The Role of Relativization in Complexity Theory



Lance Fortnow

University of Chicago

Department of Computer Science

East th Street

Chicago Illinois

Abstract

Several recent nonrelativizing results in the area of interactive pro ofs have caused many

p eople to review the imp ortance of relativization In this pap er we take a lo ok at how complexity

theorists use and misuse oracle results We pay sp ecial attention to the new interactive pro of

systems and program checking results and try to understand why they do not relativize We

give some new results that may help us to understand these questions b etter

Intro duction

The recent result IP PSPACE LFKN Sha surprised the theoretical computer science

community in more ways than one A few years earlier Fortnow and Sipser FS created an

oracle relative to which coNP did not have interactive pro ofs The IP PSPACE result was

honestly a nonrelativizing theorem

Several questions immediately p opp ed up Why didnt this result relativize What sp ecic

techniques were used that avoided relativization How can we use these techniques to prove other

nonrelativizing facts

Also much older questions resurfaced What exactly to oracle results mean What should we

infer if anything from a relativization

Such questions gained even more imp ortance when we discovered the amazing p ower of mul

tiple prover interactive pro of systems BFL transparent pro ofs BFLS and probabilistically

+

checkable pro ofs AS ALM

We may not nd satisfactory answers to these questions in the near future However in this

pap er we will give some intuition and some theorems ab out oracles that may help shed light on

some of these issues

In Section we will see that relativization is an extremely p owerful to ol in helping complexity

theorists direct their research

In Section we will lo ok at early results of provable statements with negative relativization We

will lo ok at these various examples and argue that they all lack a prop er oracle access mechanism

In section we take a close lo ok at the new unrelativizing results for interactive pro of systems

We prove some new results that may help us understand these issues b etter We also argue that the



Email fortnowcsuchicagoedu Partially supp orted by NSF grant CCR

new interactive pro of system techniques do not relativize b ecause they take advantage of certain

algebraic prop erties of complexity classes

In section we lo ok at what happ ens when we add an oracle to probabilistically checkable

+

pro ofs PCP Arora Lund Motwani Sudan and Szegedy ALM show that every language

in NP has a PCP using only a logarithmic numb er of random coin tosses and a constant numb er

of queries to the pro of We show that in a relativized world for all k such a result do es not

k

hold even if we allow the verier to use a p olynomial numb er of random bits and n pro of queries

Theorem

A A

Although Heller Hel has created an oracle A relative to which NP EXP we show that

A A

PCP log n EXP would imply that P NP Theorem Thus nding such an oracle

would b e as hard as settling the P NP question

Arora Impagliazzo and Vazirani AIV argue that the lo calcheckability prop erty of com

plexity classes is a ma jor reason that the results on interactive pro ofs do not relativize In Sec

tion we give negative evidence for this thesis by showing that under a reasonable access mech

anism lo cal checkability do es in fact relativize

We give supp ort instead to the thesis that it is the algebraic prop erties of complexity classes

that do not relativize In Section we give evidence for this prop osition by showing that IP

PSPACE holds relative to algebraic extensions of arbitrarily complicated languages

Finally in Section we give a brief arguement against infering any information from random

oracle results We argue that we should only use random oracles as a to ol to combine oracle

constructions However we b elieve that generic oracles are a much stronger and sharp er to ol for

this purp ose

Caveats

This pap er contains several opinions on the use and misuse of relativization results We must

caution the reader that other complexity theorists may have diering opinions on these matters

In this pap er we have tried to give several examples to illustrate various p oints However

this pap er is not meant to b e a survey pap er Many imp ortant works in the area have not b een

mentioned due to lack of space

Notation and Denitions

Most of the notation and denitions follow from the standard textb o oks on the eld HU GJ

We use to represent the join of two sets A and B ie A B fg A fg B We

use FP to represent the p olynomialtime computable functions

It is a misnomer to relativize a complexity class C Instead supp ose we take an enumeration

of machines for C and give them some access mechanism to an oracle set A We then say the

