On reductions to sets that avoid EXPSPACE

y z x

V Arvind J Kobler M Mundhenk

Abstract

Aset B is called EXPSPACEavoidingifevery subset of B in EXPSPACE

is sparse Sparse sets and sets of high information density called HI G H sets

in are shown to b e EXPSPACEavoiding Investigating the complexityof

sets A in EXPSPACE that honestly reduce to EXPSPACEavoiding sets we

show that if the reducibili ty used has a prop ertycalled rangeconstructibility

then A must also reduce to a sparse set under the same reducibility

Keywords Computational Complexity Reducibilities Sparse Sets

Intro duction

The study of reductions to low information content sets has received much attention

in structural complexity theory research in recentyears There is a series of results

showing that complexity classes containing intractable problems cannot b e reduced

to sets of low information content unless there is an unlikely collapse of complexity

classes The class of sparse sets is an example of a wellstudied class of

low information content sets A researc h trend is to identify dierent classes of low

information content sets and to study the consequences of the existence of hard

sets of low information contentforintractable complexity classes under dierent

kinds of reducibilities

Recently Bo ok and Lutz intro duced and studied sets whose characteristic

sequences are of very high spaceb ounded Kolmogorov complexity they call the

class of such sets HI G H The existence of HI G H sets follows from the fact that

RAN D the class of algorithmically random languages is of measure and is a

sub class of HI G H In it is shown that every set in ESPACE that is p oly

nomial time b ounded truthtable reducible to a set in HI G H is actually b ounded

A preliminary version was presented at MFCS

y

Department of Computer Science Institute of Mathematical Sciences CIT Campus Madras

India Work done at Universitat Ulm Supp orted in part by an Alexander von Humb oldt

research fellowship

z

Abt Theoretische Informatik Universitat Ulm D Ulm Germany

x

FB IV Informatik Universitat Trier D Trier Germany

Work supp orted in part by the DAAD through Acciones Integradas AIeeszk

truthtable reducible to some sparse set it is shown in that this even holds for

O log ntruthtable reducibilities The reason for considering the class ESPACE

is that most intractable complexity classes of interest like NP PSPACE etc are

contained in ESPACE and HI G H itself is dened using exp onential spaceb ounded

Kolmogorov complexity Consequently if an NP set saySAT b ounded

truthtable reduces to a set in HI G H then SAT b ounded truthtable reduces to

a sparse set and by it follows that PNP This and similar consequences for

other complexity classes eg PSPACE and PP are derived in

In this pap er we address the following question are there further p olynomial

time reducibilities such that every set in ESPACE that is reducible to a set in

HI G H is actually reducible to some sparse set

The answer weprovide to the ab ove question is based on the following obser

vation which is easy to prove see Theorem for any A HIGHit holds

that every subset of A in EXPSPACE is sparse This prop erty is captured bya

class of sets larger than HI G H whichwe call EXPSPACEavoidingAsetA is

voiding abbreviated as EAifevery subset of A in EXPSPACE is EXPSPACEa

sparse Theorem rephrased states that every set in HI G H is in EA Next we

identify a simple prop erty for reducibilities called rangeconstructibility Informally

sp eaking whenever A honestly reduces to B via a rangeconstructible reducibi

lity then Areduces to a subset C of B such that C can b e constructed byan

ESPACE machine that uses A as oracle Weshow that several natural reducibili

ties are rangeconstructible for example the manyone conjunctive and Hausdor

reducibilities

In Theorem weshow that every set in EXPSPACE which honestly reduces

to a set in EA via a rangeconstructible reducibility in fact reduces to a sparse

set From known collapse results for reductions to sparse sets under the considered

rangeconstructible reducibilities it follows that the existence of hard sets in EA

under honest rangeconstructible reductions for complexity classes like UP NP

PP or PSPACE implies unlikely collapses of complexity classes It turns out that

these collapse consequences hold for rangeconstructible reductions to sets in HI G H

without the honesty assumption

Preliminaries

Letf g b e the standard alphab et and let A b e a set The length

n n

of a string x is denoted by jxj A A denotes the set of strings in A of

length n to length n resp ectively denotes the characteristic function of

A

A

A denotes the characteristic sequence of A for all strings up to length n

n

A n A

ie j j and the ith bit of equals s where s is the ith

A i i

n n

string in in lexicographic order The cardinalityof A is denoted by jAjThe

n n

census function of a set A is census jA jAsetS is called sparse if its

A

veby a p olynomial A set T is called a tally set if census function is b ounded ab o

