In Search of an Easy Witness

Exp onential Time vs Probabilistic Polynomial Time

y

Valentine Kabanets Russell Impagliazzo

Department of Computer Science Department of Computer Science

University of California San Diego University of California San Diego

La Jolla CA La Jolla CA

kabanetscsucsdedu russellcsucsdedu

z

Avi Wigderson

Department of Computer Science

Hebrew University

Jerusalem Israel

and

Institute for Advanced Study

Princeton NJ

aviiasedu

May

Abstract

n

Restricting the search space f g to the set of truth tables of easy Bo olean functions

e establish on log n variables as well as using some known hardnessrandomness tradeos w

a numb er of results relating the complexityof exp onentialtime and probabilistic p olynomial

time complexity classes In particular we show that NEXP Ppoly NEXP MA this

can be interpreted as saying that no derandomization of MA and hence of promiseBPP is

p ossible unless NEXP contains a hard Bo olean function Wealsoprove several downward closure

results for ZPP RP BPP and MA eg weshow EXP BPP EE BPE where EE is the

doubleexp onential time class and BPE is the exp onentialtime analogue of BPP

Intro duction

One of the most imp ortant question in complexity theory is whether probabilistic algorithms are

more p owerful than their deterministic counterparts A concrete formulation is the op en question

of whether BPP P Despite growing evidence that BPP can be derandomized ie simulated

deterministically without a signicant increase in the running time so far it has not b een ruled

out that NEXP BPP

Research supp orted byNSFAward CCR and USAIsrael BSF Grant

y

This researchwas done while the author was at the Department of Computer Science UniversityofToronto and

at the Scho ol of Mathematics Institute for Advanced Study Princeton At IAS the researchwas supp orted by NSF

grant DMS

z

This research was supp orted by grant numb er of the Israel Science Foundation founded by the Israel

Academy for Sciences and Humanities and partially supp orted by NSF grants CCR and CCR

Anumb er of conditional derandomization results are known which are based on the assumption

that EXP contains hard Bo olean functions ie those of high circuit complexity NW BFNW

O n

ACR IW STV For instance it is shown in IW that BPP P if DTIME contains

n

a language that requires Bo olean circuits of size Results of this form usually called hardness

randomness tradeos are proved by showing that the truth table of a hard Bo olean function can

b e used to construct a pseudorandom generator which is then used to derandomize BPP or some

other probabilistic complexity class It is well known that such pseudorandom generators exist

if and only if there exist hard Bo olean functions in EXP However it is not known whether the

existence of hard Bo olean functions in EXP is actually necessary for derandomizing BPP That is

it is not known if BPP SUBEXP EXP Ppoly

Obtaining such an implication would yield a normal form for derandomization b ecause

hardnessvsrandomness results actually conclude that BPP can b e derandomized in a very sp ecic

way Think of a probabilistic algorithm after xing its input x as dening a Bo olean function

f r on the random bits r Since the algorithm is fast we know f is easy ie has low

x x

circuit complexity For an algorithm accepting a language L in BPP f is either almost always

x

if x L or almost always otherwise To decide which it suces to approximate the fraction

of r s with f r to within a constant additive error To do this the derandomization rst

x

computes all p ossible sequences that are outputs of a generator G say r r and tries f r

t x i

for each i If G is a pseudorandom generator the nal output is the ma jority of the bits f r

x i

Other constructions such as the hittingset derandomization from ACR are more complicated

but have the same general form

In particular

We never use the acceptance probability guarantees for the algorithm on other inputs Thus

w e can derandomize algorithms even when acceptance separations arent guaranteed for all

inputs ie we can derandomize promise BPP For KRC Intuitively this means that

randomized heuristics that only p erform well on some inputs can also be simulated by a

deterministic algorithm that p erforms well on the same inputs as the randomized algorithm

The derandomization pro cedure only uses f as an oracle Although its correctness relies on

x

the existence of a small circuit computing f the circuit itself is only used in a blackbox

x

fashion

Derandomization along the lines ab ove is equivalent to proving circuit lower b ounds which

seems dicult One might hop e to achieve derandomization unconditionally by relaxing the ab ove

restrictions In particular one could hop e that it is easier to approximate the acceptance probability

using the circuit itself than it is treating it as a black box In fact recent results of a circuit

indicate that in general having access to the circuit computing a function is stronger than having

the function as an oracle BGI Bar

However we show that this hop e is illfounded for nondeterministic algorithms solving the

approximation problem for circuit acceptance oracle access is just as powerful as access to the

circuit In particular any even nondeterministic derandomization of promise BPP yields a circuit

lower b ound for NEXP and hence a blackb ox circuit approximation algorithm running in non

deterministic sub exp onential time Thus unconditional results in derandomization require either

er b ound for NEXP making a distinction b etween BPP and promise BPP or proving a circuit low

More precisely we show that NEXP Ppoly MA NEXP and hence no derandomization

of MA is p ossible unless there are hard functions in NEXP Since derandomizing promise BPP also

allows one to derandomize MA the conclusion is that no full derandomization result is p ossible

without assuming or proving circuit lower b ounds for NEXP

Another piece of evidence that it will be dicult to show EXP BPP or NEXP MA

comes from the downward closure results for these classes It is a basic fact in computational

complexity that the equalities of complexity classes translate upwards For example if NP P

then NEXP EXP by a simple padding argument Thus a separation at a higher level implies

a separation at a lower level which suggests that higherlevel separations are probably harder

to prove We show that separating EXP from BPP is as hard as separating their higher time

complexity analogues More precisely we show that EXP BPP i EE BPE where EE is the

O n

O n

class of languages accepted in deterministic time and BPE is the time analogue of BPP

Weprove similar downward closures for ZPP RPandMA

Main Techniques One of the main ideas that we use to derive our results can be informally

describ ed as the easy witness metho d invented by Kabanets Kab It consists in searching

that have for a desired ob ject eg a witness in a NEXP search problem among those ob jects

concise descriptions eg truth tables of Bo olean functions of low circuit complexity Since there

are few binary strings with small descriptions such a search is more ecient than the exhaustive

search On the other hand if our search fails then we obtain a certain hardness test an ecient

algorithm that accepts only those binary strings which do not have small descriptions With such

a hardness test we can guess a truth table of a hard Bo olean function and then use it as a source

of pseudorandomness via known hardnessrandomness tradeos

Recall that the problem SuccinctSAT is to decide whether a prop ositional formula is satisable

when given a Bo olean circuit which enco des the formula eg the truth table of the Bo olean

function computed by the circuit is an enco ding of the prop ositional formula it is easy to see that