A

relativized class C consists of the languages recognized by the acceptance criteria for C applied to

the machines using the oracle A

Of course this denition may dep end greatly on the sp ecic enumeration of the machines of C

as well as the oracle access mechanism The usual access mechanism consists of a separate oracle

tap e that the Turing machine can write down queries and learn the answers However as we shall

see in Sections and such mo dels may unduly handicap the machines

Relativizing Formulae

It will b e useful to have relativized NPcomplete and PSPACEcomplete sets

Let a relativized CNF formula b e a CNF formula with clauses of the form

x x x Ax x

i1 i2 i3 j 1 j k

where Ax x is true if x x is in A Any of the variables or the Ax x term

j 1 j k j 1 j k j 1 j k

may b e negated

Lemma Let be a relativized CNF formula over an oracle A Let be a closed relativized

A A

CNF formula with arbitrary rstorder existential and universal quantiers over the variables

A

Determining whether is satisable is NP complete

A

A

Determining the truth value of is PSPACE complete

A

Furthermore the completeness reductions do not need access to the oracle

Goldsmith and Joseph GJ prove the rst part of the lemma An easy mo dication of their pro of

gives us the second part as well

Uses of Oracle Results

In this section we will discuss some legitimate uses of relativization results As we will see

complexity theorists have used relativization as a p owerful to ol in studying complexity theory

In Section we will see how theorists use oracles to discover what techniques will not likely

work to solve certain problems In Section we will see how relativization allows us to push at a

problem in two dierent directions In Section we will argue that that lack of nonrelativizable

techniques have caused theorists to lo ok at new directions of research Finally in Section we

will see how old relativization results help us to recognize new techniques that can b e applied to

other problems

Of course theorists must execute extreme care in how one should interpret oracle results In

Sections and we lo ok at some computational mo dels where oracle results may not have the

exp ected interpretation

Limiting techniques

Supp ose we can show for some statement S that there exists an oracle A such that S fails relative to

A in some oracle mo del Then any pro of that S hold must not relativize in that mo del or otherwise

that statement would also hold relative to A If we can also nd an oracle relative to which S holds

then no relativizable technique can decide the truth of S

Baker Gill and Solovay BGS noted this in their original oracle pap er where they give a

relativized world where P NP and another where P NP They also noted that essentially all

the known complexity techniques at that time relativize They concluded that current techniques

would not solve the P NP question

In the nearly two decades since the BakerGillSolovay pap er there have b een literally hundreds

of results in complexity theory With the exception of some results in interactive pro of systems see

Sections and all of the results in complexity theory have relativized These include several

P

imp ortant results such as PH P To d and PP is closed under intersection BRS

The techniques for interactive pro ofs have not yet proven fruitful towards proving any other

theorems ab out complexity theory Thus it really do es app ear that we still need to develop new

techniques to settle the hundreds of complexity statements that relativize b oth ways

Early on some p eople sp eculated that p erhaps the BakerGillSolovay result indicated that these

questions ab out complexity theory may fall outside the axioms of set theory see Har chapter

However most researchers no longer subscrib e to this viewp oint anymore b ecause of lack of

evidence and some of the examples in Sections and

Two directions

Often in complexity theory one has a complexity statement S where one can easily show a rela

tivized world where S holds but it is op en whether there exists a relativizable pro of of S In order

to tackle this problem many complexity theorists lo ok at trying to prove two opp osite directions

Trying to prove S or

Creating a relativized world where S fails

Often by working on a problem in two directions one can often push a failure of a pro of in one

direction into a pro of of the other direction

Goldwasser and Sipser GS used this metho d in their pro of of the equivalence of public and

private coins in interactive pro of systems Initially Goldwasser and Sipser tried to prove that the

private coin interactive pro of hierarchy was innite relative to some oracle The failure of that

attempt led to the equivalence result that implied the collapse of the private coin hierarchy

New directions

Relativization results often lead researchers in new directions that prove extremely fruitful This

may happ en in two dierent ways

A sp ecic problem may have large amount of time devoted to it b ecause of the lack of a

negative relativization result

Whole new directions of research may develop in attempts to nd new techniques to answer

problems with negative relativizations

Beigel Reingold and Spielman BRS concentrated their eorts on trying to show that PP

is closed under intersection mainly b ecause of the imp ortance of the question and the fact that no

negative relativization existed They succeeded in nding a relativizable pro of

The whole area of circuit complexity was develop ed as a p otential metho d for attacking the

hard problems like P NP Although one can easily relativize circuits Wil many researchers

b elieved the structure of circuits would allow us to nd nonrelativizable techniques to solve some

basic complexity questions

Circuit complexity still has a long way to go b efore it can fulll this promise However circuit

