Measure on Small Complexity Classes, with Applications for BPP

 

Martin Strauss Eric Allender

Department of Mathematics Department of Computer Science

Rutgers University Rutgers University

New Brunswick, NJ 08903 New Brunswick, NJ 08903

[email protected] [email protected]

Abstract not have prop erty Q has zero in E.

This sometimes presents itself as an easier

We present a notion of resource-boundedmea-

task than explicitly building a set in E having

sure for P and other subexponential-time classes.

prop erty Q. For example, Lutz and Mayor-

This generalization is based on Lutz's notion of

domo [LM] were able to use this version of

measure, but overcomes the limitations that cause

the probabilistic metho d to prove stronger

Lutz's de nitions to apply only to classes at least

results concerning the density of hard lan-

as large as E.Wepresent many of the basic prop-

guages than had b een proved previously.

erties of this measure, and use it to explore the

 Certain interesting notions of intractability

class of sets that are hard for BPP.

can b e formulated only in the context of

Bennett and Gil l showed that almost al l sets are

resource-b ounded measure. Lutz and Juedes

hard for BPP; Lutz improved this from Lebesgue

[JL] studied \weakly complete" sets in E,

measuretomeasureon ESPACE. We use our

which are sets in E that are hard for more

measure to improve this stil l further, showing that

than a measure-zero subset of E.

for al l > 0, almost every set in E is hard



S



n

for BPP, where E = DTIME2 ; which



<

 Many separation results and hierarchy the-

is the best that can be achieved without showing

orems in complexity theory have stronger

that BPP is properly containedinE. A number

versions provable in the context of resource-

of relatedresults are also obtained in this way.

p

b ounded measure. For example, the  -

m

complete sets are a measure-zero subset of

cn

E [JL], and DTIME2  is a measure-zero

1 Intro duction

subset of E [L92]. However, for many other

separations, no corresp onding measure result

Resource-b ounded measure theory was intro-

is known. Often, as in [JL], when a measure-

duced in [L90a] and has proved to b e a useful

theoretic strengthening of a separation result

to ol in complexity theory. Among the reasons for

is p ossible, the pro of gives enlightening addi-

studying resource-b ounded measure, we mention

tional information concerning the complexity

the following:

classes involved.

 The probabilistic metho d can b e applied only

 Unproven but plausible hyp otheses suchas

if there is a meaningful notion of probability

\P 6= NP" or the stronger hyp othesis \the

on the underlying space. To show that a set

p olynomial hierarchy is in nite" provide

inEhaving some prop erty Q exists, it is suf-

useful information concerning complexity-

cient to show that the class of sets that do

theoretic prop ositions. That is, \X  the



Research supp orted by NSF grant CCR-9204874. p olynomial hierarchy collapses" is taken as

evidence for :X . Lutz [L93a] argues that sketch Lutz's formulation. For more details, con-

the hyp othesis \NP is not a measure zero sult [L92].

O 1

n

subset of DTIME2 " a stronger hy- The intuition b ehind Lutz's formulation of

p othesis than P 6= NP is plausible and o ers

resource-b ounded measure is that a measure-zero

a great deal of explanatory p ower.

set is a set such that, for all k , the set can b e cov-

ered by a collection of intervals whose sizes sum

k

Unfortunately, the formulation of resource-

to 2 , where the \intervals" are easy to compute

b ounded measure studied by Lutz and others ap-

in some sense.

plies only to complexity classes at least as large

To make this more precise, x an enumeration

1

as E, which greatly limits the range of questions

fs g of all words, and let p osx denote the

i

i=0

that can b e explored in this framework. The goal

p osition of x in this ordering. A language L is

of the work presented here is to remedy this situa-

identi ed with its characteristic : an in-

tion by providing a meaningful notion of measure

nite sequence ! = ha i with a =1i s 2 L:

i i i

for P and other sub exp onential time classes, and

th

Let w [i::j ] denote the string of the i through

demonstrate its usefulness.

th

j bits of w: Let y v w indicate that y is an

Section 2 presents Lutz's de nition of resource-

initial substring of w .

b ounded measure, explains why it do es not ex-

The sets we will try to measure are sets of lan-

tend in anyobvious waytoprovide a measure

guages i.e., sets of , and the analogs of

on smaller classes, and then shows how these ob-

\intervals" in this context are the sets of the form

stacles were overcome to provide a measure on P

C = f! w xg, which is called the \cylinder at

x

and other classes. This section concludes with a

x". Thus C is the set of languages with the jxj

x

discussion of how natural and robust our de ni-

initial words according to x and the other words

tion of measure is, and a comparison of it to the

arbitrary.

work of Mayordomo [M,M2], where a measure on

The next few de nitions describ e what it

PSPACE is de ned.

means in this context for a set to b e covered, for

Section 3 presents our pro of that sets that are

all k ,by a collection of intervals whose sizes sum

hard for BPP are abundantinE for every >0,



k

to 2 .

as well as related results for PSPACE.

Section 4 presents our results concerning pseu-

De nition 1 A density-system is a function

dorandom sources for BPP.



d f0; 1g ! [0; 1; where{

i ;;i

r

1

Section 5 includes a discussion of whether the

notion of measure intro duced in this pap er rel-

1. The subscripts are natural numbers.

ativizes in a meaningful way. That is, we dis-

2. The argument is consideredapartial charac-

cuss the question of whether nonrelativizing re-

teristic sequence of a language.

sults might b e obtained using this notion of mea-

sure.

d w 0+d w 1

 

3. d w = :



2

For Lutz's measure, this de nition is equiva-

2 De ning Resource-Bounded Mea-

lent to the de nition in [L92]; see also Subsection

sure

2.5. A density-system is called an n-DS accord-

ing to the numb er of subscripts. A 0-DS is a

2.1 Background Concerning Measure

density function. Note there is no mention here

of resource b ounds.

