Technical

Report

Automated

Reasoning

Project

Research

School

of

Information

Sciences

and

Engineering

and

Centre

for

Information

Science

Research

Australian

National

University

May

Canonicity

for

Intensional

Logics

With

Even

Axioms

Timothy

J.

Surendonk

Abstract

This

paper

looks

at

the

concept

of

neighborhood

canonicity

introduced

by

BRIAN

CHELLAS

-LSB-

-RSB-

We

follow

the

lead

of

the

author

's

paper

-LSB-

-RSB-

where

it

was

shown

that

every

non-iterative

logic

is

neighborhood

canonical

and

here

we

will

show

that

all

logics

whose

axioms

have

a

simple

syntactic

form?no

intensional

operator

is

in

boolean

combination

with

a

propositional

letter?and

which

have

the

nite

model

property

are

neighborhood

canonical

One

consequence

of

this

is

that

KMcK

the

McKinsey

logic

is

nieghborhood

canonical

an

interesting

counterpoint

to

the

results

of

ROBERT

GOLDBLATT

and

XIAOPING

WANG

who

showed

respectively

that

KMcK

is

not

relational

canonical

-LSB-

-RSB-

and

that

KMcK

is

not

relationally

strongly

complete

-LSB-

-RSB-

Canonicity

for

Intensional

Logics

With

Even

Axioms

Timothy

J.

Surendonk

May

Abstract

This

paper

looks

at

the

concept

of

neighborhood

canonicity

introduced

by

BRIAN

CHELLAS

-LSB-

-RSB-

We

follow

the

lead

of

the

author

's

paper

-LSB-

-RSB-

where

it

was

shown

that

every

non-iterative

logic

is

neighborhood

canonical

and

here

we

will

show

that

all

logics

whose

axioms

have

a

simple

syntactic

form?no

intensional

operator

is

in

boolean

combination

with

a

propositional

letter?and

which

have

the

nite

model

property

are

neighborhood

canonical

One

consequence

of

this

is

that

KMcK

the

McKinsey

logic

is

nieghborhood

canonical

an

interesting

counterpoint

to

the

results

of

ROBERT

GOLDBLATT

and

XIAOPING

WANG

who

showed

respectively

that

KMcK

is

not

relational

canonical

-LSB-

-RSB-

and

that

KMcK

is

not

relationally

strongly

complete

-LSB-

-RSB-

Introduction

In

-LSB-

-RSB-

the

author

showed

that

every

non-iterative

intensional

logic

is

neighborhood

canonical

In

that

paper

there

was

no

indications

of

how

one

could

extend

this

result

to

iterative

logics

Of

course

we

did

have

well

known

relational

canonicity

results

for

some

normal

modal

logics

to

immediately

give

us

a

few

results

and

we

also

had

the

elementary

techniques

of

BRIAN

CHELLAS

-LSB-

-RSB-

and

their

re?nements

given

by

ROY

A.

