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1996 The Quantitative Structure of Exponential Time Jack H. Lutz

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Recommended Citation Lutz, Jack H., "The Quantitative Structure of Exponential Time" (1996). Computer Science Technical Reports. 115. http://lib.dr.iastate.edu/cs_techreports/115

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Abstract Department of Computer Science Iowa State University Ames, Iowa 50010 Recent results on the internal, measure-theoretic structure of the exponential time complexity classes linear polynomial E = DTIME(2 ) and E = DTIME(2 ) 2 are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time many-one reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages. Possible implications for the structure of NP are also discussed.

Disciplines Theory and Algorithms

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The Quantitative Structure of

Exp onential Time

1

Jack H. Lutz

ABSTRACT Recent results on the internal, measure-theoretic structure of

the exp onential time complexity classes E and EXP are surveyed. The mea-

sure structure of these classes is seen to interact in informativeways with

bi-immunity, complexity cores, p olynomial-time reductions, completeness,

circuit-size complexity, Kolmogorov complexity, natural pro ofs, pseudoran-

dom generators, the density of hard languages, randomized complexity, and

lowness. Possible implications for the structure of NP are also discussed.

1 Intro duction

In the past veyears, new developments in resource-b ounded measure have

op ened the way for a systematic investigation of the internal, measure-

theoretic structure of the exp onential time complexity classes E and EXP.

The investigation is very far from complete, but it has already yielded a

number of interesting insights and results. This pap er surveys the motiva-

tions, ideas, and results of the earliest phase of the investigation, i.e., the

part completed by mid-1995.

It should b e emphasized that the material surveyed here is the work of

several investigators. The ongoing e orts of these investigators, together

with the e orts of more recent participants, virtually guarantee that this

survey will be incomplete by the time it app ears. At the time of this

writing, there are already several pap ers in review and manuscripts in cir-

culation that app ear to extend the body of knowledge presented here.

Nevertheless, it is to b e hop ed that the \organized snapshot" provided by

this survey will provide context and motivation for future research.

There are three reasons for our interest in the complexity classes E and

EXP.

i E and EXP have rich, apparently well-b ehaved, internal structures.

These structures have manyinteracting facets, including a varietyof

1

Departmentof Computer Science, Iowa State University, Ames, IA 50011.

E-mail: [email protected]. This research was supp orted in part by National

Science Foundation Grant CCR-9157382, with matching funds from Ro ckwell

International, Microware Systems Corp oration, and Amo co Foundation.

1. The Quantitative Structure of Exp onential Time 2

reducibilities [LLS75], complete languages under these reducibilities

[SC79, Wat87b], measure structure [Lut92], and category structure

[Lut90,Fen91,Fen95].

ii EXP is the smallest deterministic time complexity class known to

contain NP. It also contains PSPACE, and hence the p olynomial-time

hierarchy and many other classes of interest in complexity theory.E

is a prop er subset of EXP, but it contains P and \the essential part

of NP" [Wat87b], i.e., many NP-complete problems.

iii E and EXP have b een proven to contain intractable problems [HS65].

From the standp oint of complexity theory, this existence of intractabil-

ityisavaluable resource. This is b ecause, in practice, a pro of that

a sp eci c language A is intractable pro ceeds by inferring the in-

tractabilityof A from the intractability of some language B chosen

or constructed for this purp ose.

Taken together, i, ii, and iii suggest E and EXP as appropriate

spaces in whichtoinvestigate emb ed problems involving NP, PH, PSPACE,

and other classes in this range.

Until recently, the issues addressed by research on the structure of com-

plexity classes have b een largely qualitative rather than quantitative. In-

deed, the intro duction to [Sch86b] o ered \qualitative" as a synonym for

\structural." This seemed to be an inevitable asp ect of the sub ject. A

problem is, or is not, complete for a complexity class. One complexity class

is, or is not, contained in another. This was unfortunate, since the ob jec-

tive of complexity theory is a quantitative theory of computation. However,

since the sets of interest are all countably in nite, there app eared to b e no

p ossibility of making quantitativeversions of these judgments.

The main ob jective of the work surveyed here is to remedy this situation.



Supp ose that a language A f0; 1g is chosen by a random exp eri-

ment in which an indep endent toss of a fair coin is used to decide whether

each string is in A. Then classical Leb esgue measure theory describ ed

in [Hal50, Oxt80], for example identi es certain measurable sets of lan-

guages also called events  and assigns to each measurable set X a mea-

sure X , which is the probability that A 2 X in this exp eriment. A set

X of languages is then small in the sense of measure if it has measure 0.

E ective measure theory, whichsays what it means for a set of decidable

languages to have measure 0 as a subset of the set of all such languages, has

b een investigated byFreidzon [Fre72], Mehlhorn [Meh74], and others. The

resource-bounded measure theory intro duced by Lutz [Lut92, Lutb] has the

classical and e ective theories as sp ecial cases, but also de nes measurabil-

ity and measure for subsets of many complexity classes. The smal l subsets

of such a complexity class are then the measure 0 sets; the large subsets are

the measure 1 sets complements of measure 0 sets. We say that almost

1. The Quantitative Structure of Exp onential Time 3

every language in a complexity class C has a given prop ertyif the set of

languages in C exhibiting the prop erty is a measure 1 subset of C .

Thus, resource-b ounded measure provides a means of investigating the

sizes of various subsets of E and EXP. This is a priori a hop eful devel-

opment, b oth b ecause quantitative results are more informative and be-

cause Leb esgue measure has b een so useful in analysis, probability, and

mathematical physics. However, much of the ongoing motivation for this

work arises not from a priori considerations, but rather from the fact that

resource-b ounded measure turns out to interact informatively with many

prop erties of interest in computational complexity. Suchinteractions sur-

veyed in this pap er involve bi-immunity section 4, complexity cores sec-

tions 5, 7, and 8, the structure of E and EXP under p olynomial-time

reductions sections 6,7, and 8, circuit-size complexity and time-b ounded

Kolmogorov complexity section 9, natural pro ofs and pseudorandom gen-

erators section 9, the densityof hard languages section 11, and other

prop erties that had b een extensively studied prior to the advent of resource-

b ounded measure. It is to b e hop ed that sustained, systematic investigation

along these lines will lead to a detailed, quantitative understanding of E

and EXP.

From the standp oint of classical mathematics and recursion theory, classes

likeP,NP, PH, and PSPACE are all negligibly small, hence dicult to dis-

tinguish by quantitative structural means. From the standp oint of E and

EXP, matters maybevery di erent. If EXP is, indeed, the smallest deter-

ministic time class containing NP, then there maywell b e a natural \notion

of smallness" for subsets of EXP such that P is a small subset of EXP, but

NP is not. Similarly, it may be that P is a small subset of E, but that

NP \ E is not.

It is p ossible that resource-b ounded measure already provides such a

notion of smallness. It is certainly the case that P has measure 0 in E

and EXP [Lut92]. In section 12 we discuss the reasonableness and known

consequences of the hyp othesis that NP is not small in this sense. This is a

very strong hyp othesis that app ears to havemuch more explanatory p ower

than traditional, qualitativehyp otheses, suchasP6= NP or the separation

of the p olynomial-time hierarchy. Only further investigation will determine

whether this hyp othesis is reasonable.

2 Preliminaries

In this pap er, [[ ]] denotes the Boolean value of the condition  , i.e., [[ ]]=

if  then 1 else 0.



All languages here are sets of binary strings, i.e., sets A f0; 1g . We

1

identify each language A with its characteristic sequence 2 f0; 1g A

1. The Quantitative Structure of Exp onential Time 4

de ned by

=[[s 2 A]][[s 2 A]][[s 2 A]]:::;

A 0 1 2

where s = , s = 0, s = 1, s = 00; ::: is the standard enumeration

0 1 2 3

 1

of f0; 1g . Relying on this identi cation, the set f0; 1g , consisting of all

in nite binary sequences, will b e regarded as the set of all languages.

Wesay that a condition n holds almost everywhere a.e. if it holds

for all but nitely many n 2 N. We say that n holds in nitely often

i.o. if it holds for in nitely many n 2 N.

 n

For A f0; 1g and n 2 N,we use the notations A = A \f0; 1g and

=n

n

A = A \f0; 1g . A language A is sparse if there is a p olynomial q n

n

such that jA jq n a.e. A language A is dense if there is a real number

n

"

n

">0 such that jA j > 2 a.e.

n

The symmetric di erence of languages A and B is A 4 B =A B  [

 c 

B A. The complement of a language A f0; 1g is A = f0; 1g A.

c 

The complement of a set X of languages is X = f A f0; 1g j A=2 X g.

  

We x a one-to-one pairing function h; i from f0; 1g f0; 1g onto f0; 1g

such that the pairing function and its asso ciated pro jections, hx; y i 7! x

and h x; y i7!y , are computable in p olynomial time.

 

For a function f : f0; 1g !f0; 1g and a natural number i,we de ne

  i

the function f : f0; 1g !f0; 1g by f x=f  0 ;x . We then regard f

i i

as a \uniform enumeration" of the functions f ;f ;f ;


.

