Simplicity

and

Strong

Reductions

Preliminary

Draft

Marcus

Schaefer

Department

of

Computer

Science

University

of

Southern

Maine

Falmouth

Street

Portland

Maine

USA

schaefer@cs.uchicago.edu

Stephen

Fennery

Department

of

Computer

Science

University

of

Southern

Maine

Falmouth

Street

Portland

Maine

USA

fenner@cs.usm.maine.edu

July

Abstract

A

set

is

called

NP-immune

if

it

does

not

contain

any

inonite

NP

subsets

and

NP-simple

if

it

lies

in

NP

and

its

complement

is

NP-immune

Hartmanis

Li

and

Yehsa

-LSB-

-RSB-

proved

in

that

no

m-hard

-LRB-

many-one

hard

-RRB-

set

for

NP

is

NP-immune

unless

NP

SUBEXP

On

the

other

hand

we

can

exhibit

a

relativized

world

in

which

there

is

an

NP-simple

set

that

is

complete

under

Cook

reductions

even

conjunctive

reductions

We

can

therefore

ask

to

what

extent

the

original

result

generalizes

to

reductions

which

lie

in

strength

in

between

many-one

and

Turing

reductions

We

show

that

no

positive

bounded

truth-table

complete

set

for

NP

is

NP-simple

unless

UP

SUBEXP

Under

the

stronger

assumption

that

UP

coUP

SUBEXP

it

is

even

true

that

no

bounded

truth-table

complete

set

for

NP

is

NP-simple

Further

results

for

dioeerent

reductions

are

shown

including

a

similar

theorem

about

NEXP

which

does

not

require

any

assumptions

We

derive

the

results

by

the

use

of

inseparable

sets

This

technique

turns

out

to

be

very

powerful

in

the

study

of

truth-table

and

even

-LRB-

honest

-RRB-

Turing

reductions

Introduction

In

Post

published

his

epochal

paper

on

Recursively

enumerable

sets

of

positive

integers

and

their

decision

problems

-LRB-

reprinted

in

Davis

-LSB-

-RSB-

-RRB-

With

this

paper

he

initiated

a

line

of

research

in

computability

which

has

since

become

famous

as

Post

's

program

To

settle

the

question

of

whether

there

are

incomplete

c.e.

noncomputable

degrees

he

pursued

a

simple

but

productive

idea

Among

the

c.e.

sets

reducing

to

a

complete

set

are

sets

with

very

fat

complements

Thus

c.e.

sets

with

very

thin

complements

should

not

be

complete

since

strong

can

not

squeeze

the

fat

complements

into

the

thin

complements

Partially

supported

by

NSF

Grant

yPartially

supported

by

NSF

Grant

this

paper

we

will

use

the

terminology

suggested

by

Soare

-LSB-

-RSB-

That

means

computable

instead

of

recursive

and

computably

enumerable

-LRB-

c.e.

-RRB-

instead

of

recursively

enumerable

-LRB-

r.e.

-RRB-

compliance

with

computability

rather

than

complexity

theory

we

think

of

m-reductions

as

the

strongest

reductions

whereas

Turing

reductions

are

considererd

weak

A

similar

approach

to

incompleteness

has

been

popular

in

complexity

theory

since

the

time

the

Isomorphism

Conjecture

-LRB-

IC

-RRB-

was

orst

suggested

by

Berman

and

Hartmanis

in

the

study

of

sparse

sets

Sparse

sets

are

thin

in

the

sense

that

they

contain

only

polynomially

many

strings

at

any

length

This

property

is

invariant

under

polynomial-time

isomorphisms

so

under

the

assumption

that

IC

holds

no

sparse

set

can

be

NP-complete

since

SAT

-LRB-

for

example

-RRB-

is

evidently

not

sparse

There

is

an

alternative

and

relatively

unexplored

approach

which

is

closer

in

spirit

to

Post

's

original

idea

consider

eoeectively

thin

sets

like

Post

's

simple

and

immune

sets

The

use

of

eoeective

thinness

notions

in

complexity

theory

to

obtain

results

on

strong

reductions

goes

back

to

a

paper

by

Hartmanis

Li

and

Yesha

-LSB-

-RSB-

published

in

In

this

paper

it

was

proved

that

every

m-hard

-LRB-

many-one

hard

-RRB-

set

for

NP

has

an

inonite

NP

subset

unless

NP

SUBEXP

We

could

also

express

this

as

saying

that

no

m-hard

set

for

NP

is

NP-immune

-LRB-

unless

of

course

NP

SUBEXP

-RRB-

In

analogy

with

computability

theory

we

also

call

a

set

NP-simple

if

it

lies

in

NP

and

its

complement

is

NP-immune

This

gives

us

the

two

most

important

examples

of

eoeective

thinness

we

will

study

In

the

present

paper

we

set

out

to

improve

upon

this

initial

immunity

result

in

several

directions

by

considering

weaker

reductions

under

dioeerent

hypotheses

We

can

prove

that

no

set

btt-hard

for

NP

is

NP-simple

unless

UP

coUP

SUBEXP

For

honest

reducibilities

we

have

similar

results

down

to

and

including

Turing

reductions

In

the

general

case

-LRB-

without

the

honesty

requirement

-RRB-

we

can

handle

o

-LRB-

log

n

-RRB-

tt-reductions

Dioeerent

notions

of

simplicity

are

involved

as

explained

in

the

next

section

Applying

our

techniques

to

exponential

time

we

can

prove

that

no

btt-complete

set

for

EXP

is

NEXP-simple

-LRB-

without

any

assumptions

-RRB-

a

result

also

proved

by

Buhrman

using

a

dioeerent

technique

This

paper

should

also

be

interesting

from

a

methodological

point

of

view

through

its

use

of

inseparable

sets

In

computability

theory

inseparable

sets

have

long

proved

to

be

a

rich

source

of

results

-LSB-

-RSB-

and

this

paper

hopes

to

show

that

the

same

can

be

true

for

complexity

theory

This

appears

to

be

the

orst

time

inseparable

sets

are

applied

to

the

study

to

resource

bounded

reductions

Inseparability

has

been

used

before

in

complexity

though

and

more

will

be

said

about

that

in

Section

An

overview

of

the

main

results

is

presented

in

Section

followed

by

several

sections

containing

the

proofs

The

paper

concludes

with

some

remarks

on

relativizations

and

a

list

of

challenging

open

problems

We

will

have

frequent

recourse

to

concepts

from

computability

without

actually

deoning

them

since

they

only

serve

as

illustration

and

are

not

essential

The

interested

reader

will

ond

the

missing

deonitions

in

either

Odifreddi

-LSB-

-RSB-

or

Soare

-LSB-

-RSB-

As

references

for

complexity

theory

we

mention

Balc?zar

Diaz

Gabarr

-LSB-

-RSB-

and

Papadimitriou

-LSB-

-RSB-

Some

short

remarks

on

notation

honest

reductions

will

be

marked

by

an

h

added

to

the

the

reduction

sign

For

example

A

hbtt

B

means

that

A

honest

bounded

truth-table

reduces

to

B.

Furthermore

all

reductions

in

this

paper

are

polynomial

time

reductions

which

is

not

reAEected

in

the

notation

Throughout

the

whole

text

will

denote

the

alphabet

unless

it

is

used

to

denote

a

sum

Immunity

and

Simplicity

The

concepts

of

immunity

and

simplicity

have

a

long

tradition

in

computability

theory

starting

with

Post

's

paper

-LSB-

-RSB-

Deonition

A

set

is

C-immune

if

it

is

inonite

and

does

not

contain

an

inonite

subset

which

lies

in

C

A

set

is

C-simple

if

it

is

in

C

and

its

complement

is

C-immune

If

C

is

closed

under

polynomial

time

isomorphisms

then

the

notions

of

C-immune

and

C-simple

are

invariant

under

polynomial

time

isomprhisms

In

particular

NP-immune

and

NP-simple

can

be

used

to

tackle

IC

and

these

notions

might

prove

more

AEexible

than

sparseness

Historical

background

on

NP-immunity

and

NP-simplicity

can

be

found

in

the

second

volume

of

Balc?zar

et

al.

-LSB-

-RSB-

from

which

the

above

deonition

is

taken

Note

that

this

text

-LRB-

like

some

others

-RRB-

abbreviates

NP-simple

to

simple

To

avoid

confusion

we

will

prefer

the

full

names

NP-immune

and

NP-simple

NP-immune

sets

exist

any

immune

set

is

NP-immune

The

case

with

NP-simple

sets

is

more

diOEcult

and

it

took

a

rather

long

time

-LRB-

after

several

initial

attempts

-RRB-

till

Balc?zar

-LSB-

-RSB-

proved

in

that

relative

to

some

computable

oracle

an

NP-simple

set

exists

Recently

it

was

shown

by

Vereshchagin

-LSB-

-RSB-

that

relative

to

a

random

oracle

an

NP-simple

set

exists

More

on

this

topic

can

be

found

in

Section

Apart

from

the

observation

that

the

existence

of

an

NP-simple

set

is

a

strong

indication

that

NP

coNP

our

knowledge

of

NP-simple

sets

is

very

limited

In

particular

we

do

not

know

of

any

probable

combination

of

assumptions

that

would

entail

the

existence

-LRB-

or

nonexistence

-RRB-

of

an

NP-simple

set

Homer

-LSB-

-RSB-

tries

to

circumvent

the

problem

by

introducing

k-simple

sets

in

analogy

with

the

k-creative

sets

of

Joseph

and

Young

We

will

not

pursue

this

line

of

research

here

To

deal

with

bounded

truth-table

reductions

we

will

introduce

a

variant

of

NP-immunity

Remember

that

NPSV

is

the

class

of

partial

single-valued

functions

computable

by

an

NP-machine

We

call

f

-LRB-

-RRB-

k

in

NPSV

with

inonite

domain

an

honest

np-k-array

if

there

is

a

polynomial

p

such

that

for

all

x

and

f

-LRB-

x

-RRB-

yki

then

p

-LRB-

jyij

-RRB-

jxj

for

all

i

We

will

identify

f

-LRB-

x

-RRB-

yki

with

the

set

ykg

Note

that

it

would

not

have

made

a

dioeerence

in

the

deonition

if

instead

of

NPSV

we

had

used

NPMV

or

even

PF

In

view

of

the

use

we

make

of

this

notion

the

NPSV

deonition

seems

to

be

the

most

natural

Deonition

A

set

A

is

called

NP-k-immune

if

there

is

no

honest

np-k-array

f

such

that

A

f

-LRB-

x

-RRB-

for

all

x

dom

-LRB-

f

-RRB-

Because

of

the

honesty

condition

it

is

clear

that

is

the

same

as

NP-immune

Furthermore

it

is

immediate

from

the

deonition

that

any

NP

-LRB-

k

-RRB-

immune

set

is

also

NP-k-immune

More

surprisingly

there

is

a

rather

unexpected

connection

with

NP-simplicity

The

computability

version

of

the

following

lemma

is

a

folklore

result

-LSB-

Lemma

-RSB-

The

lemma

is

implicit

in

Yamakami

's

manuscript

-LSB-

-RSB-

Lemma

If

A

is

NP-simple

then

A

is

NP-k-immune

for

every

k.