In order to present our new de nitions of

A density function covers X if X

resource-b ounded measure and explain the ob-

S

C : A nul l cover of X is a 1-DS such

stacles to extending Lutz's de nitions to sub ex- w

fw j dw 1 g

k

p onential complexity classes, it is necessary to that for all k , d covers X and d   2 : A

k k 2

set of languages has Leb esgue-measure zero i it mo dify Lutz's de nitions, merely using smaller

hasanull cover. Of course, all recursive complex- resource b ounds C . For instance, to de ne a

ity classes are countable and thus have Leb esgue- measure on P, one would consider density func-

O 1

measure zero. Thus it was necessary for Lutz to tions computed in DTIMElog n. This fails

restrict the class of density systems in order to to work, b ecause

de ne measures on complexity classes.

 The usual convention see, e.g., [AJ] for

1

In de ning notions of measure for E, ES-

having sublinear-time Turing machines com-

PACE, and other large complexity classes, Lutz

pute some function f is to have them rec-

de nes a -measure for various classes of func-

ognize the language fx; i; b : bit i of f xis

tions . Roughly, the sets with -measure zero

bg.However it is easy to see that the class

are the sets that havea null cover that can b e

of functions computed in this wayby, for

approximated very closely by a function in .

instance, p olylog-time-b ounded machines, is

Ifischosen to b e the p olynomial-time com-

not closed under addition and subtraction,

putable functions, Lutz obtains a measure on E;

which seems to b e necessary for many con-

if  is chosen to b e the functions computable

structions. Similar problems arise when one

O 1

log n

in space 2 , then he obtains a measure on

uses the usual binary notation or scienti c

O 1

n

DSPACE2 , and in general functions com-

notation for the numeric values of the den-

puted in space or time ff log n:f 2Cgfor a

sity functions. Our solution is to have the

suciently nice class C of functions yield a mea-

run time b ound the length of the output, and

sure on space or time ff n: f 2Cg.

to express numb ers as the di erence of two

In order to justify his claim that the notion

formal sums of p owers of two, which allows

he de nes do es, in fact, constitute a reasonable

us to p erform the necessary arithmetic op er-

notion of measure on a X , Lutz

ations in the restricted time available.

[L92] notes that if  is chosen to b e the class of

al l functions, then his de nitions are equivalentto

 Using this representation for numeric values,

Leb esgue measure, and he also formulates three

one obtains a system that quite p ossibly does

\measure axioms" that his system satis es:

de ne a measure on P. It unfortunately seems

quite dicult to verify that P itself do es not

M1 Easy unions of nul l sets are nul l.

have measure zero whichwould violate Mea-

sure Axiom 3. This is discussed in more

M2 Singleton sets of easy languages are nul l.

detail in [AS], but the central problem lies

with an observation made previously in [M]:

M3 X itself is not nul l.

the binary sequences that are constructible

O 1

in DTIMElog n corresp ond not to sets

Our goal in de ning a measure on small complex-

in P, but rather to \word-decreasing self-

ity classes is to formulate a system that is as sim-

reducible" sets, some of which are hard for

ple and natural as p ossible, while still satisfying

E [Ba]. This motivates our notion of limiting

these axioms.

the dependency set size, which allows us to

When faced with the task of de ning measure

obtain sub exp onential b ounds on the com-

on classes smaller than E, it is natural to try to

plexity.

1

The classes of sets that we are interested in measur-

ing are complexity classes, which are closed under nite

2.2 Formal De nitions of the Measure

variants and thus have measure zero or one if they are

measurable at all. For this reason, and for the reason that

The preceding paragraphs motivate some de-

an extra layer of complication is involved in making the

tailed de nitions of a class of functions computed

notion of measure precise for sets of nonzero measure, we

by sublinear-time Turing machines. In order for

follow Lutz's lead in de ning only what it means for a set

to have measure zero or one. sublinear-time machines to p erform interesting 3

computation, we follow the usual convention of chines running in time f log n for some f 2F,

providing these machines with random access to w  and let C  b e the class of functions d

i ;:::;i

1

l

their input; that is, the machines have an \ad- computed by machines whose runtime and dep en-

dress tap e," and if i is written in binary on the dency set size are b oth b ounded by functions of

address tap e, then the machine can in unit time the form f logi + :::+ i + jw j for some f 2F.

1 l

move its read/write head to bit p osition i of the

Note that if the functions in F are at least ex-

input. Among other things, such machines can,

p onential, then C = C , and hence our de -

in logarithmic time, compute the length of their

nitions coincide with those of [L92] for complexity

2

input [Bu]. In order to avoid uninteresting tech-

classes at least as large as E.



nicalities regarding enco ding of pairing functions,

If f is a function in C , where f : f0; 1g !

we adopt the convention that a machine comput-

f0; 1g, then f de nes a constructor  , where

ing a k -ary function is provided with k input tap es

 w = wf w . A constructor sp eci es the se-

j

with an \address tap e" for each input tap e. The

quence that is the limit as j !1 of  . This

running time of such a machine must b e p olylog-

gives rise to the class RC , which is the class

arithmic in m, where m is the sum of the lengths

of languages whose characteristic sequences are

of its k inputs. Note that bycho osing a suitable

given by some constructor in C . The de ni-

enco ding, such a machine can b e simulated bya

tions presented ab ovewere designed to maintain

machine with a single input tap e. The machines

the following imp ortant prop erty of the system

we consider will write their output on a write-only

develop ed in [L92]:

output tap e; thus the output is restricted to b e

of length b ounded by the running time.