BENTON

-LSB-

-RSB-

which

did

give

us

others

Unfortunately

the

normal

modal

logic

results

shed

no

light

on

the

situation

with

sub

normal

modal

logics

and

the

elementary

techniques

of

CHEL

LAS

and

BENTON

only

worked

for

a

narrow

class

of

arti?cially

simple

logics

Even

the

reults

for

normal

modal

logics

had

gaps

in

them

For

instance

it

is

known

from

ROBERT

GOLDBLATT

's

work

-LSB-

-RSB-

that

the

McKinsey

logic

is

not

relational

canonical

and

this

was

extended

by

XIAOPING

WANG

-LSB-

-RSB-

to

a

result

which

shows

that

the

McKinsey

logic

is

not

relationally

strongly

complete

But

while

relational

canonicity

trivially

implies

neighborhood

canonicity

the

converse

was

open

and

so

we

could

wonder

if

the

McKinsey

logic

is

neighborhood

canonical

In

this

paper

we

advance

our

understanding

of

this

concept

of

neighborhood

canonicity

by

identifying

a

large

class

of

easily

recognised

logics

for

which

the

nite

model

property

will

imply

neighborhood

canonicity

In

fact

this

class

includes

the

so

called

uniform

logics

of

KIT

FINE

-LSB-

-RSB-

and

so

we

will

quickly

get

the

canonicity

of

FINE

's

uniform

class?since

he

showed

that

these

logics

have

the

nite

model

property

As

FINE

points

out

the

McKinsey

logic

is

in

this

class

and

so

we

will

have

demonstrated

that

neighborhood

canonicity

does

not

necessarily

imply

relational

canonicity

As

we

are

dealing

with

neighborhood

canonicity

we

will

of

course

be

dealing

with

classical

logics?which

have

the

property

that

intensions

of

equivalents

are

equivalent?and

their

neighborhood

Montague-Scott

or

minimal

model

semantics

We

will

follow

-LSB-

-RSB-

quite

closely

in

our

development

repreating

without

reference

the

notational

and

background

material

presented

there

Our

solution

to

the

McKinsey

question

above

closes

one

of

the

questions

left

open

in

-LSB-

-RSB-

however

it

highlights

another

question

which

still

remains

open

The

logic

is

in

the

class

about

which

our

result

speaks

yet

its

canonicity

still

remains

open

as

we

are

unsure

as

to

whether

this

logic

has

the

nite

model

property

Before

we

launch

into

the

mathematics

it

is

worth

repeating

the

comment

of

-LSB-

-RSB-

which

justi?es

our

algebraic

rather

than

neighborhood

approach

of

this

paper

While

most

logicians

often

prefer

to

work

with

neighborhood

semantics

there

is

a

close

duality

between

these

and

the

algebraic

semantics

that

are

associated

naturally

with

our

logics

and

it

will

often

pay

to

work

with

the

latter

This

duality

explored

fully

by

KOSTA

DOSEN

-LSB-

-RSB-

is

essentially

the

duality

between

boolean

algebras

and

set

algebras

since

we

have

no

complication

induced

by

the

need

to

decompose

and

regenerate

accessibility

relations

While

the

argument

of

this

paper

can

proceed

within

the

realm

of

neighborhood

semantics

an

uncluttered

presentation

can

be

achieved

by

working

with

the

algebras

whose

equivalence

is

sanctioned

by

the

duality

above

Also

since

our

concentration

is

on

canonicity

we

shall

focus

on

power

set

algebras

which

are

the

algebras

dual

to

full

neighborhood

frames

Basic

De?nitions

Let

us

start

by

xing

our

intensional

language

This

will

be

formed

in

the

usual

way

from

a

countable

set

of

propositional

letters

P

g

the

boolean

connectives

and

a

collection

of

intensional

operators

Here

we

take

I

to

be

some

index

set

and

each

i

to

be

a

-LRB-

distinct

-RRB-

ni-ary

operator

De?ne

S

-LRB-

P

-RRB-

to

be

the

set

of

all

well

formed

formulae

over

this

language

We

will

use

the

Greek

letters

to

represent

formulae

in

our

intensional

language

i.e.

elements

of

S

-LRB-

P

-RRB-

Where

unambiguous

we

will

also

write

to

represent

a

sequence

-LRB-

n

-RRB-

so

i

will

represent

the

intensional

formula

i

-LRB-

n

-RRB-

where

n

ni

As

indicated

our

semantical

structures

are

boolean

algebras

with

operators

Note

that

while

some

authors

take

this

expression

to

mean

that

the

operators

are

additive

we

have

no

such

restriction

De?nition

An

S-algebra

is

a

pair

-LRB-

A

I

-RRB-

where

A

-LRB-

A

-RRB-

is

a

boolean

algebra

and

I

is

a

sequence

of

operators

on

A

with

each

Ii

having

arity

ni

Occasionally

we

will

suppress

the

sequence

I

of

intensions

in

an

S-algebra

-LRB-

A

I

-RRB-

by

refering

to

it

simply

as

A

with

the

double

underscore

indicating

that

it

is

a

full

S-algebra

rather

than

the

underlying

boolean

algebra

In

this

paper

we

will

be

using

the

symbols

both

in

the

language

and

to

denote

operators

in

the

boolean

algebra

A

We

will

even

go

further

to

use

these

symbols

to

denote

intersection

union

complementation

the

whole

set

and

the

emptyset

within

set

algebras

It

is

the

author

's

hope

that

the

reader

will

not

nd

this

confusing

and

will

appreciate

that

this

avoids

the

clutter

of

many

different

but

essentially

similar

symbols

We

will

use

A

or

-LRB-

A

I

-RRB-

B

or

-LRB-

B

J

-RRB-

and

C

or

-LRB-

C

K

-RRB-

to

denote

S-algebras

Similarly

we

will

use

A

B

and

C

to

denote

boolean

algebras

Within

a

boolean

algebra

A

we

will

often

consider

ult

-LRB-

A

-RRB-

the

collection

of

all

ultra?lters

in

A

that

is

ult

-LRB-

A

-RRB-

fu

A

j

u

is

an

ultra?lterg

where

an

ultra?lter

is

a

subset

of

a

boolean

algebra

which

is

closed

under

nite

meet

taking

larger

elements

and

satis?es

u

-LRB-

A

-RRB-

-LSB-

a

u

or

a

u

-RSB-

In

the

same

manner

as

for

a

is

the

sequence

-LRB-

an

-RRB-

and

for

h

a

map

between

algebras

h

-LRB-

a

-RRB-

is

the

sequence

-LRB-

h

-LRB-

-RRB-

h

-LRB-

-RRB-

h

-LRB-

an

-RRB-

-RRB-

The

notation

v

-LRB-

-RRB-

for

v

a

map

from

S

-LRB-

P

-RRB-

into

an

S-algebra

is

de?ned

analogously

By

a

A

we

mean

that

each

ak

A.

