THE POWER OF ADAPTIVENESS

AND ADDITIONAL QUERIES IN

RANDOMSELFREDUCTIONS

Joan Feigenbaum

Carsten Lund and

Abstract We study randomselfreductions from a structural com

plexitytheoretic p oint of view Sp ecically we lo ok at relationships

b etween adaptive and nonadaptive randomselfreductions We also lo ok

at what happ ens to randomselfreductions if we restrict the numb er

of queries they are allowed to make We show the following results

There exist sets that are adaptively randomselfreducible but not

nonadaptively randomselfreducible Under plausible assump

tions there exist such sets in NP

There exists a function that has a nonadaptive k n random

selfreduction but do es not have an adaptive k nrandomself

reduction

For any countable class of functions C and any unb ounded func

tion k n there exists a function that is nonadaptively k n

uniformlyrandomselfreducibl e but is not in C pol y This should

b e contrasted with Feigenbaum Kannan and Nisans theorem

that all nonadaptively uniformlyrandomselfreducibl e sets are

in NPpol y

Key words Adaptiveness randomselfreducibili ty

Sub ject classications Q

Intro duction

Informally a function f is randomselfreducible if the evaluation of f at

any given instance x can b e reduced in p olynomial time to the evaluation of f

at one or more random instances y

i

Randomselfreducible functions have many applications including average

case complexity eg lower b ounds eg interactive pro of systems

Feigenbaum et al

and program checkers testers and correctors eg and

cryptographic proto cols eg For a history and overview

of the sub ject see

In this pap er we study randomselfreductions from a structural complexi

tytheoretic p oint of view In particular we analyze the relationships b etween

adaptive and nonadaptive randomselfreducibility and the eect of changing

the numb er of queries available to the randomselfreduction Our main results

are

There exists a set that is adaptively randomselfreducible but not non

adaptively randomselfreducible We can make such a set sparse and

recognizable in slightly more than p olynomial space

If b oundederror probabilistic tripleexp onential time do es not contain

nondeterministic tripleexp onential time then there exists a set in NP

that is adaptively randomselfreducible but not nonadaptively random

selfreducible

For any p olynomially b ounded k n there exists a function that is non

adaptively k n randomselfreducible but not adaptively k nran

domselfreducible

For any countable class of functions C and any unb ounded function k n

there is a function that is nonadaptively k nuniformlyrandomself

reducible but not in C pol y For example there is a nonadaptively k n

uniformlyrandomselfreducible function that is not in FPSPACEpol y

for any unb ounded k n This should b e contrasted with Feigenbaum

Kannan and Nisans theorem that all nonadaptively uniformlyrandom

selfreducible sets are in NPpol y

Denitions and Notation

Throughout this pap er x is the input to a randomized reduction n is the

w n

size of x and r is a random variable distributed uniformly on f g where

w is some p olynomially b ounded function of n The parameter k n is also a

p olynomially b ounded function of n

Definition A nonadaptive k nrandomselfreduction for f is a pair of

p olynomialtime computable functions and with the following prop erties

Randomselfreducibil ity

n

For all n and all x f g

f x x r f x r f k n x r

with probability at least

For all x and x such that jx j jx j and for all i i k n

i x r and i x r are identically distributed

Note that the random queries x r k n x r are in general de

p endent

Observation Szegedy see We can assume without loss of gen

erality that in a nonadaptive randomselfreduction all of the random variables

x r k n x r are identically distributed

The word nonadaptive in Denition refers to the fact that all of the

random queries are pro duced in one round that is i x r do es not dep end

on f j x r for j i The next denition covers the case in which queries

are pro duced in multiple roundsthat is the reduction follows an adaptive

strategy

Definition An adaptive k nrandomselfreduction for f is a p olynomi

altime oracle Turing machine M with the following prop erties

n

For all n and all x f g

f

f x M x r

with probability at least

f

For all x and r M x r asks at most k n queries to the f oracle The

f f

x r queries are denoted q x r q

k n

f

For all x and x such that jx j jx j and all i i k n q x r

i

f

and q x r are identically distributed Note that we do not require that

i

0 0

f f

q x r and q x r b e identically distributed for f f

i i

We say that a function f is randomselfreducible if it has a nonadaptive or

adaptive k nrandomselfreduction for some p olynomially b ounded function

k n A language L is randomselfreducible if the characteristic function for

Feigenbaum et al

L is randomselfreducible We use the abbreviation rsr for b oth random

selfreducible and randomselfreduction

pn

For b oth adaptive and nonadaptive rsrs we can replace by for

any p olynomial pn at the cost of increasing k n by a p olynomial factor We

can achieve this b ound by using the standard technique of running indep endent

copies of the rsr and taking a plurality vote See for details

Prop erty in Denition and prop erty in Denition are referred

to as the instancehiding property This means that any one query of the rsr

reveals nothing ab out the instance x except its size It should b e noted that

the instance x might b e computable from a set of two queries Note that the

instancehiding prop erty need not hold for a cheating oracle f f ie an

adaptive oracle machine could in round i leak something b esides the size of

the instance if it is given wrong answers to one or more of its queries in rounds

through i

An rsr is oblivious if its queries only dep end on the input sizethat is

not only are the distributions of queries the same for all inputs of the same

size which is true of any rsr but moreover the reduction ignores the input

until after all queries have b een made It is straightforward to prove that

obliviously rsr sets are in Ppol y An rsr is uniform if for all inputs x each

jxj

query is uniformly distributed over f g We call such a reduction a k n

uniformlyrandomselfreduction abbreviated k nursr and stress that the

word uniform describ es the distribution of random queries ie it is not

meant to distinguish b etween reduction pro cedures that take advice and those

that do not Uniformlyrandomselfreductions are the typ e of rsrs studied in

Let BPE b e the class of languages recognizable in b oundederror proba

cn

bilistic exp onential time ie the union of BPTIME for all constants c

Similarly let BPEE and BPEEE denote the languages recognizable in b ounded

error probabilistic doubleexp onential and tripleexp onential time The non

deterministic time classes NE NEE and NEEE are dened analogously

n

Let n b e any unb ounded nondecreasing function such that n is time

constructible eg n log n

Adaptiveness versus Nonadaptiveness

In this section we study the dierence b etween adaptive and nonadaptive

rsrs We nd sets of relatively low complexity that are adaptively rsr but not

nonadaptively rsr

Randomselfreducibil ity

It is tempting to conjecture that any sparse adaptively rsr set is also non

adaptively rsr in the hop e that one could just lo ok for all the elements of that