Prop osition 1 RC = C .

Given a machine M and natural number n, de-

Pro of.Any language in C has a trivial construc-

ne a dependency set G f0;:::;ng to b e a

M;n

tor that ignores all but the length of its input, so

set such that for each i 2 G [fng; and each

M;n

wehave C RC :

word w of length n, M can compute M w [0::i]

On the other hand, given such a constructor ;

querying only input bits in G \f0; 1; 2; :::ig:

M;n

we can decide if x 2 R  as follows: Start run-

Note that for all M and n, there is a unique min-

posx1

ning  on w =  : When  needs bit i

imal dep endency set for M and n, which can eas-

of w; recursively call  on w [0; ::i 1] to gener-

ily b e computed by expanding the tree of queries

ate that bit. Unwinding this recursion e ectively

that one obtains by assuming b oth p ossible val-

generates the dep endency set G ; whichbyhy-

;jw j

ues for each queried bit. In what follows, we let

p othesis has size at most f log jw j  f jxj. For

G denote this minimal dep endency set. Given

M;n

a function f computed by machine M ,wemay

2

One technicality that needs to b e addressed concerns

abuse notation and sp eak of G instead of G ,

f;n M;n

the manner in which machines determine the length of

where this causes no confusion. Note our conven-

their input. If one were to use the usual binary search

approach to determine the length of the input, then the de-

tion ab out paired inputs to M requires that if

p endency sets cannot b e kept small while allowing any sort

M computes a subscripted function d ;G gets

k;r

of interesting computation. Our approach to this problem

a cadre of subscripts matching d's: G for

M;jw j;k ;r

is to have the length of the input b e given explicitly as

d w : Often in practice G is indep endent

an argument to the machine. We follow the useful con-

k;r M ;n;k ;r

vention of [L92] that \subscripts" to density functions are

of k and r ; in that case these subscripts will b e

presented in unary; however the reader should note that

suppressed.

providing the length of a unary argumentobviates any

need to provide the argument itself. We continue to de-

Given a complexity class C of the form

termine the complexity as if the subscripts were presented

DTIMEF  where we will always assume that

in unary. Since all but one of the inputs to the machine

F is a set of time-constructible functions such

are in unary, the dep endency set is only of interest for this

2

that f n 2F f n 2F, let C be

one non-unary input. However, the dep endency set size is

the class of functions computed byTuring ma- allowed to dep end on the length of the unary inputs. 4

each string y in the dep endency set, we require Theorem 4  C  6= 0.

C 

at most time f jy j, for total computation time

2

ab out f jxj , whichisinF by the assumed

Pro of. of Theorem 2. Given a language L 2C;

closure prop erties.

we will construct a null cover d w  of the sin-

k

gleton set fLg as follows: Decide memb ership of

Wenow make precise the notion of a density

eachword x such that p os xisapower of 2 and

system DS b eing easy to compute. AC -

is at most jw j. Supp ose there are s of these. Out-

computation of an n-DS d is an n + 1-

i ;;i

n

1

sk s

put 2 if bits 1; 2; 4; :::2 of w are all consistent

^

subscripted function d satisfying

i ;;i ;r

n

1

with L, and output 0 otherwise.

^

It is easy to check that this yields a cover; we

 d w  is computable in C , and nu-

i ;;i ;r

n

1

show here that d 2 C : Since L 2Cbyhy-

meric output is represented as a pair a; b

p othesis, we can check memb ership of any single

representing the dyadic rational a b, where

word x in time f jxj for some f 2C: This in-

eachofa and b are represented as a formal

volves no queries to the input w of d. We can

sum of p owers of 2.

in p olylog time check all log-manywords whose

r

^

jd w  d w j 2 .

i ;;i i ;;i ;r

n n

1 1

lexicographic p osition is a p ower of 2. Also, we

query the input at the bits whose p ositions are

Note that a C -computation do es not haveto

powers of 2, generating log-many p oints in the

satisfy the average law item 3 of De nition 1,

dep endency set. Note that the transitive closure

though it has to b e close, dep ending on r:

of this set of bits is trivial; no more p oints are

For any complexity class C ,wesay that a set X

added.

is C -null if there is a C -computation of a 1-

DS d w covering X such that for all k , d  

k k

Pro of. of Theorem 3. Let d b e a 2-DS wit-

j;k

k 3

2 : In this case we write  X = 0:

C 

nessing that X is a C -union of C -null sets. Let

If a class X has nonzero measure, we can talk

1

X

ab out another set Y having \measure 0 in X ,"

0

d d w : w =

j

j;k +2 +1

k

and we write  Y jX = 0; if  Y \ X = 0

C  C 

j =0

or \measure 1 in X ,"  Y jX = 1; if  X n

C  C 

Y =0.

0

We will show d isaC -null cover of X:

0

It is immediate that d is a 1-DS, and that

2.3 The Measure Axioms Are Satis ed

0 k 0

d   2 . Now, to see that d covers X ,

k

note that if ! is any sequence in X , then ! is

A set X is a C -union of C -nul l sets if X =

S

1

in some X , and for all k there is a w v !

j k

X ; and there is a 2-DS d so that d

j j;k j;k

j =0

and d w   1. Now note that for all k , k

j;k k

covers X with value 2 ; and d has a C -

j j;k

0 0

d w   1, and thus d isanull cover of

j

^

k +2 +1

k

computation d :

j;k ;r

0

X . It remains only to show that d has C -

computations.

