BPP HAS SUBEXPONENTIAL TIME

SIMULATIONS

UNLESS EXPTIME HAS PUBLISHABLE

PROOFS

Laszlo Babai Lance Fortnow

Noam Nisan and Avi Wigderson

Abstract We show that BPP can b e simulated in sub exp onential time

for innitely many input lengths unless exp onential time

 collapses to the second level of the p olynomialtime hierarchy

 has p olynomialsize circuits and

 has publishable pro ofs EXPTIME MA

We also show that BPP is contained in sub exp onential time unless ex

p onential time has publishable pro ofs for innitely many input lengths

In addition we show BPP can b e simulated in sub exp onential time

for innitely many input lengths unless there exist unary languages in

MA P

The pro ofs are based on the recent characterization of the p ower

of multiprover interactive proto cols and on random selfreducibili ty via

lowdegree p olynomials They exhibit an interplay b etween Bo olean

circuit simulation interactive pro ofs and classical complexity classes

An imp ortant feature of this pro of is that it do es not relativize

One of the ingredients of our pro of is a lemma that states that if

EXPTIME has p olynomial size circuits then EXPTIME MA This

extends previous work by Alb ert Meyer

Key words Complexity Classes Interactive Pro of Systems

Sub ject classications Q

Intro duction

How much time is necessary to deterministically simulate a probabilistic

machine We could simulate our machine for every p ossible choice of coin tosses

Babai et al

and then compute the probability of acceptance Simulating a probabilistic

p olynomialtime Turing machine using this metho d would require exp onential

time Do es there exist a faster way

One solution involves pseudorandom numb er generators A pseudorandom

numb er generator maps some set of random bits to a larger set of bits where

the larger set of bits is computationally indistinguishable from random One

could use a pseudorandom numb er generator to reduce the numb er of random

bits needed for a probabilistic algorithm to accept a certain language and thus

to reduce the numb er of computation paths to simulate Blum and Micali

describ ed the rst secure pseudorandom generator based on the assumption

of the hardness of the discrete log function These ideas are generalized by

Yao who showed that one can convert any oneway p ermutation into

a pseudorandom numb er generator to show that BPP has sub exp onential time

simulations A series of results show that any oneway function

is enough to create a pseudorandom generator that could b e used to show that

BPP is in sub exp onential time

Rather than making hardness assumptions on certain typ es of functions

in this pap er we shall consider the eect of complexity theoretic assumptions

Nisan and Wigderson show that if there exists a function computable in

exp onential time that cannot b e approximated by a p olynomialsize circuit then

there exists a pseudorandom numb er generator computable in sub exp onential

time that lo oks random to p olynomialsize circuits for innitely many input

lengths They use this pseudorandom numb er generator to show that if such a

hard function exists then BPP has such a weak sub exp onential simulation

We generalize this result by showing that a much weaker assumption will

still give the same consequence We show that BPP has a weak sub exp onential

n

for innitely many values of n and every simulation ie a simulation in time

unless exp onential time has p olynomial size circuits and is contained in

the class MA of languages with probabilistically checkable pro ofs Since MA is

P P

known to lie within we deduce that sub exp onential simulations for

2 2

BPP exist unless EXPTIME lies within the second level of the p olynomialtime

hierarchy

Thus the main theorems of our pap er are the following

Theorem

BPP admits weak sub exp onential time simulations unless EXPTIME

P

MA P pol y

2

BPP is contained in sub exp onential time unless MA weakly simulates EXPTIME

BPP and Sub exp onential Time

We will use Theorem to also prove the following

Theorem BPP admits weak sub exp onential time simulations unless there

exist unary languages in MA P

From the pro of of Theorem we will deduce the following corollary

Corollary If EH E then P BPP