set However the lo cation of those elements could make that pro cess dicult

In fact we show that our main results also hold for sparse sets

Theorem For any k n n there exists a sparse set L such that L

n

is in DSPACEn and has an oblivious adaptive k nrsr but for which any

k n

nonadaptive k nrsr will err with probability at least k n

for innitely many inputs

Proof First we will describ e how to construct a nonsparse L From there

the construction of a sparse L is straightforward In this pro of there will b e

n

a corresp ondence b etween strings of length n and the integers

th

ie the i string in the lexicographic ordering corresp onds to i

A stepfunction B is the language

nms

n

B fx f g js xg

nms

n

where s m ie m is a b ound on how big s can b e Hence B

nms

th

consists of all nbit strings from the s such string on We call s the step and

m the range

n

Note that any function on f g whose restriction to f g is equal to

B for some m has an oblivious adaptive dlog m ersr This is b ecause

nms

it takes dlog m e queries to nd the step using binary search and once

the step is known it is simple to determine memb ership in B

nms

The language L will b e a union of stepfunctions one B for each input

nms

k n

length n where the range m will always b e equal to and the steps will

b e determined by diagonalization against nonadaptive k nrsrs

The diagonalization is done as follows Let M M b e an enumeration

of all the nonadaptive k nrsrs in which every k nrsr o ccurs innitely many

times Assume without loss of generality that M s running time is b ounded

i

i

by n

We diagonalize against M on inputs of length n where n is the least

i i i

n

i

integer such that n i This choice is made in order to ensure that n

i

i

is greater than the running time of M on inputs of length n

i i

Since M asks at most k n queries there must b e an x x m such

i

that the probability that one of M s k n queries is equal to x is at most

i

k nm Lo ok at M s b ehavior on the rst such input x with oracles B

i n mx

i

and B Let

n mx

i

B

n mx

i

x p PrM

i

Feigenbaum et al

B

n mx+1

i

x p PrM

i

Then jp p j k nm On the other hand for M to err with probability

i

at most p for b oth B and B it must b e the case that p p

n mx n mx

i i

and p p by denition of the step functions Hence k nm p

Let s b e the one of x and x on which M errs with probability at least

i

p k nm on input x

n

The ab ove construction can b e carried out in deterministic space O n

We need to run M several times and all of these runs can b e done on the same

i

n

tap e We need O n additional space in order to compute probabilities of

acceptance and to search for an appropriate x

In order to make L sparse we delete from L any string that the k nrsr

do es not query during its binary search and denote the resulting language by

L This will allow the same pro cedure to nd the step of L and then decide

memb ership in L

In the pro of that we could cho ose a query x and a step s on which M

i

errs we only needed the fact that the set of functions that we could cho ose

from fB B g contains for every query x two functions that

n m n mm

i i

dier only in their answer for x eg B and B The languages L

n mx n mx

i i

y

where x is o dd have the same prop erty The function derived from B

n mx

i

y

y

except in its value at x is identical to the function derived from B

n mx

i

Thus we can use the sparse functions of L to diagonalize against M s as we

i

did for L

2

n

Corollary There exists a sparse set L DSPACEn such that L

is obliviously adaptively rsr but not nonadaptively rsr

Proof Let k n n and L b e the sparse language L constructed in the

pro of of Theorem Then for any p olynomial pn any nonadaptive pnrsr

pn

M will err with probability at least for innitely many inputs with

n+1

resp ect to L Hence there exists some input for which M errs with probability

greater than which means that M is not an rsr for L

2

Corollary For any k n n there exists a sparse set L such that L

n

is in DSPACEn and has an oblivious adaptive k nrsr but for which any

00

k (n)