Theorem 2 For each L 2C, the singleton set

De ne the computation

fLg is C -nul l.

log r +jw j

X

Theorem 3 If X is a C -union of C -nul l sets,

0

^ ^

d w = d w :

j j

j;k +2 +1;r +2 +2

k;r

then X is C -nul l.

j =0

3

This is a slightchange of notation from that used by

Lutz and others. For example, if C is taken to b e E, then

^

If d is computable in time f log n with dep en-

C =C  is the class of functions computed in p oly-

0

^

dency set size also b ounded by f log n, then d

nomial time, and if X is a measure-zero subset of E, we

2

is computable in time O log nf log n , and is

would express that as  X  = 0, while [L92] would

E

express this as  X = 0. in C by the closure prop erties assumed of F .

p 5

Mo difying [L92], one can let in this subsection that the class of P-measurable

sets is an extremely limited class. However, the

logr +jw j

X

following theorem and its corollaries and also the

 = d j w ;

j;k +2 +1

results of the next section show that measure-

j =0

zero results can b e established for some imp ortant

classes of sets.

0 0

^

and show jd w   j and j d w j are each

k k;r

r +1

at most 2 :

Theorem 5 Let C = DTIMEF .Iff 2F, then

 DTIMEf  = 0:

C 

Pro of. of Theorem 4. We show that if d is any

C -computable density function with d < 1

Pro of. It will suce to express DTIMEf n

then we can construct a language in C not covered

as a C -union of C -null sets. But this is

by d:

easy. Let M ;M ;::: b e a clo cked enumeration

1 2

As in [L92], nd q; l such that

of DTIMEf n machines, and let

l

8

d

if any bits of w in p osi-

>

>

>

>

tions numb ered f1, 2,

>

>

0

>

Let ax jxj + l +3; and de ne the desired

k

<

4, :::,2 g is inconsis-

language L by d w =

j;k

tent with LM 

> j

>

>



>

>

>

1aw 

^ ^

>

1 if d w 0 > d w + 2

:

aw  aw 

blog jw jck

 w =

2 otherwise.

0 otherwise.

The result follows.

Note that  is clearly a constructor in C ,

and thus de nes a language L in C .

k

Corollary 6 For al l k ,  DTIMEn  = 0:

P

Finally,wehave to show L is not covered by

n

d; i.e., for all n wehave d  < 1: Here the

Corollary 7 Let 0 < <, and let E denote



S



inductive argument of [L92] works unchanged.

n

. Then  E = 0: DTIME2

E 

<



Note that one can also de ne a measure anal-

It is also imp ortant for our applications to note

ogous to our measure on time-b ounded classes,

that the \Borel-Cantelli-Lutz Lemma" see [L92]

using space b ounds, as opp osed to time b ounds.

holds also for our measure:

In this way, one can obtain a measure 

PSPACE

for PSPACE. Mayordomo has previously de ned

Lemma8 Let C = DTIMEF , where F con-

a notion of measure for PSPACE, where, rather

tains no superexponential function. Let fX g be

j

than limiting the size of the dep endency sets for

acol lection of not-necessarily nul l sets, where

the machines computing the density functions,

X is coveredbyd for d 2 C . Suppose m a

j j

instead she requires that the density functions

\modulus of convergence" is an increasing func-

b e computed by p olylogspace-b ounded machines

tion of the form f log n for some f 2F, and for

with one-way access to the input [M]. Let us de-

al l k we have

note her measure  .

PSPACE 

1

X

k

d  < 2 :

j

2.4 Elementary Facts Concerning the

j =mk 

Measure

Then

1 1

\ [

Wehave established that our notion of measure

  X =0:

j

C 

satis es the measure axioms, but the reader may

t=0 j =t

wonder if enough sets are measurable for the mea-

sure to b e useful in any setting. Indeed, we show 6

The pro of of this is the same as in [L92],

Figure 1: Tree in Theorem 9

but it is imp ortant here to note that for sub ex-

p onential complexity classes it seems unavoid-

w



1



able that the mo dulus b e sublinear. Thus, for











the Borel-Cantelli-Lutz Lemma to b e applied to



w

3



X

sub exp onential-time classes, the sets X must

j X 

X

X 

X

X 

be covered by a \rapidly-vanishing" collection of

X

X 





covers. This restriction that the values d  tend

j



so rapidly toward zero makes it far from obvious

w

2

that any such functions in C  can exist; a con-

H

H

H

crete example is presented in the pro of of The-



 H



 H



orem 11. The restriction that F contain no su-  H



 H

H

w

4

p erexp onential functions is necessary; the claim

X

X

O n

X

2

X

is in fact false for F =2 . The pro of in

X

X

X

X

[L92] requires the resource-b ounding functions to

b e closed under comp osition. It is p ossible to

formulate conditions on m and d that allow the

lemma to apply to more general classes of re-

the average lawat w there's w at the same level

2 4

source b ounds.

as w such that dw 

3 4 2

Due to the extremely limited computation

Since G G ,itisnow easy to see that

power of sublinear-time machines, it is often quite

d;i

d;jw j

for all j 62 G , dw [0::j 1] = dw [0::j ].

easy to prove that certain sets are not P-null.

d;jw j

Since the language L 2 P constructed in the

Below, we present one concrete example.

pro of of Theorem 4 excludes x on the all-but-

Theorem 9 The set SPARSE, of languages hav-

p olylog-many places where dL[0::posx 1]0 =

ing at most a polynomial number of words of any

dL[0::posx 1]1 and therefore have common

given length, is not P-nul l in P.

value dL[0::posx 1]; we conclude L is sparse.

