The Computational Complexity Column

Eric Allender

Rutgers University Department of Computer Science

Piscataway NJ USA

allendercsrutgersedu

Are there to o many complexity classes Merely trying to understand one asp ect of computation

such as the p ower of randomness leads to a whole range of complexity classes such as ZPP RP and

BPP to name but a few Do we really need all of these classes

One of the most exciting developments in complexity theory in the past few years is the growing

b o dy of evidence that all of the aforementioned classes are merely pseudonyms for P Our guest

column this issue gives an overview of this area

Recent Advances Towards Proving P BPP

1 2 3

Andrea E F Clementi Jose D P Rolim Luca Trevisan

Abstract

Two indep endent techniques have b een developed recently that yield sucient conditions for

P BPP in terms of worstcase circuit complexity of functions computable in exp onential time

Andreev Clementi and Rolim proved that P BPP provided that a sparse eciently enumerable

language exists of suciently high circuit complexity This result has b een subsequently improved

by Impagliazzo and Wigderson by showing that either P BPP or all the decision problems solvable

O n on

in time are solvable by circuits of size In this column we discuss these results and their

relation with previously known sucient conditions for P BPP

Introduction

Randomness is very useful in the design of ecient algorithms for several imp ortant problems Proba

bilistic algorithms are often the simpler ones to solve a given problem or the most ecient or the only

eciently parallelizable ones see For some problems including primality testing and approxi

mation of Pcomplete counting problems only randomized solutions are known In computational

geometry problems such as the approximate computation of the volume of a convex b o dy only admit

randomized solutions

There are several classes of ecient probabilistic algorithms see eg that dier on the adopted

acceptance criteria In particular we will consider two classes of algorithms for decision problems

twosided bounded error p olynomial time algorithms BPP algorithms and onesided bounded error

p olynomial time algorithms RP algorithms A BPP algorithm is required to give a correct answer

with probability not smaller than for any input an RP algorithm is a BPP algorithm with the

additional guarantee that if the input is a NO instance the algorithm will give the right answer with

probability It is not known whether it is p ossible to eciently transform any BPP algorithm into

an RP algorithm Furthermore the question of whether BPP or RP is contained in P here we refer

1

Dipartimento di Scienze dellInformazione University of Rome La Sapienza clementidsiuniromait

2

Centre Universitaire dInformatique University of Geneva rolimcuiunigech

3

Lab oratory for Computer Science MIT lucatheorylcsmitedu

to BPP and RP as the class of languages decided by BPP and RP algorithms resp ectively is not even

a generally accepted conjecture as opp osed to other questions in complexity theory The results that

are describ ed in this column supp ort the conjecture P RP BPP

There is a rich area of research in complexity theory dealing with the use of randomness in computa

tions and in particular on how to eciently simulate randomized algorithms by means of deterministic

ones The rst foundational results for this line of research can b e found in the seminal works of Blum

and Micali and Yao motivated by cryptographic applications These pap ers introduced a

formal denition of PseudoRandom Generator PRG as a function

k n n

G f G f g f g n g

n

denoted by G k n n that stretches k n truly random bits into n pseudorandom bits k n is

commonly said the price of G More formally G is a PRG if for any suciently large n and for any

n

Bo olean circuit C f g f g whose size is at most n we have

jPr C y Pr C G x j

n

n

n k n

where y is chosen uniformly at random in f g and x in f g A Bo olean op erator is said to

b e quick if it is computable in time p olynomial in the length of its output In particular a PRG

k n n

G is quick if G f g f g is computable in time p olynomial in n

n

k n

The output set fG x x f g g of a PRG G for a xed n is also called a discrepancy

n

set for the class of circuits of size n The computation of a BPP algorithm on a xed input is an

easytocompute function C of the outcomes of the random coins It follows that a pseudorandom

generator can b e used to approximate the fraction of random coin outcomes that make C accept

and thus allows one to decide whether the algorithm accepts the input or not PRGs can then b e

considered the natural general metho d to derandomize BPP algorithms In particular we note that

the following simple implication holds

Theorem If a quick PRG of logarithmic price exists then P BPP

In Section we will see sucient conditions for the existence of PRGs of logarithmic price The

ab ove discussion shows that any such condition implies P BPP

The onesided version of a PRG is a Hitting Set Generator HSG a family of functions H

k n n

fH f g f g n g denoted by H k n n that for any suciently large n and for

n

any ninput Bo olean circuit C with size at most n such that

Pr C y

n

it holds

k n

there exists x f g such that C H x

n

While a PRG of price k n converts k n bit of randomness into a pseudorandom string of length n

to b e used in a BPP algorithm a HSG converts k n bits of nondeterminism into a string of length

n to b e used in an RP algorithm The following result is a consequence of the denition of HSGs

