk (w) = d(k w), is a
supermartingale.
P
(iii) The series 1
k=0 dk () is p-convergent.
Then
!
1 1
\
S 1dk ] = 0:
p
j =0 k=j
The BP-operator, introduced by Schoning 31], is dened as follows. If
C is a class of languages, then BP C is the class of languages A f0 1g
for which there exist a polynomial q and a language B 2 C such that, for all
x 2 f0 1g ,
(2:1)
Prq x x 2 A ()< x y >2 B ] > 32 :
y2f0 1g
(j
j)
(The probability here is computed according to the uniform distribution
on f0 1gq(jxj), using the string-pairing function < x y >= bd(x)01y, where
bd(x) is x with each bit doubled, e.g., bd(110) = 111100.) It is clear that the
BP-operator is monotone (C D ) BP C BP D) and commutes with
complementation (BP co ; C = co ; BP C ). For all \reasonable" classes
C { including all complexity classes discussed in this note { Schoning 31]
has shown that C BP C and that, in inequality (2.1), any real number
2 ( 21 1) can be used in place of 23 without changing the resulting class
BP C .
6
3 The classes BP
P
k
This section shows that, if P2 does not have p-measure 0, then at all levels
k 2 of the polynomial-time hierarchy, BP Pk = Pk . The proof uses the
idea of languages that are hard to approximate by circuits. The key denitions, which were introduced by Nisan and Wigderson 27], are as follows.
De nition. Let B C f0 1g.
1. For n s 2 N, C is sB -hard at n if, for every n-input oracle circuit with size(
) s,
1 ; 1s where L(
B ) is the set of inputs on which with oracle B outputs
1 and C=n = C \ f0 1gn . (If s = 0, this holds trivially because the
right-hand side is ;1)
2. The hardness of C at n relative to B is
L(
B ) 4 C=n
> 2n;1
HCB (n) = max s 2 NjC is sB -hard at n :
De nition. For 0 < < 1 and B f0 1g , the relativized hardness class
HB is dened by
n
o
HB = C f0 1g HCB (n) > 2n a.e.
where \a.e." (\almost everywhere") means that the condition holds for all
but nitely many n 2 N.
The following result was proven via explicit construction of a pseudorandom generator.
Theorem 3.1 (Nisan and Wigderson 27]). For all 0 < < 1 and all
A f0 1g, if EA \ HA 6= , then PA = BPPA .
Theorem 3.1 has been useful in several recent investigations, and has
focused some attention on the condition EA \ HA 6= . Lutz 17] showed
that, for 0 < < 13 , the (nonrelativized) class H has pspace-measure 1, so
7
if E \ H = , then E has measure 0 in ESPACE. Lutz 19] showed that, for
0 < < 13 , the set of all A satisfying EA \ HA = has pspace-measure 0.
This result was recently improved by Allender and Strauss 1], who proved
that, for 0 < < 31 , the set of all A satisfying EA \ HA = has p-measure 0.
The following lemma is a small, but useful, extension of this fact.
Lemma 3.2. For all 0 < < 13 and all S 2 E,
n
p
o
A EA \ HA S = = 0:
Proof. For brevity, the notation and calculations of 19] are followed, while
using the test language of 1].
Let 0 < < 13 and S 2 E. Without loss of generality, assume that is
rational. For each A f0 1g, dene the test language
n
o
C (A) = x pad(x) 2 A where
Let
pad(x) = x102 x :
j j
n
o
X = A C (A) 62 HA S
all A f0 1g, it suces to
Since C (A) 2 EA for
For each n 2 N, dene the sets
:
show that p(X ) = 0.
OCIRC(n) = f
j
is a novel n-input oracle circuit with size(
) 2n g n
f0 1gn
o
jDj 2n;1 (1 ; 2;n )
DELTA(n) = D
:
(An n-input oracle circuit is novel if it is functionally distinct from all those
preceding it in a standard enumeration.) It was shown in 19] that there is a
constant k0 2 N such that, for all k = 2n k0,
j OCIRC(n) j j DELTA(n) j 2;k e;k :
1
4
For each 2 OCIRC(n) and D 2 DELTA(n), dene the set
n
o
Y D = A L(
AS ) 4 D = C (A)=n 8
(3:1)
and for each k 2 N, let
8 S
<
Xk = :
D Y D
if k = 2n if k is not a power of 2
where the union is taken over all 2 OCIRC(n) and D 2 DELTA(n). It is
easy to see that
1 1
\
X=
Xk (3:2)
j =0 k=j
so Theorem 2.1 can be used to show that up(X ) = 0.
Dene d : N f0 1g ! 0 1) as follows, writing dk (w) for d(k w).
(i) If k < k0 or k is not a power of 2, then dk (w) = 0.
(ii) If k = 2n k0 and jwj < 2k+1 , then dk (w) = e;k .
(iii) If k = 2n > k0 and jwj 2k+1 , then
1
4
dk (w) =
X
D
Pr(Y D Cw )
where the sum is taken over all 2 OCIRC(n) and D 2 DELTA(n),
and the conditional probabilities Pr(Y D jCw ) are computed according
to the random experiment in which a language B f0 1g is chosen
probabilistically, using an independent toss of a fair coin to decide
membership of each string in B .