adaptive k nrsr will err with probability at least for innitely

k (n)+1

many inputs

Randomselfreducibil ity

Proof We can assume that the adaptive k nrsr M has the instancehiding

k n

prop erty for all the stepfunctions Otherwise we can diagonalize against

M by cho osing a stepfunction for which M do es not have the instancehiding

prop erty

k n

We construct a nonadaptive k nrsr M for L as follows M rst

k n

cho oses a random string r For each of the step functions M simulates

M r The list of queries to b e made by M is the concatenation of the lists of

queries of M on each step function When given the answers to these queries

M do es what M would have done if given these answers Since M has the

instancehiding prop erty for each step function M has the instancehiding

prop erty

Once r is xed the ith query of M is determined by the answers to the

k n

previous i queries Thus while the list of queries made by M has k n

00

k n

entries at most of these queries are distinct The corollary then follows

from Theorem

2

We turn next to the question of whether we can nd sets in NP that satisfy

the conditions of Corollary First we prove a result ab out a restricted form

of nonadaptive rsrs ie those that are lengthpreserving A lengthperserving

f

x r have the rsr requires that for every x the queries i x r or q

i

same length as x We then show how the pro of can b e mo died to yield a

result ab out general nonadaptive rsrs

Theorem If NE BPE is nonempty there exists a set in NP that has

a lengthpreserving adaptive rsr but do es not have a lengthpreserving non

adaptive rsr

Proof This construction uses the stepfunction idea from the pro of of

Theorem

x

Let L b e a set in NE BPE and L b e f jx Lg where x denotes

the integer whose binary representation is x Note that L is a tally set in

n

NP BPP Because L is in NP the question is in L can b e reduced to

c

a SAT instance in n variables for some constant c Let

n

c

n

L fy f g a satisfying assignment s for such that s y g

n

n

We will prove that L has the prop erties stated in the theorem This follows

from Claims through

Claim L is in NP

Feigenbaum et al

Proof On input x of length n a nondeterministic pro cedure can compute

c

l such that n l It rejects x if no such l exists It then computes guesses

l

an assignment s and accepts x if s satises and s x

2

l

Claim L has a lengthpreserving adaptive rsr

Proof This follows from the fact that L is a sequence of stepfunctions

2

Claim If L has a lengthpreserving nonadaptive rsr then L BPP

Proof Let M b e a lengthpreserving nonadaptive k nrsr for L As

sume without loss of generality that for all inputs x of length n M fails with

n

probability at most

We construct a probabilistic pro cedure M which on input x nds the step

n

among the L instances f g with high probability Once the step is known

it is trivial to decide memb ership in L

M rst generates m strings by taking m indep endent samples from M s

querydistribution D where m will b e determined later We call this set of m

queries the test queries Note that all test queries have length n b ecause M is

lengthpreserving

n

Because the restriction of L to f g is a stepfunction there are only

m consistent ways in which to answer the test queries We describ e b elow a

n

pro cedure M that nds the step s in f g with high probability when given

the correct answers to the test queries For now note that M can construct

a list s s s of p ossible steps by running M on each consistent set of

m

answers to the test queries M can then conclude that the correct value for

c

the step s is the lexicographically least s that satises where l n

i l

It remains to sp ecify M Because it is given the answers to the test queries

M knows some interval I in which the step must lie I is the interval b ounded

by the lexicographically greatest noinstance in the test set and the lexico

graphically smallest yesinstance in the test set M uses the rsr M to p erform

a binary search for the step in the interval I It gets the right answer with

high probability for any particular input it needs in its binary search if all of

