Effective

Category

and

Measure

in

Abstract

Complexity

Theory

Cristian

Caludey

and

Marius

Zimandz

August

Abstract

Strong

variants

of

the

Operator

Speed-up

Theorem

Operator

Gap

Theorem

and

Compression

Theorem

are

obtained

using

an

effective

version

of

Baire

Category

Theorem

It

is

also

shown

that

all

complexity

classes

of

recursive

predicates

have

effective

measure

zero

in

the

space

of

recursive

predicates

and

on

the

other

hand

the

class

of

predicates

with

almost

everywhere

complexity

above

an

arbitrary

recursive

threshold

has

recursive

measure

one

in

the

class

of

recursive

predicates

Introduction

The

abstract

complexity

theory

initiated

by

Blum

-LSB-

-RSB-

-LRB-

see

also

Bridges

-LSB-

-RSB-

Calude

-LSB-

-RSB-

Hartmanis

and

Hopcroft

-LSB-

-RSB-

Machtey

and

Young

-LSB-

-RSB-

Seiferas

-LSB-

-RSB-

-RRB-

has

revealed

fundamental

properties

of

complexity

measures

The

striking

importance

of

this

theory

relies

in

its

machine-independent

nature

Indeed

the

theory

is

built

on

just

two

axioms

-LRB-

Blum

axioms

-RRB-

and

virtually

any

conceivable

realistic

model

of

computation

is

bound

to

satisfy

the

axioms

Therefore

the

major

achievements

of

this

theory

are

primordial

facts

upon

which

any

theory

of

complexity

on

any

concrete

model

is

based

The

most

important

results

of

the

abstract

complexity

theory

are

of

existential

type

a

-RRB-

the

existence

of

speedable

functions

-LRB-

Blum

-LSB-

-RSB-

-RRB-

b

-RRB-

the

existence

of

computable

functions

having

an

arbirtrary

high

complexity

-LRB-

Blum

-LSB-

-RSB-

-RRB-

c

-RRB-

the

existence

of

arbitrarily

large

gaps

in

the

complexity

of

computable

functions

-LRB-

Borodin

-LSB-

-RSB-

Trakhtenbrot

-LSB-

-RSB-

-RRB-

d

-RRB-

the

existence

of

a

function

which

defines

a

complexity

class

as

the

union

of

an

r.e.

set

of

complexity

classes

with

bounds

satisfying

a

certain

monotonicity

condition

-LRB-

Union

Theorem

McCreight

and

Meyer

-LSB-

-RSB-

-RRB-

The

proofs

of

most

of

these

theorems

are

extremely

complex

and

are

based

on

quite

intricate

constructions

whose

goal

is

precisely

the

makeup

of

the

object

whose

existence

is

proved

Of

course

one

admires

the

crafty

work

and

the

powerful

techniques

involved

in

the

proofs

of

these

results

On

the

other

hand

all

this

ingenuity

spent

for

showing

the

existence

of

a

single

object

naturally

raises

some

questions

how

typical

is

the

existence

of

such

an

object

is

it

an

accident

or

in

fact

such

objects

are

abounding

The

general

mathematical

practice

provides

two

basic

lines

for

attacking

such

questions

a

topological

one

using

the

notion

of

Baire

category

and

the

measure-theoretical

one

Both

these

approaches

model

in

their

own

way

the

basic

intuitions

of

fewness

or

largeness

which

are

inherent

to

the

human

mind

and

each

of

them

contributed

with

valuable

insights

The

work

of

the

first

author

has

been

supported

by

Auckland

University

Research

Grants

The

second

author

has

been

supported

by

the

National

Science

Foundation

under

grant

NSF

and

by

the

Romanian

Department

of

Education

and

Science

under

grant

Weaker

versions

of

some

results

of

this

paper

have

been

presented

at

the

Second

International

Colloquium

on

Words

Languages

and

Combinatorics

Kyoto

Japan

August

yComputer

Science

Department

The

University

of

Auckland

Private

Bag

Auckland

New

Zealand

cristian@cs.auckland.ac.nz

zDepartment

of

Computer

Science

University

of

Rochester

New

York

USA

zimand@cs.rochester.edu

Object

Category

Measure

Where

complexity

class

I

Category

-LSB-

-RSB-

Measure

here

a.e.

complex

functions

II

Category

here

Measure

here

i.o.

speedable

functions

II

-LSB-

-RSB-

a.e.

speedable

functions

II

-LSB-

-RSB-

operator

speedable

functions

II

here

functions

yielding

gaps

II

-LSB-

-RSB-

functions

yielding

operator

gaps

II

here

measured

set

of

functions

I

Category

-LSB-

-RSB-

Measure

here

r-honest

functions

I

-LSB-

-RSB-

Table

The

current

state

of

affairs

Notes

Definitions

of

objects

are

provided

in

the

paper

Measure

refers

to

effective

measure

in

the

class

of

recursive

predicates

Category

refers

to

effective

Baire

category

marks

incompatibility

indicates

an

open

problem

in

numerous

domains

of

mathematics

There

is

one

fundamental

obstacle

in

utilizing

these

devices

for

our

questions

all

interesting

objects

in

computational

complexity

form

countable

sets

and

therefore

both

the

category

analysis

and

the

measure-theoretical

one

in

their

classical

setting

are

much

too

rough

to

distinguish

the

size

of

such

objects

Fortunately

effective

versions

of

category

and

measure

have

been

conceived

that

constitute

natural

way

of

investigating

our

problems

Due

to

the

lack

of

the

space

we

do

not

go

into

any

detail

here

The

definitions

inside

the

paper

provide

some

hints

and

also

pointers

to

literature

where

these

rather

subtle

matters

are

fully

discussed

We

mainly

use

the

effective

Baire

category

to

investigate

the

size

of

the

class

of

functions

admitting

an

operator

speed-up

the

size

of

the

class

of

functions

defining

the

gap

in

the

Operator

Gap

Theorem

and

the

size

of

the

class

of

functions

that

are

almost

everywhere

hard

to

compute

This

approach

has

already

been

followed

in

abstract

complexity

theory

-LRB-

Mehlhorn

-LSB-

-RSB-

Calude

-LSB-

-RSB-

Calude

Istrate

and

Zimand

-LSB-

-RSB-

Calude

and

Zimand

-LSB-

-RSB-

-RRB-

and

more

recently

in

structural

complexity

-LRB-

Lutz

-LSB-

-RSB-

Mayordomo

-LSB-

-RSB-

Fenner

-LSB-

-RSB-

Zimand

-LSB-

-RSB-

-RRB-

Our

results

may

be

viewed

as

strenghtenings

of

the

original

basic

theorems

For

example

we

show

that

the

class

of

functions

satisfying

the

Operator

Gap

Theorem

is

of

second

category

In

contrast

with

the

original

version

our

Speed-up

Theorem

can

not

be

deduced

from

the

Fundamental

Theorem

of

Complexity

Theory

of

Meyer

and

Winklman

-LSB-

-RSB-

In

a

separate

section

we

also

consider

the

measure-theoretical

approach

Although

this

is

an

extremely

active

topic

in

structural

complexity

theory

-LRB-

see

the

survey

of

Lutz

-LSB-

-RSB-

-RRB-

as

far

as

we

know

it

is

for

the

time

that

effective

measure

is

used

in

abstract

complexity

theory

The

proofs

are

not

deep

but

our

point

is

that

such

a

line

of

investigation

is

feasible

and

should

be

pursued

in

attacking

more

delicate

questions

in

abstract

complexity

theory

Table

summarizes

the

currently

known

results

and

marks

the

contribution

of

this

work

Notation

We

next

describe

our

notations

and

the

definitions

related

to

the

category

approach

The

definitions

afferent

to

the

measure-theoretical

approach

are

deffered

to

Section

We

shall

assume

familiarity

with

or

access

to

Bridges

-LSB-

-RSB-

Calude

-LSB-

-RSB-

Hartmanis

and

Hopcroft

-LSB-

-RSB-

Machtey

and

Young

-LSB-

-RSB-

Seiferas

-LSB-

-RSB-

Young

-LSB-

-RSB-

Let

N

g

be

the

set

of

naturals

and

let

-LRB-

i

-RRB-

be

an

acceptable

godelization

of

PR

the

set

of

unary

partial

recursive

-LRB-

p.r.

-RRB-

functions

from

N

to

N.

Denote

by

R

the

class

of

recursive

functions

and

by

RPRED

the

class

of

recursive

predicates

i.e.

the

class

of

functions

in

R

that

are

For

PR

we

put

dom

-LRB-

-RRB-

fx

N

j

-LRB-

x

-RRB-

is

definedg

In

what

follows

the

term

recursive

function

will

always

refer

to

a

unary

recursive

function

Let

N

be

a

fixed

pairing

function

The

set

FR

of

p.r.

functions

whose

domain

is

a

finite

initial

segment

of

N

is

recursive

a

fortiori

recursively

enumerable

-LRB-

r.e.