0 1 2

 

In general, complexity classes of functions from f0; 1g into f0; 1g will

be denoted by app ending an `F' to the notation for the corresp onding

complexity classes of languages. Thus, for t : N ! N, DTIMEF t is the

 

set of all functions f : f0; 1g !f0; 1g such that f xis computable in

O tjxj time.

3 Resource-b ounded measure

In this section weintro duce a fragment of resource-b ounded measure that

is sucient for understanding the meaning of the results surveyed in this

pap er. Although resource-b ounded measure is a very general theory whose

sp ecial cases include classical Leb esgue measure, the measure structure of

the class REC of all recursive languages, and measure in various complexity

classes, our discussion here will be sp eci c to the classes E and E . The

2

interested reader is referred to [Lut92, Lutb , May94b, Jue94, JL95b,ATZ]

for more discussion, examples, and technical machinery.



De nition. A martingale is a function d : f0; 1g ! [0; 1 with the



prop erty that, for all w 2f0; 1g ,

dw 0 + dw 1

dw = :  2

1. The Quantitative Structure of Exp onential Time 5



A martingale d succeeds on a language A f0; 1g if

lim sup d [0::n 1] = 1:

A

n!1

Intuitively, a martingale d is a b etting strategy that, given a language

A, starts with capital amountof money d and b ets on the memb er-

ship or nonmemb ership of the successive strings s ;s ;s ;


the standard

0 1 2



enumeration of f0; 1g inA. Prior to b etting on a string s , the strategy

n

has capital dw , where w =[[s 2 A]]


[[s 2 A]]: After b etting on the

0 n1

string s , the strategy has capital dwb, where b =[[s 2 A]]. Condition

n n

 ensures that the b etting is fair. The strategy succeeds on A if its capital

is unb ounded as the b etting progresses.

Martingales were used extensively by Schnorr [Sch70, Sch71a, Sch71b,

Sch73] in his investigation of random and pseudorandom sequences. Here

we use martingales as a way to de ne measure 0 sets.



Consider the random exp eriment in which a language A f0; 1g is

chosen probabilistically, using an indep endent toss of a fair coin to decide

memb ership of each string in A. Given a set X of languages, let PrX =

Prob [A 2 X ] denote the probability that A 2 X when A is chosen in this

A

fashion. If X is not Leb esgue measurable, then PrX  will not exist, but

this issue can b e safely ignored here. The following fact is intuitively clear

and not dicult to prove.

Prop osition 3.1. For every set X of languages, the following two condi-

tions are equivalent.

1 PrX =0.

2 There is a martingale d such that d succeeds on every elementof X .

In order to generalize Prop osition 3.1 we need to consider martingales

that are computable within some resource b ound. Since martingales are

real-valued, their computations must employ nite approximations of real

k

numb ers. For this purp ose, we consider functions of the form d : N 



f0; 1g ! Q, where Q is the set of rational numb ers. Formally, in or-

der to have uniform criteria for computational complexity, we consider

 

all such functions to map f0; 1g to f0; 1g . For example, a function d :

  

~

N f0; 1g ! Q is formally interpreted as a function d : f0; 1g !f0; 1g .

r

~

Under this interpretation, dr;w=q means that dh 0 ;wi =hu; v i, where

u and v are the binary representations of the numerator and denominator

of q , resp ectively.We also write d w  for dr;w.

r

De nition. The classes p = p and p , b oth consisting of functions f :

1 2

 

f0; 1g !f0; 1g , are de ned as follows.

p = ff j f is computable in p olynomial time g

1

O 1

log n

p = ff j f is computable in n timeg 2

1. The Quantitative Structure of Exp onential Time 6

Guided by Prop osition 3.1, the measure structures of E and EXP are

now develop ed in terms of the classes p , for i =1; 2.

i

b

De nition. A martingale d is p -computable if there is a function d :

i

 

b

N f0; 1g ! Q such that d 2 p and, for all r 2 N and w 2f0; 1g ,

i

r

b

 2 : d w  dw 

r

Ap-martingale is a martingale that is p -computable.

i i

Wenow come to the key idea of this section.

De nition. X = A set X of languages has p -measure0, and we write 

i p

i

0, if there is a p -martingale d that succeeds on every elementofX . A set

i

c

X of languages has p -measure 1, and we write  X =1,if  X =0.

i p p

i i

We now turn to the internal measure structures of the classes E =E

1

1

and E = EXP.

2

A set X has measure0in E , and we write X j E  = 0, if De nition.

i i

 X \ E =0.A setX has measure1 inE , and we write X j E =1,

p i i i

i

c

if X j E =0.If X j E =1,wesay that almost every language in

i i

E is in X .

i

We write X j E  6= 0 to indicate that X do es not have measure 0 in

i

E . Note that this do es not assert that \X j E " has some nonzero value.

i i

The following is obvious but useful.

1

Fact 3.2. For every set X f0; 1g ,

 X =0 =  X =0 = Pr[A 2 X ]= 0

p p

2

+ +

X j E=0 X j EXP=0;

where the probability Pr[A 2 X ] is computed according to the random



exp eriment in which a language A f0; 1g is chosen probabilistically,

1

The classes E and EXP are the rst two classes in a natural hierarchy

E ; E ; E ;::: of exp onential time complexity classes. In [Lut92 ], the measure

1 2 3

structures of these classes are develop ed in terms of a corresp onding hierarchy

p ; p ; p ;::: of function classes. Consequently, most pap ers on resource-b ounded

1 2 3

measure including all of the author's pap ers use the notation E in place of

2

EXP. However, in this b o ok, for the sake of consistency,we refrain from using

the E notation. The only exceptions are the present section and a brief mention

i

of the class E in section 9. 3

1. The Quantitative Structure of Exp onential Time 7

using an indep endent toss of a fair coin to decide whether each string



x 2f0; 1g is in A.

It is shown in [Lut92] that these de nitions endow E and EXP with

internal measure structure. This structure justi es the intuition that, if

X j E = 0, then X \ Eisanegligibly smal l subset of E and similarly for

EXP. The most imp ortant comp onent of this justi cation is the Measure

Conservation Theorem [Lut92], which implies the following.

Theorem 3.3 Lutz [Lut92]. EjE 6= 0 and EXPjEXP 6=0.

The following result shows that, if C is a \reasonable" complexity class

that contains almost every element of E resp ectively, EXP, then C con-

tains every element of E resp ectively, EXP.

Theorem 3.4 Regan, Sivakumar, and Cai [RSC95]. Let C be a set of

languages that is either closed under symmetric di erence or closed under

 nite union and intersection.

1. If CjE = 1, then E C.

2. If CjEXP = 1, then EXP C.

Resource-b ounded measure in E and EXP is known to be robust with

resp ect to various changes in the de nition [Lut92, Lutb, May94b, JL95b].

Recently, Buhrman and Longpr e [BL96]have shown that resource-b ounded

measure can also be characterized in terms of the compressibility and

decompressibility of languages.

4 Incompressibility and bi-immunity

P

Many results on the structure of E and EXP under  -reducibility use

m

languages that are \incompressible by many-one reductions." This idea,

originally exploited by Meyer [Mey77], is develop ed in the following de ni-

tions.

 

De nition. The col lision set of a function f : f0; 1g !f0; 1g is



C = fx 2f0; 1g j 9y

f



Here, we are using the standard ordering s

0 1 2

Note that f is one-to-one if and only if C = ;.

f

 

De nition. A function f : f0; 1g !f0; 1g is one-to-one almost every-

where or, brie y, one-to-one a.e. if its collision set C is nite. f

1. The Quantitative Structure of Exp onential Time 8

DTIMEt



De nition. Let A; B f0; 1g and let t : N ! N.A  -reduction

m

1

of AtoBis a function f 2 DTIMEFt such that A = f B , i.e., such

DTIMEt



that, for all x 2f0; 1g , x 2 A i f x 2 B .A  -reduction of A is

m

DTIMEt

a function f that is a  -reduction of A to f A.

m

DTIMEt

It is easy to see that f is a  -reduction of A if and only if there

m

DTIMEt

exists a language B such that f is a  -reduction of A to B .

m



De nition. Let t : N ! N. A language A f0; 1g is incompressible

DTIMEt DTIMEt

by  -reductions if every  -reduction of A is one-to-one

m m

 P

a.e. A language A f0; 1g is incompressible by  -reductions if it is

m

DTIMEq 

incompressible by  -reductions for all p olynomials q .

m

DTIMEt

Intuitively,iff is a  -reduction of A to B and C is large, then

m

f

f compresses many questions \x 2 A?" to fewer questions \f x 2 B ?"

P

If A is incompressible by  -reductions, then very little such compression

m

can o ccur.

Meyer [Mey77] proved that E contains languages that are incompressible

P

by  -reductions. The following result shows that almost every language

m

in E has this prop erty.