the random queries M makes for that input lie outside I Thus we must show

that there is an m that is large enough but still p olynomial to make I small

enough so that all of the random queries asked by M on all inputs used in the

binary search are outside of I with high probability

If the following inequality holds

Pr y I

D y

Randomselfreducibil ity

then the probability is at most k n that at least one of the random queries

for any given input falls in I The binary search pro cedure needs at most n

inputs Hence M fails to get a correct answer from the binary search with

probability at most

n

n k n

n

where the term is the probability that M fails So it remains to cho ose

appropriate values for and m

Let s b e the step and let I b e the smallest interval of the form s s

such that Pr y I Similarly let I b e the smallest interval of

D y

the form s s such that Pr y I Let I b e the op en interval

D y

s s From the minimality of I and I we get that Pr y I

D y

Furthermore if one test query b elongs to I and another b elongs to I then

m

I I With probability at most none of the m random queries

m

b elongs to I hence the probability that I I is at most Thus

m

with probability at least we have

Pr y I

D y

It follows that M fails with probability at most

m n

n k n

if we cho ose n k n and m

2

Claim If L BPP then L BPE

d

Proof Assume that L BPTIMEn for some constant d Then L

cd dcn

b elongs to BPTIMEn This in turn implies that L is in BPTIME

which is a subset of BPE

2

A similar but more intricate argument can b e used to prove the following

more general result

Theorem If NEEE BPEEE is nonempty there exists a set in NP that

is adaptively rsr but not nonadaptively rsr

Sketch of proof The reason that the construction in Theorem do es

not work for a general nonadaptive rsr M is that M may ask queries of either

smaller or larger sizes than the input size n This prop erty implies that the

pro of of Claim is no longer valid

Feigenbaum et al

If M only asked smaller sized queries then a simple change in the pro of of

Claim would b e enough to prove the analogous claim for Theorem We

could mo dify the M of the previous pro of so that it computed in a b ottomup

fashion the steps for all input sizes less than n

On the other hand if M asks queries of size greater than n there will

b e more than a p olynomial numb er of ways to answer the test queries For

example if each test query is of a dierent size and all sizes are greater than

m

n there will b e consistent sets of answers

We solve this problem by making sure that our NP language is trivial for

most input sizes We need that any p olynomial time rsr M only b e able to ask

queries of a constant numb er of sizes on which M do es not already know the

answers

We can assume that M knows the answers for all queries of size less than

n since M can use the b ottomup approach describ ed ab ove For a hard input

size N there will only b e a consistent ways of answering the test queries

of size N where a is the numb er of queries of size N Hence if there are only

a constant numb er of nontrivial query sizes larger than n M can enumerate

all the consistent answers in p olynomial time

The construction of a language with the desired prop erty is similar to the

previous construction except that we let L b e in NEEE BPEEE Hence L

cn

variables and is a tally set in NEE BPEE has

n

cn

2

a satisfying assignment s for such that s y g L fy f g

n

n

d

Let M b e an rsr that runs in time n for some constant d Hence M

d

can only ask queries of size at most n This implies that there are only a

constant numb er of nontrivial sizes greater than n that are represented among

M s queries ie the numb er of such sizes is at most log d c

The rest of the pro of of Theorem is the same as that of Theorem

except for small changes in the parameters

2

The ab ove construction can b e mo died to yield a sparse set L NP that