Pro of. Let d b e a P density function with

Note this says more than \no P density-

d < 1. We will construct a sparse set in P

system covers SPARSE;" since we constructed

that is not covered by d.For clarity of exp osition,

^

an uncovered language in P,we conclude that

assume here that d = d ; the general case is not

r

no P density-system covers the smaller set

signi cantly more complicated. Also, it is shown

SPARSE \ P; i.e., that SPARSE has nonzero mea-

in [AS2] that any P null cover can b e replaced

sure in P. A closer lo ok at this argument also

by one satisfying this condition; this is discussed

shows that the sparse set that is constructed is,

more in Subsection 2.5.

in fact, P-printable [HY],[AR], and hence also

First we show that for anyword w ,if i is the

the class K[log, p oly] of languages containing only

maximum elementof G , then dw [0::i] =

d;jw j

strings of low time-b ounded Kolmogorov com-

dw [0::j ] for any j in the range i  j jw j.

plexity [BB, HH] do es not have measure zero in P.

Fix w = w ; and let w = w [0::i] see Figure 1.

1 2

In contrast, a direct argument shows that for all

Let T b e the complete tree at w of height jw ji;

2 1

k

k , the set of n -sparse languages including the

we will show that d is constanton T: By de nition

set of all tally languages is P-null.

of dep endency set, d is constant on the leaves of T;

and by the average lawatw this common value Observe also that the pro of of Theorem 9 works

2

equals dw : For each w 2 T; wehave dw is equally well for space-b ounded machines, and

2 3 3

at least the average of d over the leaves succeeding thus SPARSE is not a measure zero subset of

w ; i.e., dw   dw : If dw  >dw  then by PSPACE using the PSPACE measure. On the

3 3 2 3 2 7

other hand, SPARSE is easily seen to b e null us- 2.5 Robustness, Alternative Formula-

tions and Auxiliary Axioms

ing the  measure of [M]. The sub ject

PSPACE 

of the relation b etween our PSPACE measure and

There are manychoices that must b e made in

that of [M] is taken up again in Section 2.5.

making the notion of a measure precise. The de -

Theorem 9 also has the following corollary.

nitions in the preceding subsections re ect one set

0

Theorem 10 P-uniform AC is not P-

of choices, but it is instructive to consider other

measurable.

ways a de nition could have b een formulated, to

Pro of. Since the class of P-printable sets is con- see if the class of measure-zero sets varies under

0

these changes.

tained in P-uniform AC ; we conclude that P-

0 0

uniform AC and therefore non-uniform AC is

Juedes, Lutz and Mayordomo have previously

not P-null. It remains only to show that P-

shown that their notion of resource-b ounded mea-

0

uniform AC do es not have measure one in P.

sure is robust in the face of many mo di cations

Consider any P density system covering the

of the de nition of covers. As a practical matter,

PARITY language. An argument similar to that

when trying to show that a class do es not have

of Theorem 9 shows that there is a set L in P that

measure zero in E or some larger complexity class,

di ers from PARITY only on a sparse set, such

it is very useful to know that, in that setting, a

that L is not covered. No set that di ers from

null cover can b e assumed without loss of gen-

0

PARITY only on a sparse set can b e in AC , and

erality to satisfy all of the following \niceness"

thus this shows that no P density system can

conditions [JLM][JL2][M]:

0

cover P n AC :

 A density system d is exactly computable if

k

It follows from this that P density functions

^

d = d :

k k;r

satisfy the measure axioms even on P-uniform

0

AC , and hence notions of measure can b e de-

 A density function is conservative if it sat-

ned even on intuitively small subsets of P. Re-

is es the following \conservation" prop erty:

cently and indep endently, Regan and Sivakumar

dw 0+dw 1

. Although our De ni- dw =

2

have given a very di erent argument showing that

4

tion 1 in this pap er requires density func-

0

non-uniform AC do es not have measure zero in

tions to b e conservative, other pap ers, such

P [RS].

as [L92], for example, require density func-

We do not view this limitation of our mea-

tions to satisfy only the weaker condition

sure as a drawback. Indeed, we conjecture

dw 0+dw 1

dw   .

2

that even NTIMElog n a prop er subset of

0

Dlogtime-uniform AC  is not a measure-zero

 If the density system d is of the form d =

k k

subset of P. Backing up our conjecture is the

k

2 d for some density function d, then we

O 1

fact that, although NTIMElog n is prop-

say that d is derived from the martingale

k

O 1n

log

erly contained in DTIME2 , it is easy to

d. Many authors require a martingale to b e

O 1

show that if NTIMElog n has measure zero

conservative; we will consider b oth conserva-

O 1n

log

in DTIME2 , then NP has measure zero

tive and nonconservative martingales. Note

O 1

n

that the condition that d be a null cover of , which seems to b e an unlikely in DTIME2

k

consequence [KM, L93a]. Of course, proving that

! is equivalenttosaying that there is no -

O 1

n

nite upp er b ound on fdw : w v ! g, so the

NP do es not have measure zero in DTIME2 

lim sup of this sequence is in nite.

entails proving that P 6=NP.However, weantic-

ipate that due to the severely limited computa-

4

An earlier version of this pap er did not require the

tional p ower of P machines, it should b e p os-

density functions to b e conservative; the subtle role that

sible to show that some other interesting classes

this condition plays in the pro of of Theorem 9 did not

b ecome clear until later. of sets are not P-null. 8

the results of that section hold for all the other  A set A is covered in the limit if there is a

measures mentioned here. martingale d such that, for all ! 2 A, the

sequence hd! [0::n]i has a limit of in nity,

instead of merely an in nite lim sup.