+

Theorem 4.1 Juedes and Lutz [JL95a]. Let c 2 Z and de ne the sets

cn

DTIME2 



reductionsg, X = fA f0; 1g jA is incompressible by 

m

c

n

DTIME2 



Y = fA f0; 1g jA is incompressible by  reductionsg.

m

Then  X  =  Y  = 1. Thus almost every language in E is incom-

p p

2

cn

DTIME2 

pressible by  -reductions, and almost every language in EXP is

m

c

n

DTIME2 

incompressible by  -reductions.

m

Sketch of pro of that  X =1. It suces to exhibit a p-martingale d :

p

 c

f0; 1g ! [0; 1 that succeeds on every elementofX .

c+1n cn

Let f 2 DTIMEF2  b e universal for DTIMEF2 , in the sense

cn

that DTIMEF2  = ff ji 2 Ng. For each i 2 N, de ne a set Z of

i i

languages as follows. If the collision set C is nite, then Z = ;. Otherwise,

f i

i

if C is in nite, then Z is the set of all languages A such that f is a

f i i

i

cn

DTIME2 

c

-reduction of A. Note that X is the union of the sets Z . The 

m

i

martingale d is de ned by

1

X

i

2 d w ; dw =

i

i=0



where the functions d : f0; 1g ! [0; 1 are de ned as follows. Let i 2

i



N, w 2 f0; 1g , and b 2 f0; 1g. Recall that s ;s ;s ;


is a standard

0 1 2



enumeration of f0; 1g .

1. The Quantitative Structure of Exp onential Time 9

i d =1.

i

, then d wb=d w . ii If s 2= C

i i f

jw j

i

iii If s 2 C , then x the least j 2 N such that f s =f s  and

f i j i

jw j jw j

i

set

d wb=2
d w 
[[b = w [j ]]]:

i i

It is easy to check that each d is a martingale, whence d itself is a martin-

i

gale. Intuitively, d b ets on memb ership of strings in a language A. Clause

i

i says that d starts with 1 dollar. Clause ii says that d do es not b et on

i i

the status of strings x 2= C . Clause iii says that, for strings x 2 C , d

f f i

i i

b ets all its capital that x 2 A i y 2 A, where y is the rst string such that

f x=f y . If A 2 Z , then this b et will b e correct, thereby doubling d 's

i i i i

capital, in nitely often. Thus d succeeds on every elementofZ . It follows

i i

c

from this that d succeeds on every elementof X .



b

Finally, to see that d is p-computable, de ne d : N f0; 1g ! Q by

r +jw j

X

i

b

2 d w : d w =

i r

i=0

c+1n

Since f 2 DTIMEF 2  and the computation of d w  only uses values

i

f u for strings u with juj = O log jw j, it is clear that d 2 p. Since

i

1 1

X X

i jw ji r

b

= 2 d w   2 =2 d w  dw 

i r

i=r +jw j+1 i=r +jw j+1



for all r 2 N and w 2f0; 1g , it follows that d is p-computable. 2

Corollary 4.2 Juedes and Lutz [JL95a]. Almost every language in E and

P

almost every language in EXP is incompressible by  -reductions.

m

Corollary 4.3 Meyer [Mey77]. There is a language A 2 E that is incom-

P

pressible by  -reductions.

m

We conclude this section with a brief discussion of P-bi-immunity.



De nition. A language A f0; 1g is P-immune if, for all languages



B A, B 2 P implies that B is nite. A language A f0; 1g is P-bi-

c

immune if A and A are b oth P-immune.

Intuitively, a language that is P-bi-immune \cannot b e nontrivially ap-

proximated, from inside or outside," byany language in P.

Ko and Mo ore [KM75]. Every language that is incom- Prop osition 4.4

P

pressible by  -reductions is P-bi-immune. m

1. The Quantitative Structure of Exp onential Time 10

P

In light of this prop osition, languages that are incompressible by  -

m

reductions are sometimes called \strongly P-bi-immune" [BS85, BDG90].

The following result shows that almost every language in E is P-bi-

immune.

Mayordomo [May94a]. Almost every language in E, and Theorem 4.5

almost every language in EXP, is P-bi-immune.

Although Theorem 4.5 follows immediately from Corollary 4.2 and Prop o-

sition 4.4, it should be noted that Mayordomo's pro of of this result pre-

ceded, and was indep endent of, the pro ofs of Theorem 4.1 and Corollary

4.2.

5 Complexity cores

Complexity cores, rst intro duced by Lynch [Lyn75], have b een studied ex-

tensively. See [BDG90] for an overview of suchwork. Intuitively, a com-

plexity core of a language A is a xed set K of inputs such that every

machine whose decisions are consistent with A fails to decide eciently

on all but nitely many elements of K . The meaning of \eciently" is a

parameter of the de nition that varies according to the context. In this sec-

tion we make this de nition precise and note that almost every language

in E and EXP has very large complexity cores.



Given a machine M and an input x 2 f0; 1g , we write M x = 1 if

M accepts x, M x=0 if M rejects x, and M x=? in any other case

i.e., if M fails to halt or M halts without deciding x. If M x 2f0; 1g,

we write time x for the number of steps used in the computation of

M

M x. If M x=?,we de ne time x=1.We partially order the set

M

f0; 1; ?g by ? < 0 and ? < 1, with 0 and 1 incomparable. A machine



M is consistent with a language A f0; 1g if M x  [[x 2 A]] for all



x 2f0; 1g .



De nition. Let t : N ! N b e a time b ound and let A; K f0; 1g . Then

K is a DTIMEtn-complexity core of A if, for every c 2 N and every

machine M that is consistent with A, the \fast set"

F = fx jtime x  c
tjxj+c g

M

satis es jF \ K j < 1. By our de nition of time x, M x 2f0; 1g for

M

all x 2 F .Thus F is the set of all strings that M \decides eciently."

Note that every subset of a DTIMEtn-complexity core of A is a

DTIMEtn-complexity core of A. Note also that, if sn=O tn, then

every DTIMEtn-complexity core of A is a DTIMEsn-complexity

core of A.

1. The Quantitative Structure of Exp onential Time 11



De nition. Let A; K f0; 1g .

1. K is a polynomial complexity core or, brie y,aP-complexity coreof

k

A if K is a DTIMEn -complexity core of A for all k 2 N.

2. K is an exponential complexity core of A if there is a real number >0



n

such that K is a DTIME2 -complexity core of A.

Intuitively, a P-complexity core of A is a set of infeasible instances of

A, while an exp onential complexity core of A is a set of extremely hard

instances of A.

The following observation, an obvious generalization of a result of Balc azar

and Schoning [BS85] see Corollary 5.2 b elow, relates incompressibilityto

complexity cores.

Juedes and Lutz [JL95a]. If t : N ! N is time constructible Lemma 5.1

DTIMEt

then every language that is incompressible by  -reductions has

m



f0; 1g as a DTIMEt-complexity core.

Corollary 5.2. Let c 2 N.

1. Balc azar and Schoning [BS85] Every language that is incompressible

P 

by  -reductions has f0; 1g as a P-complexity core.

m

cn

DTIME2 

2. Every language that is incompressible by  -reductions has

m

 cn

f0; 1g as a DTIME2 -complexity core.

c

n

DTIME2 

3. Every language that is incompressible by  -reductions has

m

c

 n

f0; 1g as a DTIME2 -complexity core.

Theorem 4.1 and Corollary 5.2 now tell us that almost every language

decidable in exp onential time has complexity cores of the largest p ossible

size.

+

Corollary 5.3 Juedes and Lutz [JL95a]. Let c 2 Z .

 cn

1. Almost every language in E has f0; 1g as a DTIME2 -complexity

core.

c

 n

-complexity 2. Almost every language in EXP has f0; 1g as a DTIME2

core.

6 Small span theorems

In this section we describ e research on small span theorems, which illumi-

nate key asp ects of the structure of E and EXP under p olynomial reduc-

P

-reductions. tions. We b egin with the Small Span Theorem for 

m

 P

-span of a language A f0; 1g to b e De ne the lower 

m

 P

P A=fB f0; 1g j B  Ag:

m m

1. The Quantitative Structure of Exp onential Time 12

P

Similarly, de ne the upper  -span of A to b e

m

1  P

P A=fB f0; 1g j A  B g:

m m

P

Intuitively, in the  -reducibility structure of the set of all languages,

m

1

we think of P A as lying \b elow" A, while P A lies \ab ove" A. See

m

m

Figure 1. We will b e esp ecially concerned with the size, i.e., the resource-

b ounded measure, of the upp er and lower spans of various languages. If

neither of these spans is small i.e., neither has resource-b ounded measure

0, then wehave the con guration depicted schematically in Figure 1. On

the other hand, if one or b oth of these spans is small, then we have one

of the \small-span" con gurations depicted schematically in Figure 2. The

P

Small  -Span Theorem says that, if A is in E or EXP, then at least one

m

1

of the sets P A, P A is smal l. That is, only small-span con gurations

m

m

can o ccur in E or EXP.

-1 Pm (A)

-1 deg P (A) = P (A) ∩P (A) • A m m m

Pm (A)

FIGURE 1. The upp er span, lower span shaded, and degree of A.

P

Theorem 6.1 Small  -Span Theorem|Juedes and Lutz [JL95a].

m

1. For every A 2 E,

P A j E=0

m

or

1 1

A j E = 0: A = P  P

p

m m

2. For every A 2 EXP,

P A j EXP=0

m

or

1 1

 P A = P A j EXP=0:

p

2

m m

1. The Quantitative Structure of Exp onential Time 13

1

Amb os-Spies [Amb86] has shown that P A has Leb esgue measure

m

0 whenever A 62 P. The following lemma obtains a stronger conclusion

resource-b ounded measure 0 from a stronger hyp othesis on A.