Proof

Assume

that

A

NP

and

there

is

an

honest

np-k-array

f

NPSV

such

that

A

f

-LRB-

x

-RRB-

for

all

x.

Let

m

lim

jA

f

-LRB-

x

-RRB-

j.

By

assumption

m

k.

Consider

the

set

S

fy

y

f

-LRB-

x

-RRB-

and

f

-LRB-

x

-RRB-

fyg

contains

at

least

m

elements

in

Ag

Then

S

is

inonite

because

f

is

honest

and

has

inonite

domain

S

NP

since

A

is

in

NP

and

f

is

an

honest

function

in

NPSV

Finally

S

is

a

subset

of

A

except

for

onitely

many

exceptions

hence

A

is

not

NP-simple

NP-immunity

NP-k-immunity

and

NP-simplicity

will

not

carry

us

beyond

bounded

truth-table

reductions

not

surprisingly

in

view

of

the

situation

in

computability

The

need

arises

for

a

new

concept

which

corresponds

to

hyperimmunity

from

computability

theory

To

get

this

we

just

relax

the

cardinality

condition

we

introduced

for

np-k-arrays

Call

a

partial

NPSV

function

f

-LRB-

-RRB-

with

inonite

domain

an

honest

np-array

if

there

is

a

polynomial

p

such

that

for

all

x

if

f

-LRB-

x

-RRB-

ymi

then

p

-LRB-

jyij

-RRB-

jxj

for

all

i

m

In

other

words

f

is

honest

in

all

outputs

As

usual

identify

f

-LRB-

x

-RRB-

ymi

with

the

set

ymg

Deonition

A

set

A

is

called

NP-hyperimmune

if

there

is

no

honest

np-array

f

such

that

A

f

-LRB-

x

-RRB-

for

all

x

dom

-LRB-

f

-RRB-

Call

a

set

NP-hypersimple

it

is

is

in

NP

and

its

complement

is

NP-hyperimmune

It

is

clear

from

the

deonition

that

being

NP-hyperimmune

implies

being

NP-k-immune

for

every

k

and

hence

an

NP-hypersimple

set

is

NP-simple

but

probably

not

vice

versa

as

will

be

argued

in

the

section

on

relativizations

-RRB-

To

summarize

NP-hyperimmune

-RRB-

NP

-LRB-

k

-RRB-

immune

-RRB-

NP-k-immune

-RRB-

NP-immune

and

NP-hypersimple

-RRB-

NP-simple

-RRB-

co-NP-k-immune

At

this

point

we

should

ask

how

much

farther

we

can

take

eoeective

thinness

In

analogy

with

computability

we

could

call

a

set

A

NP-maximal

if

it

has

no

superset

in

NP

which

dioeers

inonitely

often

from

both

A

and

But

NP-maximal

sets

do

no

exist

This

follows

from

a

result

of

Breidbart

's

on

splittings

by

P-sets

and

was

observed

by

Homer

and

Maass

-LSB-

-RSB-

Inseparability

Inseparability

was

orst

used

by

Rosser

-LSB-

-RSB-

in

in

connection

with

extensions

of

G?del

's

Theorem

Since

then

it

has

been

very

successful

as

a

clean

and

easy

tool

to

deal

with

reductions

-LSB-

-RSB-

This

paper

tries

to

show

that

the

same

is

true

for

the

complexity

theoretic

setting

Deonition

Two

disjoint

sets

A

and

B

are

called

C-inseparable

if

there

is

no

C

C

for

which

A

C

B

If

C

is

the

class

of

computable

sets

and

A

and

B

are

required

to

be

c.e.

then

C-inseparable

corresponds

to

the

classical

notion

of

computably

inseparable

In

this

paper

we

are

mainly

interested

in

three

cases

P-inseparable

SUBEXP-inseparable

and

P

subexp-inseparable

Whereas

the

deonitions

of

P

and

SUBEXP

are

reasonably

familiar

we

should

make

some

remarks

about

P

subexp

Traditionally

nonuniform

complexity

classes

are

deoned

in

such

a

way

that

the

advice

string

counts

towards

the

input

size

With

that

interpretation

an

algorithm

deciding

a

language

in

P

subexp

might

take

subexponential

time

rather

than

polynomial

time

To

avoid

this

anomaly

in

this

paper

we

will

follow

the

convention

that

the

advice

string

is

accessed

like

an

oracle

via

random

access

namely

the

algorithm

writes

down

the

address

of

the

bit

in

the

advice

string

it

wants

to

read

Consequently

the

advice

string

will

not

be

considered

part

of

the

input

With

this

convention

we

can

deone

P

for

a

oxed

and

thus

P

subexp

T

P

Note

that

our

new

convention

does

not

aoeect

the

popular

nonuniform

complexity

classes

like

P

poly

Deonition

We

call

a

set

E

a

C-separator

if

it

separates

a

C-inseparable

pair

of

disjoint

sets

in

NP

i.e.

there

are

disjoint

sets

A

and

B

in

NP

which

are

C-inseparable

and

A

E

B

We

will

write

p-separator

instead

of

P-separator

and

s-separator

instead

of

SUBEXP-separator

Grollmann

and

Selman

proved

that

p-separators

exist

unless

P

UP

which

is

considered

to

be

unlikely

Fact

-LRB-

Grollmann

and

Selman

-LSB-

-RSB-

-RRB-

If

P

UP

then

there

are

two

disjoint

NP-complete

sets

which

are

not

separated

by

any

set

in

P

We

can

generalize

their

result

to

hold

for

all

reasonable

classes

C

where

a

nonempty

class

of

subsets

of

is

called

reasonable

if

it

is

closed

under

polynomial

time

truth-table

reducibilities

for

which

all

queries

have

the

same

length

Lemma

If

UP

C

for

a

reasonable

class

C

then

there

are

two

disjoint

NP-complete

sets

which

are

C-inseparable

Proof

If

UP

C

then

we

can

choose

a

set

C

UP

n

C

Since

C

UP

there

is

a

set

D

P

and

a

polynomial

p

such

that

x

C

if

and

only

if

-LRB-

-RRB-

-LSB-

jyj

p

-LRB-

jxj

-RRB-

and

hx

yi

D

-RSB-

Furthermore

for

all

x

C

there

is

exactly

one

such

y

Now

deone

A

fhx

i

bi

-LRB-

-RRB-

-LSB-

jyj

p

-LRB-

jxj

-RRB-

hx

yi

D

jij

dlog

p

-LRB-

jxj

-RRB-

eand

yi

bg

and

B

fhx

i

bi

-LRB-

-RRB-

-LSB-

jyj

p

-LRB-

jxj

-RRB-

hx

yi

D

jij

dlog

p

-LRB-

jxj

-RRB-

eand

yi

bg

where

yi

is

the

ith

bit

of

y

interpreting

the

string

i

as

a

number

It

is

clear

from

the

deonition

that

A

and

B

are

disjoint

sets

in

NP

-LRB-

even

UP

-RRB-

Furthermore

any

separator

of

A

and

B

in

C

would

allow

us

to

decide

C

in

C

by

making

a

polynomial

number

of

truth-table

queries

-LRB-

all

of

the

same

length

-RRB-

to

the

separator

It

is

then

straightforward

to

see

that

A

SAT

and

B

SAT

are

two

disjoint

NP-complete

sets

which

are

not

separated

by

any

set

in

C

-LRB-

using

closure

of

C

under

m-reductions

-RRB-

Apparently

this

easy

way

of

converting

A

and

B

into

complete

sets

was

orst

observed

by

Dubhashi

-LSB-

-RSB-

Grollmann

and

Selman

-LSB-

-RSB-

use

a

dioeerent

construction

which

additionally

allows

us

to

let

A

-LRB-

or

B

-RRB-

be

any

specioc

m-complete

set

-LRB-

like

SAT

-RRB-

There

is

an

easier

way

of

getting

inseparable

sets

-LRB-

mentioned

by

Dubhashi

-RRB-

which

does

not

need

any

assumptions

on

C.

Lemma

If

NP

coNP

C

then

there

are

two

disjoint

sets

in

NP

coNP

which

are

C

inseparable

Proof

Take

A

NP

coNP

n

C

and

let

B

A.