4

The restriction to circuits with n inputs and of size n is done without loss of generality since we can see a generic

circuit with k inputs and of size n as a circuit having n inputs n k of them are not used

Theorem If a quick HSG of logarithmic price exists then P RP

Indeed given an RP algorithm and an input one can run the algorithm on each pseudorandom

sequence generated by the HSG and accept the input if and only if one of these sequences make the

algorithm accept It is apparent from the denition that a PRG is also a HSG but the converse is not

necessarily true In Section we will see sucient conditions for the existence of HSGs of logarithmic

price

Proving P BPP Under Circuit Complexity Assumptions

Using AverageCase Circuit Complexity

The main results in the theory of derandomization for general classes of probabilistic algorithms

can b e seen as general techniques to construct PRGs that rely on unproven hardness conditions

In particular Nisan and Wigderson presented a metho d to construct quick PRGs based on the

existence of Bo olean functions in EXP that have exp onential hardness The hardness condition

used by Nisan and Wigderson requires the existence of a function in EXP that has hard averagecase

n

circuit complexity More formally a function f f g f g is Lhard if for any circuit C of

size at most L

jPr C x f x j

n

Given a Bo olean function F fF f g f g n g the hardness at n of F denoted as

n

H n is dened as the maximum integer h such that F is h h hard Then F has exp onential

F n n n n

n

hardness if H n Nisan and Wigderson showed a fundamental Hardness vs Randomness

F

result

Theorem If a Boolean function F exists such that i F EXP and ii F has exponential

hardness then there exists a quick PRG G k n n where k n O log n and consequently

P BPP

p oly log n

A more general form of the ab ove theorem states that if BPP is not contained in DTIMEn

then any EXP complete function has rather low hardness This p ossible shap e of the complexity

world would b e quite dierent from what most complexity theorists exp ect Nisan and Wigdersons

work thus had a tremendous impact in the Complexity Community p eople started to think that

the gap b etween BPP and P might b e very small In another Hardnessvs Randomness trade

(1)

n

o has b een obtained that states that if there is a function in EXP having hardness then

p oly log n

BPP DTIMEn We thus have that under the hardness assumption ab ove the use of

randomness can b e just slightly helpful in sp eeding computation

Further research on the Hardness vs Randomness problem has b een fo cused on the following

asp ect of Nisan and Wigdersons work The hardness required by Nisan and Wigdersons construction

of quick PRGs refers to averagecase circuit complexity Then a consequent and natural question

is the following Do es any worstcase hardness assumption on the circuit complexity of Bo olean

functions computable in time exp onential in the input size exist which allows one to derive an ecient

derandomization metho d in particular to obtain P BPP To motivate this question consider the p otential new insights that would ow from the knowledge of the precise relationship b etween the

true computational p ower of randomness and the problem of nding lower b ounds for the worstcase

circuit complexity of classes of recursive Bo olean functions this latter b eing one of the most studied

problems in complexity theory This argument will b e the sub ject of Sections and

HSGs How to Use and Construct Them

Hitting Set Generators and BPP

Another interesting question is that concerning the real relationship b etween RP and BPP In the

context of derandomization this question leads us to a deep er comparison b etween the real p ower

of PRGs and that of HSGs can the latter replace the former to derandomize BPP algorithms As

we have observed ab ove an HSG is not in general a PRG and since the acceptance criterion of BPP

requires the presence of a twosided random space the intuition here is that HSGs do not correctly

work for BPP Furthermore in several cases the construction of combinatorial ob jects having onesided

random ie hitting prop erties has turned out to b e more ecient than that of combinatorial ob jects

having twosided random ie discrepancy prop erties for a survey of these cases see App endix

C of This is for instance the case for extractors and ORdispersers Another case in

which onesided randomness seems to b e easier to achieve is in the case of smal l linear subspaces

n

of f g It is indeed p ossible to construct small hitting sets for this class of subsets and so

for the corresp onding characteristic functions that imply some explicit exp onential lower b ounds for

the branching program mo del but no construction of nontrivial discrepancy sets for this class is

known

One general reason for the fact that onesided random ob jects seem to b e easier to construct is that

they have a monotonicity prop erty not satised by discrepancy sets if H fH n g is a family of

n

0 0 0

hitting ob jects then any other family H fH n g such that for any n H contains H

n

n n

has at least the same same hitting prop erties

In a rather surprising way Andreev et al showed that the ab ove intuition ab out the real relationship

b etween PRGs and HSGs is somewhat false

Theorem Let k n O log n If there exists a quick HSG H k n n then BPP P

The pro of of the ab ove result relies on the following lemma which is of indep endent interest