3 Hard Sets for BPP

 A density system is regular if d z =1 and

k

It was shown in [BG] that for almost every

z v w imply d w = 1:

k

A A

A, BPP =P . In [L93]itwas shown that al-

When considering measure on sub exp onential

most every set in ESPACE has this prop erty, and

complexity classes, there are additional choices

thus in particular almost every such set is hard

involved in the de nition, concerning how or if 

for BPP. Note, on the other hand, that only a

p

the length of the input is provided, questions con-

measure zero set of languages is hard for  as-

2

p

cerning how dep endency sets should b e de ned,

suming BPP 6= , and thus the \reason" that

2

etc. Several of the pro ofs of [JLM, JL2 , M] show-

a PSPACE-complete set is hard for BPP is in a

ing that their notion of measure is robust under

fundamental way quite di erent from the \rea-

these changes do not translate directly to our set-

son" that a random set is hard for BPP.

ting on small measure, which raises the sp ectre

In this section, we improve the results of [L93]

5

that each of the 2 combinations of the niceness

by showing that almost every set in PSPACE and

conditions listed ab ove not counting additional

in E for >0 is hard for BPP, b ecause almost ev-



choices concerning providing the input length,

ery such set \lo oks random enough". As in [L93],

etc. would give rise to a di erent notion of mea-

we make use of the pseudorandom generators of

sure.

[NW], although our construction di ers from that

We show in [AS2] that, in fact, only two no-

of [L93]inseveral fundamental ways.

tions of measure on P can b e de ned byvarying

Theorem 11 For almost every A 2 E we have

these parameters. It turns out that anynull set



A

BPP P : can b e covered by an exactly-computable martin-

gale, but surprisingly, assuming any of the other

Pro of. De ne H to b e the set of languages L of

niceness conditions is equivalent to assuming al l

n

hardness 2 . That is, for suciently large n any

of them.

n n1 n

circuit of size 2 errs on at least 2 1 2 

That is, it is shown in [AS2] that the no-

of the words of length n: For a xed n; we let

tion of measure de ned in this pap er is equiv-

H n denote the set of languages satisfying this

alent to the de nition that results from having

condition at least at the given n:

the measure-zero sets b e covered in the limit

Fix p ositive : We show that for almost every

by exactly-computable conservative martingales,

A

A 2 E and every



where the machines that compute the martingales

H 6= ;: It follows from Lemma 12 that BPP

are even more limited than the machines that are

A

P for any such A; and hence for almost all A 2

considered in this pap er. On the other hand,

E :

SPARSE is covered in the lim sup sense bya 

non-conservative martingale, so two distinct no-

A A

Lemma12 If E \ H 6= ;; then BPP P :

tions of measure do result.

It is also shown in [AS2] that the plogon mea-

Pro of. This is a \partially" relativized version of

sure PSPACEof[M] is strictly richer that the

[NW, Theorem 3]. |

conservativeversion of our space measure, but

PSPACE is incomparable with the nonconser-

Let 0 < 1=: Let

vativeversion of our space measure.

F A denote the language

The set studied in Section 3 is shown to b e

bjuj

2

null in the most restrictive sense of measure, so u 2 Ag: fu :0 9

A

Then F A 2 E : We will show that for almost value of PrF A 2= H n j C . If wewere to

w

n

all A 2 E ;FA has hardness 2 : set d w  to this value, it would de ne a non-

 n

Fix a circuit with n inputs. For a word u conservative density system; thus it remains only

of length n; the set fA : u 2 F Ai u= for us to patch this so that the conservation ax-

1g has Leb esgue measure 1=2: That is, if we iom is satis ed, at the one place where it may fail:

cho ose A at random, the circuit u is to o small the \seam" b etween the crude Cherno b ound for

bn

2

short w and the precise conditional probability for

to query 0 u 2 A; so with probability1=2we

large w: Let b e de ned as:

haveu 2 F A i  accepts u. The events in

fu 2 F A i  accepts ug are mutually

juj=n

=Ch PrF A 2= H n j C :

n

w [0::p1]

indep endent, so we can apply the Cherno b ound

to get

Note that dep ends on n but not on w . For

9 8

jw j p, de ne d w tobe

n

u computes F Aonat> >

= <

n1 n

A :

least 2 1+2 words

d w = + PrF A 2= H n j C 

> >

n w

; :

u

bn

2 n

Observe that if jw j > p os0 1 , then the con-

has measure less than

ditional probability is either 0 or 1, and it follows

n n 2 n n=3

2  =2 2

e  2 1

easily that d covers X .

n n

It remains to show that d w  has compu-

n

for all suciently large n: Finally, considering all

tations, whichwe do in pieces. First consider

n

circuits of size 2 and having n inputs, the set

PrF A 2= H n j C ; for jw j p: There are at

w

1=b

n n

some u computes

9 8

log jw j 2 2

circuits =2 extensions and 2 most 2

> >

= <

n

F A on at least

to consider, and each simulation takes time 2 :

X = A :

n

> >

n1 n ; :

Thus the total time to do all simulations is less

2 1+2 words



log jw j

than 2 : The dep endency set G consists

d;jw j;n

u

of the p olylogjw j-many p ositions of all words of

bjv j

2

= f A : F A 2= H n g

=n

the form 0 v; so the dep endency b ound is met

n

also.