The

class

NP

coNP

is

of

interest

since

it

contains

exactly

the

low

sets

for

NP

namely

those

sets

whose

NP-jump

is

in

NP

The

relationships

between

some

of

the

assumptions

mentioned

above

in

the

case

that

C

P

are

discussed

by

Fortnow

and

Rogers

-LSB-

-RSB-

The

Grollmann

and

Selman

paper

-LSB-

-RSB-

shows

that

a

secure

public-key

cryptosystem

exists

-LRB-

i.e.

one

which

can

not

be

cracked

in

polynomial

time

-RRB-

only

if

p-separators

exist

Homer

and

Selman

-LSB-

-RSB-

construct

an

oracle

relative

to

which

all

complete

sets

are

polynomial

time

isomorphic

but

no

p-separators

exist

Summary

of

Main

Results

In

this

section

we

collect

the

main

theorems

and

roughly

group

them

together

by

the

strength

of

the

hypotheses

First

are

the

results

that

only

assume

P

UP

With

this

hypothesis

we

can

only

capture

honest

reductions

but

we

can

settle

the

question

all

the

way

down

to

Turing

reductions

Theorem

No

set

h-btt-hard

for

NP

is

NP-simple

unless

P

UP

Theorem

No

set

h-d-hard

for

NP

is

co-NP-immune

-LRB-

let

alone

NP-simple

-RRB-

unless

P

UP

Theorem

No

set

h-T-hard

for

NP

is

NP-hyperimmune

unless

P

UP

Yamakami

-LSB-

-RSB-

proves

a

weaker

version

of

the

second

theorem

it

only

applies

to

d-reductions

that

are

forced

to

be

honest

on

all

inputs

Our

deonition

of

an

honest

disjunctive

reduction

is

more

generous

and

can

be

found

in

the

next

section

For

arbitrary

reductions

-LRB-

without

honesty

condition

-RRB-

we

need

the

stronger

assumption

that

UP

SUBEXP

Theorem

No

EXP

set

for

NP

is

NP-immune

unless

UP

SUBEXP

Disjunctive

reductions

are

just

the

inverse

of

conjunctive

reductions

which

we

would

expect

to

behave

rather

like

many-one

reductions

So

the

next

result

does

not

come

as

a

surprise

Theorem

No

EXP

set

d-hard

for

NP

can

be

co-NP-immune

-LRB-

let

alone

NP-simple

-RRB-

unless

UP

SUBEXP

More

astonishingly

a

similar

result

can

be

proved

for

positive

bounded

truth-table

reductions

The

proof

is

more

involved

Theorem

No

set

pos-btt-hard

for

NP

can

be

either

NP-simple

or

co-NP-simple

unless

UP

SUBEXP

As

a

matter

of

fact

we

can

establish

the

result

for

arbitrary

bounded

truth-table

reductions

using

a

dioeerent

technique

and

a

stronger

assumption

Theorem

No

set

btt-hard

for

NP

can

be

NP-simple

unless

UP

coUP

SUBEXP

Note

that

we

need

UP

coUP

SUBEXP

for

the

last

theorem

rather

than

UP

SUBEXP

This

will

be

explained

later

The

last

result

was

claimed

by

Yamakami

-LSB-

-RSB-

-LRB-

under

the

weaker

hypothesis

that

NP

SUBEXP

-RRB-

but

we

are

as

yet

unable

to

verify

the

proof

We

will

show

the

result

to

be

a

corollary

of

the

proof

of

the

next

theorem

Theorem

No

set

o

-LRB-

log

n

-RRB-

tt-hard

for

NP

is

NP-hypersimple

unless

UP

coUP

SUBEXP

Finally

we

have

some

results

on

EXP

and

NEXP

which

do

not

require

any

assumptions

A

slightly

weaker

form

of

the

following

theorem

-LRB-

with

NEXP

instead

of

EXP

-RRB-

was

orst

proved

by

Harry

Buhrman

Theorem

No

btt-hard

set

for

EXP

is

NEXP-simple

We

can

uniformly

improve

the

results

which

use

the

hypothesis

UP

C

by

changing

it

to

NP

coNP

C.

Under

the

latter

hypothesis

we

even

get

a

low

set

for

NP

which

does

not

reduce

to

thin

sets

using

strong

reductions

This

implies

the

original

results

-LRB-

under

the

stronger

assumption

-RRB-

since

a

low

set

m-reduces

to

every

NP-hard

set

As

an

example

we

show

what

this

would

look

like

in

the

case

of

Theorem

Theorem

There

is

a

low

set

for

NP

-LRB-

i.e.

a

set

in

NP

coNP

-RRB-

which

does

not

to

any

NP-immune

set

in

EXP

unless

NP

coNP

SUBEXP

Most

of

the

results

above

follow

by

putting

together

the

existence

theorems

on

separators

with

the

corresponding

lemma

from

the

next

sections

For

example

Theorem

is

proved

thus

if

P

UP

then

there

is

a

p-separator

E

NP

-LRB-

take

one

of

the

P-inseparable

set

-RRB-

Suppose

now

that

M

is

NP-hard

under

honest

disjunctive

reductions

Then

E

hd

M

and

by

Lemma

M

is

not

NP-immune

We

will

not

mention

this

procedure

henceforth

unless

-LRB-

as

in

the

case

of

positive

bounded

truth-table

reductions

-RRB-

additional

reasoning

is

required

Disjunctive

reductions

and

m-reductions

This

section

is

mainly

intended

for

warming

up

and

serves

to

illustrate

some

of

the

basic

ideas

The

case

for

honest

disjunctive

reductions

is

perhaps

the

easiest

result

Our

deonition

of

an

honest

disjunctive

reduction

might

not

be

considered

standard

which

is

why

we

include

it

below

Deonition

A

set

A

honest

disjunctively

reduces

-LRB-

h-d

reduces

-RRB-

to

B

in

polynomial

time

-LRB-

or

A

hd

B

for

short

-RRB-

if

there

is

a

polynomial

time

computable

function

f

such

that

f

maps

to

-LRB-

-RRB-

fTg

and

if

f

-LRB-

x

-RRB-

ymi

then

x

A

if

and

only

if

B

ymg

or

f

-LRB-

x

-RRB-

T

and

f

is

a

polynomially

honest

function

with

regard

to

all

m

output

values

-LRB-

if

m

-RRB-

Note

that

m

above

is

only

bounded

by

the

fact

that

f

has

to

be

computable

in

polynomial

time

Furthermore

the

deonition

leaves

the

reduction

the

possibility

to

make

f

-LRB-

x

-RRB-

the

empty

set

or

the

constant

T

to

get

around

the

honesty

condition

for

cases

it

can

decide

easily

by

itself

without

querying

B

This

feature

is

not

standard

Lemma

If

E

is

a

p-separator

and

E

hd

M

then

M

is

not

NP-immune

Proof

Let

f

be

the

function

witnessing

the

reduction

from

E

to

M

and

let

E

separate

the

two

NP

sets

A

and

B

Then

for

x

B

and

f

-LRB-

x

-RRB-

ymi

all

the

yi

are

in

M

Note

that

in

this

case

inonitely

often

m

since

otherwise

we

could

separate

A

and

B

by

a

set

in

P

Since

f

is

polynomially

honest

we

can

deone

an

inonite

NP

subset

of

M

To

get

rid

of

the

honesty

condition

we

will

make

use

of

the

stronger

premise

that

s-separators

exist

The

next

proof

shows

in

a

nutshell

how

SUBEXP

is

used

to

force

the

reduction

to

be

suOEciently

honest

Lemma

If

E

is

an

s-separator

and

E

<=d

M

where

M

EXP

then

M

is

not

NP-immune

Proof

Let

f

be

the

reduction

from

E

to

M

and

let

E

be

an

s-separator

with

regard

to

the

two

NP

sets

A

and

B.

Deone

g

-LRB-

x

-RRB-

to

be

the

largest

string

in

ymg

where

f

-LRB-

x

-RRB-

ymi

Then

g

-LRB-

x

-RRB-

M

for

all

x

B

There

are

two

cases

-LRB-

i

-RRB-

There

is

a

polynomial

p

and

inonitely

many

x

B

such

that

p

-LRB-

jg

-LRB-

x

-RRB-

j

-RRB-

jxj

In

this

case

the

set

S

fy

-LRB-

B

-RRB-

-LSB-

p

-LRB-

jyj

-RRB-

jxj

and

y

g

-LRB-

x

-RRB-

g

is

an

inonite

NP

subset

of

M

-LRB-

ii

-RRB-

For

every

polynomial

p

there

are

at

most

onitely

many

x

B

for

which

p

-LRB-

jg

-LRB-

x

-RRB-

j

-RRB-

jxj

This

means

that

B

SUBEXP

which

contradicts

the

assumption

We

can

apply

the

lemma

to

get

a

result

on

m-reductions

which

we

will

need

as

a

technical

lemma

for

the

case

of

Lemma

If

E

is

an

s-separator

and

E

m

M

where

M

EXP

then

neither

M

nor

M

are

NP-immune

-LRB-

in

particular

M

is

not

NP-simple

-RRB-

Proof

Since

E

m

M

implies

E

<=d

M

we

can

apply

the

last

lemma

to

get

that

M

is

not

NP

immune

Since

E

is

an

s-separator

if

E

is

and

E

m

M

implies

that

E

m

M

we

can

conclude

that

M

is

not

NP-immune

This

result

is

not

the

best

possible

for

m-reductions

Hartmanis

et

al.

-LSB-

-RSB-

showed

that

no

m-hard

set

can

be

NP-immune

unless

NP

SUBEXP

and

no

m-hard

set

can

be

co-NP-immune

unless

NP

coNP

SUBEXP

Fact

-LRB-

Hartmanis

et

al.

-LSB-

-RSB-

-RRB-

Every

m-hard

set

for

NP

has

an

inonite

dense

subset

in

NP

unless

NP

SUBEXP

A

set

is

dense

if

there

is

an

such

that

jA

nj

for

almost

all

n.

Bounded

truth-table

reductions

In

this

section

we

will

tackle

honest

and

positive

bounded

truth-table

reductions

The

general

case

of

bounded

truth-table

reductions

will

be

resolved

by

dioeerent

methods

in

the

section

on

truth-table

reductions

Let

us

orst

deone

what

we

mean

by

an

honest

bounded

truth-table

reduction

For

the

deonition

let

-LSB-

-LSB-

F

-RSB-

-RSB-

equal

if

F

is

true

and

otherwise

Deonition

A

set

A

honest

bounded

truth-table

reduces

-LRB-

h-btt

reduces

-RRB-

to

B

in

polynomial

time

-LRB-

or

A

hbtt

B

for

short

-RRB-

if

there

are

two

polynomial

time

computable

functions

f

and

ff

and

k

such

that

f

maps

to

-LRB-

-RRB-

k

ff

maps

to

and

if

f

-LRB-

x

-RRB-

yki

then

x

A

if

and

only

if

ff

-LRB-

x

-LSB-

-LSB-

B

-RSB-

-RSB-

-LSB-

-LSB-

yk

B

-RSB-

-RSB-

-RRB-

and

f

is

a

polynomially

honest

function

with

regard

to

all

k

output

values

Note

that

h-btt

reductions

can

get

around

the

honesty

condition

for

easy

instances

by

making

ff

-LRB-

x

-RRB-

the

constant

function

taking

or

Then

f

-LRB-

x

-RRB-

can

be

deoned

to

be

-LRB-

x

x

-RRB-

-LRB-

-RRB-

k.