SuccinctSATisNEXP complete Thus the idea of reducing the search space for NEXP problems

to easy witnesses is suggested by the following natural question Is it true that every satisable

prop ositional formula that is describ ed by a small Bo olean circuit must have at least one satisfying

assignment that can also b e describ ed by a small Bo olean circuit We will show that this is indeed

the case if NEXP Pp oly

This idea was applied in Kab to RP search problems in order to obtain certain uniform

setting derandomization of RP In this pap er we consider NEXP search problems which allows

us to prove our results in the standard setting

Remainder of the pap er In Section we present the necessary background In Section we

describ e our main technical to ols In particular as an application of the easy witness metho d

we show that nontrivial derandomization of AM can be achieved under the uniform complexity

assumption that NEXP EXP cf Theorem where the class AM is a probabilistic version of

NP see the next section for the denitions

In Section we proveseveral results on complexityof NEXP In particular Section contains

the pro of of the equivalence NEXP Ppoly NEXP MA In Section we show that every

polyn

NEXP search problem can be solved in deterministic time if NEXP AM we also prove

that if NEXP Ppoly then every language in NEXP has memb ership witnesses of p olynomial

circuit complexity

from Section for the Section contains several interesting implications of our main result

circuit approximation problem and natural pro ofs

In Section we establish our downward closure results for ZPP RP BPP and MA We also

prove gap theorems for ZPE BPE and MA in particular our gap theorem for ZPE states that

Such closure results were also obtained byFortnow and Miltersen Fortnow p ersonal communication July

indep endently of our work

either ZPE EEorZPE can b e simulated innitely often in deterministic subdoubleexp onential

time

Concluding remarks and op en problems are given in Section

Preliminaries

Complexity Classes

We assume that the reader is familiar with the standard complexity classes such as P NP ZPP

RPandBPP see eg Pap We will need the two exp onentialtime deterministic complexity

O n polyn

classes E DTIME and EXP DTIME and their nondeterministic analogues NE

n n

and NEXP We dene SUBEXP DTIME and NSUBEXP NTIME We will

use the exp onentialtime analogues of the probabilistic complexity classes BPP RP and ZPP

O n O n O n

BPE BPTIME RE RTIME and ZPE ZPTIME We also dene the

O n O n

doubleexp onential time complexity classes EE DTIME NEE NTIME and the

n n

classes SUBEE DTIME and NSUBEE NTIME

We shall also need the denitions of the classes MA and AM Bab BM The class MA can

b e viewed as a nondeterministic version of BPP and is dened as follows A language L f g

is in MA i there exists a p olynomialtime decidable predicate R x y z and a constant c N such

n

that for every x f g wehave

c

n

c

x L y f g Pr R x y z and

n

z fg

c

n

c

Pr x L y f g R x y z 

n

z fg

The class AM a probabilistic version of NP consists of all binary languages L for which there is a

n

p olynomialtime decidable predicate R x y z and a constant c N such that for every x f g

wehave

c

n

c

R x y z and x L Pr y f g

n

z fg

c

n

c

y x L Pr R x y z  f g

n

z fg

We shall also use the exp onentialtime version of MA denoted as MA E where the strings y and z

cn c

from the denition of MA are of length rather than n

For an arbitrary function s N N we dene the nonuniform complexity class SIZEs

to consist of all the families f ff g of nvariable Bo olean functions f such that for all

n n n

suciently large n N f can b e computed by a Bo olean circuit of size at most sn Similarly

n

A

for any oracle Awe dene the class SIZE stocontain the families of nvariable Bo olean functions

computable by oracle circuits of size at most sn with Aoracle gates

Let C be any complexity class over an alphab et We dene the class C poly to consist of

all languages L for which there is a language M C and a family of strings fy g where

n n

n polyn

y such that the following holds for all x

n

x L x y M

n

O tn

More generally for any function t N N we dene the class C t by requiring that y

n

Finally for an arbitrary complexity class C over an alphab et we dene

n n

ioC fL jM C suchthat L M innitely often g

Nondeterministic Generation of Hard Strings

As we shall see b elow the truth table of a hard Bo olean function can b e used in order to approximate

the acceptance probability of a Bo olean circuit of appropriate size Thus an ecient algorithm

for generating hard strings the truth tables of hard Bo olean functions would yield an ecient

derandomization pro cedure for probabilistic algorithms

Usually one talks ab out deterministic algorithms for generating hard strings For example the

on

existence of such algorithms follows from the assumptions suchas EXP Ppoly or E SIZE

In some cases however we can aord to use nondeterministic algorithms for generating hard strings

We formalize this with the following denition

Wesay that a Turing machine M nondeterministical ly generates the truth table of an nvariable

n

Bo olean function of circuit complexityatleastsn for some function s N N if on input

there is at least one accepting computation of M and

whenever M enters an accepting state the output tap e of M contains the truth table of some

nvariable Bo olean function of circuit complexity at least s n

The following lemma will b e useful

n

Lemma Suppose NEXP Ppoly Then there is a poly time Turing machine which given

n

an advice string of size n nondeterministical ly generates bit truth tables of nvariable Boolean

functions f satisfying the fol lowing for every d N and innitely many n N f has circuit

n n

d

complexity greater than n

Proof By a simple padding argument we have that NEXP Ppoly implies NE Pp oly Let

L NE n Ppoly be any language Supp ose also that x is the binary enco ding of the cardinality c

n n

n n

jL f g jobviously the length of x is at most log n Then we can nondeterministically

n

n

construct the truth table of the Bo olean function deciding L f g with the following algorithm

n

B Given x as advice B nondeterministically guesses c strings y L f g together with

n n i

O n

n

their certicates z f g After B veries the correctness of its guess it outputs the bit

i

binary string t which has in exactly those p ositions that corresp ond to the guessed y s and

i

elsewhere

As follows from the pro of Lemma the nondeterministic algorithm B given appropriate advice

generates a unique truth table for every n In general however we will allow our nondeterministic

generating algorithm to output dierent hard strings on dierent accepting computation paths

Hierarchy Theorems

We shall need several separation results that are provable by diagonalization

c

Theorem For any xed c N EXP ioSIZEn

Proof By counting we have that for all suciently large n N there is an nvariable Bo olean

c c c

function of circuit complexity n n The lexicographically rst circuit of size n with no

c

equivalent circuit of size n can be constructed in deterministic exp onential time by brute force

search We apply this circuit to the input

c

n c

Theorem For any xed c N EXP ioDTIME n

Proof For agiven n N let S b e the set of the truth tables of all nvariable Bo olean functions

n

c

n

computable by some deterministic time Turing machine of description of size n that uses an

c

c n

n

advice string of size at most n Note that jS j  Dene the truth table t t t of an

n

nvariable Bo olean function not in S as follows The rst bit t has the value opp osite to that of

n

the rst bit of the ma jorityof strings in S Let S be the subset of S that contains the strings

n n

n

with the rst bit equal to t the size of S is at most a half of the size of S We dene t to

n

n

have the value opp osite to that of the second bit of the ma jority of strings in S this leaves us

n

with the subset S of S of half the size After we have eliminated all the strings in S which

n

n n

c

will happ en after at most n steps we dene the remaining bits of t to be We dene

n

L EXP by for every x f g x L i the corresp onding p osition in t is By construction

c

n c

L ioDTIME n

cn

Theorem For any xed c N EE io DTIME cn

Proof Dene a language as follows On inputs of length nwe construct all truth tables of the rst

cn

n Turing machines run for time with all advice strings of length cn or smaller there are at

n n

cn

most n such truth tables Then weenumerate all p ossible truth tables of nvariable