-RRB-

and

we

fix

an

enumeration

-LRB-

i

-RRB-

of

FR

For

in

FR

we

define

the

length

of

by

j

j

maxfn

N

j

n

dom

-LRB-

-RRB-

g

We

often

consider

in

FR

as

being

a

finite

string

where

the

ith

bit

of

is

-LRB-

i

-RRB-

We

denote

the

set

of

such

strings

by

N

If

ff

N

then

denotes

their

concatenation

For

f

g

PR

we

write

f

v

g

in

case

dom

-LRB-

f

-RRB-

dom

-LRB-

g

-RRB-

and

f

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

for

every

x

in

dom

-LRB-

f

-RRB-

For

every

t

FR

put

Ut

ff

PR

j

t

v

fg

The

family

-LRB-

Ut

-RRB-

is

a

system

of

basic

neighborhoods

in

PR

we

work

with

the

topology

generated

by

this

system

-LRB-

see

Calude

-LSB-

-RSB-

Mehlhorn

-LSB-

-RSB-

Rogers

-LSB-

-RSB-

-RRB-

In

the

classical

framework

a

set

A

in

a

topological

space

is

nowhere

dense

-LRB-

or

rare

-RRB-

if

for

every

open

set

O

there

exists

an

open

subset

O

such

that

A

A

set

is

meager

-LRB-

or

of

first

Baire

category

-RRB-

if

it

is

a

finite

or

denumerable

union

of

nowhere

dense

sets

and

it

is

of

the

second

Baire

category

if

it

is

not

meager

In

the

effective

variant

of

these

notions

there

exists

a

recursive

function

f

which

for

every

basic

open

set

Ut

produces

a

witness

f

-LRB-

t

-RRB-

which

indicates

the

basic

open

set

Uf

-LRB-

t

-RRB-

which

is

disjoint

from

the

nowhere

dense

set

This

ideas

lead

to

the

following

definition

Definition

-RRB-

A

set

X

PR

is

recursively

nowhere

dense

if

there

exists

a

recursive

function

f

called

the

witness

function

such

that

i

-RRB-

n

v

f

-LRB-

n

-RRB-

for

all

n

N

ii

-RRB-

There

exists

a

natural

j

N

such

that

for

all

natural

n

j

nj

j

implies

X

U

f

-LRB-

n

-RRB-

-RRB-

A

set

X

PR

is

recursively

meager

-LRB-

or

recursively

of

first

Baire

category

-RRB-

if

there

exist

a

sequence

of

sets

-LRB-

Xi

-RRB-

and

a

recursive

function

f

such

that

i

-RRB-

X

S

Xi

and

for

all

i

N

ii

-RRB-

n

v

f

-LRB-

i

n

-RRB-

for

all

n

iii

-RRB-

there

exists

a

natural

j

such

that

for

all

n

j

nj

j

implies

Xi

U

f

-LRB-

i

n

-RRB-

-RRB-

A

set

X

PR

is

a

set

of

recursively

second

Baire

category

if

X

is

not

a

set

of

recursively

first

Baire

category

For

conciseness

we

drop

most

of

the

times

the

word

recursively

in

the

above

terminology

as

well

as

the

name

of

the

originator

of

this

topological

classification

Ren?e

Baire

The

subsets

of

PR

can

be

classified

with

respect

to

the

following

hierarchy

of

sets

of

increasing

size

nowhere

dense

meager

second

category

co-meager

and

co-nowhere

dense

sets

Although

not

all

of

these

taxonomical

notions

are

used

in

this

work

we

have

stated

them

all

for

completeness

One

can

easily

observe

that

the

extensions

v

in

the

above

definition

can

be

taken

to

be

proper

-LRB-

-RRB-

and

this

will

be

the

case

in

all

our

further

considerations

The

above

definition

can

be

stated

in

terms

of

the

relativized

topology

of

p.r.

predicates

i.e.

functions

by

simply

considering

that

-LRB-

n

-RRB-

enumerates

FPRED

the

set

of

all

p.r.

predicates

having

the

domain

equal

to

a

finite

initial

segment

of

N

In

this

case

the

topology

is

generated

by

the

basic

open

sets

-LRB-

Ut

-RRB-

where

Ut

ff

FPRED

jt

ae

fg

This

abuse

of

notation

will

always

be

clarified

by

context

See

Calude

-LSB-

-RSB-

for

a

general

treatment

A

Blum

space

-LRB-

see

-LSB-

-RSB-

-RRB-

is

a

pair

-LRB-

-LRB-

i

-RRB-

-LRB-

i

-RRB-

-RRB-

where

-LRB-

i

-RRB-

is

an

acceptable

godelization

of

PR

and

-LRB-

i

-RRB-

is

a

sequence

of

p.r.

functions

-LRB-

called

the

measure

complexity

functions

-RRB-

satisfying

the

following

two

axioms

-LRB-

called

Blum

axioms

-RRB-

i

-RRB-

dom

-LRB-

i

-RRB-

dom

-LRB-

i

-RRB-

for

all

i

N

and

ii

-RRB-

the

ternary

predicate

cost

-LRB-

i

x

y

-RRB-

if

i

-LRB-

x

-RRB-

y

and

cost

-LRB-

i

x

y

-RRB-

otherwise

is

recursive

Here

as

well

as

in

the

rest

of

the

paper

we

use

the

following

conventions

If

PR

and

x

N

are

such

that

the

-LRB-

x

-RRB-

is

undefined

we

write

-LRB-

x

-RRB-

and

we

consider

y

for

all

y

N

In

the

sequel

we

fix

a

Blum

space

-LRB-

-LRB-

i

-RRB-

-LRB-

i

-RRB-

-RRB-

If

g

is

a

recursive

function

then

the

set

C?g

ff

Rjthere

exists

an

i

such

that

OEi

f

i

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

a.e.

xg

is

called

the

complexity

class

defined

by

g

If

P

-LRB-

x

n

-RRB-

is

a

predicate

then

we

write

P

-LRB-

x

n

-RRB-

a.e.

n

in

case

P

-LRB-

x

n

-RRB-

holds

true

for

all

x

N

and

for

all

but

a

finite

set

of

n

N

similarly

P

-LRB-

x

n

-RRB-

i.o.

n

means

that

P

-LRB-

x

n

-RRB-

holds

true

for

all

x

N

and

an

infinity

of

n

N

An

operator

F

PR

o

PR

is

called

effective

if

there

exists

a

p.r.

function

N

o

N

such

that

for

every

i

in

the

domain

of

F

-LRB-

i

-RRB-

is

defined

and

F

-LRB-

i

-RRB-

-LRB-

x

-RRB-

-LRB-

i

-RRB-

-LRB-

x

-RRB-

for

every

x

in

N

The

operator

is

total

if

it

is

defined

on

every

recursive

function

and

it

preserves

total

recursiveness

-LRB-

i.e.

recursive

functions

are

mapped

to

recursive

functions

-RRB-

The

behaviour

of

effective

operators

is

governed

by

the

Kreisel-Lacombe-Shoenfield

's

Theorem

We

state

it

in

a

particular

form

which

interests

total

effective

operators

If

F

is

a

total

effective

operator

then

there

exists

a

recursive

function

g

N

N

such

that

for

every

recursive

i

Graph

-LRB-

F

-LRB-

i

-RRB-

-RRB-

-LSB-

Graph

-LRB-

g

-LRB-

j

-RRB-

-RRB-

where

Ci

fj

N

j

Graph

-LRB-

j

-RRB-

Graph

-LRB-

i

-RRB-

g.

The

Speed-Up

Phenomenon

In

this

section

we

analyse

from

a

topological

point

of

view

the

Operator

Speed-up

Theorem

in

its

strongest

form

which

involves

p.r.

predicates

Consequently

in

this

section

we

consider

the

topology

generated

by

-LRB-

Ut

-RRB-

It

is

easy

to

derive

all

results

in

this

section

for

the

case

of

functions

taking

arbitrary

integer

values

For

a

total

effective

operator

F

let

SPEED

-LRB-

F

-RRB-

denote

the

class

of

recursive

functions

having

F

speedup

almost

everywhere

More

precisely

SPEED

-LRB-

F

-RRB-

ff

R

j

for

all

i

f

there

exists

j

f

such

that

F

-LRB-

j

-RRB-

-LRB-

x

-RRB-

i

-LRB-

x

-RRB-

a.e.

xg

The

main

result

of

this

section

is

stated

in

the

following

theorem

Theorem

For

every

total

effective

operator

F

the

set

SPEED

-LRB-

F

-RRB-

is

of

second

category

Proof

Fix

a

total

effective

operator

F

for

compactness

let

denote

by

SPEED

the

set

SPEED

-LRB-

F

-RRB-

Assume

that

SPEED

is

meager

This

means

that

there

exists

a

decomposition

SPEED

-LSB-

j

SPEEDj

and

a

recursive

function

f

such

that

for

every

j

SPEEDj

U

f

-LRB-

j

n

-RRB-

a.e.

n

We

construct

a

recursive

function

g

satisfying

for

every

natural

i

the

following

two

requirements

R

-LRB-

i

-RRB-

f

-LRB-

i

m

-RRB-

ae

g

i.o.

m

Q

-LRB-

i

-RRB-

if

i

-LRB-

x

-RRB-

pi

-LRB-

x

-RRB-

i.o.

x

then

g

i

where

-LRB-

pi

-RRB-

is

a

sequence

of

functions

built

as

in

the

standard

proof

of

the

Operator

Speed-Up

Theorem

-LRB-

see

Meyer

and

Fisher

-LSB-

-RSB-

Calude

-LSB-

-RSB-

-RRB-

Conditions

Q

-LRB-

i

-RRB-

guarantee

that

g

SPEED

By

the

initial

assumption

g

SPEEDi

for

some

i.