Perhaps

the

most

amazing

fact

about

bounded

truth-table

reductions

is

that

we

can

assume

we

are

dealing

with

one

oxed

truth-table

rather

than

having

the

truth-table

depend

on

the

input

We

will

establish

this

orst

Fixing

truth-tables

For

the

sake

of

clarity

we

include

the

deonition

of

a

-LRB-

honest

-RRB-

reduction

via

a

oxed

truth-table

Deonition

Let

E

and

M

be

arbitrary

sets

let

ff

be

a

k-ary

Boolean

function

and

let

f

be

computable

in

polynomial

time

We

say

that

E

polynomial

time

ff-reduces

to

M

via

f

-LRB-

f

E

ff

M

-RRB-

if

there

is

a

polynomial

p

such

that

ffl

-LRB-

E

-RRB-

ff

-LRB-

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

-RRB-

and

ffl

-LRB-

E

-RRB-

ff

-LRB-

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

-RRB-

where

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

is

the

vector

-LRB-

M

-LRB-

-RRB-

M

-LRB-

yk

-RRB-

-RRB-

where

f

-LRB-

x

-RRB-

yki

We

say

that

E

ff

M

is

there

exists

such

an

f

If

additionally

there

is

a

polynomial

p

such

that

p

-LRB-

jyij

-RRB-

jxj

for

all

i

and

f

-LRB-

x

-RRB-

yki

we

call

the

reduction

honest

and

say

that

E

honest

polynomial

time

ff-reduces

to

M

via

f

-LRB-

f

E

hff

M

-RRB-

-LRB-

We

say

that

E

hff

M

if

there

exists

such

an

f

-RRB-

We

try

to

formulate

the

following

lemma

as

general

as

possible

so

it

applies

to

both

p-separators

and

s-separators

Call

a

class

C

oxable

if

C

contains

P

and

is

closed

under

onite

union

and

intersection

Lemma

Suppose

E

is

a

C-separator

for

a

oxable

class

C

and

M

an

arbitrary

set

If

E

k-tt

M

then

there

is

a

C-separator

eE

and

a

k-ary

Boolean

function

ff

such

that

eE

ff

M

If

the

k-tt

reduction

was

honest

or

positive

then

it

will

remain

so

Proof

The

proof

will

not

change

the

reduction

so

we

can

ignore

the

issues

of

honesty

and

positivity

They

just

carry

over

from

the

assumption

Let

A

and

B

be

two

C-inseparable

NP

sets

with

A

E

B

Since

E

k-tt

M

there

is

a

function

f

PF

and

a

k-ary

Boolean

function

ffx

computable

in

polynomial

time

in

x

and

its

arguments

such

that

f

-LRB-

x

-RRB-

yki

for

all

x

and

ffl

-LRB-

A

-RRB-

ffx

-LRB-

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

-RRB-

and

ffl

-LRB-

B

-RRB-

ffx

-LRB-

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

-RRB-

Let

and

let

o

enumerate

all

k-ary

Boolean

functions

For

i

set

Ti

fx

j

ffx

oig

All

Ti

are

in

P

If

A

Ti

and

B

Ti

are

C-separable

for

all

i

then

clearly

A

and

B

are

C-separable

so

ox

i

such

that

eA

A

Ti

and

eB

B

Ti

are

C-inseparable

and

set

ff

oi

and

eE

fx

ff

-LRB-

OM

-LRB-

Df

-LRB-

x

-RRB-

-RRB-

The

last

result

allows

us

to

apply

our

recently

gained

knowledge

about

m-reductions

Lemma

If

E

is

an

s-separator

and

E

M

where

M

EXP

then

M

is

not

NP-immune

unless

UP

SUBEXP

Proof

By

Lemma

we

can

assume

that

E

reduces

to

M

via

a

oxed

truth-table

of

arity

one

There

are

four

possible

truth-tables

We

can

exclude

the

constant

truth-tables

since

E

is

not

in

P

So

either

E

m

M

or

E

m

M

In

both

cases

neither

M

nor

M

are

NP-immune

by

Lemma

Honest

bounded

truth-tables

The

techniques

developed

so

far

are

suOEcient

to

settle

the

case

of

honest

bounded

truth-table

reductions

Lemma

If

E

is

a

p-separator

and

E

hk-tt

M

then

M

is

not

NP-k-immune

By

Lemma

we

can

state

the

result

for

NP-simplicity

as

a

corollary

Corollary

If

E

is

a

p-separator

and

E

hbtt

M

then

M

is

not

NP-simple

Lemma

is

a

consequence

of

two

results

Lemma

which

showed

that

an

h-btt

reduction

can

be

assumed

to

work

with

a

oxed

truth-table

and

Lemma

which

handles

that

case

Lemma

If

E

is

a

p-separator

and

M

is

NP-k-immune

then

there

is

no

k-ary

ff

such

that

E

hff

M

Proof

Let

A

and

B

be

a

P-inseparable

pair

of

NP

sets

with

A

E

B.

Suppose

M

is

NP

simple

and

there

is

a

k-ary

ff

such

that

E

hff

M

via

some

computable

f

We

can

assume

without

loss

of

generality

that

ff

-LRB-

-RRB-

since

otherwise

we

consider

ff

and

E

So

we

know

that

if

x

A

f

-LRB-

x

-RRB-

intersects

M

That

is

if

we

deone

the

partial

NPSV

function

f

-LRB-

x

-RRB-

f

-LRB-

x

-RRB-

if

x

A

then

f

is

an

np-k-array

intersecting

M

for

every

x

A

so

M

is

not

NP-k-immune

and

hence

M

is

not

NP-simple

by

Lemma

Positive

bounded

truth-table

reductions

Our

eventual

aim

is

to

extend

the

result

of

the

last

section

on

honest

btt-reductions

to

arbitrary

btt-reductions

Along

the

particular

path

we

have

chosen

this

seems

diOEcult

and

the

best

we

can

do

at

present

is

to

deal

with

the

class

of

positive

btt-reductions

which

contains

bounded-disjunctive

and

bounded-conjunctive

reductions

In

the

next

section

on

tt-reductions

we

take

a

slightly

dioeerent

approach

which

leads

us

to

a

result

for

general

btt-reduction

but

requires

a

stronger

hypothesis

than

UP

SUBEXP

A

reduction

is

called

positive

if

the

language

accepted

by

the

reduction

does

not

become

smaller

if

the

set

it

is

reduced

to

is

substituted

by

a

superset

For

truth-table

reductions

this

means

that

for

every

truth-table

ff

generated

on

some

input

it

holds

that

if

v

w

-LRB-

where

is

the

partial

order

that

compares

v

and

w

componentwise

-RRB-

and

ff

-LRB-

v

-RRB-

then

ff

-LRB-

w

-RRB-

Lemma

If

E

is

an

s-separator

and

E

pbtt

M

then

neither

M

nor

M

is

NP-simple

The

lemma

will

be

a

consequence

of

the

following

lemma

and

the

observation

that

if

A

pbtt

B

then

also

A

pbtt

B

-LRB-

generally

not

by

the

same

reduction

however

-RRB-

Lemma

If

E

is

an

s-separator

and

E

pbtt

M

then

M

is

not

NP-simple

Proof

Suppose

M

NP

Let

A

and

B

be

disjoint

NP

sets

which

are

SUBEXP-inseparable

such

that

A

E

B

Now

E

k-tt

M

via

a

positive

reduction

We

can

reorder

the

queries

of

that

reduction

such

that

they

are

in

length

nondecreasing

order

That

is

the

reduction

query

looks

like

yki

where

jykj

By

Lemma

we

can

assume

that

there

is

a

oxed

positive

truth-table

ff

of

arity

k

and

a

polynomial

time

computable

function

f

from

to

such

that

x

E

ioe

ff

-LRB-

OMf

-LRB-

x

-RRB-

-RRB-

The

reduction

f

also

retains

the

property

that

if

f

-LRB-

x

-RRB-

yki

then

jykj

Consider

the

set

V

fv

there

are

inonitely

many

x

A

for

which

v

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

g

We

will

use

the

partial

order

on

V

where

v

w

if

vi

-RRB-

wi

for

all

i

k.

For

v

deone

Av

fx

A

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

vg

Note

that

A

S

Av

since

Av

is

onite

for

v

V

By

one

-LRB-

v

-RRB-

we

denote

the

position

of

the

rightmost

in

v

Then

one

-LRB-

v

-RRB-

k

for

v

V

since

V

does

not

contain

k

-LRB-

otherwise

the

truth-table

is

constantly

true

-RRB-

We

will

prove

that

for

every

v

V

either

Av

is

separable

from

B

by

a

set

in

SUBEXP

or

there

is

an

inonite

subset

of

M

in

NP

If

all

of

the

Av

were

SUBEXP-separable

from

B

then

A

S

Av

would

be

SUBEXP-separable

from

B

which

contradicts

the

assumption

Hence

there

is

an

inonite

subset

of

M

in

NP

Fix

any

v

V

Deone

the

set

C

fx

A

f

-LRB-

x

-RRB-

yki

and

for

all

i

k

if

vi

then

yi

Mg

Then

C

NP

since

M

NP

and

furthermore

C

contains

Av

Let

i

k

be

minimal

such

that

there

is

a

polynomial

p

for

which

the

set

Di

;p

fx

A

f

-LRB-

x

-RRB-

yki

and

for

all

j

k

-LSB-

vj

-RRB-

yj

M

-RSB-

and

p

-LRB-

jyij

-RRB-

jxjg

is

inonite

If

no

such

i

exists

or

one

-LRB-

v

-RRB-

i

then

Done

-LRB-

v

-RRB-

;p

is

onite

for

any

choice

of

a

polynomial

p

Since

the

reduction

is

positive

and

v

has

zeroes

to

the

right

of

one

-LRB-

v

-RRB-

this

means

that

Av

and

B

are

SUBEXP-separable

Hence

we

can

assume

that

i

one

-LRB-

v

-RRB-

and

p

exist

Fix

them

and

deone

S

fhyi

yki

x

Di

;p

f

-LRB-

x

-RRB-

ykig

S

is

an

honest

np

-LRB-

k

i

-RRB-

array

If

all

but

onitely

many

of

the

sets

in

S

intersect

M

then

Lemma

assures

us

that

M

is

not

NP-simple

and

we

are

done

So

suppose

for

a

contradiction

that

there

are

inonitely

many

sets

in

S

which

do

not

intersect

M

Then

there

is

a

w

V

with

w

v

and

one

-LRB-

w

-RRB-

i

for

which

fx

A

f

-LRB-

x

-RRB-

yki

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

w

and

p

-LRB-

jyij

-RRB-

jxjg

is

inonite

Since

i

was

minimal

we

then

know

that

for

all

polynomials

q

and

for

almost

all

x

C

we

have

q

-LRB-

jyone

-LRB-

w

-RRB-

j

-RRB-

jxj

where

f

-LRB-

x

-RRB-

yki

Since

C

contains

Av

we

know

that

for

all

polynomials

q

and

for

almost

all

x

Av

it

is

true

that

q

-LRB-

jyone

-LRB-

w

-RRB-

j

-RRB-

jxj

where

f

-LRB-

x

-RRB-

yki

Since

ff

-LRB-

z

-RRB-

for

every

z

with

w

z

we

can

deone

an

s-separator

which

separates

Av

from

B.