Bo olean functions and use the rst one that is not on our list We output the value of our input

in this table

We shall need the following auxiliary lemmas whose pro of relies on the existence of universal

Turing machines

n

Lemma If NEXP Ppoly then there is a xed constant d N such that NTIME n

d

SIZEn

n n

Proof Let L NTIME n be any binary language Then there is a language M NTIME

n n

and a sequence fy g of binary strings y f g such that for every x f g

n n n

x L x y M

n

Consider the following nondeterministic Turing machine U On input i x of size n where

n

i N and x f g the machine U runs in time simulating the ith nondeterministic Turing

machine M on input x the machine U accepts i M accepts

i i

By assumption there is some constant k N such that the language of U can be decided by

k n

Bo olean circuits of size n almost everywhere It follows that every language M NTIME can

k k

be decided by Bo olean circuits of size jij n O n where i is the constantsize description

n

of a nondeterministic time Turing machine deciding M Consequently every language L

n k k

NTIME n can be decided by Bo olean circuits of size O n which is in O n The claim

follows if wetake d k

n

Lemma If NEXP EXP then there is a xed constant d N such that NTIME n

d

n

DTIME n

n n

Proof For an arbitrary L NTIME n there is a nondeterministic time Turing machine M

and a sequence of nbit advice strings a such that an nbit string x L i M x a accepts

n n

n

Let U be the universal Turing machine for the class NTIME By the assumption NEXP

k

n

EXP we get that there is a constant k N such that the language of U is in DTIME The

d

n

universalityof U implies that the language of M is in DTIME for d k

n

Lemma If NEE EE then there is a xed constant d N such that NTIME n

d n

DTIME n

Proof The pro of is virtually identical to that of Lemma

Combining the hierarchy theorems and the auxiliary lemmas ab ove we obtain the following

n

Corollary If NEXP Ppoly then EXP ioNTIME n

n

Proof If NEXP Ppoly then by Lemma there is a xed d N such that NTIME n

d

SIZEn The claim nowfollows by Theorem

n

Corollary If NEXP EXPthen NEXP ioNTIME n

n

Proof If NEXP EXP then by Lemma there is a xed constant d N such that NTIME n

d

n

n Applying Theorem concludes the pro of DTIME

n

n Corollary If NEE EE then NEE ioNTIME

n

Proof If NEE EE then by Lemma there is a xed constant d N such that NTIME n

d n

DTIME n By Theorem the conclusion is immediate

Pseudorandom Generators and Conditional Derandomization

For more background on pseudorandom generators and derandomization the reader is referred to

the b o ok by Goldreich Gol as well as the surveys by Miltersen Mil and Kabanets Kab

l n n

A generator is a function G f g f g which maps f g to f g for some

function l N N weareinterested only in the generators with l n n

A

l n n

For any oracle A we say that a generator G f g f g is SIZE npseudorandom

if for any ninput Bo olean circuit C of size n with Aoracle gates the following holds

jPr C Gx Pr n C y j  n

ln

y fg

xfg

For the case of the empty oracle A we will omit the mention of A and simply call the generator

SIZEnpseudorandom

l n n

f g f g quick if its output can be computed in Finally we call a generator G

O l n

deterministic time

Theorem BFNW KM Thereisapolynomialtime computable function F f g

f g f g with the fol lowing properties Let A be any oracle For every there exist

and d N such that

n

n n

F f g f g f g

and if r is the truth table of an n variable Boolean function of Aoracle circuit complexity at least

A

d n

n then the function G s F rs is a SIZE npseudorandom generator mapping f g

r

n

into f g

Such a circuit C may not use some of its n inputs

n n

As observed in Yao NW a quick SIZEnpseudorandom generator G f g f g

k

n

allowsonetosimulate every BPP algorithm in deterministic time for some k N Goldreich

n n

and Zuckerman GZ show that aquick SIZEnpseudorandom generator G f g f g

k

n

allows one to decide every MA language in nondeterministic time for some k N Thus

sup erp olynomial circuit if we can eciently generate the truth tables of Bo olean functions of

complexity then we can derandomize MA by placing it in nondeterministic sub exp onential time

Note that for the case of BPP we need a deterministic algorithm for generating hard Bo olean

functions but for the case of MA a nondeterministic algorithm suces

Theorem readily implies the following

n

Theorem Suppose that there is a poly time Turing machine which given an advice

n

string of size an for some a N N nondeterministical ly generates bit truth tables of

nvariable Boolean functions f satisfying the fol lowing for every d N and al l suciently

n

d

large n N f has circuit complexity greater than n Then for every MA

n

n

NTIME an

If the Boolean functions f from Statement above are such that for every d N and

n

d

innitely many n N f has circuit complexity greater than n then for every

n

n

MA ioNTIME an

SAT

Klivans and Van Melkeb eek KM show that a quick SIZE npseudorandom generator

k

n n n

G f g f g allowsonetosimulate every language in AM in nondeterministic time

for some k N Thus if the truth tables of Bo olean functions of sup erp olynomial SAToracle

circuit complexity can b e generated nondeterministically in time p olynomial in their length then

AM NSUBEXP see also MV for derandomization of AM under weaker assumptions More

preciselywehave the following

n

Theorem following KM Suppose thereisap oly time algorithm which given

an advice string of length at most an for some a N N nondeterministical ly generates

n

bit truth tables of nvariable Boolean functions f satisfying the fol lowing for every d N

n

d

and al l suciently large n N f has SAToracle circuit complexity greater than n Then

n

n

for every AM NTIME an

from Statement above are such that for every d N and innitely If the functions f

n

d

many n N f has SAToracle circuit complexity greater than n then for every

n

n

an AM ioNTIME

Stronger derandomization results hold for BPP MA and AM under stronger complexity as

sumptions In particular Impagliazzo and Wigderson IW show that a quick SIZEnpseudorandom

O log n n

generator G f g f g can be constructed from a given truth table of a O log n

variable Bo olean function of circuit complexity at least n Since this result relativizes see KM

we get the following

Theorem IW KM There is a polynomialtime computable function F f g

f g f g with the fol lowing properties Let A be any oracle For every there exist

c d N such that

c

d log n n n

f g f g F f g

and if r is the truth table of an c log nvariable Boolean function of Aoracle circuit complexity