n

Lemma Let q n be any positive function such that n q n There is a deterministic

algorithm A that given access to a quick HSG H k n n with k n log n and given in

input any circuit C x x of size at most q n computes in time polynomial in

n

O (1)

k q n

a value AC such that

jPr C AC j

q n

The algorithm A uses the hitting set generated by the HSG to construct a discrepancy set for C The

main novelty is that the obtained discrepancy set dep ends on C and thus on the BPP algorithm that

we want to derandomize This implies that the derandomization metho d is not oblivious Roughly

sp eaking the algorithm tradeo uniformity versus twosidedness

Actually Theorem gives a more general consequence by considering the price k n with

k n log n of the HSG as a parameter we have the following

Corollary If a quick HSG H k n n exists then for any timebound tn we have

 

O (1)

O k t

BPTIMEt DTIME

where BPTIMEt is the class of languages accepted by probabilistic twosided error Turing machines

running in time t

Notice that this result is comparable to the one in stating that the existence of a quick PRG

2

O k t

G k n n implies BPTIMEt DTIME

We emphasize that the same technique used to prove Lemma has b een recently used to solve

the onesided vs twosided problem in other frameworks see Section

Using WorstCase Hardness to construct HSGs

Lets return now to the question of nding worstcase hardness conditions sucient to derandomize

BPPalgorithms Theorem shows a dierent way to achieve this goal This is used in Theorem

to show that it is indeed sucient to to use a worstcase hard Bo olean function as opp osed to

averagecase hardness to construct a HSG

Intuitively sp eaking the worstcase hardness of a Bo olean function seems to b e much closer to its

hitting prop erties than to its discrepancy prop erties In and successively in Andreev et al indeed

gave the rst worstcase hardness condition which is sucient to construct quick HSGs that satisfy

Theorem thus obtaining P BPP A more formal description of this result follows The circuit

op op

complexity of a Bo olean op erator H will b e denoted as L H Observe that if L k n denotes the

worstcase circuit complexity of Bo olean op erators H k n n then it is known that for any

log n k n

op k

L k n o nk log n

Furthermore for almost every Bo olean op erator H k n we have

op k

L H nk log n

The following theorem gives a sucient condition for P BPP in terms of the worstcase circuit

complexity of characteristic functions of sets generated by Bo olean op erators as dened ab ove

Theorem Let k n O log n A constant c exists such that if there exists

a quick operator H k n n such that the characteristic function of its output sets

n k n

H H H

F f g f g where F x i y f g st H y x n g fF

n

n n

satises

H k n c

0

LF n

n

0 0 0

then it is possible to construct a quick HSG H k n n where k n log n thus obtaining

P BPP

Another way to state the ab ove theorem is the following Assume that there exists a sparse

n

language S fS f g n g that can b e generated by a uniform algorithm that runs in

n

time p olynomial in n so in time p olynomial in the length of its output and such that the worst

case circuit complexity of deciding S is not much smaller up to some p olynomial factor than the

worstcase circuit complexity of generating S Then P BPP

The pro of of Theorem relies on the following fact There is a precise tradeo b etween the

worstcase circuit complexity of partial Bo olean functions and the number of s in their output table

In particular we give a precise mathematical form of the intuitive fact that a partial Bo olean function

having a hard worstcase circuit complexity cannot return for a large number of inputs So

according to the denition of HSGs any function that has a hard worstcase circuit complexity turns

out to have also a go o d hitting prop erty This prop erty is used to construct the preliminary version of

the HSG which is then combined with a convenient use of OR Dispersers a family of particular

expander graphs

A Stronger Result Via the Derandomization of the XOR Lemma

Theorem requires the existence of a Bo olean function f in EXP such that for some any

n n

circuit C of size can only achieve success probability while trying to predict g on a

random input of length n Babai et al later proved that if a function f in EXP exists having

n

circuit complexity then there exists another function g in EXP such that for some xed

n

any circuit of size can only achieve success probability n while trying to predict f on a

random input of size n The diculty of predicting g on a random input can b e increased by dening

a function hx x g x g x from Yaos Xor Lemma it follows that the success

k k

n

probability of a circuit for h of size go es down to exp onentially fast in k If we take k O n

n

the success probability will b e as low as however h do es not yet satisfy the conditions of

n

Theorem since it has O n inputs and the hardness is only From this assumption it is

only p ossible to infer from the techniques of that BPP can b e deterministically simulated in

p oly log n

time n

Impagliazzo and Wigderson recently made further progress by showing how to derandomize

the Xor Lemma Their main result is a pro cedure that given a function g with n inputs and a certain

unpredictability constructs another function h with O n k inputs whose unpredictability decreases

n

exp onentially in k Thus assuming that a function f in EXP exists with circuit complexity one

n

can deduce that a function g exists in EXP such that circuits of size only have success probability