complexity has provided us with several interesting combinatorial problems and some other appli

cations for machinebased complexity theory In fact circuit complexity has given us the to ols to

prove some imp ortant relativization results such as a relativized world where the p olynomialtime

hierarchy is innite see Has

Recognizing new techniques

Supp ose we have a pro of of a statement S but we also know that there exists a relativized world

where S do es not hold We can then analyze the pro of of S to nd the technique used in that

pro of that do es not relativize We can then hop efully apply this technique to other negatively

relativized problems

In Noam Nisan found a multipleprover interactive pro of for SAT a problem with a

known negative relativization FRS Lund Fortnow and Karlo analyzed this pro of and using

Nisans techniques combined with some new ones showed a singleprover interactive pro of system

SAT LFKN Several other imp ortant pap ers on interactive pro of theory followed from for

+

extensions of these techniques eg Sha BFL BFLS ALM Babai Bab go es into

more detail ab out the history of these developments

When Oracles Fail

Almost since the rst relativization results of Baker Gill and Solovay BGS complexity theorists

have lo ok for true mathematical statements that fail in some relativized world Though researchers

have published several pap ers along these lines eg Mor Kur Har HCKM Cha

most of these results hold b ecause the machine mo del do es not have prop er access to the oracle

By understanding the various typ es of failures it will enhance our ability to interpret relativiza

tion results One should not simply ignore relativization results that fall into the categories b elow

but one should cautiously draw inferences from such results

In the section we discuss several dierent categories of examples where oracles fail In Section

we will lo ok at the sp ecial case of interactive pro ofs

SpaceBounded Computation

Should the oracle tap e count towards the space b ound Either way can cause problems If we do

count the tap e that would prevent a result like ASPACEpol y EXP from relativizing b ecause

the spaceb ounded machine could not ask exp onentially long queries If we do not count the

tap e then a result like ATIMEpol y PSPACE will not relativize b ecause the spaceb ounded

machine could ask exp onentially long queries Hartmanis Chang Kadin and Mitchell HCKM

have further discussion on these oracle mo dels

Ladner and Lynch LL suggest a reasonable alternative Do not count the space on the

oracle tap e but make the oracle tap e a writeonly tap e that is automatically erased after each

query While this suggestion works well b elow P it do es not solve the quandary describ ed in the

previous paragraph

Buss Bus creates an oracle mo del that seems to get around this problem but is to o cum

b ersome for use in practice If one must relativize a space class we suggest to either use the

corresp onding relativized alternating time class or to use the Landner and Lynch mo del with a

careful eye

Partial Relativizations

In these results some but not all of the ob jects involved are allowed to have access to the oracle

SAT R SATR

Kurtz Kur showed that NP P but for a random R NP P giving a counterex

ample to the Random Oracle Hyp othesis See Section However it uses heavily the fact that

A

A SAT

queries to the SAT oracle do not have access to R In fact for all A NP P Hartmanis

Har has a similar example

Chang Cha has a dierent example where a function sn is not spaceconstructible in the

unrelativized world but sn is spaceconstructible in a relativized world However if the function

sn was computed using a relativized Turing machine the theorem Chang states would relativize

Insucient Oracle Access

One of the basic theorems in complexity theory is linear sp eedup if a program takes tn time with

tn sup erlinear then for every c for all but a nite n there is a another Turing machine that takes

only tnc time Moran Mor noted that this result do es not relativize we need only make the

language dep endent on strings in the oracle of length say tn

This oracle failure o ccurs b ecause of insucient oracle access The pro of of the linear sp eedup

theorem works by enco ding several tap e square into single square by using a larger set of tap e

symb ols However the relativized mo del do es not allow this compression on the oracle tap e If we

change the mo del to allow compressed queries to the oracle then we will indeed have a relativized

linear sp eedup theorem We will show another example of this typ e of failure in Section

When interpreting a relativization result one must b e extremely careful in lo oking at how the

computational mo del may access the oracle If we can prove a statement S false relative to an

oracle in a certain mo del then techniques that relativize only in the model will not work to prove

S

Recent nonrelativizing results on interactive pro ofs do not app ear to t in any of the ab ove

categories In Section we will explore as b est we can why these techniques do not relativize

Relativizations of Interactive Pro ofs

Interactive pro ofs were invented in simultaneously by Babai Bab BM and Goldwasser