Condition

R

-LRB-

i

-RRB-

implies

that

g

U

f

-LRB-

i

m

-RRB-

i.o.

m

which

contradicts

the

fact

that

SPEEDi

U

f

-LRB-

i

m

-RRB-

a.e.

m

The

proof

proceeds

by

defining

in

a

construction

by

stages

a

family

of

p.r.

functions

-LRB-

zs

-RRB-

s

zs

o

N

The

function

we

are

interested

in

will

be

obtained

in

two

steps

first

we

take

the

function

z

to

be

the

limit

-LRB-

as

s

goes

to

-RRB-

of

this

construction

-LRB-

we

assure

that

the

limit

exists

-RRB-

then

in

the

second

step

we

fix

to

convenient

values

the

first

three

input

variables

of

z

-LRB-

denoted

by

n

w

and

v

-RRB-

and

finally

obtain

the

desired

function

g

More

precisely

n

will

be

the

fixed

point

deduced

from

the

use

of

the

Recursion

Theorem

w

will

be

selected

in

such

a

way

as

to

achieve

the

desired

rate

of

speed-up

and

v

will

be

chosen

such

that

v

will

patch

the

initial

segment

of

the

faster

program

so

that

it

exactly

computes

the

desired

function

At

stage

s

we

construct

the

p.r.

function

zs

such

that

x

zs

-LRB-

n

w

v

x

-RRB-

-RRB-

tries

to

properly

extend

x

zs

-LRB-

n

w

v

x

-RRB-

-RRB-

It

may

be

the

case

that

some

subcomputations

in

stage

s

can

not

be

performed

-LRB-

more

precisely

the

fourth

condition

in

what

is

denoted

Test

below

-RRB-

In

this

case

the

computation

loops

forever

at

some

point

in

stage

s

In

such

a

situation

we

stipulate

that

x

zt

-LRB-

n

w

v

x

-RRB-

x

zs

-LRB-

n

w

v

x

-RRB-

for

all

t

s.

However

for

the

value

obtained

through

the

use

of

the

Recursion

Theorem

this

will

not

happen

and

consequently

for

all

w

and

v

and

for

all

s

x

zs

-LRB-

w

v

x

-RRB-

properly

extends

x

zs

-LRB-

w

v

x

-RRB-

The

particular

feature

of

the

construction

is

that

for

all

w

and

v

the

extended

part

of

x

zs

-LRB-

n

w

v

x

-RRB-

uses

from

the

previous

stages

information

related

to

the

construction

of

x

zs

-LRB-

n

x

-RRB-

-LRB-

and

not

x

zs

-LRB-

n

w

v

x

-RRB-

as

one

might

suspect

-RRB-

In

this

way

at

all

stages

s

we

extend

x

zs

-LRB-

n

w

v

x

-RRB-

for

all

w

and

v

by

the

same

amount

More

precisely

we

define

at

each

stage

s

the

integer

value

Lhs

-LRB-

n

-RRB-

such

that

for

every

w

v

dom

-LRB-

x

zs

-LRB-

n

w

v

x

-RRB-

-RRB-

Lhs

-LRB-

n

-RRB-

We

also

use

the

sets

DIAGs

-LRB-

n

w

v

-RRB-

fi

Nj

zs

-LRB-

n

w

v

x

-RRB-

i

-LRB-

x

-RRB-

for

some

x

Lhs

-LRB-

n

-RRB-

g

We

permanently

focus

on

a

pair

of

naturals

ACT

IV

Es

-LRB-

j

k

-RRB-

called

the

active

pair

with

the

intention

to

fulfill

R

-LRB-

j

-RRB-

However

if

at

some

stage

we

discover

an

index

i

with

i

j

k

such

that

Q

-LRB-

i

-RRB-

can

be

satisfied

we

prefer

to

do

it

-LRB-

this

is

a

simple

form

of

the

priority

method

-RRB-

When

R

-LRB-

j

-RRB-

is

satisfied

the

next

pair

of

naturals

in

a

standard

ordering

of

becomes

the

new

active

pair

The

construction

of

z

-LRB-

n

w

v

x

-RRB-

Stage

s

Put

z

-LRB-

n

w

v

x

-RRB-

for

all

n

w

v

x

Lh

-LRB-

n

-RRB-

for

every

n

DIAG

-LRB-

n

w

v

-RRB-

for

all

n

w

v

ACT

IV

E

-LRB-

-RRB-

Stage

s

Take

m

such

that

m

x

zs

-LRB-

n

x

-RRB-

and

let

-LRB-

j

k

-RRB-

ACT

IV

Es

-LRB-

n

-RRB-

For

x

with

x

Lhs

-LRB-

n

-RRB-

define

zs

-LRB-

n

w

v

x

-RRB-

zs

-LRB-

n

w

v

x

-RRB-

for

each

x

Lhs

-LRB-

n

-RRB-

x

j

f

-LRB-

j

m

-RRB-

j

do

if

x

dom

-LRB-

v

-RRB-

then

zs

-LRB-

n

w

v

x

-RRB-

v

-LRB-

x

-RRB-

else

if

there

exists

i

such

that

-RRB-

i

j

k

-RRB-

i

DIAGs

-LRB-

n

-RRB-

-RRB-

w

i

x

-RRB-

i

-LRB-

x

-RRB-

n

-LRB-

i

x

-RRB-

-LRB-

Conditions

-RRB-

-RRB-

will

be

further

on

denoted

as

the

Test

-RRB-

The

Test

is

checked

in

increasing

order

of

i

in

the

range

w

w

min

-LRB-

x

j

k

-RRB-

If

for

some

i

as

above

the

Test

can

not

be

evaluated

-LRB-

because

n

-LRB-

i

x

-RRB-

is

not

defined

-RRB-

then

of

course

zs

-LRB-

n

w

v

x

-RRB-

is

not

defined

and

the

procedure

loops

forever

-RRB-

then

-LRB-

Satisfy

a

bit

of

Q

-LRB-

i

-RRB-

-RRB-

choose

the

least

such

i

zs

-LRB-

n

w

v

x

-RRB-

i

-LRB-

x

-RRB-

g

DIAGs

-LRB-

n

w

v

-RRB-

DIAGs

-LRB-

n

w

v

-RRB-

-LSB-

fig

else

-LRB-

Satisfy

R

-LRB-

j

-RRB-

-RRB-

zs

-LRB-

n

w

v

x

-RRB-

f

-LRB-

j

m

-RRB-

-LRB-

x

-RRB-

end

if

end

for

if

for

all

x

-LRB-

Lhs

-LRB-

n

-RRB-

x

j

f

-LRB-

j

m

-RRB-

j

no

i

satisfies

the

Test

or

x

dom

-LRB-

v

-RRB-

-RRB-

then

ACT

IV

Es

next

-LRB-

ACT

IV

Es

-RRB-

else

ACT

IV

Es

ACT

IV

Es

end

if

DIAGs

-LRB-

n

w

v

-RRB-

DIAGs

-LRB-

n

w

v

-RRB-

Lhs

-LRB-

n

-RRB-

j

f

-LRB-

j

m

-RRB-

j

End

of

construction

We

denote

z

-LRB-

n

w

v

x

-RRB-

lims

zs

-LRB-

n

w

v

x

-RRB-

-LRB-

the

limit

exists

by

the

way

the

function

zs

extends

zs

for

each

s

-RRB-

Let

t

be

a

recursive

function

such

that

t

-LRB-

n

w

v

-RRB-

-LRB-

x

-RRB-

z

-LRB-

n

w

v

x

-RRB-

Now

we

turn

to

the

definition

of

functions

-LRB-

pi

-RRB-

i

First

let

be

the

p.r.

function

defined

by

the

following

clauses

ffl

-LRB-

n

i

x

-RRB-

if

x

i

or

there

exists

m

i

with

n

-LRB-

m

-RRB-

x

ffl

-LRB-

n

i

x

-RRB-

maxfF

-LRB-

t

-LRB-

n

i

v

-RRB-

-RRB-

-LRB-

x

-RRB-

jv

xg

if

the

first

condition

fails

to

hold

but

n

-LRB-

j

y

-RRB-

is

defined

for

all

y

x

and

all

i

j

x

ffl

-LRB-

n

i

x

-RRB-

in

the

remaining

situations

By

the

Recursion

Theorem

there

exists

such

that

-LRB-

u

-RRB-

-LRB-

u

-RRB-

for

all

u.