0

n on g Then using a result of Impagliazzo it follows that a function g in EXP exists

n

such that circuits of size have success probability at most and eventually that a function

n n

h exists in EXP such that circuits of size only have success probability In turn the latter

statement implies the existence of PRG of logarithmic price and thus P BPP

n

Theorem If a function in f exists having circuit complexity then P BPP

Related Results and Conclusion

Derandomization is not the only research direction that has b een explored ab out the use of randomness

in computation An alternative approach deals with the use of weak sources of randomness see

Even in this case there is a dierence b etween onesided pseudorandom structures and twosided pseudorandom structures and Theorem is a useful to ol

A natural question to ask is whether the results describ ed in this column or at least part of

them extend to parallel and spaceb ounded classes The issue of parallelization is not addressed by

Impagliazzo and Wigderson and some steps in their construction app ear to b e hard to parallelize

On the other hand a b ottleneck for the parallelization of the techniques of Andreev et al was the

use of Theorem whose pro of in was inherently sequential A new pro of of Theorem app eared

in extends to parallel and spaceb ounded classes and so do to a certain extent the techniques of

The current state of the art on this topic is that a reasonable worstcase sucient condition exists

implying BPNC NC but no worstcase circuit complexity condition is known to imply BPL L or

even RL L

While the main goal of derandomization theory is to prove P BPP some weaker but still

extremely interesting results may b e within reach Theorem already states that BPP is not much

more p owerful than RP A recent result due to Fortnow establishes that any BPP problem is solvable

by an RP algorithm making one oracle query to a promiseRP oracle promiseRP is the extension

of RP to promise problems It is an op en question to prove the same result without using promise

problems Solving this question may b e an imp ortant intermediate step towards proving BPP RP

a result that would have several complexitytheoretic consequences eg RP would b e closed under

complement BPP would b e contained in NP and so on

References

Andreev A Clementi A and Rolim J A New General DeRandomization Metho d

J of the ACM to app ear Extended Abstract in th Annual International Col loquium on

Algorithms Logic and Programming ICALP LNCS pp

Andreev A Clementi A and Rolim J Hitting Prop erties of Hard Bo olean Op erators

and Their Consequences on BPP ECCC Report TR

Andreev A Clementi A and Rolim J Worstcase Hardness Suces for Derandom

ization a New metho d for HardnessRandomness TradeOs in th Annual International

Col loquium on Algorithms Logic and Programming ICALP LNCS pp

Andreev A Baskakov J Clementi A and Rolim J Ecient Construction of Biased

Sample Spaces for Systems of Linear Tests and Applications Technical Report in ECCC TR

Andreev A Clementi A Rolim J and Trevisan L Weak Random Sources Hitting Sets

and BPP Simulation SIAM J of Comput to app ear Extended abstract in th Annual IEEE

Symposium on Foundations on Computer Science pp

Babai L Fortnow L Nisan N and Wigderson A BPP has Sub exp onential Time Simu

lations unless EXPTIME has Publishable Pro ofs Computational Complexity pp

Blum M and Micali S How to generate cryptographically strong sequences of pseudo

random bits SIAM J of Computing pp

Impagliazzo R and Wigderson A P BPP if E requires exp onential circuits Deran

domizing the XOR lemma In th Annual ACM Symposium on Theory of Computing pp

Motwani R and Raghavan P Randomized Algorithms Cambridge University Press

Goldreich O A Sample of Samplers A Computational Perspective on Sampling ECCC

TR

Impagliazzo R Hardcore distributions for somewhat hard problems in Proceedings of

the th IEEE Symposium on Foundations of Computer Science pp

Nisan N Using Hard Problems to Create Pseudorandom Generators ACM Distinguished

Dissertation MIT Press

Nisan N Extracting randomness How and why In Proceedings of the th Annual

IEEE Conference on Computational Complexity pp

Nisan N and Wigderson A Hardness vs Randomness J Comput System Sci pp

Papadimitriou CH Computational Complexity AddisonWesley

Saks M Srinivasan A and Zhou S Explicit disp ersers with p olylog degree In Pro

ceedings of the th Annual ACM Symposium on Theory of Computing pp

Wegener I The complexity of nite Boolean functions WileyTeubner Series in Computer

Science

TaShma A On extracting randomness from weak random sources In th Annual

ACM Symposium on Theory of Computing pp

Yao A Theory and applications of trap do or functions in th Annual IEEE Symposium

on Foundations on Computer Science pp