Juedes and Lutz [JL95a]. Let A be a language that is Lemma 6.2

P

incompressible by  -reductions.

m

1 1

1. If A 2 E, then  P A = 0; and hence P A j E=0.

p

m m

1 1

P A = 0; and hence P A j EXP=0. 2. If A 2 EXP, then 

p

2

m m

We do not prove this lemma here, but we use it to prove the Small Span

Theorem.

Pro of of Theorem 6.1. To prove 1, let A 2 E and let X b e the set of all

P

languages that are incompressible by  -reductions. Wehavetwo cases.

m

Case I. If P A \ E \ X = ;, then Corollary 4.2 tells us that

m

P A j E=0.

m

Case I I. If P A \ E \ X 6= ;, then x a language B 2 P A \ E \ X .

m m

Since B 2 E \ X , Lemma 6.2 tells us that

1 1

B  j E = 0: B  = P  P

p

m m

1 1

Since P A P B , it follows that

m m

1 1

 P A = P A j E = 0:

p

m m

This proves 1. The pro of of 2 is identical. 2

P

Using the Small Span Theorem, we note that  -hard languages for E

m

are extremely rare.

Theorem 6.3 Juedes and Lutz [JL95a]. Let H E be the set of all

m

P

languages that are  -hard for E. Then  H E = 0.

p m

m

1

Let A b e as in Corollary 4.3. Then H E P A, so Lemma Pro of.

m

m

6.2 tells us that

1

 H E =  P A = 0:

p m p

m

2

Recently, Amb os-Spies, Neis, and Terwijn [ANT ] have used resource-

b ounded genericity to prove the extension of Lemma 6.2 obtained by sub-

1

P P 1

stituting  -reductions and P A for  -reductions and P A,

m m

k tt

ktt

resp ectively, where k is a xed p ositive integer. From this they have ob-

tained the following extension of Theorem 6.1.

P

Theorem 6.4 Small  -Span Theorem - Amb os-Spies, Neis, and Ter-

k tt

wijn [ANT ]. Let k b e a p ositiveinteger.

1. The Quantitative Structure of Exp onential Time 14

-1 -1 Pm (A) Pm (A)

Pm (A) Pm (A)

(a) (b)

-1 Pm (A)

Pm (A)

(c)

FIGURE 2. Small-span con gurations. Narrow regions depict measure 0 spans.

1. For every A 2 E,

P AjE = 0

k tt

or

1 1

 P A = P AjE = 0

p

ktt ktt

2. For every A 2 EXP,

P AjEXP = 0

ktt

or

1 1

P A = P AjEXP = 0 

p

2

ktt ktt

This immediately yields the following extension of Theorem 6.3.

Theorem 6.5 Amb os-Spies, Neis, and Terwijn [ANT ]. Let k b e a p osi-

P

tiveinteger. If H E is the set of all languages that are  -hard for

k tt

k tt

E, then  H E = 0.

p k tt

1. The Quantitative Structure of Exp onential Time 15

At the time of this writing, it is not known whether Theorems 6.4 and

1

6.5 remain true when P A, P A, and H E are substituted for

btt btt

btt

1

P A, P A, and H E, resp ectively. Buhrman and Mayordomo

ktt ktt

ktt

[BM95b] and, indep endently,Amb os-Spies, Neis, and Terwijn [ANT ], have

shown that the class H E has p -measure 0.

btt 2

P P

The Small  -Span Theorem has immediate consequences for the  -

m m

degree structure of E and EXP.

P 

The  -degree of a language A f0; 1g is the set

m

P

1

deg A=P A \ P A:

m m

m



Theorem 6.6 Juedes and Lutz [JL95a]. For all A f0; 1g ,

P P

deg A j E = deg A j EXP=0:

m m

This follows immediately from Theorem 6.1. 2 Pro of.

Theorem 6.7 Mayordomo [May94a]. Let C E, C EXP b e the sets of

m m

P

languages that are  -complete for E, EXP, resp ectively. Then C E j

m

m

E = C EXP j EXP=0.

m

Mayordomo's original pro of of this result used Theorem 4.5 and Berman's

P

result [Ber76] that no  -complete language for E or EXP is P-immune.

m

We now see that Mayordomo's result also follows from Theorem 6.3 and

from Theorem 6.6.

Using Theorem 6.4 in place of Theorem 6.1 gives the following extension

of Theorem 6.6.

Theorem 6.8 Amb os-Spies, Neis, and Terwijn [ANT ]. For all A



f0; 1g and all p ositiveintegers k ,

P P

deg AjE = deg AjEXP  = 0

ktt ktt

P

It is not currently known whether all  -degrees have measure 0 in E

btt

P

-degree. or EXP, but this at least holds for the complete 

btt

Theorem 6.9 Amb os-Spies, Neis, and Terwijn [ANT]. Let C E; C EXP

btt btt

P

b e the sets of languages that are  -complete for E, EXP, resp ectively.

btt

Then C EjE = C EXPjEXP = 0.

btt btt

7 Weakly hard problems

To date, our principal means of establishing the intractability of a sp eci c

computational problem has b een to prove that the problem is hard for

1. The Quantitative Structure of Exp onential Time 16

some complexity class with resp ect to some class of ecient reductions.

P

For example, a problem that is  -hard for NP, PSPACE, or some class in

m

between is presumably intractable b ecause we are inclined to b elieve that

P

P 6=NP. A problem that is  -hard for E is provably intractable by the

m

time hierarchy theorem of Hartmanis and Stearns [HS65]. In fact, problems

P

that are  -hard for E are now known to havevery strong intractability

m

prop erties [BS85, Huy86, KOSW94, OS86,Sch86a].

In order to extend the class of provably intractable problems, Lutz [Lut90]

prop osed investigation of the following measure-theoretic generalization of

P

 -hardness.

m

 P

De nition. A language A f0; 1g is weakly  -hard for E resp ectively,

m

for EXPifP AjE 6= 0 resp ectively, P AjEXP 6= 0.

m m

P

Thus a language A is weakly  -hard for E if a nonnegligible subset of

m

P

the languages in E are  -reducible to A. Clearly,every language that is

m

P P

 -hard for E is also weakly  -hard for E.

m m

P

-hardness Weak hardness under other classes of reductions e.g. weak 

T

is de ned analogously.

The rst thing to note ab out weakly hard problems is that they are,

indeed, intractable. Sp eci cally, it is easy to see that PjE = PjEXP=

0 [Lut92], so wehave the following.

P

Observation 7.1. If A is weakly  -hard for E, then A 62 P.

T

P

In fact, languages that are weakly  -hard for E are intractable in a

m

much stronger sense. For example, consider the following strong intractabil-

ity result.

Orp onen and Schoning [OS86]. Every language that is Theorem 7.2

P

 -hard for E has a dense P-complexity core.

m

The following theorem extends Theorem 7.2 in somewhat stronger form

P

to all weakly  -hard languages for E.

m

Theorem 7.3 Juedes and Lutz [JL95a]. Every language that is weakly

P

 -hard for E or EXP has a dense exp onential complexity core.

m

P P

Thus the weakly  -hard problems for E and EXP are, like the  -hard

m m

problems, provably strongly intractable. It is then natural to ask whether

P P

-hard, for these -hard, but not  there are problems that are weakly 

m m

classes. Wenow discuss this question and, more generally, the distribution

of the weakly hard languages.

 P

De nition. A language A f0; 1g is weakly  -complete for E resp ec-

m

P

-hard for E resp ectively, for EXP and tively, for EXP if A is weakly 

m

A 2 E resp ectively, A 2 EXP.

1. The Quantitative Structure of Exp onential Time 17

As in section 6, we use the notations H E; H EXP; C E, and

m m m

P P

C EXP to denote the classes of languages that are  -hard for E,  -

m

m m

P P

hard for EXP,  -complete for E, and  -complete for EXP, resp ec-

m m

tively. We also use the notations WH E; WH EXP; WC E, and

m m m

P

WC EXP to denote the classes of languages that are weakly  -hard

m

m

P P

for E, weakly  -hard for EXP, weakly  -complete for E, and weakly

m m

P

 -complete for EXP, resp ectively.

m

We rst discuss the known inclusions among the ab ove-de ned hardness

classes. We then discuss the non-inclusions. This was not the chronologi-

cal order of discovery. It is well known that H E = H EXP, whence

m m

C E=E\C EXP. This is clear b ecause EXP = P E. Also, The-

m m m

orem 3.3 implies that H E WH E and H EXP WH EXP.

m m m

Using the martingale dilation technique develop ed by Amb os-Spies, Ter-

wijn, and Zheng [ATZ ], Juedes and Lutz proved the following.

Lemma 7.4 Juedes and Lutz [JL95b]. Let X b e a set of languages.

1. If  P X  = 0, then  X =0.

p m p

2

2. If P X jEXP = 0, then X jE = 0.

m

This yields the following.