Corollary is a rare instance of a collapse at the exp onentialtime level

implying a collapse at the p olynomialtime level

Background and Denitions

Nisan and Wigderson showed a general theorem on how certain hard

functions could b e used to create various pseudorandom numb er generators

We will use a sp ecic corollary of their work

k

n

and the notation E for We use the notation EXPTIME for DTIME

k 0

P

cn 

k

DTIME We dene EH the exp onentialtime hierarchy as E

c0 k 0

The class MA dened by Babai cf denotes the MerlinArthur class

the class of languages accepted by an interactive pro of system consisting of a

single message from the prover followed by probabilistic verication Formally

we say that L MA if there exists a probabilistic p olynomial time machine M

and a p olynomial q n such that

For all x L there exists a y jy j q jxj and M x y accepts with

probability at least twothirds and

For all x L and for all y such that jy j q jxj M x y accepts with

probability at most onethird

Arguably this represents the class of languages with publishable proofs of mem

b ership not requiring direct interaction b etween prover and verier the verier

P P

can ip coins at any later date Babai has shown that contains

2 2

MA

Let f g Let f b e a function mapping to We say f is tn

approximated by circuits of size sn if for all suciently large n there exists a

circuit C of size sn on inputs of length n such that

n

PrC x f x

n

tn

Babai et al

n

where x is chosen uniformly over

We say that f is weakly tnapproximated by circuits of size sn if the

ab ove statement holds for innitely many n Nisan and Wigderson use

j

approximated by circuits of size sn to mean weakly n approximated by

circuits of size sn for any xed j in our terminology

A function f cannot b e weakly approximated by p olynomialsize circuits

j j

if it cannot b e weakly n approximated by circuits of size n for any xed

j A language cannot b e approximated if the characteristic function of that

language cannot b e approximated A language class cannot b e approximated

if some language in that class cannot b e approximated

The class P pol y consists of all languages recognizable by a nonuniform

family of p olynomialsize circuits

We say that a machine M weakly computes the language L if for

innitely many values of n

n n

L LM

We say that L admits weak subexponential simulations if for every

n

time b ounded Turing machine which weakly computes L We there exists a

say that a language class C admits weak sub exp onential simulations if each

memb er of C do es A class D weakly simulates a class C if for every language

in C there is a language in D that agrees with C for innitely many input

lengths

We will use and signicantly extend the following theorems due to Nisan

and Wigderson

Theorem

If EXPTIME cannot b e approximated by p olynomialsize circuits then

BPP admits weak sub exp onential simulations

If EXPTIME cannot b e weakly approximated by p olynomialsize circuits

n

then BPP DTIME

0

To extend these theorems we will use some of the recent work by Babai

Fortnow and Lund on multiple prover interactive pro of systems Interac

tive pro of systems were intro duced by Babai and Goldwasser Micali and

Racko as a probabilistic extension of NP The mo del consists of an in

nitely p owerful but untrustworthy prover that tries to convince a probabilistic

p olynomialtime verier that a string is in a certain language A recent series

of results by Lund Fortnow Karlo and Nisan and Shamir show that

this mo del accepts exactly those languages recognizable in p olynomial space

BPP and Sub exp onential Time

BenOr Goldwasser Kilian and Wigderson generalize this mo del to have

many provers that cannot communicate with each other or see the communi

cation of the others with the verier Babai Fortnow and Lund show that

this mo del accepts exactly the languages recognizable in nondeterministic ex

p onential time We will use the following theorem from regarding the p ower

of EXPTIME provers

Theorem Any language in EXPTIME has a multiprover interactive

pro of system where the honest provers are limited to computing within de

terministic exp onential time

We note that this result implies that EXPTIME admits instance checking in

the sense of Blum and Kannan cf