De?nition

-LRB-

B

J

-RRB-

is

an

S

-LRB-

P

-RRB-

subalgebra

of

-LRB-

A

I

-RRB-

iff

it

is

a

subalgebra

in

the

usual

sense

i.e.

B

is

a

boolean

subalgebra

of

A

and

for

each

b

B

i

I

Ji

-LRB-

b

-RRB-

Ii

-LRB-

b

-RRB-

B.

De?nition

A

valuation

v

on

-LRB-

A

I

-RRB-

is

a

map

which

assigns

to

each

propositional

letter

p

P

v

-LRB-

p

-RRB-

A.

De?nition

For

a

valuation

v

on

-LRB-

A

I

-RRB-

we

extend

v

to

a

map

v

S

-LRB-

P

-RRB-

A

as

follows

by

de?ning

v

-LRB-

-RRB-

v

-LRB-

-RRB-

v

-LRB-

-RRB-

v

-LRB-

-RRB-

v

-LRB-

-RRB-

and

v

i

Ii

v

In

the

event

that

v

-LRB-

-RRB-

A

we

write

-LRB-

A

I

v

-RRB-

ffl

and

in

the

event

that

-LRB-

A

I

v

-RRB-

ffl

regardless

of

our

choice

of

v

we

write

-LRB-

A

I

-RRB-

ffl

or

A

ffl

The

notation

-LRB-

A

I

v

-RRB-

ffl

for

a

subset

of

S

-LRB-

P

-RRB-

will

mean

that

-LRB-

A

I

v

-RRB-

ffl

for

each

To

complete

this

section

we

shall

de?ne

a

logic

state

what

it

means

for

that

logic

to

be

even

provide

some

examples

and

compare

this

notion

to

that

of

KIT

FINE

's

uniform

logics

of

-LSB-

-RSB-

De?nition

A

logic

is

a

set

L

ae

S

-LRB-

P

-RRB-

with

the

property

that

it

is

closed

under

substitution

-LRB-

replacement

of

propositional

letters

by

arbitrary

formulae

-RRB-

modus

ponens

and

replacement

by

provable

equivalents

-LRB-

replacing

subformula

of

a

thesis

of

L

with

another

formula

to

the

original

subformula

-RRB-

Further

we

will

require

that

every

tautology

be

a

member

of

L.

De?nition

A

set

S

-LRB-

P

-RRB-

is

said

to

axiomatise

a

logic

L

iff

L

is

the

closure

of

together

with

the

set

of

tautologies

under

modus

ponens

substitution

and

replacement

by

provable

equivalents

We

should

note

in

passing

that

each

S

-LRB-

P

-RRB-

algebra

de?nes

a

logic

Proposition

f

j

-LRB-

A

I

-RRB-

ffl

g

is

a

logic

De?nition

The

set

of

purely

boolean

formulae

is

the

set

-LRB-

P

-RRB-

f

j

contains

no

intensional

operatorsg

De?nition

We

de?ne

Se

-LRB-

P

-RRB-

S

-LRB-

P

-RRB-

the

set

of

even

formulae

to

be

the

smallest

subset

of

S

-LRB-

P

-RRB-

containing

f?i

j

i

I

-LRB-

P

-RRB-

g

and

which

is

closed

under

all

logical

operations

A

logic

is

called

even

iff

it

has

an

axiomatisation

that

consists

entirely

of

even

formulae

Thus

we

can

easily

spot

an

even

formula

just

by

seeing

if

it

has

a

subformula

which

is

a

boolean

combination

of

intensional

and

propositional

letters

We

have

the

following

examples

Even

formulae

-LRB-

p

q

-RRB-

-LRB-

p

q

-RRB-

p

p

p

p

and

Uneven

formulae

formulae

S

-LRB-

P

-RRB-

are

L-equivalent

iff

L.

p

p

-LRB-

p

p

-RRB-

p

and

p

p

In

-LSB-

-RSB-

FINE

de?nes

the

following

classes?where

we

are

dealing

with

mono-modal

logics

i.e.