2

has measure at most 2 times 1, which is less

n=4

Next we consider Ch : It is not the case that

n

2

than 2 for all large n. Let Ch denote the

n=4

n

2

n=4

Ch is exactly computable, since Ch =2

n n

2

value 2 .

requires n=4+ O 1 bits to write down as a \for-

The set X = lim sup X contains the A for

n

mal sum of p owers of two." However, we can

which F A 2= H n for in nitely many n, and

write

we wish to show X is null.



n=4

We de ne a density system d w  that for all

n

Ch if r  2 ;

n

c

Ch w =

n;r

large n is an upp er b ound for the conditional

0 otherwise.

probability that a random A satis es F A 2=

n=4

n=4 2

H n given A 2 C : For short w; i.e., jw j <

If r  2 then expressing 2 in the desired

w

bn

n=4

2 n

format requires only writing the exp onent 2 ,

p os0 0 =p, de ne d w tobeCh .Ifjw j

n n

and hence requires at most log r bits to write

p; then we will exhaustively generate all exten-

n=4

down, and if r<2 then

sions of w corresp onding to elements of F Aof

bn

2 n

length n i.e., all extenstions up to p os0 1 ,

n=4

2 r

c

jCh Ch w j =Ch =2  2 :

n n;r n

and for each such extension simulate all circuits

n

of size 2 on all strings inputs of length n; and

Then set

count the numb er of extensions that can b e ap-



c

Ch w  if jw j

proximated in that way, to compute the condi-

n;r

^

d w =

n;r

d w  otherwise.

tional probability. That is, we compute the exact

n 10

Finally, we apply the Borel-Cantelli-Lutz 4 Pseudorandom Sources

Lemma to the density system d ; using mo dulus

n

n=4

mn = 1+4logn: That is, putting j =2 ;

In [L90], Lutz prop osed a notion of source for

X X X

BPP. He gave a criterion for a particular sequence

n=4

2 j k

d = 2  2  2 ;

n

to b e useful as a substitute for the sequence of in-

n>4 log k j>k

n=mk 

dep endentunbiased coin ips used by a BPP ma-

chine. Based on this work, we formulate three in-

so we conclude  lim sup X = 0:

n

E 



tuitive prop erties a notion of source should have:

Note that any improvement to a time class

smaller than all E 's would involve showing that



Universality A single source should \work" for

each language in BPP has sub exp onential time

all BPP languages.

complexity.

A similar theorem holds for space b ounds:

Abundance The set of sources should have mea-

sure 1 at some level of resource-b oundedness.

Theorem 13 For almost every A 2 PSPACE we

This implies that a random sequence of coin

A

have BPP P :

ips is a source with probability1.

Pro of. The pro of is similar to the pro of in the

Hardness If the bits of a source can b e obtained

time-b ounded case. We note that a machine for

n

2 n

in p olynomial time, then P = BPP:

d only needs bits in p ositions p os0 0  through

n

2 n

p os0 1 ; and a space-b ounded machine can

The de nition in [L90] captures the rst two prop-

store all of these bits.

erties, but lacks the third, as one can construct

The following corollary of Theorems 11 and 13

0

sources in AC without showing P = BPP [S].

was p ointed out to us by Jack Lutz:

We seek an alternate criterion for a particu-

lar computable sequence to b e \random enough."

Corollary 14 Let C be any of the classes E,

We will capture universality, abundance, and

EXP, PSPACE or E : If  NP jC 6=0then



C 

hardness.

NP

BPP P :

Intuitively, a sequence A is a pseudorandom

sequence if it is p ossible to use A in place of a

We also show that almost every set A in E sat-

sequence of random bits to recognize each BPP

A A

is es BPP =P , improving the result of [L93]

language where it is crucial to the de nition

from ESPACE to E.

that this simulation would work if A were re-

A

placed with a random sequence. Formally,we

Theorem 15 For almost every A 2 E; BPP =

A

will say that a sequence A = ha i is a pseudo-

P .

i

random sequence for BPP if for each L 2 BPP,

Pro of. The pro of of this theorem is essentially

there is a b ounded-error probabilistic p olynomial-

the same as the pro ofs of Theorems 13 and 11; the

time machine accepting L such that, for each x,

only mo di cation is to replace H in those pro ofs

M x; A= 1 , x 2 L, where \M x; A" de-

A

with H : the set of languages L of \relativized"

notes the result of running machine M on input

n

hardness 2 ; using oracle A: That is, for any

x along the path given by taking the result of the

A

A 2 H , for suciently large n any circuit of size

th

i coin ip to b e a . That is, on each input x,

i

n n1 n

2 agrees with F A on at most 2 1 + 2 

the rst p olynomially-many bits of the sequence

of the words of length n; where the circuits are

A are used in place of probabilistic bits.

now allowed to have \oracle" gates that query the

Stated another way, A is a pseudorandom se-

oracle A.

i

quence for BPP if the set T A=f0 ja =1g

i

p

reductions, where the is hard for BPP under  We do not know if the condition of Theorem

T

machine M that reduces BPP language L to T A 15 holds also for PSPACE or anyE .

 11

also accepts L when M is viewed as a BPP ma- further, this line of reasoning suggests that non-

chine i.e., where queries to the oracle are an- relativizing results might b e obtained by studying

swered randomly. That is, A is not only hard for  -measure, and similar observations ap-

PSPACE 

BPP, but it is hard b ecause it \lo oks random." ply to the similarly-limited  -measure. Might

P

It follows from [BG] that almost all sequences are one b e able to show that most sets in P are hard

pseudorandom sequences for BPP. for BPP? Such a result would, of course, require

nonrelativizing pro of techniques, but if the notion

Theorem 16 Almost every A 2 ESPACE is a of measure on P do es not relativize in a meaning-

pseudorandom sequence for BPP. ful way, then p erhaps it could b e used to obtain

results of this sort.

