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 measure 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

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.

b ounded measure. For example, the -

complete sets are a measure-zero subset of

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

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

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

as E, which greatly limits the range of questions

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

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 sequence: 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

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

demonstrate its usefulness.

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 sequences, 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

and other classes. This section concludes with a

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

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,

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

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 = :

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-

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

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

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

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 complexity class 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.

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

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

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-

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

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

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.

at most time f jy j, for total computation time

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-

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

sk s

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

subscripted function d satisfying

i ;;i ;r

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

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

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:

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

If a class X has nonzero measure, we can talk

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

d d w : w =

j;k +2 +1

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.

We will show d isaC -null cover of X:

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 ,

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

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

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

k +2 +1

computation d :

j;k ;r

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

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

^ ^

d w = d w :

j j

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

k;r

then X is C -nul l.

j =0

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-

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

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

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

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

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:

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

if any bits of w in p osi-

tions numb ered f1, 2,

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

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 .

Corollary 6 For al l k , DTIMEn = 0:

Finally,wehave to show L is not covered by

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

Corollary 7 Let 0 < <, and let E denote

inductive argument of [L92] works unchanged.

. Then E = 0: DTIME2

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

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

d < 2 :

2.4 Elementary Facts Concerning the

j =mk

Measure

Then

1 1

\ [

Wehave established that our notion of measure

X =0:

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-

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

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

sub exp onential-time classes, the sets X must

j X

be covered by a \rapidly-vanishing" collection of

covers. This restriction that the values d tend

so rapidly toward zero makes it far from obvious

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

crete example is presented in the pro of of The-

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

p erexp onential functions is necessary; the claim

O n

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

[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

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,