is adaptively rsr but not nonadaptively rsr using the same transformation from

nonsparse to sparse set as in the pro of of Theorem

Furthermore any nonadaptive rsr M for L can b e transformed into a non

adaptive rsr for L To prove this it is enough to show that L is p olynomial

time truthtable equivalent to L Given an input x consider the elements

encountered during a binary search for x If we know whether each of these

elements is in L it is easy to determine whether x is in L The same is true

Randomselfreducibil ity

the other wayin this case we need only the answers for x and the query that

is encountered just b efore x in the binary search

Numb er of queries

In this section we investigate how much each additional query increases

the p ower of a randomselfreduction We nd a function that has relatively

low complexity for which one additional query makes the dierence b etween

b eing rsr and not b eing rsr Recall that n is an unb ounded nondecreasing

n

function such that n is time constructible eg log n

Theorem Let k n b e p olynomially b ounded There is a function in

n

FDSPACEn that is obliviously nonadaptively k n ursr but not

adaptively k nrsr

Proof In this pro of we use a corresp ondence b etween strings of length n

n

and the elements of the nite eld GF The function f will b e a sequence

n

of univariate p olynomials over GF of degree at most k n one for each

input size n It is clear that such a function has an oblivious nonadaptive

k n ursr b ecause for each input size knowing the value of f at k n

p oints allows the reduction to interp olate the p olynomial and determine the

value at any other p oint

As in the pro of of Theorem we construct f by diagonalizing against all

k nrsrs so let M b e the ith k nrsr We diagonalize against M on inputs

i i

of length n i

i

We need to nd a p olynomial p of degree k n such that if f p on inputs

of length n then M fails on some such input with high probability Assume

i i

that M has the instancehiding prop erty for all p olynomials p if it do es not

i

we can diagonalize against M by cho osing a p for which M do es not have the

i i

prop erty and hence is not an rsr We show that even for a random p and a

random input M fails with high probability Sp ecically we show that

i

k n

p

x r px Pr M

xrp

i

n

where r is the random string used by M on this run This follows from the

i

following inequalities

p p

x r Pr M x r pxjM x r do es not query x

p

i i

n

Feigenbaum et al

k n

p

x r do es query x p Pr M

xr

i

n

We prove Inequality by cho osing p in the following way Assume

without loss of generality that no two of M s queries are equal For each

i

query cho ose a random value for p at that p oint After k n queries M x r s

i

answer on input x is determined However all values for px are still equally

n

likely after k n queriesjust cho ose a random element of GF as the value

of px

To prove Inequality we use the fact that M has the instancehiding

i

p

prop erty for p Let D b e the distribution for M s ith question ie

i

i

p

x r queries x in the ith step D x x Pr M

i r

i

Hence we have the following relations

k n

X

p

D x x r queries x in any step x Pr M

i r

i

i

k n

X X

p

D x x r queries x Pr M

i xr

i

n

n

i

xGF

k n

X X

k n

D x

2

i

n n

n

i

xGF

Uniform RandomSelfReductions with

an Unb ounded Numb er of Queries

Feigenbaum Kannan and Nisan showed that if S has a nonadaptive

ursr that never errs then S is in NPpoly We show by contrast that for any

unb ounded k n there exist functions f that are nonadaptively k nursr but

not even recursive with p olynomialsized advice

Theorem Let C b e any countable class of functions If k n then

there exists a function f that is nonadaptively k nursr but not in C pol y

Proof The pro of is a counting argument For each function in C there are

only an exp onential numb er of functions that can b e created by varying the

p olynomial advice string over all p ossibilities For each input size we construct