Theorem 7.5 Juedes and Lutz [JL95b]. WH E WH EXP.

m m

Pro of. Let H 2WH E. Then P HjE 6= 0, so Lemma 7.42 with

m m

X =P H tells us that P HjEXP = P P HjEXP 6=0.Thus

m m m m

H 2WH EXP. 2

m

The foregoing discussion, in combination with obvious facts, yields the

inclusion structure depicted in Figure 3. FIGURE 3. Inclusion structure of hardness classes.

1. The Quantitative Structure of Exp onential Time 18

Wenow turn to the non-inclusions. Lutz [Lut95] develop ed the martin-

gale diagonalization technique and used it to prove the following.



Theorem 7.6 Lutz [Lut95]. C E WC E.

m m

6=



Corollary 7.7. C EXP WC EXP.

m m

6=

Pro of. By Theorems 7.6, 7.5, and elementary facts,



E \C EXP=C E WC E E \WC EXP:

m m m m

6=

2

Theorem 7.6 is signi cant b ecause, in combination with Observation 7.1

P

and Theorem 7.3, it implies that the class of weakly  -hard problems for

m

E is, indeed, a strictly larger class of provably strongly intractable problems

P

than the class of  -hard problems for E. In fact, much more is true. Juedes

m

[Jue95] re ned the martingale diagonalization of [Lut95] to prove that the

class WC E do es not have measure 0 in E. By Theorem 6.7, this result

m

implies Theorem 7.6. More signi cantly,Amb os-Spies, Terwijn, and Zheng

develop ed martingale dilation a padding technique and used it to prove

the following.

Theorem 7.8 Amb os-Spies, Terwijn, and Zheng [ATZ ].

 WH E = 1:

p m

Corollary 7.9 Amb os-Spies, Terwijn, and Zheng [ATZ].

 WH EXP = 1:

p m

2

This follows immediately from Theorems 7.8 and 7.5. 2 Pro of.

By Theorems 7.8 and 6.7, then, almost every language in E is weakly

P P

-complete, for E. -complete, but not  

m m

Finally,we note that the converse of Theorem 7.5 do es not hold, even if

we restrict attention to languages in E.

Theorem 7.10 Juedes and Lutz [JL95b]. E \WC EXP 6 W C E.

m m

By Theorems 7.6, 7.10, and elementary observations, Figure 3 is com-

plete, in the sense that it depicts either directly or via transitivity all the

inclusions that hold among these eight hardness classes.

1. The Quantitative Structure of Exp onential Time 19

8 Upp er b ounds for hard problems

P

Wesaw in Theorem 6.3 that  -hard languages for E are very rare. As we

m

see in this section, this is b ecause there is a nontrivial upp er b ound on the

sizes of complexity cores of such languages.



Recall that a language D f0; 1g is dense if there is a real number

"

n

">0 such that jD j > 2 a.e.

n

P

The following result states that every  -hard language for E can be

m

4n

decided in time 2 on a dense set of instances that can itself b e decided

4n

in time 2 .

P

Theorem 8.1 Juedes and Lutz [JL95a]. For every  -hard language H

m

4n

for E, there exist B; D 2 DTIME2  such that D is dense and B = H \ D .

P

It is straightforward to use Theorem 8.1 to prove that  -hard languages

m

for E ob ey the following upp er b ound on the sizes of complexity cores.

4n

Theorem 8.2 Juedes and Lutz [JL95a]. Every DTIME2 -complexity

P

core of every  -hard language for E has a dense complement.

m

 4n

By Corollary 5.3, almost every language in E has f0; 1g as a DTIME2 -

P

complexity core. Thus, Theorem 8.2 says that  -hard languages for E are

m

unusual ly simple, in the sense that they have unusual ly smal l complexity

cores, for languages in E. This immediately implies, and also explains, The-

orems 6.3 and 6.7.

P

Lutz [Lut95] constructed a weakly  -hard language H for E that has

m

 4n

f0; 1g as a DTIME2 -complexity core, so Theorem 8.2 is a prop erty

P P

of  -hard languages that do es not extend to weakly  -hard languages.

m m

In fact, by Corollary 4.3 and Theorem 7.8, almost every language in E

P

isaweakly  -complete language that do es not satisfy the conclusion of

m

Theorem 8.2.

9 Nonuniform complexity, natural pro ofs, and

pseudorandom generators

Much remains to b e discovered ab out the nonuniform complexities of lan-

guages in E and EXP.For example, it is a long-standing conjecture that

E 6 P/Poly, i.e., that E do es not have p olynomial-size circuits, but it has

not b een proven that E do es not have linear -size circuits, or that EXP do es

not have p olynomial-size circuits. It is known, however, that the highest

levels of circuit-size and time-b ounded Kolmogorov complexity known or

provable by relativizable metho ds to b e exceeded in nitely often by any

problem in EXP are in fact exceeded almost everywhere by almost every

1. The Quantitative Structure of Exp onential Time 20

problem in the class. Moreover, recentwork of Regan, Sivakumar, and Cai

[RSC95], exploiting the \natural pro ofs" idea of Razb orov and Rudich

[RR94], has shown that, if suciently secure pseudorandom generators

exist, then these results are optimal for circuit-size complexity. We now

describ e these developments more fully.

Some terminology and notation will be useful. For a xed machine M



and \program"  2f0; 1g for M ,wesay that \M ; n=w in  t time"



if M , on input ; n, outputs the string w 2f0; 1g and halts in at most t

execution steps. We are esp ecially interested in situations where the output

n

string is of the form w = , i.e., the 2 -bit characteristic string of A ,

A =n

=n



for some language A f0; 1g .



Given a machine M , a time-b ound t : N ! N, a language A f0; 1g ,

and a natural number n, the tn-time-bounded Kolmogorov complexity of

A relativetoM is

=n

 

tn

M ; n= in  tn time : K A  = min j j

A =n

=n

M

Well-known simulation techniques show that there is a machine U that is

optimal in the sense that for each machine M there is a constant c such

that, for all t; A; and n,

ctn log tn+c tn

K A   K A +c:

=n =n

U M

As is standard in this sub ject, we x an optimal machine and omit it

from the notation. See [LV93] for a thorough treatment of Kolmogorov

complexity.

Theorem 9.1 Lutz [Lut92]. If t and q are xed p olynomials, then the

set of all languages A satisfying

tn

K A  >qn a.e.

=n

has measure 1 in EXP.

Wenow consider circuit-size complexity.Following standard usage see

[BDG95], for example, we de ne a Boolean  circuit to b e a directed acyclic

graph with vertex set I [ G, where I = fw ;:::;w g is the set of inputs

1 n

n  0 and G = fg ;:::;g g is the set of gates s  1. Each input has

1 s

indegree 0 and each gate has indegree 0, 1, or 2. Each gate of indegree

0 is lab eled either by the constant 0 or by the constant 1. Each gate of

indegree 1 is lab eled either by the identity function ID: f0; 1g ! f0; 1g

or by the negation function NOT: f0; 1g! f0; 1g. Each gate of indegree

2

2 is lab eled either by the conjunction AND: f0; 1g ! f0; 1g or by the

2

disjunction OR: f0; 1g !f0; 1g. The output gate g has outdegree 0. The

s

other gates and the inputs have unrestricted outdegree. The size of sucha

circuit is size =jGj = s, the numb er of gates.

1. The Quantitative Structure of Exp onential Time 21

n

An n-input circuit computes a Bo olean function : f0; 1g !f0; 1g in

n

the usual way.For w 2f0; 1g ; w  is the value computed at the output

gate g when the inputs are assigned the bits w ;:::;w of w . The set

s 1 n

n

computedbyan n-input circuit is then the set of all w 2f0; 1g such that

w =1.



The circuit-size complexity of a language A f0; 1g is the function

CS : N ! N de ned by

A

CS n = min f size  j computes A g :

A =n

For each function f : N ! N,we de ne the circuit-size complexity classes

SIZEf  = f A j CS n  f n a.e. g ;

A

i.o.

SIZE f  = f A j CS n  f n i.o. g :

A

The class P/Poly is then de ned by

1

[

k

P/Poly = SIZEn ;

k =0

and we write

1

[

i:o:

k

i.o.

P/Poly = SIZE n :

k =0

Using a known quantitative relationship b etween circuit size and time-

b ounded Kolmogorov complexity, the following result can b e derived from

Theorem 9.1.

i.o. k

Theorem 9.2 Lutz [Lut92]. For each xed k 2 N, the set SIZE n 

has measure 0 in EXP.

A similar argument proves the following.

i:o:

Theorem 9.3 Lutz [Lut92]. The set P/Poly has measure 0 in the

p olylog n

n

class E = DTIME2 .

3

As noted earlier, it is a long-standing conjecture that E 6 P/Poly.Intu-

itively, this conjecture says that E contains problems that are combinatori-

al ly,aswell as computationally,intractable. In light of the various strong

intractability results of sections 4, 5, and 8, the stronger conjecture that

P/Poly has measure 0 in E and in EXP seems to suggest itself. However,

as wenow explain, there is reason to b e cautious ab out such a conjecture.

We rst note that Wilson [Wil85] has exhibited oracles relative to which

E SIZE3n and EXP P/Poly, so nonrelativizable techniques will b e

required to prove EXP 6 P/Poly, let alone the stronger conjecture that

P/PolyjEXP=0.