Micali and Racko GMR We refer the reader to these pap ers for descriptions and formal

denitions of interactive pro ofs

In Fortnow and Sipser FS created a relativized world where some language in coNP

do es not have an interactive pro of Fortnow and Sipser then conjectured that there exist languages

in coNP that do not have interactive pro ofs

In Lund Fortnow Karlo and Nisan LFKN showed that every coNP language has

an interactive pro of system By the Fortnow and Sipser oracle their pro of could not relativize In

this section we will try to examine why that pro of do es not relativize

In Section we discuss a recent imp ortant result ab out probabilistically checkable pro ofs by

+

Arora Lund Motwani Sudan and Szegedy ALM and show that that result do es not relativize

in a strong way We also discuss the p ossibility that one can use the probabilistically checkable

pro of result to show that NP EXP In Section we discuss lo cal checkability develop ed by

Arora Impagliazzo and Vazirani AIV and argue that this notion do es not give a satisfactory

answer to the question of why the interactive pro of results do not relativize In Section we argue

that a more algebraic prop erty of complexity classes is the ro ot of this nonrelativization Finally

in Section we discuss some further questions ab out interactive pro ofs and relativization

Probabilistically Checkable Pro ofs

In this section we will show two results ab out probabilisticall y checkable pro of systems First we

+

show that the result of Arora Lund Motwani Sudan and Szegedy ALM do es not relativize in

a strong way We then lo ok at what happ ens if we lo ok at oracles trying to relate PCP and EXP

Arora and Safra AS dene a hierarchy of complexity classes PCP corresp onding to the

numb er of random and query bits required to verify a pro of of memb ership in the language as

follows

A verier M is a probabilistic p olynomialtime Turing machine with random access to a string

representing a memb ership pro of M can query any bit of Call M an r n q nrestricted

verier if on an input of size n it is allowed to use at most O r n random bits for its computation

and query at most O q n bits of the pro of

A language L is in PCPr n q n if there exists an r n q nrestricted verier M such

that for every input x

If x L there is a pro of which causes M to accept for every random string ie with

x

probability

If x L then for all pro ofs the probability that M using pro of accepts is b ounded by

In BenOr Goldwasser Kilian and Wigderson BGKW dened multiple prover interac

tive proof systems where the verier communicates with several provers that cannot communicate

among themselves Fortnow Romp el and Sipser FRS show that the languages accepted by

k k

multiple provers MIP and PCPn n are equivalent Babai Fortnow and Lund BFL

k 0

k k

building on the work of LFKN show that NEXP MIP PCPn n

k 0

+

Arora Lund Motwani Sudan and Szegedy ALM building on techniques of Arora and

Safra AS show the following surprising and p owerful theorem

Theorem NP PCPlogn

One can easily create an oracle relative to which this result do es not relativize b ecause the verier

has only a p olynomial numb er of computation paths We use techniques of Fortnow and Sipser

FS and Fortnow Romp el and Sipser FRS to show a much stronger negative oracle result

A A

j k

Theorem For some oracle A NP is not contained in PCP n n for any xed k

j 0

A

n 2k

Pro of Let L A f jThere exists a string x of length n in Ag Clearly L A is in NP

k k

for every A

i

Let M M b e an enumerate of probabilistic veriers where M runs in time n

1 2 i

Terminology M makes two kinds of queries A pro of query is a query to the memb ership pro of

i

An oracle query is a query to A

A A k

Requirement R L A is not accepted by verier M where M makes at most n pro of

ik k

i i

queries

We will handle these countably many requirements one at a time Initially A is the empty

oracle We will add a nite numb er of strings to A in each stage The nal A is the union of all

the strings added in each stage

Stage i k

n 2ik

Pick n large enough so that it do es not conict with earlier stages and such that n

A n A n

Lo ok at M If there exists a memb ership pro of that causes M to accept with

i i

probability greater than then we have fullled R Go on to the next stage

ik

A n k

If some nite extension to A would cause M to ask more than n pro of queries then

i

make that extension and R is fullled Go on to the next stage

ik

Put x from Lemma into A Supp ose there existed some memb ership pro of such that the