Random SelfReducibil i ty in High Complexity

Classes

Following an idea of Beaver and Feigenbaum Lipton observed that

the p ermanent function has the following random selfreducibility prop erty if

p is a prime greater than n and an oracle tells the value of n n p ermanents

1

over Z correctly for a p ortion of the set of inputs then one can use this

p

3n

oracle to nd the correct value with exp onentially small probability of error on

every input Blum Luby and Rubinfeld used similar ideas

The idea of the pro of is that the value of p ermanent of A can b e computed

using interp olation from the p ermanents of any n matrices A B where

i

B is a random matrix and Z

i p

A similar idea works for arbitrary p olynomials of low degree In particular

by extending a Bo olean function f to a multilinear function g over Z we

p

f

obtain a random selfreducible f hard and PSPACE easy function g

We prove the following lemmas from for completeness

s

Lemma Let A f g Q b e a function Then A has a unique multi

s

e

linear extension A Q Q

e

Proof Dene A by

s

X Y

e

Ab Ax

i

i

s

i=1

bf01g

where x b is the bitstring and

1 s 1 s 0 1

e

Clearly A p ossesses the required prop erties

Babai et al

s

To prove the uniqueness assume f Q Q is multilinear and its restric

s s

tion to f g is zero For x Q let k x denote the numb er of co ordinates

dierent from We prove by induction on k x that f x Indeed this

is true by assumption for k x Now for some k x supp ose eg that

f g Replacing by either or we obtain places where f vanishes

1 1

by the induction hyp othesis but then by the linearity in it vanishes at x

1

as well

2

L

L

Let us call a language L PSPACErobust if P PSPACE The oracle

PSPACEmachine is restricted to p olynomial length queries We say that L

has a tnrandom selfreduction if L has a random selfreduction that makes

at most tn queries

Lemma Every PSPACErobust language has a Turingequivalent family

of n random selfreducible multilinear functions over nite elds

Proof Let L b e a PSPACErobust language Let g x x b e the

n 1 n

n

unique multilinear extension of the characteristic function of L L f g

n

Lemma The function g is n random selfreducible via the same

n

pro of that shows the p ermanent is n random selfreducible

g

Clearly L P where g fg n g We will describ e an alternating

n

p olynomialtime Turing machine with access to L computing g First guess the

value z g x x Then existentially guess the linear function h y

n 1 n 1

g y x x and verify that h x z Then universally cho ose t f g

2 n 1 1 1

and existentially guess the linear function h y g t y x x Keep

2 1 3 n

rep eating this pro cess until we have sp ecied t t and then verify that

1 n

t t L Since a PSPACE machine can simulate an alternating p olynomial

1 n

time Turing machine if L is PSPACErobust then g is Turingreducible to L

2

In particular we have random selfreducible PSPACEcomplete functions

EXPTIMEcomplete functions etc This observation inspired by and

sp elled out simultaneously by the authors of and has signicant con

sequences as we shall see b elow

Deterministic Simulation of BPP

In this section we complete the pro of of Theorem The pro of consists of

proving the following two lemmas

BPP and Sub exp onential Time

Lemma

BPP admits weak sub exp onential time simulations unless EXPTIME is

in P pol y

BPP is in sub exp onential time unless EXPTIME can b e weakly simulated

by p olynomialsize circuits

Lemma

If EXPTIME P pol y then EXPTIME MA

If EXPTIME is weakly simulated by p olynomialsize circuits then MA

weakly simulates EXPTIME

Lemma extends a result of Alb ert Meyer see If EXPTIME

P

We use entirely dierent techniques to prove P pol y then EXPTIME

2

this stronger lemma This lemma is also an extension of a corollary in

Proof of Lemma By Theorem we know that to prove a language

L EXPTIME with multiple provers we only need EXPTIMEstrong provers

Now the MA proto col pro ceeds as follows Merlin gives Arthur C and C which

1 2

are the p olynomial size circuits for the input lengths for which such circuits

exist computing the two provers P and P resp ectively Arthur then simulates

1 2

the verier V using C and C for P and P resp ectively

2

1 2 1 2

Proof of Lemma Let L b e an EXPTIMEcomplete language We enco de