A

c

at least n then the function G s F rs is a SIZE npseudorandom generator mapping

r

d log n n

f g into f g

n

Note that if there is a deterministic poly time algorithm that generates the truth tables

n

of nvariable Bo olean functions of circuit complexity at least then BPP P and if this

algorithm is zeroerror probabilistic then BPP ZPP

Wealsohave the following version of Theorem

n

Theorem KM Suppose there is a constant and a poly time algorithm

which given an advice string of length at most an for some a N N nondeterministical ly

n

generates bit truth tables of nvariable Bo olean functions f satisfying the fol lowing for

n

n

al l suciently large n N f has SAToracle circuit complexity at least Then AM

n

NP aO log n

If the functions f from Statement above are such that for innitely many n N f has

n n

n

SAToracle circuit complexity at least then we have AM ioNP aO log n

Our Main Tools

Easy Witnesses and Hard Functions

In several applications b elow we will need to decide whether a p olynomialtime checkable relation

R x y has a satisfying assignment witness y f g for a given input x f g where

jy j l jxj for some function l N N That is we need to compute the Bo olean function f x

R

dened by

l jxj

f x i y f g R x y holds

R

n

To simplify the notation we shall assume that l n ie that f x is the characteristic

R

function of a language in NE Our approach will be to enumerate all p ossible truth tables y of

Bo olean functions on n jxj variables that are computable by Aoracle circuits of size sn for

some oracle A EXP and a function s N N where sn nandcheck whether R x y holds

for at least one of them

Let T n denote the set of truth tables of nvariable Bo olean functions computable by A

As

oracle circuits of size sn Then instead of computing f x we will b e computing the following

R

Bo olean function f x

RAs

f x i y T jxj R x y holds

RAs As

The following easy lemma shows that the set T n can b e eciently enumerated

As

Lemma For any xedoracle A EXPthereisaconstant c N such that the set T n can

As

c

sn

for any function s N N be enumerated in deterministic time

d

n

Proof Let A DTIME for some d N Then the value of an Aoracle circuit on an nbit input

d

sn

can b e computed in deterministic time polysn since the circuit of size sn can query the

oracle A on strings of size at most sn and these oracle queries can be answered by running the

d

n

deterministic time Turing machine deciding A Thus the truth table of an nvariable Bo olean

d

n s n

polysn function computed by such a circuit can be found in deterministic time by

evaluating the circuit on each nbit input Since the total number of Aoracle circuits of size s is

O s log s

at most the lemma follows

It follows that the Bo olean function f dened ab ove is computable in deterministic time

RAs

d n

sn O n

for some d N which is less than the trivial upp er bound of a bruteforce

on

deterministic algorithm for f x whenever sn For example if sn polyn then the

R

polyn

function f is computable in deterministic time ie f is the characteristic function

RAs RAs

of a language in EXP If f f then wegetanontrivial deterministic algorithm for computing

R RAs

n

f If f f thenweget a nondeterministic poly time algorithm which given a short

R R RAs

advice string generates the truth table of an nvariable Bo olean function of high Aoracle circuit

complexity More precisely the following is true

n

n

Lemma Let R x y be any polynomialtime decidable relation dened on f g f g

let A EXP be any language and let s N N be any function Let f x and f x be

R RAs

n

the Boolean functions dened above If f f then there is a nondeterministic p oly

R RAs

time algorithm B and a family fx g of nbit strings with the fol lowing property for innitely

n n

many n N the algorithm B on advice x nondeterministical ly generates the truth table of an

n

nvariable Boolean function of Aoracle circuit complexity greater than sn

n

Proof If f f then for innitely many n N there exists a string z f g such that

R RAs n

f z but f z For those n N where such a string z exists we dene x z

R n RAs n n n n

n

ie the string z preceded with a for the remaining n N we dene x

n n

It is easy to see that the following nondeterministic algorithm B is the required one on input

n

n

z f g nondeterministically guess a y f g verify that R z y holds output y and

n

n

and halt in the accepting state halt in the accepting state on input output

Using the relationship b etween Bo olean functions of high circuit complexity and pseudorandom

generators that was describ ed in Section we obtain that if f f for some A EXP and

R RAs

sn n then certain derandomization of probabilistic algorithms is p ossible For example

Lemma yields the following derandomization result for AM based on the assumption that

NEXP EXP

n

Theorem If NEXP EXPthen for every we have AM ioNTIME n

Proof It follows by a simple padding argumentthatifforevery p olynomialtime decidable relation

n

n

then NEXP EXP f g f g there is a d N such that f f R x y dened on

d

R

RSATn

n

Hence our assumption that NEXP EXP implies by Lemma that there is a poly time

algorithm which given an advice string of length ann nondeterministically generates the

truth table of an nvariable Bo olean function f such that for every d N there are innitely

n

d

many n where f has SAToracle circuit complexity greater than n The claim now follows by

n

Theorem statement

Under a stronger assumption we show that AM NP The same conclusion is known to hold

under certain nonuniform hardness assumptions KM MV and the assumption that NP is

hard in a certain uniform setting Lu

n

for some then AM NP Theorem If NE co NE ioDTIME

n

Proof Consider all pairs R R of p olynomialtime decidable relations dened on f g

n

n

f g suchthat f xf x for all x f g If for every suchpairR R and every

R R

We should note that this is a very weak conditional derandomization result for AM since it is known uncondi

tionally that AM NPpoly and obviously AM  EXP  NEXP

n

n

x for all x f g then we get there are innitely many n where f xf

R R SAT

n

by a simple padding argument that for every NE co NE io DTIME Thus under

the assumption of the theorem there is a pair R R of p olynomialtime decidable relations

n

n

dened on f g f g such that for some and all suciently large nwehave f x

R

n n

n

f x for at least one x f g This implies that there is a poly time algorithm

R SAT

n

B that nondeterministically generates bit truth tables of nvariable Bo olean functions f such

n

n

that for all suciently large n f has SAToracle circuit complexity

n

n

n

n n

y f g be any strings suchthatR x y Indeed let f g fx x g let y

i i

n

n

or R x y for all  i  and let Y y y be the concatenation of all the y s

i i i

O n

Note that such a Y can b e found nondeterministically in time It is clear that for all su

ciently large nsuch a string Y isthetruthtableofannvariable Bo olean function of SAToracle

n

circuit complexity greater than Hence the existence of the required algorithm B follows

we conclude that AM NP Applying Theorem statement with an

Essentially the same argument as in Theorem but using Theorem statement instead

of Theorem statementwealsoget the following

n

Theorem If NEXP co NEXP EXP then AM ioNTIME for every

PSampleable Distributions and Padding

A family of probability distributions f g is Psampleable if there is a p olynomial pn