The following four claims are veried below.
Claim 1. d is p-computable.
Claim 2. For each k 2 N, dk is a supermartingale with dk () e;k .
1
4
Claim 3. For all k k0 , Xk S 1 dk ].
Claim 4. X
T1j=0 S1k=j S 1dk ].
9
Assume for a moment that Claims 1-4 are true. By Claim 2, the series
k=0 dk () is p-convergent. It follows by Claim 1, Claim 4, and Theorem
2.1 that p(X ) = 0, completing the proof of Lemma 3.2. Thus it suces to
prove Claims 1-4.
Proof of Claim 1. In the denition of d, it is clear that cases (i),
(ii), and (iii) can be distinguished in time polynomial in k + jwj. In case (i),
the computation of dk (w) is then trivial, and in case (ii), standard numerical
techniques suce to compute an approximation of dk (w) to within 2;r in
time polynomial in r + k + jwj. Attention is thus focused on case (iii).
In case (iii), for each xed and D, all oracle queries in the computation
of L(
AS )4D concern strings si with jsij 2n k, whence i < 2k+1 jwj.
If A 2 Cw and such a query concerns membership in A, then the answer is
already determined by w. If such a query concerns membership in S , then,
since S 2 E, the answer can be computed in 2O(jsi j) = 2O(k) = jwjO(1) time.
Thus, for each xed and D, the conditional probability Pr(Y D jCw ) can
be exactly computed in time polynomial in jwj as follows: If Y D \ Cw = ,
i.e., the condition A 2 Cw forces L(
AS ) 4 D 6= C (A)=n, then this is
determined in time polynomial in jwj, and Pr(Y D jCw ) = 0. Otherwise,
Pr(Y D jCw ) = 2;m , where m is the number of strings x such that w does
not determine membership of pad(x) in A, and this, too, can be determined
in time polynomial in jwj. Thus Pr(Y D jCw ) can be computed in jwjO(1)
time for each and D. By (3.1), there are fewer than jwj dierent values of
and D, so it follows that dk (w) can be computed in time polynomial in jwj
in case (iii). This completes the proof of Claim 1.
Proof of Claim 2. Let k 2 N. If k < k0 or k is not a power of 2,
then dk is trivially a supermartingale, so assume that k = 2n k0. Let
w 2 f0 1g. There are three cases.
P1
1. If jwj < 2k+1 ; 1, then dk (w) = dk (w0) = dk (w1) = e;k , so dk (w) =
1
2 dk (w0) + dk (w1)].
2. If jwj 2k+1, then a routine calculation with conditional probabilities
shows that dk (w) = 12 dk (w0) + dk (w1)].
1
4
3. If jwj = 2k+1 ; 1, then dk (w) = e;k and, for b 2 f0 1g,
1
4
dk (wb) =
X
D
10
Pr(Y D Cwb ):
(3:3)
In this case, the length of wb ensures that, for A 2 Cwb and each xed
and D, the bits of wb completely determine the set L(
AS ) 4 D,
while determining none of the 2n = k bits of C (A)=n. Thus, for each
D, and b 2 f0 1g, Pr(Y D jCwb ) = 2;k . It follows by (3.1) that
dk (w) = e;k 21 dk (w0) + dk (w1)]:
1
4
The above three cases conrm that dk is a supermartingale. It is clear
that dk () e;k .
Proof of Claim 3. Let k k0 . If k is not a power of 2, then Claim 3 is
trivially armed, so assume that k = 2n . Let A 2 Xk , and x 0 2 OCIRC(n)
and D0 2 DELTA(n) such that A 2 Y D . Fix l 2 N suciently large
that L(
AS ) 4 D0 and C (A)=n are completely determined by the string
wl = A0::l ; 1]. Then l 2k+1, so
1
4
0
dk (wl) =
X
0
Pr(Y D jCwl ) Pr(Y D jCwl ) = 1:
0
D
0
Thus A 2 S 1dk ].
2 X . Then A 2 Xk for innitely many
k. It follows by Claim 3 that A 2 S 1dk ] for innitely many k, whence
A 2 \1j=0 1k=j S 1dk ].
Proof of Claim 4. Let A
This completes the proof of Lemma 3.2.
The following result is the main observation of this paper.
Theorem 3.3. If 2 j k and p(Pj ) 6= 0, then BP Pk = Pk .
Proof. Assume the hypothesis, and let
n
o
X = A EA \ HA QBFk = ;1
where = 14 . By Lemma 3.2 and the hypothesis, p (X ) = 0 and (Pj ) 6= 0,
so there exists a language A 2 Pj ; X Pk ; X . Since A 62 X ,
p
6= EA \ HA QBFk EAQBFk \ HA QBFk ;1
;1
11
;1
so by Theorem 3.1,
PAQBFk = BPPAQBFk :
;1
;1
It follows that
Pk BP Pk = BPPQBFk
BPPAQBFk = PAQBFk
= Pk
;1
;1
;1
Corollary 3.4. If (P2) 6= 0, then for all k 2, BP Pk = Pk .