n

the set L as a Bo olean function f Let p b e a prime greater than n and let

n

Z We require a lemma g b e the unique multilinear extension of f to Z

p

p

involving g

Lemma

If BPP do es not have a weak sub exp onential time simulation then there

is a family of p olyn size nonuniform circuits computing g for all but

1

a fraction of the inputs of length n

3n

If BPP is not in sub exp onential time then for an innite numb er of input

lengths n there is a p olyn size nonuniform circuit computing g for all

1

but a fraction of the inputs of length n

3n

Babai et al

Proof Assume there is no p olynomialsize circuit family computing g for

1

fraction of the inputs Goldreich and Levin show how to all but a

3n

construct from g a onebit function h that is computable in exp onential time

but cannot b e approximated by p olynomialsize circuits The lemma follows

by Theorem

2

We continue the pro of of Lemma Assume now that BPP do es not have a

weak sub exp onential time simulation By Lemma we have a family D of

n

1

p olyn size circuits computing g for all but a fraction of the inputs Since

3n

g is n random selfreducible create the following randomized p olynomial

size circuit family for g C will use random inputs to generate the randomself

n

reduction of g and use the D circuit for those queries The probability that

n

the random selfreduction queries one of the strings that D fails to compute

n

correctly is b ounded by n n for almost every n

Using techniques of we can replace the randomness with nonuniformity

We can use the usual amplication techniques to reduce the error to less than

n

Then there must b e a single random sequence that gives a correct answer

for all inputs We enco de this string into the circuit

This proves the rst part of Lemma The second part has virtually the

same pro of Theorem follows

2

Unary Languages

In this section we will prove Theorem We will use the following theorem

due to Nisan and Wigderson

Theorem If there exists a function f computable in exp onential time

n n

size circuits for some approximated by such that f cannot b e weakly

then P BPP

First we prove the following lemma

Lemma If every unary language in the p olynomialtime hierarchy is de

cidable in p olynomial time then P BPP

Proof By counting arguments for some c there exists a function

on strings of N c log n bits that cannot b e weakly napproximated by nsize

circuits We can enco de one of these functions as a unary language decidable in

P N

by cho osing the lexicographically rst function f from to that is not

4

approximated by a circuit of size n If we assume that every unary language

BPP and Sub exp onential Time

in the p olynomialtime hierarchy is decidable in p olynomial time then we can

N

compute f in p olynomial time The function f is computable in

1c 1c

N N

exp onential time in N and cannot b e weakly approximated by any

size circuit The lemma now follows from Theorem

2

Corollary follows from this lemma by noticing that a simple padding

argument see shows that EH E if and only if all unary languages in

the p olynomialtime hierarchy are decidable in p olynomial time

Corollary has the interesting prop erty of showing that a collapse of two

higher complexity classes imply a collapse at a lower level Usually one sees

the other direction for example P NP implies E NE see

Proof of Theorem Supp ose BPP do es not have a weak sub exp onential

time simulation By Theorem we have that EXPTIME MA and thus

P

PH MA since MA Also we have that BPP P so there exist unary

2

languages in PH P and thus MA P

2

Conclusions

Our pro of utilizes a p owerful new technique in complexity theory namely

the use of multilinear functions in the simulation of certain complexity classes

The signicance of this technique makes it worth studying its applications and

limits see

If BPP has a weak sub exp onential simulation then EXPTIME prop erly

contains BPP since one can easily create languages in EXPTIME that do not

cn

have a weak sub exp onential or even weak DTIME simulation Ideally

we would like to prove that EXPTIME prop erly contains BPP without any

assumptions While an oracle making these two classes equal is known to