Truth-table

and

Turing

reductions

We

have

already

dealt

with

the

special

cases

of

disjunctive

and

conjunctive

reductions

earlier

on

In

this

section

we

will

study

truth-table

and

Turing

reductions

For

these

we

do

not

expect

results

regarding

NP-simplicity

or

even

NP-immunity

-LRB-

Section

contains

a

contrary

relativization

-RRB-

Instead

we

will

make

use

of

NP-hyperimmunity

For

truth-table

reductions

we

can

not

hope

to

prove

the

very

convenient

Lemma

which

allowed

us

to

consider

oxed

truth-tables

only

This

makes

a

new

approach

necessary

As

a

matter

of

fact

the

main

inspiration

comes

from

Denisov

's

proof

-LSB-

Exercise

III

-RSB-

that

if

B

is

part

of

a

computably

inseparable

pair

of

c.e.

sets

and

B

-LRB-

computably

-RRB-

truth-table

reduces

to

A

then

A

is

not

hyperimmune

As

in

the

bounded

case

the

proof

for

the

honest

case

is

easy

and

goes

through

with

Turing

reductions

We

will

present

this

result

next

The

case

for

general

truth-table

reductions

is

still

open

but

we

present

a

result

for

o

-LRB-

log

n

-RRB-

tt

reductions

which

also

has

some

consequences

for

the

bounded

truth-table

case

Honest

Turing

reductions

The

deonition

of

a

honest

Turing

reduction

is

quite

straightforward

Deonition

A

set

A

honest

Turing

reduces

-LRB-

h-T

reduces

-RRB-

to

B

in

polynomial

time

-LRB-

or

A

hT

B

for

short

-RRB-

if

there

is

a

polynomial

p

and

an

oracle

Turing

machine

T

-LRB-

:-RRB-

running

in

time

p

-LRB-

jxj

-RRB-

on

input

x

such

that

x

A

if

and

only

if

TB

-LRB-

x

-RRB-

Furthermore

if

Q

-LRB-

x

-RRB-

denotes

the

set

of

queries

made

by

T

to

B

in

the

computation

of

TB

-LRB-

x

-RRB-

then

p

-LRB-

jwj

-RRB-

jxj

for

every

w

Q

-LRB-

x

-RRB-

i.e.

the

machine

only

makes

honest

queries

Since

we

have

a

bound

on

the

running

time

we

can

assume

that

T

-LRB-

:-RRB-

always

halts

in

at

most

p

-LRB-

jxj

-RRB-

steps

on

input

x

with

output

or

and

only

makes

honest

queries

regardless

of

the

oracle

The

bound

on

the

running

time

is

the

main

dioeerence

between

Turing

reductions

in

computability

and

complexity

and

it

leads

to

the

fact

that

in

complexity

we

can

deal

with

Turing

reductions

using

only

hyperimmunity

whereas

in

computability

not

even

maximality

is

good

enough

-LRB-

there

are

Turing

complete

maximal

sets

-LSB-

-RSB-

-RRB-

Theorem

No

NP-hyperimmune

set

can

be

h-T-hard

for

NP

unless

P

UP

The

proof

is?as

usual?split

up

into

several

parts

It

follows

from

the

following

lemma

together

with

Lemma

and

the

observation

that

P

UP

if

and

only

if

UP

P

-LRB-

using

self-reducibility

-RRB-

Note

that

to

apply

Lemma

the

class

P

has

to

be

closed

under

tt-reductions

making

queries

of

one

length

only

This

is

the

case

since

the

advice

string

only

depends

on

the

length

of

the

input

Lemma

If

E

is

a

P

and

E

hT

M

then

M

is

not

NP-hyperimmune

Proof

Let

E

separate

A

B

NP

where

A

and

B

are

not

separated

by

any

set

in

P

and

E

hT

M

Let

T

be

an

oracle

Turing

machine

running

in

polynomial

time

p

-LRB-

n

-RRB-

which

witnesses

the

reduction

i.e.

x

E

-RRB-

TM

-LRB-

x

-RRB-

and

x

E

-RRB-

TM

-LRB-

x

-RRB-

Furthermore

we

know

that

T

-LRB-

:-RRB-

always

halts

in

at

most

p

-LRB-

jxj

-RRB-

steps

on

input

x

is

honest

and

outputs

only

or

regardless

of

the

oracle

For

an

input

x

let

Q

-LRB-

x

-RRB-

be

the

set

of

queries

made

by

T

-LRB-

x

-RRB-

Now

there

are

two

cases

First

consider

the

case

that

there

are

inonitely

many

n

with

x

A

and

y

B

of

length

n

such

that

T

-LRB-

x

-RRB-

T

-LRB-

y

-RRB-

Fix

such

an

n

x

and

y

Since

TM

-LRB-

x

-RRB-

TM

-LRB-

y

-RRB-

one

of

these

values

has

to

be

dioeerent

from

T

-LRB-

x

-RRB-

T

-LRB-

y

-RRB-

Therefore

M

-LRB-

Q

-LRB-

x

-RRB-

-LSB-

Q

-LRB-

y

-RRB-

-RRB-

can

not

be

empty

With

an

NPSV

function

on

an

input

string

z

xy

where

jxj

jyj

we

can

verify

that

x

A

y

B

and

T

-LRB-

x

-RRB-

T

-LRB-

y

-RRB-

and

then

output

Q

-LRB-

x

-RRB-

-LSB-

Q

-LRB-

y

-RRB-

which

works

since

the

queries

in

Q

are

honest

In

the

other

case

we

know

that

for

almost

all

n

and

x

A

and

y

B

of

length

n

it

is

true

that

T

-LRB-

x

-RRB-

T

-LRB-

y

-RRB-

We

can

use

this

to

separate

A

from

B

by

coding

for

each

length

n

the

possible

values

of

T

-LRB-

x

-RRB-

for

x

A

of

length

n

There

are

four

possible

cases

no

value

-LRB-

if

A

is

empty

at

length

n

-RRB-

or

-LRB-

in

this

case

B

is

empty

at

length

n

-RRB-

Two

bits

are

suOEcient

to

code

which

case

we

are

in

o

-LRB-

log

n

-RRB-

tt

reductions

A

o

-LRB-

log

n

-RRB-

tt

reduction

is

a

truth-table

reduction

which

on

input

x

makes

at

most

o

-LRB-

log

jxj

-RRB-

queries

In

particular

this

will

have

consequences

for

bounded

truth

table

reductions

As

seen

already

in

the

case

of

honest

Turing

reductions

we

need

assumptions

on

nonuniform

complexity

classes

for

our

proof

technique

to

be

successful

We

can

then

establish

the

following

result

Theorem

No

set

o

-LRB-

log

n

-RRB-

tt-hard

for

NP

can

be

NP-hyperimmune

unless

UP

coUP

SUBEXP

The

most

important

step

in

the

proof

is

the

next

lemma

Lemma

If

E

is

a

P

subexp-separator

and

E

o

-LRB-

log

n

-RRB-

tt

M

then

M

is

not

NP-hyperimmune

Proof

Since

E

o

-LRB-

log

n

-RRB-

tt

M

there

are

polynomial

time

computable

functions

f

ff

and

m

such

that

for

all

x

x

E

-LRB-

-RRB-

ffx

-LRB-

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

-RRB-

We

use

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

as

a

shorthand

for

the

vector

-LRB-

-LSB-

-LSB-

M

-RSB-

-RSB-

-LSB-

-LSB-

ym

-LRB-

x

-RRB-

M

-RSB-

-RSB-

-RRB-

where

f

-LRB-

x

-RRB-

ym

-LRB-

x

-RRB-

i.

Remember

that

we

identify

f

-LRB-

x

-RRB-

ym

-LRB-

x

-RRB-

i

with

the

set

ym

-LRB-

x

-RRB-

g

when

convenient

Furthermore

let

-LRB-

-RRB-

n

denote

the

set

of

all

strings

of

length

at

most

n

Since

E

is

a

P

subexp-separator

there

are

two

P

subexp-inseparable

sets

A

and

B

in

NP

with

A

E

B

There

are

two

cases

to

consider

First

suppose

that

there

is

an

and

inonitely

many

n

for

which

there

are

x

A

and

y

B

with

jxj

jyj

n

such

that

f

-LRB-

x

-RRB-

-LRB-

-RRB-

n

f

-LRB-

y

-RRB-

-LRB-

-RRB-

n

and

-LRB-

-LRB-

-RRB-

n

-RRB-

-LSB-

ffx

-LRB-

OF

-LRB-

f

-LRB-

x

-RRB-

-RRB-

-RRB-

ffy

-LRB-

OF

-LRB-

f

-LRB-

y

-RRB-

-RRB-

-RRB-

-RSB-

-LRB-

-RRB-

Then

we

can

deone

an

NPSV

function

as

follows

On

input

z

xy

where

jxj

jyj

n

verify

that

x

A

and

y

B

and

that

f

-LRB-

x

-RRB-

-LRB-

-RRB-

n

and

f

-LRB-

y

-RRB-

-LRB-

-RRB-

n

are

equal

Let

D

be

the

set

of

those

queries

i.e.