A n

probability of M accepts is one Then this same memb ership pro of will cause A fxg

i

to accept with probability at least b ecause of Lemma This contradicts step Thus

we have fullled R Go on to the next stage

ik

2k

Lemma There exists a string x of length n such that

A n

PrThere exists a membership proof where M has an oracle query to x

i

A n

The probability is taken over the random strings of M

i

A n

Pro of Fix a random coin toss r The numb er of oracle queries that M could make is

i

k

i n k

at most n for all p ossible memb ership pro ofs b ecause there are at most n pro of queries So

2k

a randomly chosen string of length n will have extremely low probability of b eing in this set of

oracle queries The lemma follows from the usual averaging argument 2

The oracle constructed by this algorithm will fulll all the requirements and thus LA is not

A

j k

contained in PCP n n for any xed k 2

j 0

A A

Since clearly PCP log n NP for all A We can interpret Theorem as a weak

characterization of NP Perhaps we can use this characterization to separate NP from higher

complexity classes like PSPACE and EXP by separating PCPlog n from these classes

A A B B

However P PCP log n for all A and for some B we have P PSPACE BGS for

B B B B

this B we will have P PCP log n NP PSPACE Thus we would need additional

nonrelativizable techniques to separate PCPlog n from PSPACE

A A

The class EXP do es not fall into the same trap For every oracle A we have P EXP

since the deterministic time hierarchy theorem relativizes HS Heller Hel showed that there

A A

exists an oracle A where NP EXP Since Theorem do es not relativize Hellers theorem

A A

do es not necessarily imply that PCP log n EXP In fact any such oracle will b e hard to

nd

A A

Theorem If there exists an oracle A such that PCP log n EXP then P NP in the

unrelativized world

A

Pro of Since a PCP log n verier has only a p olynomial numb er of computation paths a

p olynomial time machine could query all of the oracle queries on these paths Then the p olynomial

time machine could determine whether a pro of exists by a single unrelativized NP question Thus

A ASAT

we have for every A that PCP log n P

A A

Assume that P NP and for some oracle A PCP log n EXP We then have

A A ASAT A

EXP PCP log n P P which contradicts the fact that the deterministic time

hierarchy relativizes 2

Lo cal Checkability

Arora Impagliazzo and Vazirani AIV dene the notion of pro of checker as follows

A proofchecker is a Turing machine M that uses universal quantication and which is provided

in addition to the input a proof string It is allowed random access to b oth the input and the pro of



string It is said to accept an input x using pro ofstring denoted M x i all branches

created by its universal branching accept

A language L is in the class PFCHKtn i there is a pro ofchecker M that runs in O tn

time and has the prop erty



x L there exists a such that M x



x L M x for every

Using the Co okLevin theorem Co o Lev Arora Impagliazzo and Vazirani show that

NP PFCHKlog n Arora Impagliazzo and Vazirani call this fact the Lo cal Checkability

Theorem

Now supp ose we relativize PFCHK to an oracle A by giving the pro ofchecker an oracle tap e

by writing a full oracle query z on the tap e and magically entering a state q if z A and a state

y

A

q if z A However since only queries of length O log n can b e asked by a PFCHK log n

n

A A

pro of checker it is easy to construct oracles A such that A PFCHK log n and thus NP

A

PFCHK log n

Arora Impagliazzo and Vazirani conclude that lo cal checkability do es not relativize However

we feel that any oracle access mechanism that prevents a machine from querying its own input is an

extremely weak access mo del We will present a more robust access mo del and show that relative

to this mo del lo cal checkability do es relativize

We will allow M to query the oracle as follows When M wants to make an oracle query M

writes two p ointers on the oracle tap e M will now go to state q if the string lo cated b etween

y

those two p ointers on the pro of tap e is in the oracle and will go to state q otherwise

n

A A

Theorem For al l oracles A NP PFCHK log n

Pro of Let b e a relativized CNF formula as describ ed in Section A pro of will

A

consist of a satisfying assignment as well as a list of x x for each Ax x o ccurring in

i1 ik i1 ik

a clause Such a pro of can b e universally veried in O log n time using the oracle access mechanism

describ ed ab ove

A

Let L NP By Lemma there is an unrelativized reduction f mapping an input x to

some relativized CNF formula Part of will contain the formula as well as as pro of that