exist see the new metho ds indicated do not relativize and therefore a

relativized collapse no longer seems as intimidating as it used to b e

Acknowledgments

The rst author was partially supp orted by NSF Grant CCR The

second author was partially supp orted by NSF Grant CCR The third

author p erformed some of this work at MIT and was partially supp orted by

NSF CCR and ARO DLLK The third and fourth authors

were partly supp orted by a grant from the Wolfson foundation administered

by the Israel Academy of Sciences and Humanities

Babai et al

References

L Adleman Two theorems on random p olynomial time in Pro ceedings

of the th IEEE Symp osium on Foundations of Computer Science IEEE

New York

L Babai Trading group theory for randomness in Pro ceedings of the

th ACM Symp osium on the Theory of Computing ACM New York

L Babai and L Fortnow Arithmetization A new metho d in struc

tural complexity theory Computational Complexity

L Babai L Fortnow and C Lund Nondeterministic exp onential

time has twoprover interactive proto cols Computational Complexity

L Babai and S Moran ArthurMerlin games a randomized pro of

system and a hierarchy of complexity classes Journal of Computer and

System Sciences

D Beaver and J Feigenbaum Hiding instances in multioracle queries

in Pro ceedings of the th Symp osium on Theoretical Asp ects of Com

puter Science volume of Lecture Notes in Computer Science Springer

Berlin

M BenOr S Goldwasser J Kilian and A Wigderson Multi

prover interactive pro ofs How to remove intractability assumptions in

Pro ceedings of the th ACM Symp osium on the Theory of Computing

ACM New York

A A

C Bennet and J Gill Relative to a random oracle P NP

A

co NP with probability one SIAM Journal on Computing

M Blum and S Kannan Designing programs that check their work

in Pro ceedings of the st ACM Symp osium on the Theory of Computing

ACM New York

M Blum M Luby and R Rubinfeld Selftesting and selfcorrecting

programs with applications to numerical programs in Pro ceedings of the

nd ACM Symp osium on the Theory of Computing ACM New York

BPP and Sub exp onential Time

M Blum and S Micali How to generate cryptographically strong

sequences of pseudorandom bits SIAM Journal on Computing

R Boppana and R Hirschfeld Pseudorandom generators and com

plexity classes in Randomness and Computation volume of Advances

in Computing Research S Micali ed JAI Press Greenwich

O Goldreich H Krawczyk and M Luby On the existence of

pseudorandom generators in Pro ceedings of the th IEEE Symp osium

on Foundations of Computer Science IEEE New York

O Goldreich and L Levin A hardcore predicate for all oneway

functions in Pro ceedings of the st ACM Symp osium on the Theory of

Computing ACM New York

S Goldwasser S Micali and C Rackoff The knowledge com

plexity of interactive pro ofsystems SIAM Journal on Computing

J Hastad Pseudorandom generators under uniform assumptions in

Pro ceedings of the nd ACM Symp osium on the Theory of Computing

ACM New York

J Hartmanis N Immerman and V Sewelson Sparse sets in NP

P EXPTIME versus NEXPTIME Information and Control

H Heller On relativized exp onential and probabilistic complexity

classes Information and Computation

R Impagliazzo L Levin and M Luby Pseudorandom numb er

generation from oneway functions in Pro ceedings of the st ACM Sym

p osium on the Theory of Computing ACM New York

R Karp and R Lipton Some connections b etween nonuniform and

uniform complexity classes in Pro ceedings of the th ACM Symp osium

on the Theory of Computing ACM New York

L Levin Oneway functions and pseudorandom generators Combina

torica

Babai et al

R Lipton New directions in testing in Distributed Computing and

Cryptography volume of DIMACS Series in Discrete Mathematics and

Theoretical Computer Science J Feigenbaum and M Merritt eds Amer

ican Mathematical So ciety Providence

C Lund L Fortnow H Karloff and N Nisan Algebraic meth

o ds for interactive pro of systems Journal of the ACM

N Nisan and A Wigderson Hardness vs randomness in Pro ceedings

of the th IEEE Symp osium on Foundations of Computer Science IEEE

New York

A Shamir IP PSPACE Journal of the ACM

A Yao Theory and applications of trap do or functions in Pro ceedings of

the rd IEEE Symp osium on Foundations of Computer Science IEEE

New York