n n

pn

and a p olynomialtime Turing machine M such that the following holds if r f g is chosen

uniformly at random then the output of M n r isannbit string distributed according to

n

or any language L f g we dene its characteristic function f g f g so that F

L

x i x L

L

Lemma Suppose that for every language L BPP every and every Psampleable distri

n

time algorithm A such that Pr Ax bution family f g there is a deterministic

x n n

n

n

x n for innitely many n N Then for every n BPE ioDTIME

L

Proof Let be arbitrary We dene a padded version of any given language L BPE by

jxj

jxji jxj

L fx j x L  i g Clearly L BPP

pad pad

n

n ni

for Note that for every n N and  i the number of interesting strings y x

n n n

some x f g is which is at most their length m i Hence the uniform distribution

on the set of such y s will assign each y the probability at least m It is easy to see that this

m

n n n

probability distribution is Psampleable for m i where  i and r f g we

mn

dene M m r to output r

m

By the assumption there is a time algorithm A such that for innitely many m N

n n

Ay y m For innitely many m i where  i this Pr

y L

m

pad

n

ni

algorithm A must be correct on every string y x since each such y has probability at

least m according to Thus there are innitely many lengths n N such that for some

m

n

n n ni

 i we have for every x f g that Ax x Using the nbit enco dings

L

of such is as advice we obtain a deterministic algorithm with linearlength advice that runs in

subdoubleexp onential time and correctly decides L innitely often

Complexity of NEXP

In this section we proveseveral theorems relating uniform and nonuniform complexityofNEXP

NEXP versus MA

Babai Fortnow and Lund BFL Corollary based on an observation by Nisan improved a

result of Alb ert Meyer KL by showing the following

Theorem BFL EXP Ppoly EXP MA

Here we will prove

Theorem NEXP Ppoly NEXP MA

NP NP

Buhrman and Homer BH proved that EXP Ppoly EXP EXP but left op en

the question whether NEXP Pp oly NEXP EXP Resolving this question is the main step in

our pro of of Theorem

Theorem If NEXP Ppoly then NEXP EXP

Proof Our pro of is b ycontradiction Supp ose that

NEXP Pp oly

but

NEXP EXP

By Theorem we get that assumption implies that EXP AM MA By Theorem

n

n Combining the two we get from assumption that for every AM ioNTIME

n

implications we get that EXP ioNTIME n This and assumption contradict Corollary

Corollary If NEXP Ppoly then NEXP MA

Proof If NEXP Ppoly then NEXP EXP by Theorem and EXP MA by Theorem

Remark Buhrman Fortnow and Thierauf BFT show that MAE Ppoly Combined

with a simple padding argument their result yields the following implication MA NP NEXP

Ppoly Our Corollary is a signicant strengthening of this implication

The other direction of Theorem was proved by Dieter van Melkeb eek van Melkeb eek p er

sonal communication September

Theorem van Melkeb eek If NEXP MA then NEXP Ppoly

Proof Supp ose that

NEXP MA

but

NEXP Ppoly

EXP and so by we get that EXP Pp oly By The assumption implies that NEXP

n

Theorem the latter yields that MA ioNTIME Applying Corollary concludes the

pro of

Proof of Theorem The pro of follows immediately from Corollary and Theorem

NP NP

Actually their result is even stronger EXP EXP p oly  EXP EXP

Search versus Decision for NEXP

It is well known that if NP P then every NP search problem can be solved in deterministic

p olynomial time Here by an NP search problem we mean the problem of nding for a given

input string x a witness string y of length at most p olynomial in the length of x such that R x y

holds where R x y is a p olynomialtime decidable binary relation Assuming that NP P we

can nd sucha string y in p olynomial time xing it bit by bit That is we nd y by asking a

series of NP questions of the form Is there a y with a prex y suchthat R x y

The same approach fails in the case of NEXP search problems Supp ose that NEXP EXP Let

polyjxj

R x y b e a predicate decidable in time and the NEXP search problem is to nd given a

polyjxj

string x a witness string y of length at most such that R x y holds When we attempt

to nd a y satisfying R x y by enco ding prexes y of y as part of the instance weeventually get

an instance whose size is exponential in jx j the size of the original instance Being able to solve

such an instance in deterministic exp onential time would only give us a doubleexp onential time

algorithm for solving the original search problem which is not b etter than solving it by brute

force

Thus apparently the assumption NEXP EXP do es not suce to conclude that every NEXP

polyn

search problem is solvable in deterministic time The following theorem of Impagliazzo and

Tardos IT gives some evidence to this eect

Theorem IT There is an oracle relative to which NEXP EXP and yet there is a

NEXP search problem that cannot be solved deterministical ly in less than double exponential time

Under the stronger assumption that NEXP AM we obtain the desired conclusion for NEXP

search problems

Theorem If NEXP AM then every NEXP search problem can be solved in deterministic

polyn

time

The pro of will follow from the next theorem

Theorem If NEXP AM then for every language L NEXP there is a constant d such that

poly n

x L has at least one witness y f g that can be every suciently large nbit string

d

described by a SAToracle circuit of size at most n

Proof The pro of is by contradiction It is easy to see by a simple padding argument that if for

n

n

every p olynomialtime decidable relation R x y denedonf g f g there is a d N such

that f f then the conclusion of the Theorem is true So let us supp ose that there is

d

R

RSATn

n

n

a p olynomialtime decidable relation R x y on f g f g such that for every d N we

have f f

d

R

RSATn

n

Applying Lemma and Theorem we obtain that for every AM io NTIME n

Together with our assumption that NEXP EXP AMthiscontradicts Corollary

Proof of Theorem By Theorem witnesses for any language in NEXP can b e found in deter

ministic exp onential time byenumerating all SAToracle circuits of some xed p olynomial size and

checking whether any of these circuits enco des a witness

We conclude this section by showing that if NEXP Ppolythenevery language in NEXP has

memb ership witnesses of p olynomial circuit complexity

Theorem If NEXP Pp oly then for every language L EXP thereisaconstant d N such

that every suciently large nbit string x L has at least one witness that can be described by a

d

Boolean circuit of size at most n

Proof The assumption NEXP Pp oly implies by Theorem that NEXP MA For the sakeof

contradiction supp ose that the conclusion of our theorem do es not hold Then similarly to the

pro of of Theorem ab ove we conclude that there is a p olynomialtime decidable relation R x y

n

n

on f g f g such that for every d N wehave f f

d

R

Rn

n

Applying Lemma and Theorem we obtain that for every MA io NTIME n

Combined with our assumption that NEXP EXP MAthiscontradicts Corollary

Implications for Circuit Approximation and Natural Prop erties

In this section we present two implications of our Theorem for the problem of circuit approx

imation and natural prop erties of Razb orov and Rudich RR In Section we show that

for nondeterministic Turing machines with sublinear amount of advice if the problem of circuit

approximation can b e solved eciently at all then it can also b e solved eciently with only oracle

access to the Bo olean circuit to b e approximated In Section we show that the mere existence

of an NPnatural prop erty useful against Ppoly already implies the existence of a hard Bo olean

function in NEXP

Circuit Appro ximation

Recall that the Circuit Acceptance Probability Problem CAPP is the problem of computing the

fraction of inputs accepted byagiven Bo olean circuit This problem is easily solvable in probabilistic

p olynomial time and in a certain sense is complete for promise BPP see eg KRC For