A A

f x 2

A

Algebraic Oracles

Babai and Fortnow BF give an algebraic characterization of various complexity classes and argue

that the interactive pro of take advantage of this characterization This algebraic characterization

also do es not seem to relativize

In this section we will give additional evidence that it is the algebraic prop erties of complexity

classes that prevent the relativization of interactive pro of results

 n

Let A b e any function mapping f g to f g Let A b e that function restricted to f g

n

Let f to b e the unique multilinear extension to A

n n

Let hy y i b e a standard pairing function such that jhy y ij jy jjy jjy j jy j

1 k 1 k 1 2 3 k



For any set L we dene the algebraic extension A of L inductively in n as follows

Let f x x b e the unique multilinear extension of Ax x

1 n 1 n

Let h y y i b e in A i y y is in L

1 n 1 n

Let h x x i b e in A if f x x

1 n 1 n

Let hi x x i b e in A if the ith bit of f x x is one

1 n 1 n

Note that L is manyone reducible to A

Lund Fortnow Karlo and Nisan LFKN and Shamir Sha show that every language in

PSPACE has an interactive pro of This result do es not relativize Fortnow and Sipser FS

show that relative to some oracle A even coNP do es not have interactive pro ofs However the

IP PSPACE result do es hold for algebraic extensions

A A



Theorem For A an algebraic extension for any set L IP PSPACE

Pro of Sketch

Instead of rep eating the entire pro of in Sha we will just describ e how to mo dify it

We use the relativized formulae describ ed in Section We arithmetized the formulas in the

same way as in Sha replacing Ax x with f x x Thus the arithmetized degree

j 1 j k j 1 j k

remains low as required At the end of the proto col the verier can read o the values of f using

the enco ding in the oracle A 2

From Theorem we immediately have the have the following corollary

Corollary For any set L there is an oracle A such that

L is manyone reducible to A and

A A

IP PSPACE

Further Questions ab out Interactive Pro ofs

We would like to see the ideas of Section applied to other classes based on the interactive pro of

system such as multiple prover interactive pro of systems and probabilistically checkable pro ofs This

may lead to even more evidence of an algebraic prop erty that is the main cause of the nonrelativizing

nature of these results

However given what we have seen in Section we may question the usual oracle access

mechanisms used in these mo dels We think the oracle access mechanism used in PCPlog n

works ne b ecause the verier is allowed to run in p olynomial time However time may prove us

wrong

The access mechanism used by Fortnow Romp el and Sipser FRS is almost surely the wrong

j j

one If one thinks ab out multipleprovers as PCPn n then we see the pro of might have

j 0

exp onential size and thus describ e exp onential strings in the oracle It is not clear how to extend

the access mechanism One simple but workable suggestion is to have the verier run in exp onential

time

It should b e noted however that even if the verier is given access to exp onentially long strings

A A

in the oracle there will still b e an oracle A such that coNP MIP We can easily mo dify the

pro of of Fortnow Romp el and Sipser FRS so all the diagonalizations o ccur on exp onentially

distant lengths with the oracle empty in b etween

A

A

Because of the diculty in creating an oracle A such that PCPlog n EXP we should

continue to try to show that PCPlog n EXP and thus NP EXP We should also see if

some dierent assumption like a suitably strong oneway function would allow us to nd an oracle

A

A

relative to which PCPlog n EXP

Random and Generic Oracles

In Bennet and Gill BG lo oked at what happ ens when we cho ose the oracles randomly

decide for each string whether or not it should b e in the oracle indep endently with probability

onehalf We say a statement S holds with probability one if the set of oracles relative to which S

holds has measure one From measure theory we have a wonderful zeroone law for virtually any

complexity theory statement S we know that S holds with either probability zero or probability

one

Bennet and Gill conjectured the random oracle hypothesis roughly stated as if a complexity

statement holds with a random oracle with probability one then it holds in the unrelativized world

The random oracle hyp othesis has great app eal Intuitively it seems right We ought to b e able to

simulate a random oracle with a suitably strong pseudorandom numb er generator

Kurtz Kur showed that the formulation of the random oracle hyp othesis given by Bennet and

Gill was false Other counterexamples come from the area of interactive pro ofs CGH HCRR

Despite these examples many complexity theorists still b elieve that some version of the random

oracle hyp othesis We however would like to argue that we should not have ever b elieved the

random oracle hyp othesis in the rst place

Kolmogorov complexity tells us that a random set R will contain lots of information It is

true that for a xed set L lo oking at a random R may not greatly aect the complexity of L