D

f

-LRB-

x

-RRB-

-LRB-

-RRB-

n

f

-LRB-

y

-RRB-

-LRB-

-RRB-

n

Verify

that

for

all

F

D

it

is

true

that

ffx

-LRB-

OF

-LRB-

f

-LRB-

x

-RRB-

-RRB-

-RRB-

ffy

-LRB-

OF

-LRB-

f

-LRB-

y

-RRB-

-RRB-

-RRB-

If

this

is

the

case

output

-LRB-

f

-LRB-

x

-RRB-

-LSB-

f

-LRB-

y

-RRB-

-RRB-

n

-LRB-

-RRB-

n

Note

that

the

algorithm

described

above

is

indeed

in

NPSV

as

D

is

of

size

at

most

log

n

and

we

have

to

check

no

more

than

n

subsets

each

in

polynomial

time

By

assumption

the

algorithm

will

try

inonitely

often

to

output

-LRB-

f

-LRB-

x

-RRB-

-LSB-

f

-LRB-

y

-RRB-

-RRB-

n

-LRB-

-RRB-

n

which

is

honest

if

nonempty

So

we

only

have

to

prove

that

-LRB-

f

-LRB-

x

-RRB-

-LSB-

f

-LRB-

y

-RRB-

-RRB-

n

-LRB-

-RRB-

n

contains

a

string

in

M

We

know

that

for

all

F

D

it

is

true

that

ffx

-LRB-

OF

-LRB-

f

-LRB-

x

-RRB-

-RRB-

-RRB-

ffy

-LRB-

OF

-LRB-

f

-LRB-

y

-RRB-

-RRB-

-RRB-

But

then

-LRB-

-RRB-

has

to

hold

since

D

contains

all

the

queries

f

makes

on

x

and

y

of

length

at

most

n

Hence

if

we

let

M

M

-LRB-

-RRB-

n

it

follows

that

ffx

-LRB-

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

-RRB-

ffy

-LRB-

OM

-LRB-

f

-LRB-

y

-RRB-

-RRB-

-RRB-

At

the

same

time

ffx

-LRB-

OM

-LRB-

f

-LRB-

x

-RRB-

-RRB-

-RRB-

ffy

-LRB-

OM

-LRB-

f

-LRB-

y

-RRB-

-RRB-

-RRB-

since

x

belongs

to

A

and

y

to

B

So

M

and

M

have

to

disagree

on

a

query

in

-LRB-

f

-LRB-

x

-RRB-

-LSB-

f

-LRB-

y

-RRB-

-RRB-

n

-LRB-

-RRB-

n

Since

M

is

empty

above

length

n

it

follows

that

M

contains

a

string

in

-LRB-

f

-LRB-

x

-RRB-

-LSB-

f

-LRB-

y

-RRB-

-RRB-

n

-LRB-

-RRB-

n

So

in

this

case

M

is

not

NP-hyperimmune

If

the

orst

assumption

does

not

apply

it

must

be

true

that

for

every

and

for

almost

all

n

and

all

x

A

y

B

with

jxj

jyj

n

either

the

sets

f

-LRB-

x

-RRB-

-LRB-

-RRB-

n

and

f

-LRB-

y

-RRB-

-LRB-

-RRB-

n

are

not

the

same

or

the

truth-tables

ffx

and

ffy

are

dioeerent

on

a

set

of

strings

F

D

where

D

f

-LRB-

x

-RRB-

-LRB-

-RRB-

n

f

-LRB-

y

-RRB-

-LRB-

-RRB-

n

We

can

use

this

knowledge

to

distinguish

between

strings

from

A

and

B.

Call

two

strings

x

and

y

of

the

same

length

n

equivalent

if

f

-LRB-

x

-RRB-

-LRB-

-RRB-

n

f

-LRB-

y

-RRB-

-LRB-

-RRB-

n

and

ffx

-LRB-

OF

-LRB-

f

-LRB-

x

-RRB-

-RRB-

-RRB-

ffy

-LRB-

OF

-LRB-

f

-LRB-

y

-RRB-

-RRB-

-RRB-

for

all

F

D

f

-LRB-

x

-RRB-

-LRB-

-RRB-

n

By

assumption

it

will

not

be

possible

that

there

are

two

equivalent

strings

one

of

which

is

in

A

and

the

other

one

in

B

So

the

strings

of

length

n

will

be

partitioned

into

a

onite

number

of

equivalence

classes

of

which

intersects

both

A

and

B.

Every

equivalence

class

is

uniquely

determined

by

the

set

D

and

a

truth-table

on

the

elements

of

D.

Each

equivalence

class

can

be

described

by

at

most

log

n

strings

of

length

at

most

n

and

a

truth-table

of

length

log

n

n

Putting

it

together

the

description

will

have

length

log

n

which

is

smaller

than

for

large

enough

n.

Using

this

description

as

an

index

into

an

advice

string

we

can

code

for

each

equivalence

class

the

information

whether

it

contains

a

string

in

A

in

one

random

access

advice

string

of

length

at

most

This

gives

us

a

separator

of

A

and

B

in

P

subexp

Note

that

even

stronger

we

only

have

to

query

one

single

bit

of

the

advice

string

To

onish

the

proof

of

the

theorem

we

need

a

-LRB-

highly

nontrivial

-RRB-

result

from

Homer

and

Longpr

-LSB-

-RSB-

which

we

will

state

in

the

form

we

need

it

Let

cA

-LRB-

n

-RRB-

jA

nj

be

the

census

function

of

A.

For

a

set

E

NP

there

is

a

D

P

and

a

polynomial

p

such

that

x

E

if

and

only

if

there

is

y

with

jyj

p

-LRB-

jxj

-RRB-

and

hx

yi

D

Then

the

left

set

L

of

E

is

deoned

to

be

fhx

yi

-LRB-

-RRB-

-LSB-

jzj

jyj

p

-LRB-

jxj

-RRB-

y

z

and

hx

zi

D

-RSB-

g.

Call

a

class

C

NP

a

left

class

if

it

is

closed

under

the

left

set

construction

Lemma

-LRB-

Homer

et

al.

-LSB-

Theorem

-RSB-

-RRB-

Let

TO

f

-LRB-

n

-RRB-

-LRB-

n

-RRB-

-LRB-

f

-LRB-

n

-RRB-

-RRB-

-RRB-

-LRB-

cO

-LRB-

n

-RRB-

-RRB-

f

-LRB-

n

-RRB-

n

If

O

is

f

-LRB-

n

-RRB-

tt-hard

for

a

left

class

C

then

for

each

C

C

there

is

a

t

such

that

C

DTIME

-LRB-

TO

f

-LRB-

nt

-RRB-

-RRB-

Furthermore

t

depends

only

on

C

and

the

running

time

of

an

f

-LRB-

n

-RRB-

tt

reduction

from

C

to

O

in

particular

it

does

not

depend

on

O

The

lemma

can

be

verioed

by

carefully

checking

the

proof

presented

in

the

cited

paper

Proof

of

Theorem

Let

M

be

a

o

-LRB-

log

n

-RRB-

tt-hard

set

for

NP

Suppose

M

is

not

NP

hyperimmune

Then

the

conclusion

of

Lemma

fails

That

means

for

every

set

A

UP

coUP

and

B

A

there

is

a

separator

in

P

subexp

More

exactly

as

we

mentioned

at

then

end

of

the

proof

only

one

query

to

the

advice

string

is

necessary

We

can

also

focus

our

attention

on

the

case

that

In

this

case

there

is

one

reduction

algorithm

-LRB-

parameterized

by

-RRB-

whose

running

time

does

not

depend

on

and

is

polynomial

This

allows

us

to

apply

Lemma

with

f

-LRB-

n

-RRB-

O

an

oracle

with

cO

-LRB-

n

-RRB-

that

codes

the

advice

strings

and

the

left

class

UP

coUP

Hence

A

DTIME

-LRB-

t

-RRB-

for

some

oxed

t

not

depending

on

O

or

Since

can

be

chosen

arbitrarily

small

this

implies

A

SUBEXP

and

hence

UP

coUP

SUBEXP

Remark

We

would

have

liked

to

prove

the

theorem

under

the

hypothesis

that

UP

SUBEXP

Unfortunately

the

orst

existence

proof

for

separators

-LRB-

Lemma

-RRB-

requires

C

to

be

closed

under

polynomial

time

truth-table

reducibilities

This

we

can

not

guarantee

if

only

one

query

to

the

oracle

is

allowed

On

the

other

hand

if

we

use

the

full

power

of

P

subexp

instead

we

can

not

apply

the

result

by

Homer

and

Longpr

any

longer

So

we

have

to

use

the

second

existence

proof

for

separators

-LRB-

Lemma

-RRB-

which

did

not

pose

any

requirements

on

C.