Wesay that CAPP can b e nontrivial ly approximated if for every there is a nondetermin

n

istic time algorithm which using advice of size n approximates the acceptance probabilityof

anygiven Bo olean circuit of size n to within an additive error for innitely many input sizes

n Here wesaythata nondeterministic algorithm M approximates a realvalued function g xto

within for inputs of size n if

n

for every x f g there is an accepting computation of M on x and

ery accepting computation of M on x outputs a rational number q g x gx ev

Wesay that an algorithm M for approximating CAPP is blackbox if M is given only oracle

access to an input Bo olean function f computable by a circuit of size n That is M is allowed

to query the value of f on any binary string but M is not allowed to view the actual syntactic

representation of any circuit computing f

Finallywesay that a blackb ox algorithm M for approximating CAPP is nonadaptive if the

queries asked by M on a given input Bo olean function f dep end only on n and all of these queries

are computed before obtaining the value of f on any one of them

Theorem The fol lowing assumptions are equivalent

NEXP Ppoly

CAPP can be nontrivial ly approximated

CAPP can be nontrivial ly approximated by a blackbox nonadaptive algorithm

Proof Sketch Trivial

It is not dicult to see that if CAPP can be nontrivially approximated then for

every

n

MA io NTIME n

This implies that NEXP MA since otherwise we would contradict Corollary Hence by

Theorem we conclude that NEXP Ppoly

This follows immediately from Lemma and Theorem

Remark This raises the op en question of whether an analogue of Theorem can b e proved

where all nondeterministic assumptions are replaced by the corresp onding deterministic as

sumptions In particular we want to know if the existence of a deterministic ecient algorithm

for approximating CAPP is equivalent to the existence of a deterministic ecient algorithm for the

same problem with the additional prop erty of b eing blackb ox and nonadaptive

Note that the existence of a deterministic p olynomialtime algorithm that approximates the

acceptance probabilityof a given Bo olean circuit to within an additive error is equivalent to

the statement that promise BPP promise P which means the following for every probabilistic

p olynomialtime algorithm M there is a deterministic p olynomialtime algorithm A such that A

accepts every element in the set

fx f g Pr M x accepts g

and A rejects every element in the set

fx f g PrM x accepts g

The statement promise BPP promise SUBEXP is interpreted similarly with the deterministic

algorithm A running in sub exp onential time

As an immediate consequence of Theorem we obtain the following

BPP promise SUBEXP NEXP Ppoly Corollary promise

Obviouslyifpromise BPP promise P then BPP P However the converse is not known to

hold If the converse were to hold then Theorem would yield that BPP P NEXP Pp oly

and hence derandomizing BPP would b e as hard as proving circuit lower b ounds for NEXP

Natural Prop erties

Razb orov and Rudich RR argue that all known pro ofs of circuit lower b ounds for nonmonotone

Bo olean functions consist of two parts First one denes a certain natural prop erty of Bo olean

functions or such a prop erty is implicit in the pro of so that any family of Bo olean functions

that satises this prop ertymust require large circuits Then one shows that a particular explicit

family of Bo olean functions satises this natural prop erty

We consider the scenario where one has made the rst step dened an appropriate prop ertyof

Bo olean functions but cannot do es not knowhow to prove that some explicit Bo olean function

satises this prop erty Do es the existence of such a prop erty alone yield any circuit lower b ounds for

explicit Bo olean functions We will argue that the answer is yes if one considers a NEXPcomplete

function explicit

Recall that a family F fF g of nonempty subsets F of nvariable Bo olean functions is

n n n

called Pnatural if it satises the following conditions

constructiveness the language T consisting of the truth tables of Bo olean functions in F is

in Pand

n N c

largeness there is a c N such that for every N we have jT j N where

N

N

T T f g

N

By replacing P with NP in the constructiv eness condition ab ove weobtainanNPnatural prop erty

Finally a prop erty F is called useful against Ppoly if for every family of Bo olean functions

f ff g the following holds if f F for innitely many nthenf Ppoly

n n n n

Theorem If thereexistsanNPnatural property even without the largeness condition that is

useful against Ppoly then NEXP Ppoly

Proof Sketch The existence of an NPnatural prop erty allows us to guess and certify Bo olean

functions of sup erp olynomial circuit complexity nondeterministically in time p olynomial in the

size of their truth tables note that this do es not require the largeness condition By Theorem

these hard Bo olean functions can then be used to derandomize MA yielding NEXP MA Now

the claim follows by Theorem

Remark Note the following subtlety in our pro of of Theorem Although we conclude that

NEXP Pp oly we do not prove that any Bo olean function in NEXP actually satises the given

natural prop erty

Remark Here the interesting op en problem is to try to prove a deterministic version of

Theorem That is do es the existence of a Pnatural prop erty useful against Ppoly imply that

EXP Ppoly

Downward Closures and Gap Theorems

The results showing that a collapse of higher complexity classes implies a collapse of lower complex

ity classes are known as downward closure results Very few such results are known For example

IN prove that P NP DTIMEp olylog n NTIME p olylog n Impagliazzo and Naor

co NTIMEp olylog n RTIMEp olylog n see also BFNW and HIS We prove several

downward closure results for probabilistic complexity classes Along the waywe also obtain gap

theorems for the complexityof BPE ZPEandMA

Note Fortnow For gives much simpler pro ofs of the downward closures presented in this

section However our techniques also allowusto establish the gap theorems that do not seem to

followfromFor

Case of BPP

Here we establish the following

Theorem EXP BPP EE BPE

Usuallybyanexplicit Bo olean function one means a function in NP

Our pro of will rely on the following result by Impagliazzo and Wigderson IW on the deran

domization of BPP under a uniform hardness assumption

Theorem IW Suppose that EXP BPP Then for every binary language L BPP

n

and every there is a deterministic time algorithm A satisfying the fol lowing condition

for every Psampleable distribution family f g there are innitely many n N such that