However the theory of random oracles works in the other direction First we x the set R Then we

can lo ok at a language L designed to take advantage of the information in R For example Bennet

R R

and Gill BG create a language L NP P that takes advantage of the fact that some of

the information in R can b e accessed nondeterministically but not deterministically

Since an empty oracle do es not contain any information there is no reason to b elieve that results

ab out random oracles should carry over to the unrelativized world If we know that a statement S

holds with probability one we should not infer anything ab out the statement S other than what

we can infer from the fact that there exists an oracle where S holds see Section

We know that all the denable statements that hold with probability one all hold simultaneously

with probability one Thus random oracles give us a nice relativized world where several interesting

complexity theory statements all hold at the same time However a much more p owerful to ol for

such purp oses is the theory of generic oracles

Generic oracles allow us to combine dierent oracle requirements in a clean manner They give

us very p owerful to ols in developing oracles Fenner Fortnow and Kurtz FFK use generic oracles

to develop a relativized world where the isomorphism conjecture holds answering a longstanding

op en question Space limitations prevent us from giving more details ab out generic oracles here

but for the interested reader we recommend BI FFKL FR

Conclusions and Other Questions

Hop efully this pap er will convince the reader of the many and varied uses of relativization results

if done prop erly The area of relativization remains a very imp ortant and active area of complexity

theory We caution researchers in the area though to keep in mind the limitations mentioned in

Sections and Also theorists must rememb er that oracles results are a to ol Theorems strictly

ab out the structure of oracles should b e discouraged

Although we do not know how to settle many imp ortant complexity theory statements the

opp osite is true in relativized worlds For most imp ortant complexity theory statements S we

either know how to prove S or show that S do es not hold in some relativized world We thus would

like to end this pap er with two interesting exceptions

Do es P UP and NP coNP imply that P NP See HS

Do es the isomorphism conjecture imply that there are no oneway functions See FFK

Acknowledgments

This pap er grew out of an informal debate with Russell Impagliazzo on relativization results held

at the Eighth Annual Structures in Complexity Theory Conference that was part of the Federated

Computing Research Conference in San Diego in May I thank the Structures program and

conference committees particularly Steve Homer in organizing the debate I would also like to

thank Russell Impagliazzo for interesting discussion b efore and during the debate

I have also had several interesting discussion ab out oracles and pro of checking with many p eople

including Sanjeev Arora Laszlo Babai Richard Beigel Joan Feigenbaum Stuart Kurtz Carsten

Lund Muli Safra and Mike Sipser Stuart Kurtz was particularly helpful in Section The author

based Section on discussions with Mike Sipser

References

AIV S Arora R Impagliazzo and U Vazirani Relativizing versus nonrelativizing tech

niques The role of lo cal checkability Manuscript University of California Berkeley

+

ALM S Arora C Lund R Motwani M Sudan and M Szegedy Pro of verication and

hardness of approximation problems In Proceedings of the rd IEEE Symposium on

Foundations of Computer Science pages IEEE New York

AS S Arora and S Safra Probabilistic checking of pro ofs A new characterization of NP

In Proceedings of the rd IEEE Symposium on Foundations of Computer Science

pages IEEE New York

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

Symposium on the Theory of Computing pages ACM New York

Bab L Babai Email and the unexp ected p ower of interaction In Proceedings of the th

IEEE Structure in Complexity Theory Conference pages IEEE New York

BF L Babai and L Fortnow Arithmetization A new metho d in structural complexity

theory Computational Complexity

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

interactive proto cols Computational Complexity

BFLS L Babai L Fortnow L Levin and M Szegedy Checking computations in p olyloga

rithmic time In Proceedings of the rd ACM Symposium on the Theory of Computing

pages ACM New York

A A A

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

probability one SIAM Journal on Computing

BGKW M BenOr S Goldwasser J Kilian and A Wigderson Multiprover interactive

pro ofs How to remove intractability assumptions In Proceedings of the th ACM

Symposium on the Theory of Computing pages ACM New York

BGS T Baker J Gill and R Solovay Relativizations of the P NP question SIAM

Journal on Computing

BI M Blum and R Impagliazzo Generic oracles and oracle classes In Proceedings of the

th IEEE Symposium on Foundations of Computer Science pages IEEE

New York

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

BRS R Beigel N Reingold and D Spielman PP is closed under intersection In Proceedings

of the rd ACM Symposium on the Theory of Computing pages ACM New York

Bus J Buss Relativized alternation and spaceb ounded computations Journal of Com

puter and System Sciences

CGH B Chor O Goldreich and J Hastad The random oracle hyp othesis is false

Manuscript Technion Haifa Israel

Cha R Chang An example of a theorem that has contradictory relativizations and a

diagonalization pro of Bul letin of the European Association for Theoretical Computer