Fix

such

an

and

denote

pi

-LRB-

x

-RRB-

-LRB-

i

x

-RRB-

the

complexity

of

pi

-LRB-

x

-RRB-

is

then

-LRB-

i

x

-RRB-

One

has

pi

-LRB-

x

-RRB-

if

x

i

or

there

exists

m

i

with

-LRB-

m

-RRB-

x

maxfF

-LRB-

t

-LRB-

i

v

-RRB-

-RRB-

-LRB-

x

-RRB-

jv

xg

if

pj

-LRB-

y

-RRB-

is

defined

for

all

y

x

and

all

j

i

j

x

otherwise

-LRB-

-RRB-

We

continue

our

proof

with

a

series

of

intermediate

results

Fact

For

all

natural

i

x

-LRB-

i

x

-RRB-

is

defined

i.e.

pi

is

recursive

for

all

i.

Proof

We

show

first

that

-LRB-

x

-RRB-

is

defined

for

all

x

i.e.

is

recursive

Suppose

there

exists

w

such

that

-LRB-

w

-RRB-

is

undefined

and

so

is

-LRB-

w

-RRB-

Hence

-LRB-

w

-RRB-

x

Then

for

all

i

w

pi

-LRB-

x

-RRB-

for

all

x

by

the

first

clause

in

the

definition

of

pi

We

retain

that

pi

is

recursive

for

all

i

w

It

follows

that

for

all

v

x

z

-LRB-

w

v

x

-RRB-

or

t

-LRB-

w

v

-RRB-

are

recursive

Indeed

this

is

a

consequence

of

the

fact

that

condition

in

the

Test

can

be

effectively

checked

-LRB-

because

-LRB-

i

x

-RRB-

pi

-LRB-

x

-RRB-

is

defined

for

all

i

w

-RRB-

and

if

needed

i

-LRB-

x

-RRB-

can

be

computed

in

order

to

fulfill

Q

-LRB-

i

-RRB-

From

-LRB-

-RRB-

we

notice

that

pw

-LRB-

x

-RRB-

is

defined

for

all

x

In

the

same

manner

we

deduce

that

pw

pw

are

recursive

so

is

recursive

Next

we

observe

that

-LRB-

x

-RRB-

is

defined

by

the

second

clause

for

almost

every

x

which

implies

that

for

all

j

pj

is

recursive

Fact

For

all

v

pi

-LRB-

x

-RRB-

F

-LRB-

t

-LRB-

i

v

-RRB-

-RRB-

-LRB-

x

-RRB-

a.e.

x.

Proof

Since

pi

is

recursive

it

follows

that

pi

is

defined

by

the

second

clause

for

almost

every

x

The

conclusion

follows

Fact

For

all

i

there

exists

v

such

that

t

-LRB-

i

v

-RRB-

t

-LRB-

-RRB-

Proof

There

exists

a

stage

s

when

DIAGs

-LRB-

-RRB-

i

-LSB-

t

DIAGt

-LRB-

-RRB-

i

Take

v

such

that

v

x

zs

-LRB-

x

-RRB-

For

x

j

vj

one

has

t

-LRB-

i

v

-RRB-

-LRB-

x

-RRB-

v

-LRB-

x

-RRB-

zs

-LRB-

x

-RRB-

t

-LRB-

-RRB-

-LRB-

x

-RRB-

if

x

j

vj

then

t

-LRB-

i

v

-RRB-

-LRB-

x

-RRB-

t

-LRB-

-RRB-

-LRB-

x

-RRB-

by

the

construction

of

zs

Indeed

t

-LRB-

i

v

-RRB-

and

t

-LRB-

-RRB-

could

differ

only

because

at

some

stage

t

the

procedures

computing

them

satisfy

Q

-LRB-

h

-RRB-

and

respectively

Q

-LRB-

-RRB-

with

h

However

after

stage

s

the

procedure

computing

t

-LRB-

-RRB-

does

not

satisfy

any

Q

-LRB-

-RRB-

with

i

-LRB-

by

the

definition

of

s

-RRB-

It

follows

-LRB-

from

our

strategy

according

to

which

stage

s

of

the

construction

of

x

zs

-LRB-

n

w

v

x

-RRB-

uses

information

related

to

x

zs

-LRB-

n

x

-RRB-

namely

ACT

IV

Es

-LRB-

n

-RRB-

and

DIAGs

-LRB-

n

-RRB-

-RRB-

that

at

each

stage

t

s

whenever

a

condition

Q

-LRB-

-RRB-

is

satisfied

the

procedures

computing

t

-LRB-

i

v

-RRB-

and

t

-LRB-

-RRB-

select

the

same

value

h

for

satisfying

Q

-LRB-

h

-RRB-

In

conclusion

after

stage

s

the

procedures

computing

t

-LRB-

i

v

-RRB-

and

t

-LRB-

-RRB-

produce

the

same

output

in

spite

of

the

fact

that

the

procedure

for

t

-LRB-

-RRB-

spends

exceedingly

more

resources

-LRB-

in

fact

this

is

the

source

of

speed-up

-RRB-

Fact

Define

the

recursive

function

g

by

g

-LRB-

x

-RRB-

t

-LRB-

-RRB-

-LRB-

x

-RRB-

x

N

For

every

natural

j

there

exist

infinitely

many

m

such

that

f

-LRB-

j

m

-RRB-

v

g

i.e.

R

-LRB-

j

-RRB-

is

satisfied

for

every

j.

Proof

We

show

first

that

every

pair

of

naturals

-LRB-

j

k

-RRB-

becomes

eventually

the

active

pair

in

the

algorithm

computing

t

-LRB-

-RRB-

x

z

-LRB-

-RRB-

We

proceed

by

induction

on

j

k

For

j

k

the

statement

clearly

holds

-LRB-

see

stage

s

-RRB-

There

are

only

finitely

many

naturals

i

j

k

which

can

satisfy

the

Test

and

once

such

an

i

has

satisfied

the

Test

it

will

not

attempt

doing

this

any

more

since

i

is

inserted

in

DIAG

Eventually

no

such

i

is

found

so

the

construction

satisfies

R

-LRB-

j

-RRB-

and

takes

the

next

pair

of

naturals

as

the

active

pair

Consider

now

the

stage

s

when

R

-LRB-

j

-RRB-

is

satisfied

and

the

value

m

as

defined

at

this

stage

It

is

easy

to

see

that

zs

-LRB-

x

-RRB-

zs

-LRB-

x

-RRB-

m

-LRB-

x

-RRB-

f

-LRB-

j

m

-RRB-

-LRB-

x

-RRB-

for

all

x

with

x

Lhs

-LRB-

-RRB-

Moreover

the

corresponding

values

m

for

the

active

pair

-LRB-

j

k

-RRB-

are

distinct

Since

at

stage

s

R

-LRB-

j

-RRB-

is

satisfied

we

conclude

that

g

t

-LRB-

-RRB-

-LRB-

-RRB-

z

-LRB-

-RRB-

w

zs

-LRB-

-RRB-

f

-LRB-

j

m

-RRB-

i.o.

m

Fact

If

g

i

then

i

-LRB-

x

-RRB-

pi

-LRB-

x

-RRB-

a.e.

x.

Proof

There

is

a

stage

s

such

that

from

that

moment

on

all

active

pairs

-LRB-

j

k

-RRB-

satisfy

j

k

i

and

all

Q

-LRB-

-RRB-

for

i

have

been

satisfied

In

case

i

DIAGs

-LRB-

-RRB-

i

was

already

diagonalized

Hence

i

z

-LRB-

-RRB-

g

a

contradiction

So

i

DIAGs

-LRB-

-RRB-

Suppose

now

i

-LRB-

x

-RRB-

pi

-LRB-

x

-RRB-

-LRB-

i

x

-RRB-

for

some

x

Lhs

-LRB-

n

-RRB-

There

exists

a

stage

t

s

such

that

Lht

-LRB-

n

-RRB-

x

Lhs

-LRB-

n

-RRB-

It

follows

that

at

stage

t

the

relation

zt

-LRB-

-RRB-

i

-LRB-

x

-RRB-

is

realized

and

hence

g

t

-LRB-

-RRB-

z

-LRB-

-RRB-

i

again

a

contradiction

We

finish

the

proof

of

the

theorem

by

showing

that

g

SPEED

Indeed

let

i

g

Then

i

-LRB-

x

-RRB-

pi

-LRB-

x

-RRB-

F

-LRB-

t

-LRB-

i

v

-RRB-

-RRB-

-LRB-

x

-RRB-

a.e.

x

for

all

v

N

The

first

inequality

comes

from

Fact

and

the

second

one

from

-LRB-

-RRB-

Take

v

such

that

t

-LRB-

i

v

-RRB-

t

-LRB-

-RRB-

and

j

t

-LRB-

i

v

-RRB-

Then

j

g

and

i

-LRB-

x

-RRB-

F

-LRB-

j

-RRB-

-LRB-

x

-RRB-

a.e.

x.