Randomselfreducibil ity

sup erexp onentially many functions that all have the same nonadaptive k n

ursr The existence of an f with the desired prop erty then follows by a standard

diagonalization argument

2

Lemma Given n and k n log log n let m dlog k ne There

n m

exists a family F of functions g f g f g and a probabilistic

n

n

p olynomialtime machine M such that M when restricted to inputs in f g

k (n)2

n

is a nonadaptive k nursr for any function in F and jF j for large

n n

n

Proof Any g in F will corresp ond to an l variable p olynomial p of degree

n

at most k n over some nite eld GF q This typ e of function is known to

have a k nursr as long as k n q To prove Lemma we need to

l n

count the numb er of such functions and to emb ed GF q into f g The

n

purp ose of the emb edding is to create a function dened on f g and to have

n

the rsr pro duce queries that are distributed uniformly over f g the natural

l

domain for the p olynomials is GF q The emb edding will determine the

parameters q and l

m

Let q and l bnmc Note that q k n which is required by the

nlm l

standard reduction in Write down copies of GF q There

l lm l n

are q elements in GF q so there are a total of elements in this

nlm

multiset consisting of copies Assign an nbit string to each element in the

l n

multiset This assignment is the emb edding of GF q into f g We now

dene the rsr Given an l variable p olynomial p we evaluate the corresp onding

n n

function g on f g as follows To nd g x where x f g rst nd the

l

element y GF q that is mapp ed to x by the emb edding Evaluate py

m

to get an element z of GF q z corresp onds to an element of f g in the

natural way used in Theorem To p erform the rsr of g on input x rst

nd y and p erform the ursr of p on input y this pro duces uniformly

l

random queries y y For each y pick a copy of GF q uniformly

i

k n

n

at random take the x f g that corresp onds to y in that copy in the

i i

emb edding and evaluate g x This pro cedure results in g queries that are

i

n

distributed uniformly over f g

In order to estimate the numb er of such functions note rst that two dif

t

ferent p olynomials p and p dene dierent functions We use the expression q

as a lower b ound on the numb er of l variable p olynomials over GF q of degree

at most k n where t is the numb er of multilinear monomials of degree

Feigenbaum et al

l

such monomial and k n There are exactly t

k n

n

k n

b c k n

l

dlog k ne

k n

n

k n k n

n

k n

b ecause k n log log n Thus k n n and b c k n

dlog k ne

k n

n Hence our lower b ound on the numb er of l variable p olynomials is

k (n)2 k (n)2

n n

q

2

Op en Questions and Subsequent Related Work

Theorem shows that there are functions that are k n rsr but not

k nrsr Is this also true of sets Similarly do es Theorem hold for sets as

well as functions

Theorem and Feigenbaum Kannan and Nisans result ab out ursrs

together suggest the following question Is there a set that is O ursr but not

in NPpol y

In Section we provided one hyp othesis that guarantees the existence of

sets in NP that are adaptively rsr but not nonadaptively rsr namely NEEE

BPEEE Subsequently Hemaspaandra Naik Ogiwara and Selman using

a suggestion of Beigel improved this result by showing that if NE BPE then

such sets exist

Acknowledgements

These results rst app eared in our Technical Memorandum They were

presented in preliminary form at the th IEEE Structure in Complexity Theory

Conference Boston MA June

Lance Fortnows work was supp orted in part by NSF Grant CCR

Daniel Spielmans work was done while he was a student at Yale College and

a summer intern at ATT Bell Lab oratories

Randomselfreducibil ity

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Manuscript received January

Joan Feigenbaum Lance Fortnow

ATT Bell Lab oratories University of Chicago

Mountain Avenue Computer Science Department

Murray Hill NJ USA E th St

jfresearchattcom Chicago IL USA

fortnowcsuchicagoedu

Carsten Lund Daniel Spielman

ATT Bell Lab oratories Massachusetts Institute of Technology

Mountain Avenue Mathematics Department

Murray Hill NJ USA Cambridge MA USA

lundresearchattcom spielmanmathmitedu