Pro of. Theorem 13 shows that for most A in

Unfortunately, it turns out that these notions

juj

2

PSPACE, the language F A= fu :0 u 2 Ag

of measure do relativize in a natural way. When

m

has hardness 2 . An essentially equivalentway

one is dealing with extremely weak mo dels of

of restating this is to say that for almost all A

computation, the question of howtoprovide

in PSPACE, for almost all m, the function f :

access to the oracle is often rather controver-

2

m m +u

 !f0; 1g de ned by f u= 0  has

A

sial. A survey of pap ers discussing the issues

m

hardness 2 .Thus, informally, almost all tally

involved may b e found in [A90]. It is easy

sets in PSPACE lo ok \random enough" to serve

to see that, for instance, if one were to use

as sources for BPP. Since tally sets in PSPACE

the so-called \Ruzzo-Simon-Tompa" relativiza-

are essentially the same thing as sets in ESPACE,

tion metho d [RST], then there are oracles A rela-

a direct argument patterned after the pro of of

A A

tive to whichP would have measure zero in P ,

Theorem 13 can b e used to show that almost all

and thus this do es not constitute a meaningful no-

sets in ESPACE are pseudorandom sequences for

A

tion of measure on P . On the other hand, if one

BPP.

O 1

5

adopts the convention that a DTIMElog n

Theorem 16 is in some sense analogous to the machine is p ermitted to write only queries of

O 1

length log n on its query tap e, then it is

main result of [L90], but note that in contrast to

straightforward to show that one obtains a mea-

[L90], we can make a limited claim of optimal-

A

sure on P . Similar observations hold for the

ity, in the following sense. Theorem 16 shows

O n

2

measures on other sub exp onential time classes

that there are sources in DTIME 2 . If

and on PSPACE.

there is any source A in DTIMEF n for some

o1 on

n 2

, and , then T A 2 DTIME 2 F n 2 2

Although this rules out the prosp ect of obtain-

k

ing nonrelativizing results via this notion of mea-

hence simulating any n -time b ounded BPP ma-

k o1

k n 

sure, a comp ensating factor is that one obtains

chine would require at most time n 2 , and

SAT

measures for P and for all of the other 

i

thus every language in BPP would have time com-

o1

n

levels of the p olynomial hierarchy. One can also

plexity2 .

P P

classes, by using \  de ne a measure on the 

k k

L

machines such that some path covers d having 

k

outputs, and all non-ab orting paths output dw :

5 Conclusions

For related observations concerning exp onential-

5.1 Do es this Relativize?

time classes, see also [M].

What is \relativized one-way p olylogarithmic

space"? One's initial resp onse is likely to b e that

5

In most other settings, allowing only short queries to

this is a ludicrous notion, and that attaching ora-

the oracle do es not provide a satisfactory notion of rela-

cles to such limited automata would probably not

tivization, b ecause one cannot reduce the set A to itself

give rise to a very meaningful class. If followed b ecause one cannot write the input on the query tap e. 12

5.2 Summary [AJ] C. Alvarez and B. Jenner, On Dlogtime

and Polylogtime reductions, Pro c. 11th

Lutz's resource-b ounded measure forms the ba-

STACS, Lecture Notes in Computer Sci-

sis for a large and growing b o dy of interesting

ence 775, 1994.

work. A limitation of Lutz's notion of measure is

[Ba] J. Balc azar, Self-Reducibility, Journal

that it do es not apply to P and other imp ortant

of Computer and System Sciences 41

sub exp onential time classes. Wehave remedied

1990 367-388.

that situation by providing a notion of measure

that do es apply to these classes, and wehave used

[BB] J. Balc azar and R. Bo ok, Sets with

this measure to show, among other things, that

small generalized Kolmogorov complex-

almost all sets in E are hard for BPP, substan-



ity, Acta Informatica 23 1986 679{688.

tially improving a result of [L93].

It is worth noting that Lutz's de nitions of

[BG] C. Bennett and J. Gill, Relative to a ran-

resource-b ounded measure haveevolved some-

dom oracle, PA 6= NPA 6= Co-NPA

what over the years. Similarly,wemay exp ect

with probability1, SIAM J. Comput. 10

that as exp erience is gained, alternative formula-

1981 96-113.

tions of measure on small time classes may arise.

However, wehave established that interesting re-

[Bu] S. R. Buss, The Bo olean formula value

sults can b e obtained with our current notion of

problem is in ALOGTIME, Proc. 19th

measure, and we lo ok forward to further work in

ACM Symposium on Theory of Comput-

this area.

ing, 1987, pp. 123{131.

Acknowledgments

[HH] J. Hartmanis and L. Hemachandra, On

sparse oracles separating feasible com-

We gratefully acknowledge helpful discussions

plexity classes, Information Processing

and corresp ondence with Lane Hemaspaandra,

Letters 28 1988 291{296.

Elvira Mayordomo, Jack Lutz, David Juedes, and

Ken Regan.

[HY] J. Hartmanis and Y. Yesha, Computa-

tion times of NP sets of di erent den-

sities, Theoretical Computer Science 34

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th

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[NW] N. Nisan and A. Widgerson, Hard-

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[RS] K. Regan and D. Sivakumar, On

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