Assuming that (P2) =
6 0, the inclusion relations depicted in Figure
2 follow from Corollary 3.4 and the inclusion relations in Figure 1. The
operators P 9P and 8P also behave as one would expect in Figure 2. That is
(still assuming that (P2) =
6 0), the identities P(AM) = P2, 8P AM = P2,
p
p
p
etc. all hold. It should be noted, however, that Figure 2 cannot be used as
\casually" as Figure 1, because Figure 2 does not relativize. For example,
even if (P2) 6= 0 in the unrelativized case, Ko 10] has shown that there is
an oracle A such that, for all k 0, BP Pk (A) 6 Pk+1(A).
p
4 Lowness of AM co AM
\
;
In this section it is shown that, if P2 does not have p-measure 0, then
AM \ co ; AM is low for P2. The demonstration is easy, using Theorem 3.3
and the following known result.
Theorem 4.1 (Kobler, Schoning, and Toran 13]). AM \ co-AM is low for
AM.
The following observation is now easily established.
Theorem 4.2. If (P2) =6 0, then AM \ co-AM is low for P2.
Proof. Assume the hypothesis. Then, by Theorems 4.1 and 3.3,
NP(AM \ co-AM) AM(AM \ co-AM) = AM
= BP NP BP P2 = P2
p
12
so
P2 P2(AM \ co-AM) = P(NP(AM \ co-AM))
P(P2) = P2:
It was recently shown by Allender and Strauss 1] that, if (P2) 6= 0,
then BPP P2. The following corollary extends this result.
p
Corollary 4.3. If (P2) 6= 0, then BPP is low for P2.
Proof. This follows immediately from Theorem 4.2 and the fact that BPP AM \ co-AM.
The graph isomorphism problem is known to be in NP \ co-AM 13],
which is contained in AM \ co-AM. This gives the following corollaries.
Corollary 4.4. If (P2) =6 0, then the graph isomorphism problem is low
p
for P2.
p
Corollary 4.5. If (P2) 6= 0 and P2 6= PH, then the graph isomorphism
problem is not Pm- complete, PT-complete, or SNP
T -complete for NP.
p
p
Note that, in each of Corollaries 4.3, 4.4, and 4.5, the added hypothesis
(P2) 6= 0 has allowed P2 to replace P2 in a previously known result.
5 Lowness and Polynomial Advice
The relationship between uniform and nonuniform complexity is one of the
greatest enigmas of computational complexity. A principal component of
current understanding of this relationship is the proof by Karp and Lipton
8] that, if P2 6= PH, then NP 6 P=Poly. That is, if the polynomial-time
hierarchy does not collapse to P2, then NP does not have polynomial-size
circuits. The following recent result allows a signicant weakening of Karp
and Lipton's hypothesis.
Theorem 5.1 (Kobler and Watanabe 14]).
1. Every self-reducible language in (NP\co-NP)=Poly is low for ZPP(NP).
13
2. If k 1, and ZPP(Pk ) 6= PH, then Pk 6 (Pk \ Pk )=Poly.
Corollary 5.2. If ZPP(NP) 6= PH, then NP 6 (NP \ co-NP)=Poly.
In this brief section, it is noted that the hypothesis p(P2) 6= 0 allows
P
2 to replace ZPP(NP) here.
Theorem 5.3.
1. If (P2) 6= 0, then every self-reducible language in (NP \ co-NP)=Poly
is low for P2.
2. If k 1, p(Pk+1) 6= 0, and Pk+1 6= PH, then Pk 6 (Pk \ Pk )=Poly.
p
Proof.
1. Assume the hypothesis, and let A 2 (NP\co-NP)=Poly be self-reducible.
Then by Theorems 5.1(1) and 3.3 (in that order), P2(A) ZPP(NP(A))
= ZPP(NP) BPP(NP) = BP P2 = P2.
2. Assume the hypothesis. Then, by Theorem 3.3, ZPP(Pk ) BPP(Pk+1 )
= BPPk+1 = Pk+1 6= PH, so by Theorem 5.1(2), Pk 6 (Pk \Pk )=Poly.
Corollary 5.4. If (P2) 6= 0 and P2 6= PH, then NP 6 (NP\co-NP)=Poly.
Thus, if (P2) =
6 0 and the polynomial-time hierarchy does not collapse
p
to
P2,
p
then NP does not have polynomial-size circuits.
6 Conclusion
The following two questions arise immediately from the observations presented here.
1. Assuming that (NP) 6= 0, can P2 be replaced by a smaller class, e.g.,
$P2, in any or all of the above observations?
2. What is the relationship between the hypotheses (NP) 6= 0 and
(P2) 6= 0? Are they equivalent, or is the latter in some sense weaker?
It is to be hoped that this paper is a small rst step toward a comprehensive understanding of lowness properties under strong, measure-theoretic
hypotheses.
p
p
p
14
Acknowledgement
I thank two referees and the SUNY-Bualo complexity theory seminar for
several remarks that improved this paper.
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