1. The Quantitative Structure of Exp onential Time 22

In recentyears, many nonrelativizable combinatorial techniques for prov-

ing lower b ounds on nonuniform complexity have b een develop ed see

[RR94] for references to such developments, so Wilson's oracle construc-

tions are not as daunting to day as they were when he discovered them.

However, all such techniques develop ed to date are for proving lower b ounds

with resp ect to restricted nonuniform mo dels b ounded-depth circuits, mono-

tone circuits, etc., and do not seem to yield to lower b ound techniques for

general circuit-size complexity.

Razb orov and Rudich [RR94] develop ed the notion of natural proofs

in order to b etter understand these limitations on known techniques. The

central idea in their work is that of a \natural combinatoral prop erty,"

whichwenow describ e, not in full generality, but in terms of the present

discussion.

De nition 9.4 Razb orov and Rudich [RR94].

1. A combinatorial property is a sequence P =P ; P ; P ;:::, where

0 1 2

n

each P is a set of subsets of f0; 1g .

n



2. A language A f0; 1g is drawn from a combinatorial prop erty P if,

for all n 2 N;A 2P .

=n n

As part 2 of the ab ove de nition suggests, we regard each comp onent P

n

of a combinatorial prop erty P as a \set of candidate n-slices" A for lan-

=n

n

guages A that are drawn from P .We identify each set S 2P with its 2 -bit

n

characteristic string , and we regard the \complexity" of P as the com-

S

n

plexity of deciding memb ership of 2 -bit strings in P . As with martingales

n

and measure, we will use the lower-case notations p; p , etc. for complexity

2

classes of functions whose inputs are characteristic sequences. At present,

we are interested in nonuniform p -complexity, where the nonuniformityis

2

provided by an advice function.



A nonuniform p -advice function is a function h : N !f0; 1g De nition.

2

for which there is a constant k 2 N such that, for all n>0; jhnjk +

k

log n

2 . Note that h need not b e computable; this is the nonuniformity.

A combinatorial prop erty P is nonuniformly p -decidable if De nition.

2

there exist a nonuniform p -advice function h and a function f 2 p such

2 2

n

that, for all n 2 N and S f0; 1g ,

n

f  ;h2  = [[ 2P ]]:

S S n

In a lower b ound argument, the typical role of a combinatorial prop erty

is to reliably diagonalize against some complexity class.

De nition 9.5 Razb orov and Rudich [RR94]. Let P b e acombinatorial

prop erty, and let C b e a class of languages.

1. The Quantitative Structure of Exp onential Time 23

1. P is useful in nitely often useful i.o. against C if, for every A 2C,

there exist in nitely many n 2 N such that A 62 P .

=n n

2. P is useful almost everywhere useful a.e. against C if, for every

A 2C, for all suciently large n 2 N, A 62 P .

=n n

The crucial thing that Razb orov and Rudich observed is that the com-

binatorial prop erties used in lower b ound pro ofs are typically large, in the

following sense.

De nition 9.6 Razb orov and Rudich [RR94]. A combinatorial prop erty

P is large if there is a constant k 2 N such that, for all suciently large

n 2 N,

n

2 kn

jP j 2 :

n

To rephrase the de nition probabilistically, let PrP  denote the prob-

n

n

ability that S 2 P when S f0; 1g is chosen according to a random

n

n

exp eriment in which all subsets of f0; 1g are equally probable. Then P is

kn

large if there exists k such that, for all suciently large n,PrP   2 .

n

Wenowhave all the elements of the notion of a \natural combinatorial

prop erty."

Razb orov and Rudich [RR94]. A combinatorial prop erty De nition 9.7

P is nonuniformly p -natural i.o. against a class C of languages if P is

2

nonuniformly p -decidable, P is useful i.o. against C , and P is large.

2

Here we are sp eci cally interested in the EXP versus P/Poly question,

so wehave sp ecialized the ab ove de nition to nonuniform p -decidability

2

and i.o. diagonalization. The interested reader is referred to [RR94, Raz]

for more general asp ects of Razb orov and Rudich's work.

Regan, Sivakumar, and Cai's work on natural combinatorial prop erties

involves the existence of a certain kind of secure pseudorandom generator.

Wenow develop the required de nitions.

De nition. Let p b e a p olynomial such that pn  n +1. A pn-generator



is a function g 2 PF such that, for all x 2f0; 1g , j g x j= pjxj.

Intuitively a generator g , given a short, random seed x, outputs a long,

hop efully pseudorandom, string g x. The desired notion of pseudorandom-

ness, also called \security," is given by the following de nition, due to Yao

[Yao82].

De nition. Let s = N ! N. A pn-generator g is nonuniformly sn-

secure if, for every suciently large n 2 N, for every pn-input, 1-output

circuit with siz e   sn,

1

j Pr[ g x = 1] Pr[ y =1]j < ;

sn

1. The Quantitative Structure of Exp onential Time 24

where the probabilities are computed according to the uniform distributions

n pn

on x 2f0; 1g and y 2f0; 1g , resp ectively.

Intuitively, a generator g is sn-secure if no sn-gate circuit can statisti-

pn

cally distinguish the uniform distribution on f0; 1g from the distribution

pn

induced on f0; 1g by the generator g with the uniform distribution on

n

the seed space f0; 1g .

De nition.

1. A pn-generator g is nonuniformly polynomial ly secure if g is sn-

secure for every p olynomial s.

2. A pn-generator g is nonuniformly exponential ly secure if there is a



n

real constant >0 such that g is 2 -secure.

It is easy to show that, if there exists a nonuniformly p olynomially secure

pn-generator g , then NP 6 P/Poly whence P 6= NP. In fact, such gener-

ators are widely conjectured to exist. The existence of pn-generators that

are nonuniformly exp onentially secure is an even stronger conjecture, but

not entirely implausible. For example, Regan, Sivakumar, and Cai [RSC95]

have p ointed out that the smallest circuits known to break pseudorandom

generators that are based on the discrete logarithm problem have nearly

p

n

2 gates.

In any case, the following result shows that the existence of nonuni-

formly exp onentially secure generators is not consistent with the existence

of nonuniformly p -natural prop erties against P/Poly.

2

Theorem 9.8 Razb orov [Raz]; see also [RSC95]. If there is a nonuni-

formly exp onentially secure 2n-generator, then there is no combinatorial

prop erty that is nonuniformly p -natural i.o. against P/Poly.

2

The following result relates these issues to the measure of P/Poly in

EXP.

Regan, Sivakumar, and Cai [RSC95]. If P/PolyjEXP = Theorem 9.9

0, then there is a combinatorial prop erty that is nonuniformly p -natural

2

i.o. against P/Poly.

By Theorems 9.4 and 9.5, wehave the following.

Theorem 9.10 Regan, Sivakumar, and Cai [RSC95]. If there is a

nonuniformly exp onentially secure 2n-generator, then P/PolyjEXP 6=0.

By Lemma 7.4, P/PolyjEXP = 0 implies that P/PolyjE=0.At

the time of this writing, the converse is not known to hold, nor is an ana-

logue of Theorem 9.6 known to hold for E. It is thus conceivable that

P/Poly has measure 0 in E and by Theorem 9.3 in E , but not in E =

3 2

EXP.

1. The Quantitative Structure of Exp onential Time 25

10 Weak sto chasticity

It is now known that almost every language in E, and almost every language

in E , is statistically unpredictable by feasible deterministic algorithms,

2

even with some nonuniform advice. This result, which app ears to b e very

useful, is explained in this section.

Prop erties de ned in terms of limiting frequencies of failure of predic-

tion schemes are called stochasticity prop erties in the terminology of Kol-

mogorov [KU87, USS90]. Such prop erties were originally prop osed byvon

Mises [vM39] and Church [Chu40] in their e orts to de ne randomness.

Because the prediction schemes allowed in this section are of a restricted

sort, the prop erty discussed here is a weak stochasticity prop erty.

We now make our terminology precise. Our notion of advice classes is



standard [KL80]. An advice function is a function h : N !f0; 1g : Given

a function q : N ! N,we write ADVq  for the set of all advice functions



h such that jhnjq n for all n 2 N. Given a language A f0; 1g and

an advice function h,we de ne the language A=h \A with advice h" by



A=h = fx 2f0; 1g jhx; hjxji2Ag:

Given functions t; q : N ! N,we de ne the advice class

DTIMEt=ADVq =fA=h j A 2 DTIMEt;h 2 ADVq g:



De nition. Let t; q ;  : N ! N and let A f0; 1g . Then A is weakly

t; q ;  -stochastic if, for all B; C 2 DTIMEt=ADVq  such that jC j

=n

 n for all suciently large n,

jA 4 B  \ C j 1

=n

lim = :

n!1

jC j 2

=n

Intuitively, B and C together form a \prediction scheme" in which B tries

to guess the b ehavior of A on the set C . A is weakly t; q ;  -sto chastic if

no suchscheme is b etter in the limit than guessing by random tosses of a

fair coin. This de nition is slightly stronger than the weak sto chasticity

de ned in [LM94], in that the language C is allowed advice here.