Science Octob er

Co o S Co ok The complexity of theoremproving pro cedures In Proceedings of the rd

ACM Symposium on the Theory of Computing pages ACM New York

FFK S Fenner L Fortnow and S Kurtz The isomorphism conjecture holds relative to

an oracle In Proceedings of the rd IEEE Symposium on Foundations of Computer

Science pages IEEE New York

FFKL S Fenner L Fortnow S Kurtz and L Li An oracle builders to olkit In Proceedings

of the th IEEE Structure in Complexity Theory Conference pages IEEE

New York

FR L Fortnow and J Rogers Separability and oneway functions Technical Rep ort CS

University of Chicago Department of Computer Science

FRS L Fortnow J Romp el and M Sipser On the p ower of multiprover interactive

proto cols In Proceedings of the rd IEEE Structure in Complexity Theory Conference

pages IEEE New York

FS L Fortnow and M Sipser Are there interactive proto cols for coNP languages In

formation Processing Letters

GJ M R Garey and D S Johnson Computers and intractability A Guide to the theory

of NPcompleteness W H Freeman and Company New York

GJ J Goldsmith and D Joseph Three results on the p olynomial isomorphism of complete

sets In Proceedings of the th IEEE Symposium on Foundations of Computer Science

pages IEEE New York

GMR S Goldwasser S Micali and C Racko The knowledge complexity of interactive

pro ofsystems SIAM Journal on Computing

GS S Goldwasser and M Sipser Private coins versus public coins in interactive pro of

systems In S Micali editor Randomness and Computation volume of Advances in

Computing Research pages JAI Press Greenwich

Har J Hartmanis Feasible Computations and Provable Complexity Properties volume

of CBMSNSF Regional Conference Series in Mathematics So ciety for Industrial and

Applied Mathematics Philadelphi a

Har J Hartmanis Solvable problems with conicting relativizations Bul letin of the Euro

pean Association for Theoretical Computer Science Octob er

Has J Hastad Almost optimal lower b ounds for small depth circuits In S Micali editor

Randomness and Computation volume of Advances in Computing Research pages

JAI Press Greenwich

HCKM J Hartmanis R Chang J Kadin and S Mitchell Some observations ab out rela

tivizations of space b ounded computations Bul letin of the European Association for

Theoretical Computer Science June

HCRR J Hartmanis R Chang D Ranjan and P Rohatgi Structural complexity theory

Recent surprises In SWAT nd Scandinavian Workshop on Algorithm Theory

volume of Lecture Notes in Computer Science pages Springer Berlin

Hel H Heller Relativized polynomial hierarchy extending two levels PhD thesis Universitat

Munc hen

HS J Hartmanis and R Stearns On the computational complexity of algorithms Trans

actions of the American Mathematical Society

HS S Homer and A Selman Oracles for structural prop erties The isomorphism problem

and publickey cryptography Journal of Computer and System Sciences

HU J E Hop croft and J D Ullman Introduction to Automata Theory Languages and

Computation AddisonWesley Reading Mass

Kur S Kurtz On the random oracle hyp othesis Information and Control

April

Lev L Levin Universalnye p ereb ornye zadachi Universal search problems in Russian

Problemy Peredachi Informatsii Corrected English translation in

Tra

LFKN C Lund L Fortnow H Karlo and N Nisan Algebraic metho ds for interactive pro of

systems Journal of the ACM

LL R Ladner and N Lynch Relativization of questions ab out logspace reducibili ty

Mathematical Systems Theory

Mor S Moran Some results on relativized deterministic and nondeterministic time hierar

chies Journal of Computer and System Sciences

Sha A Shamir IP PSPACE Journal of the ACM

To d S To da PP is as hard as the p olynomialtime hierarchy SIAM Journal on Computing

Tra R Trakhtenbrot A survey of Russian approaches to Perebor bruteforce search

algorithms Annals of the History of Computing

Wil C Wilson Relativized circuit complexity Journal of Computer and System Sciences