Let

g

be

an

increasing

recursive

function

and

define

the

set

HARD

-LRB-

g

-RRB-

ff

Rj

for

every

i

f

i

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

a.e.

xg

of

all

recursive

functions

requiring

at

least

g

-LRB-

x

-RRB-

complexity

An

well

known

result

due

to

Rabin

-LSB-

-RSB-

-LRB-

see

also

Calude

-LSB-

-RSB-

-RRB-

asserts

that

HARD

-LRB-

g

-RRB-

is

non-empty

Here

we

can

easily

derive

a

stronger

result

Theorem

For

every

increasing

recursive

function

g

HARD

-LRB-

g

-RRB-

is

of

the

second

category

Proof

Consider

the

total

effective

operator

F

defined

by

F

-LRB-

i

-RRB-

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

for

all

i

x

N

and

apply

Theorem

to

F

as

SPEED

-LRB-

F

-RRB-

HARD

-LRB-

g

-RRB-

It

is

common

to

derive

the

Speed-up

Theorem

-LRB-

and

its

Operator

variant

-RRB-

by

using

a

complexity

sequence

and

the

so

called

Fundamental

Theorem

of

Computational

Complexity

-LRB-

see

Meyer

and

Winklman

-LSB-

-RSB-

Seiferas

-LSB-

-RSB-

Seiferas

and

Meyer

-LSB-

-RSB-

-RRB-

A

sequence

of

recursive

functions

is

a

complexity

sequence

for

a

recursive

function

f

if

ffl

for

all

i

with

i

f

there

exists

j

such

that

pj

-LRB-

x

-RRB-

i

-LRB-

x

-RRB-

a.e.

x

and

ffl

for

all

i

there

exists

j

with

j

f

and

j

-LRB-

x

-RRB-

pi

-LRB-

x

-RRB-

a.e.

x

The

Fundamental

Theorem

states

that

each

sequence

of

recursive

functions

satisfying

some

very

weak

conditions

is

the

complexity

sequence

of

some

recursive

function

For

each

total

effective

operator

F

one

can

find

a

sequence

of

recursive

functions

such

that

for

all

i

F

-LRB-

pi

-RRB-

-LRB-

x

-RRB-

F

-LRB-

pi

-RRB-

-LRB-

x

-RRB-

a.e.

x

It

follows

immediately

that

the

function

f

for

which

the

sequence

is

a

complexity

sequence

is

F

speedable

It

is

natural

to

ask

if

Theorem

-LRB-

which

in

fact

is

a

stronger

version

of

the

Operator

Speed-Up

Theorem

since

it

asserts

the

existence

of

many

f-speedable

functions

-RRB-

can

also

be

obtained

in

this

way

The

answer

is

negative

Proposition

Let

be

a

sequence

of

recursive

functions

Then

the

set

A

ff

is

a

complexity

sequence

for

fg

is

of

the

first

category

Proof

The

set

A

is

included

in

the

union

S

i

j

A

i

j

k

where

A

i

j

k

ae

j

if

j

-LRB-

x

-RRB-

pi

-LRB-

x

-RRB-

for

all

x

with

x

k

otherwise

Each

set

A

i

j

k

is

nowhere

dense

via

the

witness

function

f

i

j

k

which

on

input

v

acts

as

follows

ffl

If

jvj

k

then

f

i

j

k

-LRB-

v

-RRB-

f

i

j

k

-LRB-

-RRB-

ffl

If

jvj

k

and

j

-LRB-

jvj

-RRB-

pi

-LRB-

jvj

-RRB-

then

f

i

j

k

-LRB-

v

-RRB-

-LRB-

In

this

case

A

i

j

k

and

there

is

nothing

to

worry

about

-RRB-

ffl

If

jvj

k

and

j

-LRB-

jvj

-RRB-

pi

-LRB-

jvj

-RRB-

then

f

i

j

k

-LRB-

v

-RRB-

vy

where

y

j

-LRB-

jvj

-RRB-

g.

-LRB-

In

this

case

we

diagonalize

over

j

insuring

that

Uf

i

j

k

-LRB-

v

-RRB-

A

i

j

k

:-RRB-

Since

the

functions

f

i

j

k

are

obtained

in

an

uniform

way

from

i

j

k

the

conclusion

follows

So

Theorem

is

not

a

corollary

of

the

Fundamental

Theorem

as

is

the

case

of

the

Operator

Speed-Up

Theorem

It

is

known

that

the

Speed-Up

Theorem

is

ineffective

in

many

aspects

Blum

has

shown

in

-LSB-

-RSB-

that

for

no

speedable

function

i

can

one

find

algorithmically

from

i

the

faster

program

Although

for

some

speedable

functions

it

is

possible

to

recursively

bound

the

size

of

the

index

of

the

faster

program

-LRB-

see

Meyer

and

Fischer

-LSB-

-RSB-

Helm

and

Young

-LSB-

-RSB-

-RRB-

Schnorr

has

shown

in

-LSB-

-RSB-

that

it

is

not

possible

for

any

speedable

function

to

simultaneosly

bound

the

size

of

the

faster

program

and

the

threshold

value

starting

from

which

the

faster

program

is

indeed

faster

A

recent

result

due

to

Bridges

and

Calude

-LSB-

-RSB-

states

that

in

general

there

is

no

recursive

function

of

the

initial

index

that

gives

a

bound

for

the

exceptional

values

on

speed-up

but

that

if

the

bounding

function

is

taken

as

a

function

of

the

speed-up

index

then

it

can

be

chosen

to

be

recursive

The

topological

analysis

of

the

Speed-Up

Theorem

easily

yields

another

facet

of

this

phenomenon

Given

any

sound

formal

system

it

is

not

possible

to

detect

with

the

exception

of

a

tiny

meager

set

of

functions

that

a

function

is

speedable

Proposition

Let

T

be

any

sound

formal

system

and

F

a

total

effective

operator

There

exists

a

function

h

SPEED

SPEED

-LRB-

F

-RRB-

such

that

for

each

machine

M

computing

h

the

sentence

The

function

computed

by

M

belongs

to

SPEED

is

not

a

theorem

of

T

Moreover

the

set

of

such

functions

h

is

of

the

second

category

Proof

Suppose

there

exists

a

formal

system

T

such

that

for

all

h

SPEED

there

exists

a

Turing

machine

m

computing

f

and

such

that

the

sentence

The

function

computed

by

M

belongs

to

SPEED

is

a

theorem

of

T

By

extracting

from

the

theorems

of

T

the

ones

having

the

above

form

we

obtain

an

r.e.

sequence

of

machines

Mi

such

that

fh

PRj

is

computed

by

some

Mig

SPEED

The

set

Ai

fh

PRj

is

computed

by

some

Mig

is

nowhere

dense

via

the

witness

function

fi

which

is

defined

by

fi

-LRB-

v

-RRB-

vy

where

y

h

-LRB-

jvj

-RRB-

g

It

follows

that

SPEED

is

meager

which

contradicts

Theorem

The

second

-LRB-

stronger

-RRB-

assertion

follows

also

by

the

same

reasoning

Gap

and

Compression

A

natural

problem

in

computational

complexity

is

to

investigate

to

what

extent

by

allocating

more

resources

we

get

more

computational

power

If

F

is

a

total

effective

operator

such

that

F

-LRB-

f

-RRB-

is

much

bigger

than

f

are

we

guaranteed

that

C?f

ae

C?F

-LRB-

f

-RRB-

The

Operator

Gap

Theorem

-LRB-

see

Constable

-LSB-

-RSB-

Calude

-LSB-

-RSB-

and

Young

-LSB-

-RSB-

-RRB-

gives

a

surprisingly

negative

result

to

this

question

There

are

functions

f

such

that

C?f

C?F

-LRB-

f

-RRB-

Our

next

result

shows

that

there

are

many

such

functions

Theorem

Let

F

be

a

total

effective

operator

such

that

for

all

i

and

x

F

-LRB-

i

-RRB-

-LRB-

x

-RRB-

i

-LRB-

x

-RRB-

Then

the

set

GAP

-LRB-

F

-RRB-

ft

R

j

C?t

C?F

-LRB-

t

-RRB-

g

is

of

second

category

Proof

Suppose

by

contradiction

that

GAP

-LRB-

F

-RRB-

S

j

GAPj

and

there

exists

a

recursive

function

f

such

that

for

all

natural

j

i

-RRB-

n

f

-LRB-

j

n

-RRB-

ii

-RRB-

GAPj

U

f

-LRB-

j

n

-RRB-

for

sufficiently

large

n

We

construct

a

function

t

GAP

-LRB-

F

-RRB-

such

that

for

all

j

there

are

infinitely

many

n

with

f

-LRB-

j

n

-RRB-

v

t

It

will

follow

that

for

some

i

t

GAPi

U

f

-LRB-

i

n

-RRB-

i.o.

n

a

contradiction

By

Kreisel-Lacombe-Shoenfield

's

Theorem

there

exists

a

recursive

function

g

N

N

such

that

for

every

recursive

i

Graph

-LRB-

F

-LRB-

i

-RRB-

-RRB-

-LSB-

Graph

-LRB-

g

-LRB-

j

-RRB-

-RRB-

where

Ci

fj

N

j

Graph

-LRB-

j

-RRB-

Graph

-LRB-

i

-RRB-

g

The

function

t

will

be

defined

in

stages

At

stage

s

we

construct

a

finite

initial

segment

ts

of

t

and

keep

track

of

the

value

Lhs

such

that

dom

-LRB-

ts

-RRB-

Lhs

Construction

of

t.