n n

Pr Ax x n

x L

n

This allows to prove the following

n

Theorem If EXP BPP then for every we have BPE io DTIME n

Proof If EXP BPP then by Theorem the assumption of Lemma is satised and hence

our claim follows

Proof of Theorem If EXP BPP then by padding we conclude EE BPE

n

Assume BPE EE butBPP EXP By Theorem BPE ioDTIME n But then

so is EE contradicting Theorem

As a corollary to Theorem we obtain the following

Theorem Gap Theorem for BPE Exactly one of the fol lowing holds

BPE EE or

n

for every BPE ioDTIME n

Proof First by Theorem statements and cannot both hold at the same time Now if

statement do es not hold then by padding we get EXP BPP which implies statement

via Theorem

Cases of ZPP and RP

In this section we prove the following results

Theorem EXP ZPP EE ZPE

Theorem EXP RP EE RE

The pro of of Theorem will rely on the following result implicit in IW

Theorem IW Suppose that EXP BPP Then for every binary language L ZPP

n

and every there is a deterministic time algorithm A satisfying the fol lowing conditions

for every x f g wehaveAx f x gwhere x is if x Landis if x L

L L

ie Ax either outputs the correct answer or says dont know and

for every Psampleable distribution family f g thereare innitely many n N such

n n

Ax n that Pr

x

n

As a corollarywe can prove

n

Theorem If EXP BPP then for every we have ZPE io DTIME

Proof If EXP BPP then the conclusion of Theorem holds Pro ceeding exactly as in the pro of

of Lemma we obtain that for every language L BPP and every there is a deterministic

n

time algorithm A satisfying the following there are innitely many n N such that for some

n

n ni n

 i wehave Ax xforevery x f g

L

At that p oint in the pro of of Lemma we to ok the binary enco dings of such good is as

advice However in the present case we know that by condition of Theorem our algorithm

A never gives a wrong answer though it may output Hence we can simply try all p ossible is

and check if A outputs or on any of them That is our new algorithm B is the following

n

n n ni

On input x f g accept x if there is a  i such that Ax and reject

otherwise It is easy to see that B correctly decides L innitely often and that the running time

of B is subdoubleexp onential

Before we can prove our downward closure result we need to show that the assumption of

Theorem can b e weakened to say EXP ZPP To this end weprove the following

n

Lemma If for some ZPE ioDTIME then BPP ZPP

Proof The pro of is very similar to that of Theorem For agiven language L ZPE there are

two p olynomialtime decidable predicates R x y andR x y such that for some c N wehave

n

for ev ery x f g that

cn cn

x L Pr R x y and Pr R x y

y fg y fg

cn cn

x L Pr R x y and Pr R x y

y fg y fg

Without loss of generality wemay assume that c

R R and every there are innitely many n where f x If for all such pairs

R

n

n

f x for every x f g then it follows by a simple padding argument that ZPE

R

n

ioDTIME for every Hence by our assumption we have some pair R R and

n

n

f x f x for at least one x f g some such that for all suciently large n

R R

n

Pro ceeding as in the pro of of Theorem we obtain the existence of a poly time algorithm

that nondeterministically generates the truth tables of nvariable Bo olean functions of circuit

n

n

n

where y f g such that complexity This algorithm outputs the string Y y y

i

n

n

f g either R x y or R x for each x x y However in our case this

i i i i

algorithm can b e viewed as zeroerror probabilistic b ecause of the abundance of witnesses for x L

and for x L Once we have such an algorithm we conclude that BPP ZPP by applying

Theorem

Nowwe can strengthen Theorem

n

Theorem If EXP ZPP then for every we have ZPE io DTIME

n

Proof We prove the contrap ositive Supp ose that for some ZPE ioDTIME Then

by Theorem we get EXP BPP and by Lemma wegetBPP ZPP

Proof of Theorem This follows by a simple padding argument

Supp ose that EE ZPE but EXP ZPP Then by Theorem we have EE ZPE

n

ioDTIME contrary to Theorem

The pro of of Theorem is now immediate

Proof of Theorem This follows by a simple padding argument

If EE REthenEE ZPE and hence by Theorem wegetEXP ZPP RP

Theorem yields the following

Theorem Gap Theorem for ZPE Exactly one of the fol lowing holds

ZPE EE or

n

for every ZPE ioDTIME

Proof The pro of is very similar to that of Theorem

Case of MA

For MA we only know how to prove the following downward closure statement which is weaker

than what we exp ect to b e true

Theorem NEE MAE NEXP co NEXP MA

Proof The pro of is bycontradiction Supp ose that NEE MAE but that NEXP coNEXP MA

The latter assumption implies that

either NEXP co NEXP EXP

or EXP MA

We will show that in each of these two cases one gets that MA ioNSUBEXP

Indeed if NEXP co NEXP EXP then it follows by Theorem that MA AM ioNSUBEXP

On the other hand if EXP MA then it follows by Theorem that EXP Ppoly That is

one can generate deterministically in p olynomial time without any advice the truth tables of

Bo olean functions of sup erp olynomial circuit complexity innitely often and hence by Theo

rem statement we again obtain that MA ioNSUBEXP

Now it follows by a simple padding argument that if MA ioNSUBEXP then MAE

ioNSUBEEn where the advice of length n is used to point to the correct length as in the pro of

of Lemma

Finally we observe that our assumptions NEE MAE and MA E ioNSUBEEn contradict

Corollary

We conclude this section with the following gap theorem for MA

Theorem Gap Theorem for MA Exactly one of the fol lowing holds

MA NEXPor

n

for every MA ioNTIME n

Proof If MA NEXP then by Theorem NEXP Ppoly Applying Lemma and Theorem

n

statement implies that for every MA ioNTIME n

n

On the other hand if b oth MA NEXP and MA ioNTIME n then wegetacontradic

tion by Corollary

The statement that we actually wish to prove is the following NEE MA E  NEXP MA

Concluding Remarks and Op en Problems

As we mentioned in the Intro duction our result that hard Bo olean functions are required for de

randomizing MA Corollary has the following consequence If there is an ecient deterministic

algorithm for estimating the acceptance probability of a given Bo olean circuit and hence MA

can b e derandomized then NEXP requires sup erp olynomial circuit size Thus hard Bo olean func

tions are also required for derandomizing promise RP promise BPP and the class APP intro duced

in KRC

Wewould like to p oint out which of our theorems relativize and whichdonot It follows from

the results in BFT that the collapse of NEXP to MA when NEXP Ppoly Corollary

do es not relativize although the only nonrelativizing ingredientinourproofisthe the old result

EXP Ppoly EXP MA The converse implication Theorem rela from BFL that

tivizes The pro of of NEXP Pp oly NEXP EXP Theorem uses the same nonrelativizing

result from BFL but we do not know whether the statement of Theorem itself relativizes

The pro of of Theorem uses only relativizing techniques and hence the statement relativizes