T We use T ALLY and SPARSE to denote the classes of tally and sparse

O n

sets resp ectivelyESPACE denotes the DSPACE whereas

O 

n

EXPSPACE DSPACE

The deterministic and nondeterministic reducibilities discussed in this pap er

are the standard p olynomialtime reducibilities dened in and the Hausdor

reducibilityintro duced in

p

Denition A is Hausdor reducible to B in symbols A B

hd

if there exists a polynomialtime f mapping every string

x toasequence of queries such that for al x iff xhy y i

 k

then

y B implies y B for al l i k and

i i

A maxfj j j k y B gfg is odd x

j

In this context the ith query y i k xcomputedbyf x is also

i

p

denotedbyf x i We cal l f abounded Hausdor reduction A B if

bhd

the number k x of queries producedbyf on x is bounded byaconstant for

al l x

co

B A set A is conp manyone reducible to a set B denoted A

m

if there exists a polynomialtime nondeterministic Turing transducer M such

that for every x x A if and only if al l outputs of M on input x are

members of B

corp

cf A set A is corp manyone reducible to a set B denoted A B

m

if thereexistapolynomialtime computable function f and a polynomial q

such that for al l x

f x w B and x A Prob

q jxj

w 

x A Prob f x w B q jxj

q jxj

w 

q jxj

Here the string w is chosen uniformly at random from the set

corp corp p

B B andA B implies A Note that for all sets A and B A

m m c

conp

implies A B A reduction f of some typ e fromasetA to a set B is

m

called honest if there is a p olynomial p such that for every x and for every query



jy j For any reducibilitytyp e y generated by f x it holds that jxjp

Note that for a reduction f of any reducibilitytyp e discussed in this pap er it is meaningful

to talk of queries generated by f on input x

EXPSPACEavoiding sets

Bo ok and Lutz measured the information contentofasetby the Kolmogorov

complexity of its characteristic sequences Let M be a Turing machine z b e a string

and let d s b e natural numbers Wesay that z KS d s if M on some input

M

of length at most d outputs z using space at most s In other words KS d s

M

is the set of strings whose sspacebounded Kolmogorov complexity relativetoM

is b ounded by dWell known simulationtechniques see show that there is a

Universal Turing machine U such that for every machine M there is a constant c

c cs c Henceforth wex U as such such that for all d s KS d s KS d

M U

a Universal Turing machine and omit the subscript Note that there is a constant

A n

c such that for every set A and for all n the characteristic sequence of A is

n

n cn

in KS c A set A is said to have maximum information content if

for every c and almost every n

A n cn

KS n

n

As in HI G H is used to denote the class of sets of maximum information con

tent Intuitively there is a large gap b etween the information contentofsetsin

EXPSPACE and sets in HI G H We capture this prop erty in the following deni

tion

CEavoiding in short B is in EA if every Denition A set B is EXPSPA

subset of B in EXPSPACE is sparse

This can b e seen as a weaker form of EXPSPACEimmunitycf since only

nonsparse subsets in EXPSPACE are forbidden in EXPSPACEavoiding sets We

show next that only sparse subsets of a HI G H set can b e in EXPSPACE and



therefore HI G H E A Since sparse sets are in EA and not in HI G H the next

theorem implies that HI G H is a prop er subset of EA

Theorem Every set A in HI G H is in EAMoreover even the padded version

n

pad Afx j x A n g of A is in EA

Proof Assume that A is in HI G H but pad A is not in EA Then there exists a

nonsparse subset B of pad A which is in EXPSPACE Since pad A is paddable

it follows that there exists a nonsparse subset C of pad A whichisinESPACE

k n

For every nlet A b e the subset fx jk n jxjx C g of A and

n

n A

let v b e the length jA j substring of obtained by deleting all s of

n n

n

A

which corresp ond to some x A Since C is nonsparse it follows that for all

n

n

qn Then it is clear p olynomials q there exist innitely many n such that jA j

n

A O n

that the following outputs and can b e implemented in space

n

denotes the empty string



This and subsequent results concerning EXPSPACE in the pap er can b e easily extended to

EXPSPACEpoly

n

input v

n

i

n

for x to in lexicographic order do

if x A then output

n

else i i output the ith bit of v

n

n n A

Since jv j jA j n it follows that there is a constant c such that

n n

n

n cn

KS q n for every p olynomial q and innitely many n contradicting the

fact that A HIGH

It can b e similarly shown that if pad AisinEA then also A is in EA

Rangeconstructible reducibilities and collapse

implications

We rst formally dene the notion of rangeconstructibility

Denition Areducibility is rangeconstructible if for al l sets A B such

n

that A B thereisanEXPSPACE oracle transducer which on input and

n

with oracle A outputs a subset B of B to which A correctly reduces via the given

n

reduction for al l inputs of length up to n

Using this denition the pro of of the following theorem is straightforward

Theorem Every set A in EXPSPACE which honestly reduces to a set B in EA

viaarangeconstructible reducibility also reduces via the same typeofreducibility

toasparse set

Proof Let p b e the p olynomial witnessing the honesty of the reduction from A

h query y on input x fullls pjy j jxj According to the denition to B ie eac

of rangeconstructibility let B b e the set pro duced by the EXPSPACE oracle

n

n n

transducer on input with oracle A At rst we dene for every n a subset

of B to which A correctly reduces via the given reduction for all inputs of length

n

fy j y B pjy j ng has this exactly n Since the reduction is honest B

n

n

prop ertyThus it is straightforward to see that by replacing in the reduction from