Stage

s

Put

-LRB-

-RRB-

Stage

s

Let

s

i

k

-LRB-

The

pair

-LRB-

i

k

-RRB-

acts

like

the

active

pair

in

the

previous

proofs

-RRB-

Let

m

such

that

ts

m

We

put

ts

-LRB-

x

-RRB-

ts

-LRB-

x

-RRB-

for

x

Lhs

and

ts

-LRB-

x

-RRB-

f

-LRB-

i

m

-RRB-

for

Lhs

x

j

f

-LRB-

i

m

-RRB-

j

-LRB-

Note

that

we

have

insured

that

f

-LRB-

i

m

-RRB-

v

ts

-RRB-

Next

we

proceed

like

in

Young

's

proof

of

the

Operator

Gap

Theorem

-LRB-

Young

-LSB-

-RSB-

Calude

-LSB-

-RSB-

-RRB-

We

define

s

recursive

extensions

of

the

function

ts

defined

so

far

namely

t

-LRB-

s

-RRB-

t

-LRB-

s

-RRB-

t

-LRB-

-RRB-

N

N

in

the

following

way

t

-LRB-

s

-RRB-

-LRB-

z

-RRB-

ae

ts

-LRB-

z

-RRB-

if

z

j

f

-LRB-

j

n

-RRB-

j

otherwise

and

for

i

s

s

-LRB-

in

this

order

-RRB-

t

-LRB-

i

-RRB-

-LRB-

z

-RRB-

ae

ts

-LRB-

z

-RRB-

if

z

j

f

-LRB-

j

n

-RRB-

j

F

-LRB-

t

-LRB-

i

-RRB-

-RRB-

-LRB-

z

-RRB-

otherwise

Next

we

construct

the

following

-LRB-

x

-RRB-

finite

initial

segment

functions

u

-LRB-

s

-RRB-

u

-LRB-

s

-RRB-

u

-LRB-

-RRB-

u

-LRB-

i

-RRB-

-LRB-

z

-RRB-

ae

t

-LRB-

i

-RRB-

-LRB-

z

-RRB-

if

z

Ai

otherwise

for

every

i

sg

Here

j

f

-LRB-

j

n

-RRB-

j

and

Ai

min

-LSB-

z

Ai

and

Graph

-LRB-

u

-LRB-

i

-RRB-

-RRB-

-LSB-

i

Graph

-LRB-

g

-LRB-

y

-RRB-

-RRB-

-RSB-

where

Bz

i

fy

N

j

y

-LRB-

n

-RRB-

t

-LRB-

i

-RRB-

-LRB-

n

-RRB-

for

every

n

zg

-LRB-

Recall

that

g

is

the

function

corresponding

to

the

operator

F

by

Kreisel-Lacombe-Shoenfield

's

Theorem

-RRB-

The

idea

in

defining

u

-LRB-

i

-RRB-

from

t

-LRB-

i

-RRB-

is

to

retain

enough

information

such

that

if

N

N

is

an

arbitrary

extension

of

u

-LRB-

i

-RRB-

then

F

-LRB-

-RRB-

-LRB-

z

-RRB-

t

-LRB-

i

-RRB-

-LRB-

z

-RRB-

for

every

z

such

that

z

Ai

-LRB-

Observe

that

F

-LRB-

t

-LRB-

i

-RRB-

-RRB-

-LRB-

z

-RRB-

t

-LRB-

i

-RRB-

-LRB-

z

-RRB-

for

every

z

-RRB-

For

all

j

s

and

i

sg

we

say

that

u

-LRB-

i

-RRB-

is

unsafe

for

j

if

i

-RRB-

u

-LRB-

i

-RRB-

-LRB-

z

-RRB-

j

-LRB-

z

-RRB-

for

some

z

with

Lhs

z

Ai

and

ii

-RRB-

i

or

-LRB-

j

-LRB-

z

-RRB-

u

-LRB-

i

-RRB-

-LRB-

z

-RRB-

i

-RRB-

for

all

z

with

Lhs

z

Ai

Keeping

in

mind

that

Lhs

As

and

u

-LRB-

i

-RRB-

-LRB-

z

-RRB-

u

-LRB-

i

-RRB-

-LRB-

z

-RRB-

for

all

z

with

Lhs

z

Ai

for

all

i

sg

one

deduce

that

for

every

j

s

at

most

one

u

-LRB-

i

-RRB-

is

unsafe

for

j

Since

there

are

s

such

extensions

u

-LRB-

i

-RRB-

at

least

one

of

them

is

safe

for

all

j

s

and

such

an

u

-LRB-

i

-RRB-

can

be

found

in

a

recursive

way

We

extend

ts

to

a

certain

u

-LRB-

-RRB-

which

is

safe

for

every

j

s

and

set

Lhs

End

of

stage

s.

Finally

let

t

lims

ts

End

of

construction

of

t.

For

each

i

there

are

infinitely

many

k

such

that

f

-LRB-

i

k

-RRB-

v

t

since

at

each

step

s

i

k

we

make

f

-LRB-

i

k

-RRB-

v

ts

and

ts

v

t

It

remains

to

show

that

t

GAP

-LRB-

F

-RRB-

Suppose

there

exists

j

such

that

j

-LRB-

x

-RRB-

F

-LRB-

t

-RRB-

-LRB-

x

-RRB-

a.e.

x

but

j

-LRB-

x

-RRB-

t

-LRB-

x

-RRB-

i.o.

x

There

exists

a

stage

s

j

such

that

j

-LRB-

x

-RRB-

F

-LRB-

t

-RRB-

-LRB-

x

-RRB-

for

all

x

with

Lhs

x

Lhs

and

j

-LRB-

x

-RRB-

t

-LRB-

x

-RRB-

for

some

x

with

Lhs

x

Lhs

This

contradicts

the

choice

of

a

safe

initial

segment

at

stage

s.

Indeed

if

Lhs

x

Lhs

then

t

-LRB-

x

-RRB-

u

-LRB-

-RRB-

-LRB-

x

-RRB-

where

u

-LRB-

-RRB-

is

the

safe

segment

selected

at

stage

s

If

then

u

-LRB-

-RRB-

is

a

safe

segment

if

and

only

if

u

-LRB-

-RRB-

-LRB-

x

-RRB-

j

-LRB-

x

-RRB-

for

all

x

with

Lhs

z

Lhs

If

then

for

all

x

F

-LRB-

t

-RRB-

-LRB-

x

-RRB-

F

-LRB-

u

-LRB-

-RRB-

-RRB-

-LRB-

x

-RRB-

u

-LRB-

-RRB-

-LRB-

x

-RRB-

u

-LRB-

-RRB-

-LRB-

x

-RRB-

and

Lhs

So

our

assumptions

on

j

would

imply

that

u

-LRB-

-RRB-

is

not

safe

The

Compression

Theorem

-LSB-

-RSB-

assures

us

that

for

nice

families

of

functions

T

one

can

indeed

get

more

computational

power

by

raising

the

resource

bound

from

t

-LRB-

x

-RRB-

to

g

-LRB-

x

t

-LRB-

x

-RRB-

-RRB-

for

an

appropriate

g

and

for

any

t

T

Nice

means

here

a

measured

set

i.e.

a

r.e.

set

of

partial

functions

i

N

o

N

for

which

the

ternary

predicate

i

-LRB-

n

-RRB-

m

is

recursive

The

Compression

Theorem

states

that

if

-LRB-

i

-RRB-

is

a

measured

set

then

there

exist

two

recursive

function

g

N

and

k

N

N

such

that

for

each

recursive

i

i

-RRB-

if

j

k

-LRB-

i

-RRB-

then

j

-LRB-

x

-RRB-

i

-LRB-

x

-RRB-

a.e.

x

and

ii

-RRB-

k

-LRB-

i

-RRB-

-LRB-

x

-RRB-

g

-LRB-

x

i

-LRB-

x

-RRB-

-RRB-

a.e.

x

We

would

like

to

know

what

properties

should

a

set

of

functions

have

in

order

to

possess

a

Compression

Theorem

-LRB-

being

a

measured

set

is

probably

not

a

necessary

condition

-RRB-

It

is

known

from

Calude

-LSB-

-RSB-

-LRB-

see

also

Calude

-LSB-

-RSB-

-RRB-

that

every

measured

set

is

meager

-LRB-

In

-LSB-

-RSB-

the

result

is

shown

in

a

different

topology

the

superset

topology

which

is

adequate

for

investigating

the

size

of

sets

of

partial

recursive

functions

However

the

proof

from

-LSB-

-RSB-

can

easily

be

adapted

for

the

Cantor

topology

-RRB-

Our

next

result

shows

that

the

meagerness

of

a

measured

set

seems

to

be

essential

and

gives

a

partial

answer

to

the

above

question

If

we

fix

the

increasing

factor

g

and

consider

a

second

category

set

A

of

recursive

functions

then

the

compression

property

does

not

hold

for

any

f

in

A.