Note

that

the

NPSV-function

in

the

proof

of

Lemma

outputs

f

-LRB-

x

-RRB-

-LSB-

f

-LRB-

y

-RRB-

In

case

we

start

with

a

bounded

truth-table

this

means

that

jf

-LRB-

x

-RRB-

-LSB-

f

-LRB-

y

-RRB-

j

and

we

can

conclude

that

M

is

not

Hence

M

is

not

NP-simple

This

establishes

the

following

two

corollaries

of

the

proof

Corollary

If

E

is

a

P

subexp-separator

and

E

btt

M

then

M

is

not

NP-simple

Corollary

No

set

btt-hard

for

NP

is

NP-simple

unless

UP

coUP

SUBEXP

Also

the

truth-table

result

has

the

obvious

consequences

for

o

-LRB-

log

log

n

-RRB-

Turing

reductions

Exponential

time

In

this

section

we

will

test

our

techniques

on

EXP

and

NEXP

One

advantage

is

that

we

can

dispense

with

hypotheses

But

let

us

orst

recall

what

is

already

known

The

following

account

is

based

on

Buhrman

-LSB-

-RSB-

In

his

PhD

thesis

Berman

proved

that

all

m-complete

sets

for

EXP

are

in

fact

one-to-one

length-increasing

m-complete

Hence

an

m-complete

set

for

EXP

will

not

be

P

immune

since

for

example

the

tally

set

n

g

reduces

to

it

via

a

one-to-one

length-increasing

reduction

Fact

-LRB-

Berman

-RRB-

No

m-complete

set

for

EXP

is

P-immune

On

the

other

hand

it

is

easy

to

construct

a

counterexample

for

Fact

-LRB-

Buhrman

-LSB-

-RSB-

-RRB-

There

is

a

set

for

EXP

which

is

P-immune

The

case

for

NEXP

seemed

more

diOEcult

It

was

proved

by

Homer

and

Wang

-LSB-

-RSB-

that

every

m-complete

set

for

NE

and

its

complement

contain

dense

E

UP

subsets

Finally

Tran

-LSB-

-RSB-

extended

the

Berman

result

to

NEXP

Fact

-LRB-

Tran

-LSB-

-RSB-

-RRB-

No

m-complete

set

for

NEXP

is

P-immune

Since

it

had

been

proven

earlier

by

Buhrman

Spaan

and

Torenvliet

that

a

set

for

NEXP

is

also

m-complete

the

following

corollary

was

immediate

Fact

-LRB-

Tran

-LSB-

-RSB-

-RRB-

No

set

for

NEXP

is

P-immune

Again

this

seems

to

be

the

limit

Fact

-LRB-

Buhrman

-LSB-

-RSB-

-RRB-

There

is

a

set

for

NEXP

which

is

P-immune

But

not

quite

It

should

be

observed

that

Tran

's

proof

actually

yields

results

for

conjunctive

and

disjunctive

reductions

as

well

Theorem

A

conjunctively

complete

set

for

NEXP

is

not

P-immune

Theorem

The

complement

of

a

disjunctively

complete

set

for

NEXP

is

not

P-immune

But

what

can

we

say

about

truth-table

reductions

Fortunately

the

proof

of

Lemma

can

be

adapted

for

the

exponential

time

case

We

orst

have

to

deone

a

new

version

of

hyperimmunity

An

honest

exp-array

is

a

partial

function

f

-LRB-

-RRB-

computable

in

exponential

time

and

with

inonite

domain

with

the

following

property

there

is

a

polynomial

p

such

that

for

all

x

ffl

p

-LRB-

jyij

-RRB-

jxj

for

all

i

m

and

ffl

jf

-LRB-

x

-RRB-

j

p

-LRB-

jxj

-RRB-

where

f

-LRB-

x

-RRB-

ymi

Note

that

compared

to

np-arrays

we

only

increased

the

computational

power

of

the

function

The

output

is

still

restricted

to

polynomially

many

strings

of

polynomial

size

in

the

input

We

call

a

set

EXP-hyperimmune

if

there

is

no

honest

exp-array

f

such

that

f

-LRB-

x

-RRB-

A

for

all

x

dom

-LRB-

f

-RRB-

To

prove

the

next

two

theorems

we

need

a

result

from

Fu

-LSB-

-RSB-

which

can

be

verioed

by

a

careful

reading

of

his

proof

Lemma

-LRB-

Fu

-LSB-

Theorem

-RSB-

-RRB-

There

is

a

set

E

EXP

such

that

for

any

set

S

for

which

E

S

there

is

an

such

that

cS

-LRB-

n

-RRB-

for

almost

all

n.

Furthermore

only

depends

upon

an

upper

bound

on

the

runtime

of

the

not

on

S.

Theorem

No

set

o

-LRB-

log

n

-RRB-

tt-hard

for

EXP

is

EXP-hyperimmune

Proof

The

proof

is

an

easy

adaptation

of

Lemma

Start

with

the

set

E

from

the

result

of

Fu

's

and

let

A

E

B.

Let

M

be

o

-LRB-

log

n

-RRB-

tt-hard

for

EXP

Then

E

o

-LRB-

log

n

-RRB-

tt

M

Assume

that

M

is

EXP-hyperimmune

Then

the

orst

case

in

the

proof

of

Lemma

can

not

apply

So

by

the

second

case

for

every

there

is

an

oracle

S

such

that

E

S

and

cS

-LRB-

n

-RRB-

Furthermore

we

can

assume

that

and

we

have

one

upper

bound

on

the

running

time

of

the

regardless

of

the

actual

oracle

S

This

contradicts

the

last

lemma

As

with

Lemma

the

theorem

allows

us

to

draw

a

conclusion

for

btt-reductions

We

should

note

that

Lemma

works

for

NEXP

too

Theorem

A

btt-hard

set

for

EXP

is

not

NEXP-simple

Corollary

-LRB-

Buhrman

-RRB-

No

btt-complete

set

for

NEXP

is

NEXP-simple

The

corollary

was

orst

reported

by

Buhrman

with

a

proof

using

a

oxed-point

construction

Buhrman

improved

his

construction

to

yield

our

Theorem

by

incorporating

the

Denisov

idea

-LSB-

-RSB-

Relativizations

We

do

not

expect

NP-simplicity

to

carry

us

far

beyond

bounded

truth-table

reductions

and

this

expectation

is

made

precise

by

constructing

a

relativized

world

in

which

there

is

an

-LRB-

honest

-RRB-

conjunctively

complete

NP-simple

set

although

UP

SUBEXP

Let

f

-LRB-

n

-RRB-

be

any

positive

monotone

polynomially-bounded

ptime

computable

function

such

that

limn

f

-LRB-

n

-RRB-

For

every

x

deone

Query

-LRB-

x

-RRB-

-LRB-

jxj

-RRB-

g.

Lemma

Relative

to

some

oracle

A

there

is

a

language

LA

coUPA

which

is

NE-immune

and

for

all

x

LA

Query

-LRB-

x

-RRB-

Assuming

the

lemma

we

can

prove

the

oracle

result

Theorem

There

is

an

oracle

A

such

that

UPA

SUBEXPA

and

there

exists

an

NPA-simple

set

which

is

-LRB-

honest

-RRB-

f

-LRB-

n

-RRB-

c-complete

for

NPA

Proof

Let

A

and

LA

be

as

in

Lemma

Note

that

LA

UPA

SUBEXPA

so

UPA

SUBEXPA

Note

also

that

LA

is

NPA-simple

and

Query

-LRB-

x

-RRB-

LA

for

any

x.

Let

C

be

some

m-complete

set

for

NPA

and

let

SA

LA

-LSB-

-LSB-

Query

-LRB-

x

-RRB-

Since

SA

is

a

coinonite

superset

of

LA

it

is

also

NPA-simple

Moreover

SA

is

f

-LRB-

n

-RRB-

c-complete

via

the

reduction

x

Query

-LRB-

x

-RRB-

Proof

of

Lemma

Our

construction

is

loosely

based

on

one

by

Balc?zar

-LSB-

Theorem

-RSB-

but

ours

is

complicated

by

the

fact

that

our

LA

must

be

quite

dense

whereas

Balc?zar

et

al.