Also Fortnow For shows that all of our downward closure results from Section have pro ofs

that relativize On the other hand the gap theorems for BPE ZPE and MA Theorems

and are proved using nonrelativizing techniques However we do not know if these statements

themselves relativize

As we mentioned in Section one op en problem is to decide if the assumption promise BPP

promise P is equivalent to the existence of a deterministic p olynomialtime algorithm for CAPP

which is blackb ox and nonadaptive Another op en problem is to decide if the existence of a

Pnatural prop erty useful against Ppoly yields EXP Ppoly

We also would like to mention a few other op en questions One question is to show that

Theorem do es or do es not relativize Another question is whether Theorem can b e improved

to have the conclusion NEXP MA rather than NEXP coNEXP MA Finallyitisinteresting

to try to generalize our downward closures to higher time complexity classes the techniques in

this pap er as well as those used by Lance Fortnow for the relativizing pro ofs fail to show that

EEE BPEE EE BPE where EEE is the class of languages decidable in tripleexp onential

time and BPEE is the doubleexp onential version of BPP

Of course the largest op en problem on derandomization is to prove unconditional derandom

ization results Our results indicate that this is likely to require proving circuit lower b ounds

However it is not clear whether sustained eort has been put into proving circuit lower b ounds

for classes of very high complexitysuchas NEXP suchlower b ounds might b e quite a bit easier to

obtain than those for problems in NP or PSPACE

Acknowledgements The authors would like to thank Lance Fortnow Dieter van Melkeb eek

and Salil Vadhan for their comments sp ecial thanks are due to Dieter van Melkeb eek for p ointing

pap er as well as for allowing us to include his theorem out an error in an early version of this

Theorem in our pap er We want to thank an anonymous referee of the conference version

of this pap er for bringing BH to our attention The second author also wishes to thank Steve

Co ok and Charlie Racko for many helpful discussions Finally we thank anonymous referees of

the journal version of this pap er for many helpful comments and suggestions

References

ACR AE Andreev AEF Clementi and JDP Rolim A new general derandomization

metho d Journal of the Association for Computing Machinery

preliminary version in ICALP

Bab L Babai Trading group theory for randomness In Proceedings of the Seventeenth

Annual ACM Symposium on Theory of Computing pages

Bar B Barak How to go beyond the blackb ox simulation barrier In Proceedings of the

FortySecond Annual IEEE Symposium on Foundations of Computer Science pages

BFL L Babai L Fortnow and C Lund Nondeterministic exp onential time has twoprover

interactive proto cols Computational Complexity

BFNW L Babai L Fortnow N Nisan and A Wigderson BPP has sub exp onential time

simulations unless EXPTIME has publishable pro ofs Complexity

BFT H Buhrman L Fortnow and L Thierauf Nonrelativizing separations In Proceedings

of the Thirteenth Annual IEEE Conference on Computational Complexity pages

BGI B Barak O Goldreich R Impagliazzo S Rudich A Sahai S Vadhan and K Yang

On the imp ossibility of obfuscating programs In J Kilian editor Advances in Cryp

tology CRYPTO volume of Lecture Notes in Computer Science pages

SpringerVerlag Berlin Germany

BH H Buhrman and S Homer Sup erp olynomial circuits almost sparse oracles and the ex

ponential hierarchyInRShyamasundar editor Proceedings of the Twelfth Conference

on Foundations of SoftwareTechnology and Theoretical Computer Sciencevolume

of Lecture Notes in Computer Science pages Berlin Germany Springer

Verlag

BM L Babai and S Moran ArthurMerlin games a randomized pro of system and a

hierarchy of complexity classes Journal of Computer and System Sciences

For L Fortnow Comparing notions of full derandomization In Proceedings of the Sixteenth

Annual IEEE Conference on Computational Complexity pages

Gol O Goldreich Modern Cryptography Probabilistic Proofs and Pseudorandomness vol

ume of Algorithms and Combinatorics series Springer Verlag

GZ O Goldreich and D Zuckerman Another pro of that BPPPH and more Electronic

Col loquium on Computational Complexity TR

HIS J Hartmanis N Immerman and V Sewelson Sparse sets in NPP EXPTIME versus

NEXPTIME Information and Control

IN R Impagliazzo and M Naor Decision trees and downward closures In Proceedings of

the Third Annual IEEE Conference on Structure in Complexity Theory pages

IT R Impagliazzo and G Tardos Decision versus search problems in sup erp olynomial

time In Proceedings of the Thirtieth Annual IEEE Symposium on Foundations of

Computer Science pages

IW R Impagliazzo and A Wigderson PBPP if E requires exp onential circuits Deran

domizing the XOR Lemma In Proceedings of the TwentyNinth Annual ACM Sympo

sium on Theory of Computing pages

IW R Impagliazzo and A Wigderson Randomness vs time Derandomization under a

uniform assumption In Proceedings of the ThirtyNinth Annual IEEE Symposium on

F oundations of Computer Science pages

Kab V Kabanets Easiness assumptions and hardness tests Trading time for zero error

Journal of Computer and System Sciences preliminary version

in CCC

Kab V Kabanets Derandomization A brief overview Bul letin of the European Association

for Theoretical Computer Science also available as ECCC TR

KL RM Karp and RJ Lipton Turing machines that take advice LEnseignement

Mathematique preliminary version in STOC

KM A Klivans and D van Melkeb eek Graph nonisomorphism has sub exp onential size

pro ofs unless the p olynomial hierarchy collapses In Proceedings of the ThirtyFirst

Annual ACM Symposium on Theory of Computing pages

KRC V Kabanets C Racko and S Co ok Eciently approximable realvalued functions

Electronic Col loquium on Computational Complexity TR

Lu CJ Lu Derandomizing ArthurMerlin games under uniform assumptions Computa

tional Complexity preliminary version in ISAAC

Mil PB Miltersen Derandomizing complexity classes In S Ra jasekaran P Pardalos

J Reif and J Rolim editors Handbook of Randomized Computingvolume I I Kluwer

Academic Publishers a draft is available at wwwbricsdkbromille

MV PB Miltersen and NV Vino dchandran Derandomizing ArthurMerlin games using

In Proceedings of the Fortieth Annual IEEE Symposium on Foundations hitting sets

of Computer Science pages

NW N Nisan and A Wigderson Hardness vs randomness Journal of Computer and

System Sciences

Pap CH Papadimitriou Computational Complexity AddisonWesley Reading Mas

sachusetts

RR AA Razb orov and S Rudich Natural pro ofs Journal of Computer and System

Sciences

STV M Sudan L Trevisan and S Vadhan Pseudorandom generators without the XOR

lemma In Procee dings of the ThirtyFirst Annual ACM Symposium on Theory of

Computing pages

Yao AC Yao Theory and applications of trap do or functions In Proceedings of the Twenty

Third Annual IEEE Symposium on Foundations of Computer Science pages