Proposition

Let

g

N

be

a

recursive

function

Then

the

set

Ag

fs

R

j

there

exists

an

i

with

i

-LRB-

x

-RRB-

g

-LRB-

x

s

-LRB-

x

-RRB-

-RRB-

a.e.

x

and

for

all

j

with

j

i

one

has

j

-LRB-

x

-RRB-

s

-LRB-

x

-RRB-

a.e.

xg

is

meager

Proof

We

start

noticing

that

Ag

-LSB-

i

k

A

i

k

where

A

i

k

fs

R

j

i

-LRB-

x

-RRB-

g

-LRB-

x

s

-LRB-

x

-RRB-

-RRB-

and

i

-LRB-

x

-RRB-

s

-LRB-

x

-RRB-

a.e.

xg

The

set

A

i

k

is

nowhere

dense

via

the

following

witness

function

f

i

k

N

N

If

v

N

has

length

less

than

k

then

we

-LRB-

arbitrarily

-RRB-

put

f

i

k

-LRB-

v

-RRB-

If

jvj

k

then

let

x

jvj

ffl

If

i

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

put

f

i

k

-LRB-

v

-RRB-

vy

where

y

i

-LRB-

x

-RRB-

-LRB-

In

this

way

if

f

i

k

-LRB-

v

-RRB-

is

a

prefix

of

s

then

i

-LRB-

x

-RRB-

s

-LRB-

x

-RRB-

-RRB-

ffl

If

i

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

put

f

i

k

-LRB-

v

-RRB-

-LRB-

In

this

way

if

f

i

k

-LRB-

v

-RRB-

is

a

prefix

of

s

then

i

-LRB-

x

-RRB-

g

-LRB-

x

s

-LRB-

x

-RRB-

-RRB-

-RRB-

It

is

clear

from

the

above

remarks

that

for

any

v

N

with

jvj

k

Uf

i

k

-LRB-

v

-RRB-

A

i

k

and

consequently

Ag

is

meager

It

is

not

hard

to

prove

that

any

function

in

a

measured

set

is

r-honest

for

an

appropriate

recursive

function

r.

-LRB-

We

recall

that

in

a

Blum

space

-LRB-

i

i

-RRB-

a

recursive

function

f

is

r-honest

if

there

is

an

i

such

that

i

f

and

i

-LRB-

x

-RRB-

r

-LRB-

x

f

-LRB-

x

-RRB-

-RRB-

a.e.

x

here

r

N

is

a

recursive

function

-RRB-

Consequently

the

functions

in

the

Compression

Theorem

are

all

honest

It

is

also

well

known

that

the

classical

time

and

space

hierarchy

theorems

for

Turing

machines

-LRB-

see

Hopcroft

and

Ullman

-LSB-

-RSB-

-RRB-

require

time

and

respectively

space

constructibility

of

the

involved

functions

which

is

a

strong

variant

of

honesty

One

may

suspect

that

functions

defining

the

gaps

in

the

anti-hierarchy

Operator

Gap

theorem

are

not

honest

Indeed

we

can

prove

that

many

of

them

are

not

honest

in

an

extremely

strong

sense

Fix

a

uniform

sequence

of

Blum

spaces

-LRB-

-LRB-

i

-RRB-

-LRB-

i

-RRB-

-RRB-

in

the

sense

that

there

exists

a

recursive

function

V

N

such

that

V

-LRB-

i

x

-RRB-

-LRB-

i

-RRB-

-LRB-

x

-RRB-

for

all

i

and

x.

Proposition

Let

r

g

be

recursive

functions

and

-LRB-

-LRB-

i

-RRB-

-LRB-

i

-RRB-

-RRB-

as

above

Then

the

set

of

functions

in

GAP

-LRB-

g

-RRB-

which

are

not

r-honest

in

any

Blum

space

-LRB-

-LRB-

i

-RRB-

-LRB-

i

-RRB-

-RRB-

is

of

second

category

Proof

By

a

result

of

Mehlhorn

-LSB-

-RSB-

the

class

of

r-honest

functions

is

meager

It

follows

that

the

set

A

ff

PR

j

f

is

r-honest

in

-LRB-

-LRB-

i

-RRB-

-LRB-

i

-RRB-

-RRB-

for

some

natural

ig

is

also

meager

since

it

is

the

uniform

union

of

a

countable

set

of

meager

sets

Since

by

Theorem

GAP

-LRB-

g

-RRB-

is

of

second

category

we

conclude

that

GAP

-LRB-

g

-RRB-

n

A

is

of

second

category

as

well

Effective

Measure

To

the

best

of

our

knowledge

there

has

been

no

prior

investigation

of

the

size

of

objects

in

abstract

computational

complexity

from

an

effective

measure

theoretical

point

of

view

It

is

the

aim

of

this

section

to

illustrate

that

such

a

study

is

quite

feasible

In

fact

we

easily

classify

from

the

effective

measure-theoretical

point

of

view

all

classes

of

predicates

that

were

analyzed

in

the

topological

setting

in

the

early

works

of

Mehlhorn

-LSB-

-RSB-

and

Calude

-LSB-

-RSB-

The

recursive

and

resource-bounded

variants

of

measure

theory

have

been

developed

by

Freidzon

-LSB-

-RSB-

Mehlhorn

-LSB-

-RSB-

Schnorr

-LSB-

-RSB-

and

exhaustively

by

Lutz

-LSB-

-RSB-

It

is

important

to

retain

that

this

theory

is

applicable

only

to

functions

i.e.

to

predicates

-LRB-

and

this

is

a

major

drawback

of

effective

measure

when

compared

to

effective

category

for

example

the

size

of

the

class

of

r-honest

functions

can

not

be

analyzed

in

the

measure-theoretical

setting

-RRB-

The

following

definitions

come

from

the

latter

paper

restricted

to

the

necessities

of

the

current

work

For

motivations

and

connections

with

classical

measure

theory

we

direct

the

reader

to

Lutz

-LSB-

-RSB-

Allender

and

Strauss

-LSB-

-RSB-

and

to

Mayordomo

-LSB-

-RSB-

Let

and

be

the

sets

of

finite

and

respectively

infinite

binary

strings

and

let

denote

the

empty

string

If

w

then

w

can

be

identified

with

a

function

in

FPRED

which

for

all

i

jwj

maps

i

into

the

-LRB-

i

-RRB-

th

bit

of

w

The

basic

neighborhoods

Uw

are

defined

as

in

Section

-LRB-

in

the

topology

of

p.r.

predicates

-RRB-

Definition

-LRB-

i

-RRB-

A

density

function

is

a

function

d

-LSB-

-RRB-

satisfying

d

-LRB-

w

-RRB-

d

-LRB-

-RRB-

d

-LRB-

-RRB-

for

all

strings

w

-LRB-

ii

-RRB-

The

global

value

of

a

density

function

d

is

d

-LRB->=-RRB-

The

set

covered

by

a

density

function

d

is

S

-LSB-

d

-RSB-

-LSB-

;d

-LRB-

w

-RRB-

Uw

A

density

function

d

covers

a

set

X

if

X

S

-LSB-

d

-RSB-

-LRB-

iii

-RRB-

An

density

system

is

a

function

d

N

-LSB-

-RRB-

such

that

for

all

i

N

di

is

a

density

function

where

di

is

defined

by

di

-LRB-

x

-RRB-

d

-LRB-

i

x

-RRB-

Definition

-LRB-

i

-RRB-

A

set

X

has

recursive

measure

zero

if

there

exists

an

recursive

density

system

d

-LRB-

i.e.

d

is

a

recursive

real-valued

function

-RRB-

such

that

for

all

k

dk

covers

X

with

global

value

dk

-LRB->=-RRB-

k.