relied

heavily

on

their

LA

being

very

sparse

We

let

be

an

eoeective

enumeration

of

all

linear

exptime

nondeterministic

oracle

TMs

each

Ni

running

in

time

for

all

oracles

For

all

y

deone

Region

-LRB-

y

-RRB-

fyz

j

jzj

For

all

X

deone

LX

fy

j

Region

-LRB-

y

-RRB-

Xg

Clearly

LX

coNPX

We

construct

an

A

to

make

LA

satisfy

the

conditions

of

the

lemma

Initially

we

set

A

-LSB-

-LSB-

-LRB-

x

-RRB-

Region

-LRB-

y

-RRB-

thus

making

LA

S

Query

-LRB-

x

-RRB-

and

we

commit

all

strings

currently

not

in

A

to

stay

out

of

A

This

commitment

guarantees

that

LA

S

Query

-LRB-

x

-RRB-

We

modify

A

in

stages

one

stage

for

each

x

At

each

stage

we

may

extend

our

current

commitments

by

committing

a

onite

number

of

additional

strings

always

respecting

our

previous

commitments

Thus

when

a

string

is

committed

its

membership

status

in

A

is

frozen

at

its

current

value

and

can

not

be

changed

later

The

onal

A

will

extend

all

our

commitments

Stage

x

Let

n

jxj

First

commit

all

strings

of

length

n

-LRB-

if

they

have

n't

been

committed

already

-RRB-

to

their

current

membership

status

in

A

Now

let

oe

encode

all

of

our

commitments

so

far

Find

the

least

i

n

such

that

Ni

has

not

yet

been

killed

There

is

a

way

to

extend

oe

by

committing

a

minimal

number

of

currently

uncommitted

strings

to

guarantee

-LRB-

with

the

resulting

change

in

A

-RRB-

that

-LRB-

a

-RRB-

NAi

-LRB-

y

-RRB-

accepts

for

some

y

Query

-LRB-

x

-RRB-

LA

-LRB-

b

-RRB-

For

any

at

most

one

string

in

Region

-LRB-

-RRB-

is

committed

to

be

out

of

A

and

-LRB-

c

-RRB-

-LRB-

x

-RRB-

-LSB-

jLA

Query

-LRB-

x

-RRB-

j

i

-RSB-

If

there

is

no

such

i

then

do

nothing

and

go

on

to

the

next

stage

Otherwise

make

the

minimal

commitment

described

above

Then

commit

all

strings

in

S

-LRB-

x

-RRB-

Region

-LRB-

-RRB-

to

their

current

status

regarding

A

This

kills

Ni

End

of

Stage

x

The

theorem

is

proved

via

the

three

claims

below

Claim

LA

coUPA

Proof

For

any

x

of

length

n

and

Query

-LRB-

x

-RRB-

we

never

remove

more

than

one

string

in

Region

-LRB-

-RRB-

from

A

thus

it

is

easy

to

see

that

LA

coUPA

Claim

For

all

x

LA

Query

-LRB-

x

-RRB-

Proof

Fix

x

At

the

start

of

the

construction

jLA

Query

-LRB-

x

-RRB-

j

jQuery

-LRB-

x

-RRB-

j

f

-LRB-

jxj

-RRB-

Since

LA

Query

-LRB-

x

-RRB-

is

frozen

after

Stage

x

the

only

times

we

reduce

the

size

of

this

set

is

when

we

kill

some

Ni

for

i

at

some

Stage

x

But

whenever

we

do

so

we

explicitly

ensure

that

jLA

Query

-LRB-

x

-RRB-

j

i

so

LA

Query

-LRB-

x

-RRB-

never

becomes

empty

Claim

LA

is

NEA-immune

Proof

Fix

any

i

and

suppose

that

L

-LRB-

NAi

-RRB-

is

inonite

-LRB-

Here

we

use

A

to

denote

the

onal

oracle

-RRB-

We

want

to

show

that

LA

L

-LRB-

NAi

-RRB-

If

L

-LRB-

NAi

-RRB-

contains

a

string

not

in

Query

-LRB-

x

-RRB-

for

any

x

then

we

are

done

so

assume

otherwise

We

show

that

Ni

eventually

gets

killed

and

so

we

are

done

Choose

an

x

big

enough

so

that

jQuery

-LRB-

x

-RRB-

j

f

-LRB-

jxj

-RRB-

i

NAi

-LRB-

y

-RRB-

accepts

for

some

y

Query

-LRB-

x

-RRB-

jxj

i

-LRB-

this

implies

that

Ni

-LRB-

y

-RRB-

in

item

can

not

query

all

of

Region

-LRB-

y

-RRB-

because

jRegion

-LRB-

y

-RRB-

j

-RRB-

and

for

any

Nj

with

j

i

killed

at

any

Stage

we

have

x

and

jxj

where

n

is

the

length

of

the

longest

string

in

Query

-LRB-

-RRB-

-LRB-

that

is

x

is

longer

than

any

strings

committed

on

behalf

of

killing

Nj

-RRB-

Let

denote

the

oracle

at

the

beginning

of

Stage

x

If

Ni

is

still

alive

by

the

start

of

Stage

x

then

Query

-LRB-

x

-RRB-

j

i

This

is

because

any

commitments

we

made

previously

that

could

aoeect

Query

-LRB-

x

-RRB-

were

to

kill

some

Nj

for

j

i

Suppose

Ni

is

left

alive

after

Stage

x

Then

either

nothing

gets

killed

at

Stage

x

or

some

Nj

gets

killed

for

j

i

In

either

case

we

freeze

all

strings

in

S

-LRB-

x

-RRB-

Region

-LRB-

y

-RRB-

and

so

for

the

onal

oracle

A

we

have

that

NAi

-LRB-

y

-RRB-

accepts

no

more

than

one

string

in

each

Region

-LRB-

-RRB-

is

missing

from

A

and

jLA

Query

-LRB-

x

-RRB-

j

i

Furthermore

y

LA

for

the

y

in

item

above

so

Region

-LRB-

y

-RRB-

A

But

all

this

just

means

that

we

could

have

killed

Ni

at

stage

x

remove

from

A

a

single

string

in

Region

-LRB-

y

-RRB-

that

is

not

queried

by

Ni

-LRB-

y

-RRB-

This

keeps

NAi

-LRB-

y

-RRB-

accepting

and

decreases

the

size

of

LA

Query

-LRB-

x

-RRB-

by

one

so

it

is

still

has

at

least

i

elements

Thus

Ni

is

killed

at

or

before

Stage

x.

Open

Questions

Simplicity

is

a

notion

which

is

as

natural

as

it

is

elusive

We

would

like

to

know

about

any

combination

of

natural

assumptions

that

implies

the

existence

of

an

NP-simple

set

This

might

in

particular

yield

an

easier

proof

of

Vereshchagin

's

relativization

result

The

study

of

NP-simple

sets

has

been

less

vigorous

than

one

might

hope

for

and

there

seems

to

be

plenty

of

work

left

to

be

done

Establishing

complexity

versions

of

computability

results

would

be

the

least

of

them

We

claimed?and

we

hope

we

have

shown?that

inseparable

sets

are

an

important

technique

But

many

questions

regarding

inseparable

sets

are

still

open

-LRB-

this

by

the

way

is

true

for

the

computable

case

as

well

-RRB-

Are

there

other

assumptions

under

which

inseparable

sets

exist

other

constructions

which

allow

better

hypotheses

Or

the

onal

challenge

is

there

a

necessary

and

suOEcient

condition

for

the

existence

of

inseparable

sets

In

this

context

we

should

mention

a

paper

by

Fenner

Fortnow

Naik

and

Rogers

-LSB-

-RSB-

which

investigates

two

propositions

Q

and

Q

which

might

be

closely

related

to

inseparable

sets

On

the

other

hand

an

even

stronger

case

could

be

made

for

inseparable

sets

if

further

applications

can

be

found

A

possible

candidate

would

be

Selman

's

p-selectivity

-LRB-

or

superterseness

-RRB-

Inseparable

sets

have

been

successfully

used

in

computability

to

that

eoeect

-LSB-

-RSB-

We

already

mentioned

that

better

constructions

might

allow

us

to

improve

upon

the

hypotheses

But

improvement

of

the

conclusions

seems

to

be

just

as

important

what

can

we

say

about

general

truth-table

reductions

what

about

Turing

reductions

Finally

where

do

we

go

from

here

Some

initial

results

in

the

realm

of

exponential

time

have

shown

that

our

techniques

can

be

useful

higher

up

There

seems

to

be

room

for

improvement

We

might

also

ask

what

happens

below

NP

Do

our

techniques

apply

at

all

to

classes

below

P

How

could

we

use

them

to

prove

results

on

P-immunity

for

example

Since

the

reductions

run

in

polynomial

time

there

does

not

seem

to

be

a

way

out

unless

we

restrict

ourselves

to

logspace

reductions

Furthermore

one

might

ask

whether

our

method

yields

new

assumptions

that

will

allow

separations

of

reducibilities

in

the

complete

degrees

Acknowledgment

We

would

like

to

thank

Harry

Buhrman

and

Lance

Fortnow

for

their

insights

into

truth-tables

and

Steve

Homer

for

his

ready

supply

of

papers

Further

thanks

go

to

Dieter

Van

Melkebeek

and

Janos

Simon

for

proofreading

References

-LSB-

-RSB-

J.

L.

Balc?zar

Simplicity

Relativizations

and

Nondeterminism

SIAM

J.

Comp.

vol

pp.

-LSB-

-RSB-

J.

L.

Balc?zar

J.

Diaz

J.

Gabarr

Structural

Complexity

vs.

I

and

II

Springer

Berlin

-LSB-

-RSB-

Harry

Buhrman

Complete

Sets

are

not

Simple

Unpublished

manuscript

February

-LSB-

-RSB-

Harry

Buhrman

Resource

Bounded

Reductions

Ph.D.

Thesis

at

the

University

of

Amsterdam

-LSB-

-RSB-

Martin

Davis

The

Undecidable

Raven

Press

-LSB-

-RSB-

Devdatt

Dubhashi

On

p-Separability

Technical

Report

Cornell

University

-LSB-

-RSB-

Stephen

Fenner

Lance

Fortnow

Ashish

Naik

John

Rogers

Inverting

Onto

Functions

Proceedings

of

the

Annual

Conference

on

Computational

Complexity

Theory

-LSB-

-RSB-

Stephen

Fenner

Marcus

Schaefer

Bounded

Immunity

and

Btt-Reductions

Unpublished

manuscript

a

preliminary

version

appeared

as

A

note

on

a

variant

of

immunity

btt-reducibility

and

minimal

indices

Technical

Report

University

of

Southern

Maine

-LSB-

-RSB-

Lance

Fortnow

John

Rogers

Separability

and

one-way

Functions

Proc

of

the

International

Symposium

on

Algorithms

and

Computation

Springer

LNCS

-LSB-

-RSB-

Bin

Fu

With

Quasi-linear

Queries

EXP

is

not

Polynomial

Time

Turing

Reducible

to

Sparse

Sets

Proceedings

of

the

Annual

Conference

on

Structure

in

Complexity

Theory

-LSB-

-RSB-

J.

Grollmann

A.

Selman

Complexity

measures

for

public-key

cryptography

SIAM

J.

Comp.

vol

-LSB-

-RSB-

J.

Hartmanis

M.

Li

Y.

Yesha

Containment

separation

complete

sets

and

immunity

of

complexity

classes

Proceedings

of

the

International

Colloquium

on

Automata

Languages

and

Programming

Springer

LNCS

-LSB-

-RSB-

Steven

Homer

On

Simple

and

Creative

Sets

in

NP

Theoretical

Computer

Science

-LSB-

-RSB-

Steven

Homer

Stuart

Kurtz

James

S.

Royer

A

Note

on

Hard

Languages

Theoretical

Computer

Science

-LSB-

-RSB-

Steven

Homer

Luc

Longpr

On

Reductions

of

NP

Sets

to

Sparse

Sets

Proceedings

of

the

Annual

Conference

on

Structure

in

Complexity

Theory

-LSB-

-RSB-

Steven

Homer

Wolfgang

Maass

Oracle-Dependent

Properties

of

the

Lattice

of

NP-sets

Theoretical

Computer

Science

-LSB-

-RSB-

Steven

Homer

Alan

L.

Selman

Oracles

for

Structural

Properties

The

Isomorphism

Problem

and

Public-Key

Cryptography

Journal

for

Computer

and

System

Sciences

-LSB-

-RSB-

Steven

Homer

Jie

Wang

Immunity

of

Complete

Problems

Information

and

Computation

-LSB-

-RSB-

Piergiorgio

Odifreddi

Classical

recursion

theory

North-Holland

Amsterdam

-LSB-

-RSB-

Christos

H.

Papadimitriou

Computational

Complexity

Addison-Wesley

Reading

-LSB-

-RSB-

Marcus

Schaefer

A

Short

History

of

Minimal

Indices

Technical

Report

University

of

Southern

Maine

-LSB-

-RSB-

Robert

I.

Soare

Recursively

Enumerable

Sets

and

Degrees

Springer

New

York

-LSB-

-RSB-

Robert

I.

Soare

Computability

and

Recursiveness

Bulletin

of

Symbolic

Logic

-LSB-

-RSB-

Nicholas

Tran

On

P-immunity

of

Nondeterministic

Complete

Sets

Proceedings

of

the

Annual

Conference

on

Structure

in

Complexity

Theory

-LSB-

-RSB-

Nikolai

K.

Vereshchagin

NP-Sets

are

Co-NP-Immune

Relative

to

a

Random

Oracle

Technical

Report

TR

University

of

Rochester

Computer

Science

Department

-LSB-

-RSB-

Tomoyuki

Yamakami

Simplicity

Unpublished

manuscript

August