Weak Sto chasticity Theorem|Lutz and Mayordomo [LM94]. Theorem 10.1

For every xed k 2 N and every xed real number >0,

kn n kn n

WS 2 ;kn;2 jE =  WS2 ;kn;2  = 1

p

and

k  k 

n k n n k n

 WS2 ;n ; 2  = WS 2 ;n ; 2 jEXP = 1:

p

2

That is, almost every language in E, and almost every language in EXP,

is weakly sto chastic with the indicated parameters.

1. The Quantitative Structure of Exp onential Time 26

Regan and Sivakumar [RS95]have recently given a more precise analysis

of the Weak Sto chasticity Theorem, esp ecially with resp ect to the rate of

convergence.

11 Density of hard languages

As noted in section 9 ab ove, it is a long-standing op en conjecture that

E 6 P/Poly, i.e., that not every language in E has p olynomial circuit-size

complexity. Many ongoing e orts to prove this conjecture follow a program

that b egan with the following results of Meyer.



Recall that a language A f0; 1g is sparse if there is a p olynomial q such

that jA jq n a.e., and dense if there is a real number ">0 such that

n

"

n

jA j > 2 a.e. We write SPARSE for the set of all sparse languages and

n

c

DENSE for the set of all dense languages. Note that SPARSE $ DENSE ,

c

P

where DENSE is the complement of DENSE. For each reducibility  ,

r

each language A, and each set S of languages, we write



P

B  A P A= B

r

r

and

[

P S = P S :

r r

A2S

Theorem 11.1 Meyer [Mey77]. P/Poly = P SPARSE.

T

P

Theorem 11.2  Meyer [Mey77]. Every  -hard language for E or any

m

c

larger class is dense. That is, E 6 P DENSE :

m

Meyer [Mey77]. E 6 P SPARSE. Corollary 11.3

m

Meyer's results suggest proving theorems of the form

E 6 P SPARSE

r

for successively larger classes P SPARSE in the range

r

P SPARSE P SPARSE P SPARSE:

m r T

Along the way, we should try to make our results as strong as p ossible.

For example, results of Nisan and Wigderson [NW88, Nis92] indicate

that suciently strong lower b ounds on the nonuniform complexity of E

could lead to the construction of useful pseudorandom generators.

The next big step in this program was taken byWatanab e, who proved

P

-reducibility p olynomial-time truth- the following result concerning 

q ntt

table reducibility with q n queries on inputs of length n.

1. The Quantitative Structure of Exp onential Time 27

P

Theorem 11.4 Watanab e [Wat87b]. Every  -hard language

O log ntt

c

for E is dense. That is, E 6 P DENSE :

O log n-tt

Recently, a measure-theoretic attack on this problem has led to the fol-

lowing strengthening of Theorem 11.4.

Lutz and Mayordomo [LM94]. For every real number Theorem 11.5

c c

< 1 e.g., = 0:99, P DENSE  j E = P DENSE  j

n tt n tt

EXP=0:

Corollary 11.6 Lutz and Mayordomo [LM94]. For every real number

c

P

< 1 e.g., = 0:99, E 6 P DENSE ; i.e., every  -hard

n tt

n tt

language for E is dense.

The pro of of Theorem 11.5 uses a simple combinatorial technique|the

sequential ly most frequent query selection |to show that every language

c

in P DENSE  is predictable, i.e., fails to b e weakly sto chastic with

n tt

suitable parameters. The result then follows immediately from Theorem

10.1, the Weak Sto chasticity Theorem.

Given the Weak Sto chasticity Theorem, whichisavery general princi-

ple, this pro of of Corollary 11.6 via Theorem 11.5 is much simpler than

the stage construction originally used to prove Theorem 11.4. This is not

surprising, once it is noted that our pro of of Corollary 11.6 is an appli-

cation of a resource-b ounded generalization of  the probabilistic metho d

[Erd47, Sha48, Sha49, ES74, Sp e87, AS92], which exploits the fact that it

is often easier to establish the abundance of ob jects of a given typ e than

to construct a speci c ob ject of that typ e.

It should b e emphasized here that Theorem 11.5 is more than a means of

proving Corollary 11.6. By analogy, the value of classical Leb esgue measure

and probability far surpasses their role as to ols for existence pro ofs. The

c

quantitative content of Theorem 11.5|that the set P DENSE  \ E

n tt

is a negligibly smal l subset of E|is much stronger than the qualitative

separation of Corollary 11.6.

Recently,Fu has indep endently proven the following, related result.

Theorem 11.7 Fu[Fu95].

c

1

,E6 P DENSE . 1. For every real <

n T

4

c

2. For every real <1, EXP 6 P DENSE .

n T

Note that the reducibilities here are Turing, i.e., adaptive, as opp osed to

the nonadaptive truth-table reducibilities of Corollary 11.6.

1. The Quantitative Structure of Exp onential Time 28

NP j EXP 6=0 = NP j E 6=0

m +

 NP 6=0 =  NP 6=0

p p

2

+ +

k

n cn

8k NP 6 DTIME2  = 8cNP 6 DTIME2 

+

P 6=NP

FIGURE 4. Non-smallness conditions for NP.

12 Strong hyp otheses

At our present state of knowledge i.e., lack thereof , many results in com-

plexity theory contain strong, unproven hyp otheses. Here are just three

examples.

P P

Theorem 12.1 Karp and Lipton [KL80]. If  6=  , then NP 6

2 2

P SPARSE.

T

Mahaney [Mah82]. If P 6=NP, then NP 6 P SPARSE. Theorem 12.2

m

Ogiwara and Watanab e [OW91]. If P 6= NP, then NP 6 Theorem 12.3

P SPARSE.

btt

This last result refers to p olynomial-time truth-table reducibility with

an arbitrary but xed numb er of queries.

The pro ofs of the ab ove three theorems have given complexity theory

some its most b eautiful and useful techniques. However, the conclusions of

P

these theorems are far weaker than the observation that al l known  -hard

T

languages for NP are dense. In this sense, relative to our current knowledge,

P P

the hyp otheses P 6= NP and  6= lack explanatory p ower.

2

In order to make progress on matters of this typ e, we have prop osed

investigation of various strong measure-theoretic hyp otheses. For example,

Figure 4 gives the implications among various conditions asserting the non-

smallness of NP. In this section we brie y discuss the reasonableness and

known consequences of the weakest measure-theoretic hyp othesis in Figure

4, namely, the hyp othesis that NP do es not have p-measure 0.

This hyp othesis is b est understo o d by considering the meaning of its

negation, that NP has p-measure 0. This latter condition o ccurs if and

only if there is a p-martingale that succeeds b ets successfully on every

language A 2 NP. The fact that the strategy d is p-computable means

cjxj

that, when b etting on the condition \x 2 A", d requires only 2 time for

some xed constant c. This is b ecause the running time of d for this b et is

1. The Quantitative Structure of Exp onential Time 29

p olynomial in the numb er of predecessors of x in the standard ordering of



f0; 1g . On the other hand, for all k 2 N, there exist languages A 2 NP

with the prop erty that the apparent search space space of witnesses for

k k

jxj cn n

each input x has 2 elements. Since c is xed, we have x  x for

large values of k . Such a martingale d would thus be a very remarkable

algorithm! It would bet successfully on al l NP languages, using far less

than enough time to examine the search spaces of most such languages.

It is reasonable to conjecture that no such martingale exists, i.e., that NP

do es not have p-measure 0.

Kautz and Miltersen [KM94]have shown that, if A is an algorithmically

A

random oracle, then  A NP  6= 0. This pro of, though interesting for its

p

analysis of indep endence and randomness, gives no evidence for the truth

of the unrelativized  NP 6= 0 conjecture.

p

Since  NP 6= 0 implies P 6=NP, and  NP = 0 implies NP 6= EXP,

p p

we are unable to prove or disprove the  NP 6= 0 conjecture at this time.

p

Until such a mathematical resolution is available, the condition  NP 6=0

p

is b est investigated as a scienti c hypothesis, to b e evaluated in terms of

the extent and credibility of its consequences.

Wenow survey known consequences of the hyp othesis that NP do es not

have p-measure 0. The rst follows immediately from Theorem 4.5.

Theorem 12.4 Mayordomo [May94a]. If NP do es not have measure 0,

then NP contains a P-bi-immune language.

Using standard techniques, the following result has b een derived from

Theorem 12.4.

Lutz and Mayordomo [LM]. If NP do es not have p- Theorem 12.5

measure 0, then E 6= NE and EE 6= NEE.

Corollary 12.6 Lutz and Mayordomo [LM]. If NP do es not have p-

measure 0, then there is an NP search problem that do es not reduce to the

corresp onding decision problem.

Pro of. Bellare and Goldwasser [BG94] have shown that, if EE 6= NEE,

then there is an NP search problem that do es not reduce to the corresp ond-

ing decision problem. The present corollary follows immediately from this

and Theorem 12.5. 2

P

We now consider complexity cores of languages that are  -hard for

m

NP. The following result is well-known.

Theorem 12.7 Orp onen and Schoning [OS86]. If P 6= NP, then every

P

language that is  -hard for NP has a nonsparse P-complexity core. m

1. The Quantitative Structure of Exp onential Time 30

Strengthening the hyp othesis of Theorem 12.7 gives a stronger conclu-

sion. This essentially follows from Theorem 7.3.