-LRB-

ii

-RRB-

A

set

X

has

recursive

measure

one

if

the

complement

of

X

has

recursive

measure

zero

The

general

idea

is

to

give

an

effective

touch

to

the

standard

method

in

measure

theory

consisting

in

the

covering

of

sets

by

intervals

This

is

realized

by

using

recursive

martingales

-LRB-

called

here

density

functions

-RRB-

betting

on

the

cylinders

Uw

The

initial

investment

is

the

global

value

Thus

roughly

speaking

a

set

has

recursive

measure

zero

if

there

is

a

recursive

winning

betting

strategy

covering

the

set

that

starts

with

an

arbitrary

small

amount

of

money

In

fact

one

could

weaken

condition

-LRB-

i

-RRB-

in

Definition

by

requiring

that

the

density

system

has

only

arbitrarily

close

recursive

approximations

but

practically

in

all

proofs

the

density

system

is

recursive

itself

A

function

f

RPRED

will

be

identified

with

the

infinite

binary

string

f

-LRB-

-RRB-

f

-LRB-

-RRB-

Lutz

-LSB-

-RSB-

has

shown

that

RPRED

does

not

have

recursive

measure

zero

This

important

feature

makes

a

recursive

measure

meaningful

for

studying

the

size

of

classes

of

functions

in

RPRED

with

respect

to

the

whole

space

RPRED

Definition

-LRB-

i

-RRB-

A

set

X

has

recursive

measure

zero

in

RPRED

if

X

RPRED

has

recursive

measure

zero

-LRB-

ii

-RRB-

A

set

X

has

recursive

measure

one

in

RPRED

if

the

complement

of

X

has

recursive

measure

zero

in

RPRED

The

following

useful

result

is

proved

by

Lutz

-LSB-

-RSB-

A

recursive

union

of

recursive

measure

zero

sets

has

recursive

measure

zero

Formally

let

d

-LSB-

-RRB-

be

a

recursive

function

such

that

for

each

j

dj

is

a

density

system

-LRB-

where

dj

-LRB-

i

x

-RRB-

d

-LRB-

j

i

x

-RRB-

for

all

i

N

and

x

-RRB-

If

X

X

and

for

all

i

N

di

is

a

witness

that

Xi

has

recursive

measure

zero

then

X

has

recursive

measure

zero

The

analogous

result

from

the

point

of

view

of

recursive

measure

theory

holds

too

-LRB-

and

surprisingly

the

proof

is

much

simpler

-RRB-

Theorem

For

every

recursive

function

g

Hard

-LRB-

g

-RRB-

RPRED

has

recursive

measure

one

in

RPRED

Proof

We

have

to

show

that

the

set

HARDc

-LRB-

g

-RRB-

ff

RPRED

j

there

exists

i

f

i

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

i.

o.

xg

has

recursive

measure

zero

Let

-LRB-

-LRB-

i

-RRB-

-LRB-

i

-RRB-

-RRB-

be

a

Blum

space

of

valued

p.r.

functions

It

is

immediate

that

HARDc

-LRB-

g

-RRB-

SHi

where

Hi

ff

RPRED

j

i

fg

if

there

exist

infinitely

many

x

such

that

i

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

otherwise

We

describe

a

recursive

function

d

-LSB-

-RRB-

such

that

di

witnesses

that

Hi

has

efectively

measure

zero

where

di

-LRB-

k

x

-RRB-

d

-LRB-

i

k

x

-RRB-

for

all

i

N

and

x

In

fact

we

define

the

projections

di

k

as

follows

di

k

-LRB->=-RRB-

k

and

inductively

di

k

-LRB-

-RRB-

if

i

-LRB-

jxj

-RRB-

g

-LRB-

jxj

-RRB-

and

i

-LRB-

jxj

-RRB-

k

-LRB-

x

-RRB-

if

i

-LRB-

jxj

-RRB-

g

-LRB-

jxj

-RRB-

and

i

-LRB-

jxj

-RRB-

di

k

-LRB-

x

-RRB-

otherwise

di

k

-LRB-

-RRB-

k

-LRB-

x

-RRB-

if

i

-LRB-

jxj

-RRB-

g

-LRB-

jxj

-RRB-

and

i

-LRB-

jxj

-RRB-

if

i

-LRB-

jxj

-RRB-

g

-LRB-

jxj

-RRB-

and

i

-LRB-

jxj

-RRB-

di

k

-LRB-

x

-RRB-

otherwise

It

can

readily

be

checked

that

di

k

is

a

density

function

Also

by

induction

on

h

one

can

see

that

di

k

-LRB-

x

-RRB-

k

if

and

only

if

there

exist

more

than

h

integers

y

with

y

jxj

and

i

-LRB-

y

-RRB-

g

-LRB-

y

-RRB-

Therefore

it

follows

that

Hi

S

-LSB-

di

k

-RSB-

for

all

k

N.

The

above

proof

also

implies

the

effective

measure-theoretical

analogue

of

the

basic

result

of

Mehlhorn

-LSB-

-RSB-

Calude

-LSB-

-RSB-

stating

that

all

complexity

classes

are

meagre

Indeed

if

-LRB-

-LRB-

i

-RRB-

-LRB-

i

-RRB-

-RRB-

is

a

Blum

space

of

p.r.

functions

and

g

is

a

recursive

function

then

the

complexity

class

defined

by

g

C?g

is

included

in

HARDc

-LRB-

g

-RRB-

the

set

defined

in

the

above

proof

and

shown

to

have

effective

measure

zero

in

RPRED

Therefore

Theorem

Let

-LRB-

-LRB-

i

-RRB-

-LRB-

i

-RRB-

-RRB-

be

a

Blum

space

of

p.r.

functions

and

g

a

recursive

function

Then

C?g

has

effective

measure

zero

in

RPRED

We

pass

to

the

study

of

measured

set

of

predicates

-LRB-

see

the

definition

in

the

previous

section

-RRB-

We

are

interested

in

measured

sets

of

recursive

predicates

i.e.

sets

of

recursive

functions

i

Many

natural

classes

of

recursive

predicates

form

measured

sets

e.g.

the

class

of

primitive

recursive

predicates

every

r.e.

complexity

class

of

predicates

and

in

fact

any

r.e.

set

of

recursive

predicates

It

follows

from

the

next

theorem

that

all

these

classes

-LRB-

and

all

their

subclasses

like

all

levels

in

Grzegorczyk

's

hierarchy

-RRB-

have

effective

measure

zero

Theorem

If

-LRB-

i

-RRB-

is

a

measured

set

of

recursive

predicates

then

has

effective

measure

zero

Proof

We

decompose

S?i

where

i

f

ig

It

is

immediate

to

build

for

each

i

a

density

system

di

witnessing

that

i

has

recursive

measure

zero

Namely

di

k

-LRB->=-RRB-

k

and

di

k

-LRB-

-RRB-

ae

if

i

-LRB-

jxj

-RRB-

k

-LRB-

x

-RRB-

if

i

-LRB-

jxj

-RRB-

di

k

-LRB-

-RRB-

ae

k

-LRB-

x

-RRB-

if

i

-LRB-

jxj

-RRB-

if

i

-LRB-

jxj

-RRB-

Final

Discussion

The

marked

entries

in

Table

constitute

as

many

interesting

open

questions

The

status

of

the

class

of

non-speeadable

functions

remains

open

It

seems

that

the

exact

Baire

classification

of

the

set

of

non-F

speedable

functions

depends

upon

F

We

believe

that

if

F

is

a

fast

growing

function

then

the

corresponding

set

of

non-F

speedable

functions

might

be

of

second

category

This

claim

can

be

proved

for

instance

in

case

we

restrict

the

class

of

witness

functions

to

an

r.e.

class

of

recursive

functions

All

our

results

involved

Cantor

's

topology

which

is

based

on

extension

of

initial

finite

segments

of

functions

With

some

care

the

proofs

can

be

adapted

to

work

for

the

case

of

the

superset

topology

which

is

based

on

extension

of

finite

sets

-LRB-

see

Calude

-LSB-

-RSB-

-RRB-

It

would

be

interesting

to

find

the

most

general

class

of

topologies

for

which

our

results

hold

Such

an

analysis

has

been

pursuited

for

the

case

of

independent

statements

in

a

formal

theory

in

Calude

Jurgensen

and

Zimand

-LSB-

-RSB-

It

is

interesting

to

note

folowing

Marcus

-LSB-

-RSB-

that

in

real

analysis

the

property

of

meagerness

of

a

class

of

functions

is

-LCB-

to

some

extent

-LCB-

conditioned

by

the

degree

of

effectiveness

of

the

definition

of

functions

This

situation

extends

over

all

previous

studied

sets

in

abstract

complexity

theory

and

includes

most

of

the

present

results

Fulk

-LSB-

-RSB-

has

shown

that

every

proof

of

the

existence

of

almost

everywhere

arbitrarily

complex

predicates

should

have

a

nonconstructive

component

The

proof

is

partially

constructive

in

that

it

effectively

gives

a

program

for

the

function

the

nonconstructive

argument

is

used

for

the

verification

that

the

constructed

function

has

the

desired

properties

In

case

of

speedable

functions

the

situation

is

from

the

constructive

point

of

view

worse

The

construction

of

the

function

itself

uses

a

sequence

of

programs

from

which

the

desired

function

is

selected

noneffectively

-LRB-

see

Blum

-LSB-

-RSB-

-RRB-

The

natural

question

is

whether

there

is

a

more

profound

connection

between

non-effectiveness

and

the

property

of

being

topologically

large

Acknowledgement

The

authors

thank

the

anonymous

referee

whose

valuable

comments

led

them

to

get

stronger

results

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compression theorem
effective measure
operator gap theorem

complexity measure
effective baire classification
operator speed-up theorem