jxj

A to B on input x each query y by the query y we obtain a reduction from

n

A to B fy j y B g We will shownow that B is sparse Since A is in

n

S

also is in EXPSPACE Since EXPSPACE it follows that the set B B

n

n

B B and B EA B is sparse B is a subset of pad B thus it nally follows

that B is sparse

Next we givevarious examples of reducibilities that are rangeconstructible and

apply Theorem to these reducibilities to derive strong collapse consequences

for various complexity classes under the assumption that they honestly reduce to

conp corp

sets in EAWestartbyshowing the rangeconstructibility of the and

m m

reducibilities

Prop osition The conp manyone and the corp manyone reducibilities are

rangeconstructible

conp

Proof Let A B via a p olynomial time nondeterministic Turing transducer

m

M that is for every x x A if and only if all output strings pro duced by M

n

on input x are in B LetB fy jx A Mx outputs y g It is clear that B

n n

n

can b e computed in exp onential space with oracle A Also B is a subset of B

n

to which A reduces via M for all instances up to length n The pro of for the corp

manyone reducibility pro ceeds analogously

Theorem The composition of the Hausdor reducibility and the conp

manyone reducibility is rangeconstructible

The composition of the Hausdor and the corp manyone reducibilities is

also rangeconstructible

p

Proof Weprove the rst part The second parts pro of is analogous Let A B

hd

via a p olynomial time computable function f and let Bconp manyone reduce to

C via a p olynomial time nondeterministic Turing transducer M Let k xbethe

numb er of queries in the list f x and let M y denote the set of all outputs pro

duced by M on input y Dene C as the set computed by the following algorithm

n

n n

on input with oracle A

n

input

C

rep eat

n

for all x do

maxfgfj j j k x and M f x j C g l

S

C C fM f x j j j l g

if x A l is even then C C M f x l

until C remained unchanged during the last execution of the lo op

output C

At rst weshowthatC is a subset of C for all n We pro ceed by induction

n

on the numb er of iterations of the rep eatlo op Assume that C C after the

m st iteration and that C is extended by some set M f x j during the

mth iteration There are two cases In the rst case there exists an index lj

such that M f x l C and since C C this implies M f x j C bythe

monotonicity of the Hausdor reduction In the second case it holds that j l

where l maxfj j j k x and for all i j M f x i C g is

even if and only if x is in A It is clear that lkx since C C Bywayofa

contradiction assume M f l x C Because C C this contradicts the

fact that A reduces to C via f and M since it would imply that x A maxfj j

j k jxj and for all i j M f x i C g is even

Let p b e a p olynomial b ounding the length of the queries pro duced by the com

pn

p osition of f and M Observe that C implying that the algorithm

n

O n O n

terminates Moreover since jC j the algorithm stops after steps Fi

n

nally it is clear from the denition of the algorithm that A reduces to C via f

n

n

and M for all instances x

p p

It follows that the manyone conjunctive b ounded Hausdor and

m c

the comp osition of the b ounded Hausdor and conjunctive reducibilities are also

rangeconstructible b ecause they are sp ecial cases of the reducibilities considered in

Theorem As a consequence of Theorem and the ab ove rangeconstructibility

results we obtain the following theorem

Theorem Let A beinEXPSPACE and let be one of the fol lowing reducibility

types

conjunctive corp manyone conp many one or the composition of

the bounded Hausdor reducibility with one of these reducibilities

Then the fol lowing conditions areequivalent

i A is honestly reducible to some set in EA

ii A SPARSE

iii A R HI G H

Proof The implication i ii directly follows from Corollary and from the

rangeconstructibility of the considered reducibilities Toshow ii iii observe

p

that R S P ARS E R T ALLY follows from S P ARS E R TALLY

c

p

and the fact that R R C R C for all the reducibilities considered here

c

Since for every tally set T there exists a set B in HI G H such that T B

it follows that R SPARSE R HI G H Finally consider A EXPSPACE

which reduces to a set B HIGH Then A also honestly reduces to the set

pad B whichisin EA by Theorem This sho ws iii i

Wenowhave the corollary of collapse consequences

Corollary Let C be any complexity class from fUP NP C P PPg If

p

p

CR R HI G H then C P

c

bhd

Let C be any of the complexity classes from fUP NP C P PPg If C

p

R HI G H then C P

d

p p

p

For CfNP PSPACEgifCR R HI G H then C is low for



hd c

p

corp

R HI G H then NP RP NP is containedinR If

bhd m

Proof The rst part is a direct consequence of Theorem and the result that

p

p

CR R SPARSE implies C P for any of the ab ove complexity classes

bhd c

p p

p

The second part follows from the rst since R HI G H R R HI G H The

c

d bhd

third holds since the existence of a sparse hard set for NP or PSPACE with resp ect

to the comp osed Hausdor and conjunctive reducibility implies the collapse of the

p p

p olynomial time hierarchyto resp ectively PSPACE The fourth part

 

p

corp

follows from the result that NP is not contained in R R SPARSE unless

bhd m

NP RP cf

It is easy to see from Theorem that the ab ove corollary also holds for honest

reductions to EA sets for the considered reducibilities

Acknowledgments We thank Montse Hermo and Elvira Mayordomo for helpful

comments

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