Theorem 12.8 Juedes and Lutz [JL95a]. If NP do es not have p-measure

P

0, then every language that is  -hard for NP has a dense exp onential

m

complexity core.

Concerning the density of hard languages for NP, Theorem 11.5 gives

us the following result. Note that the hyp othesis and conclusion are b oth

stronger than in Theorem 12.6.

Theorem 12.9 Lutz and Mayordomo [LM94]. If NP do es not havep-

c

measure 0, then for every real number <1, NP 6 P DENSE , i.e.,

n tt

P

every  -hard language for NP is dense.

n tt

The next result concerns NP-completeness. The NP-completeness of de-

cision problems has two principal, well-known formulations. These are the

P P

-completeness -completeness intro duced by Co ok [Co o71] and the  

m

T

intro duced by Karp [Kar72] and Levin [Lev73]. It is widely conjectured

[LLS75,You83,LY90, Hom90] that these two notions are distinct:

\Co ok versus Karp-Levin". There exists a language CvKL Conjecture.

P P

that is  -complete, but not  -complete, for NP.

m

T

The CvKL Conjecture is very ambitious, since it implies that P 6= NP.

The question has thus b een raised [LLS75, Sel79, Hom90, BHT91] whether

the CvKL Conjecture can be derived from some reasonable complexity-

theoretic hyp othesis, suchasP6= NP or the separation of the p olynomial-

time hierarchyinto in nitely many levels. To date, despite extensivework

[Sel79, KM75, Wat87a, Wat87b, WT92, Wat87b, BHT91, LY90, LLS75,

Sel79, Hom90, BHT91] , even this more mo dest ob jective has not b een

achieved.

The following result shows that the CvKL Conjecture holds under our

strong measure-theoretic hyp othesis.

Lutz and Mayordomo [LM]. If NP do es not have p- Theorem 12.10

P P

- measure 0, then there is a language C that is  -complete, but not 

m

T

complete for NP.

Of the measure-theoretic results mentioned thus far in this section, The-

orems 12.4, 12.8, and 12.9 hold with NP replaced byany class whatso ever.

Theorem 12.5, Corollary 12.6, and Theorem 12.10 are more sp eci c to NP.

The hyp othesis  NP 6= 0 also has consequences involving the com-

p

P

plexity classes BPP and BPP  for k  1. In fact, these consequences

k

P

all follow from the hyp othesis that the class
do es not have p-measure 2

1. The Quantitative Structure of Exp onential Time 31

P P

0. Since NP
, the hyp othesis  
 6= 0 follows from, and is thus at

p

2 2

least as plausible as, the hyp othesis  NP 6=0.

p

P

The rst consequence of  
 6= 0 is a tightening of the result, due to

p

2

P P

Lautemann [Lau83] and Sipser and G acs [Sip83], that BPP  \  .

2 2

P

Allender and Strauss [AS94]. If  
 6= 0, then Theorem 12.11

p

2

P

BPP
.

2

A slight strengthening of the pro of of Theorem 12.11 yields the following.

P P

Theorem 12.12 Lutz [Luta]. If  
 6= 0, then for all k  1, BPP =

p

2

k

P

.

k +1

Theorem 12.12 has consequences for lowness and p olynomial advice. If

C and L are classes of languages, then L is low for C if C L C . The

following corollary follows easily from Theorem 12.12 and the fact, due to

Kobler, Schoning, and Tor an [KST93], that AM \ co-AM is low for AM.

See [KST93] for the de nition and basic prop erties of the \Arthur-Merlin"

class AM.

P

Lutz [Luta]. If 
 6= 0, then AM \ co-AM is low Corollary 12.13



2

P

for
.

2

P P

Corollary 12.14 Lutz [Luta]. If  
 6= 0, then BPP is low for
.

p

2 2

P

Lutz [Luta]. Assume that  
 6= 0. Then the graph Corollary 12.15

p

2

P P

isomorphism problem is low for
. Thus, if
6= PH, then the graph

2 2

P P SNP

isomorphism problem is not  -complete,  -complete, or even  -

m

T T

complete for NP.

SNP

is de- The strong nondeterministic p olynomial-time reducibility 

T

SNP

B if and only if A 2 NPB  \ co-NPB . ned by A 

T

P

By Theorem 11.1, Theorem 12.1 says that, if  6= PH, then NP 6

2

P/Poly. A recent, signi cant improvement of this result is the following.

Theorem 12.16. [BCKT94, KW95] If ZPPNP 6= PH then NP 6 P/Poly.

See [BDG95] for the de nition and basic prop erties of the zero-error

probabilistic p olynomial-time complexity class ZPP.Itiswell-known that

P P P

ZPP  \  .

2 2 2

The following result, which follows immediately from Theorems 12.12

and 12.16, derives the same conclusion as Theorems 12.1 and 12.16 from a

somewhat di erenthyp othesis.

P P

Corollary 12.17 Lutz [Luta]. If  
 6= 0 and
6= PH, then NP 6

p

2 2

P/Poly.

1. The Quantitative Structure of Exp onential Time 32

13 Conclusion

Resource-b ounded measure has b een shown to interact in informative,

quantitativeways with p olynomial-time reductions, bi-immunity, complex-

ity cores, completeness, circuit complexity, Kolmogorov complexity, the

density of hard languages, randomized complexity,lowness, and other much-

studied structural asp ects of the exp onential time complexity classes E and

EXP. This work has expanded the class of provably intractable problems

section 7, and there are indications throughout that it mayhave profound

implications for the structure of NP and other classes that characterize im-

p ortant computational problems.

Ultimately, the ob jective of this work is a detailed account of the quan-

titative structure of E and EXP, with sucient resolution to yield useful

b ounds on the complexities of natural computational problems. The results

achieved to date are only a very small b eginning. Here we mention just a

few directions for further work.

1. One of the most signi cantchallenges is to nd natural examples of

P P

languages that are weakly  -complete, but not  -complete, for

m m

EXP. Theorem 7.8 suggests the existence of such natural examples,

and Theorem 7.3 underscores the imp ortance of nding them.

2. Most of the results mentioned in sections 4-8 concern the structure

P

of E and EXP under  -reducibility. It will b e worthwhile to inves-

m

P

tigate how far in the direction of  -reducibility these results can

T

P

b e extended. For example, a Small Span Theorem for  -reductions

T

P

or even for  -reductions in EXP would imply that EXP 6 BPP

tt

[JL95a, ANT ].

3. In light of Theorem 11.4 and Corollary 11.6, it may well be that

measure arguments can b e used to simplify or replace other known

stage constructions. Such simpli cation might clarify issues, leading

to further progress.

4. Many other structural asp ects of E and EXP remain to be investi-

gated from the standp oint of resource-b ounded measure. For exam-

ple, it seems likely that resource-b ounded measure will shed lighton

the theory of average-case complexity. Cai and Selman [CS96]have

made one observation in this regard, but we hop e that this is only a

b eginning.

5. Work to date has fo cused on the measure-theoretic structure of classes

of languages, i.e., decision problems. Classes of functions, search prob-

lems, optimization problems, approximation problems, etc., should

also b e investigated in this light.

6. The reasonableness and consequences of strong hyp otheses such as

those mentioned in section 12 require further investigation. Are the

1. The Quantitative Structure of Exp onential Time 33

P

hyp otheses  NP 6= 0 and  
 6= 0 equivalent, or is the latter

p p

2

in some sense weaker? Do es  NP 6= 0 imply that there is a lan-

p

P P

guage that is  -complete, but not  -complete, for NP? Do these

tt

T

hyp otheses have unreasonable consequences? Many signi cant ques-

tions remain.

7. Amb os-Spies, Neis, and Terwijn [ANT] have recently shown that the

notion of resource-b ounded genericity intro duced by Amb os-Spies,

Fleischhack, and Huwig [AFH87, AFH88] interacts very usefully

with resource-b ounded measure. See [Amb95] for a survey of this

and other typ es of resource-b ounded genericity. Balc azar and May-

ordomo [BM95a]havecharacterized this genericity as a strong kind

of bi-immunity, and Amb os-Spies, Mayordomo, Wang, and Zheng

[AMWZ96]have further investigated the relationships b etween gener-

icity and measure, but more investigation is needed to fully under-

stand the relativepower of these two metho ds.

8. One of the most challenging tasks remaining is the development of

measure in sub exp onential complexity classes. Mayordomo [May94c,

May94b] and Allender and Strauss [AS94]have prop osed inequiv-

alent de nitions of measure in PSPACE, and Allender and Strauss

[AS94, AS95] have investigated various formulations of measure in

P and other sub exp onential classes, but at the time of this writing,

many issues are unresolved.

Resource-b ounded measure is a p owerful generalization of Leb esgue mea-

sure. There is reason to hop e that it will b e as fruitful in complexity the-

ory as Leb esgue measure has b een in analysis and mathematical physics.

Many investigators will haveto ask and answer many questions in order

for resource-b ounded measure to achieve its full p otential.

Acknowledgments

I thank many colleagues and students for fun and helpful discussions over

the past few years. I esp ecially thank my colleagues David Juedes and

Elvira Mayordomo for working with me in this area, and for allowing me

to include so much of our work in this survey. I also thank a referee for

numerous valuable improvments in the exp osition.

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