I

is

of

cardinality

one

De?nition

-LRB-

Fine

-RRB-

We

de?ne

the

sets

unif

-LRB-

n

-RRB-

for

n

as

follows

unif

-LRB-

-RRB-

f?i

j

i

I

-LRB-

P

-RRB-

g

and

unif

-LRB-

n

-RRB-

is

the

closure

of

f?i

j

i

I

unif

-LRB-

n

-RRB-

g

in

S

-LRB-

P

-RRB-

under

the

boolean

operations

The

collection

of

all

uniform

formulae

is

then

the

set

unif

-LRB-

-RRB-

-LSB-

unif

-LRB-

n

-RRB-

A

uniform

logic

is

one

which

has

a

uniform

axiomatisation

Note

that

the

axiom

p

p

is

not

uniform

yet

it

is

even

As

FINE

points

out

a

uniform

formula

is

one

where

the

propositional

letters

are

all

nested

within

a

uniform

number

of

intensional

operators

Clearly

this

rules

out

the

prospect

of

having

a

subformula

where

a

propositional

letter

is

in

boolean

combination

with

an

intensional

operator

so

we

get

Proposition

Every

uniform

formula

is

also

even

and

every

uniform

logic

is

also

even

De?nition

We

say

that

A

is

a

power

set

boolean

algebra

iff

it

is

isomorphic

to

-LRB-

P

-LRB-

X

-RRB-

-RRB-

where

X

is

and

arbitrary

set

and

P

-LRB-

X

-RRB-

is

the

power

set

of

X

One

nal

piece

of

notation

If

f

W

Y

and

U

W

then

we

take

f

-LSB-

U

-RSB-

ff

-LRB-

u

-RRB-

j

u

Ug

and

if

Z

Y

then

we

take

f

-LSB-

Z

-RSB-

fw

W

j

f

-LRB-

w

-RRB-

Zg

Canonical

Algebras

In

this

section

we

will

de?ne

the

canonical

algebra

induced

by

a

logic

Fix

for

a

moment

a

logic

L.

De?ne

s

to

be

an

equivalence

relation

on

S

-LRB-

P

-RRB-

so

that

s

iff

L

In

the

natural

and

usual

way

this

de?nes

an

algebra

S

-LRB-

P

-RRB-

s

While

this

is

arguably

the

canonical

algebra

for

L

it

is

convenient

to

use

the

following

isomorphic

algebra

De?nition

Let

XL

fx

ae

S

-LRB-

P

-RRB-

j

x

is

a

maximal

set

g

For

each

S

-LRB-

P

-RRB-

let

k

k

fx

XL

j

xg

AL

is

then

the

boolean

-LRB-

set

-RRB-

algebra

with

elements

fk

k

j

S

-LRB-

P

-RRB-

g.

For

each

i

I

de?ne

ILi

Ani

A

by

setting

ILi

-LRB-

k

k

-RRB-

k?i

-LRB-

-RRB-

k.

Set

IL

ILi

ff

The

canonical

valuation

vL

is

de?ned

by

vL

-LRB-

p

-RRB-

kpk

AL

IL

is

called

the

canonical

algebra

of

L

It

takes

little

work

to

see

that

AL

IL

is

indeed

well

de?ned

Each

x

XL

corresponds

directly

to

a

member

of

ult

AL

via

the

maps

o

XL

ult

AL

and

its

inverse

ult

AL

XL

where

for

x

XL

and

u

ult

AL

xo

fk

k

j

xg

and

u

f

j

k

k

ug

Completeness

and

Canonicity

We

could

say

that

L

is

weakly

-LSB-

strongly

-RSB-

complete

iff

for

every

L-consistent

formula

-LSB-

set

-RSB-

there

is

a

valuation

v

on

an

S

algebra

-LRB-

A

I

-RRB-

-LRB-

A

I

-RRB-

ffl

L

such

that

v

-LRB-

-RRB-

A

-LSB-

v

-LRB-

-RRB-

A

for

all

-RSB-

Then

the

weak

and

strong

completeness

of

every

intensional

logic

L

would

be

immediate

as

AL

IL

and

vL

would

witness

this

fact

As

will

be

seen

in

the

next

section

it

is

more

interesting

to

de?ne

completeness

in

terms

of

frames

De?nition

A

frame

is

an

S-algebra

-LRB-

A

I

-RRB-

with

the

property

that

A

is

a

power

set

boolean

algebra

Here

if

A

-LRB-

P

-LRB-

X

-RRB-

-RRB-

then

each

element

of

X

can

be

considered

to

be

a

point

in

the

frame

De?nition

An

intensional

logic

L

is

weakly

complete

iff

for

every

L-consistent

formula

S

-LRB-

P

-RRB-

there

is

a

frame

-LRB-

A

I

-RRB-

-LRB-

A

I

-RRB-

ffl

L

A

-LRB-

P

-LRB-

X

-RRB-

-RRB-

a

valuation

v

and

an

x

X

such

that

x

v

-LRB-

-RRB-

De?nition

An

intensional

logic

L

is

strongly

complete

iff

for

every

L-consistent

set

ae

S

-LRB-

P

-RRB-

there

is

a

frame

-LRB-

A

I

-RRB-

-LRB-

A

I

-RRB-

ffl

L

A

-LRB-

P

-LRB-

X

-RRB-

-RRB-

a

valuation

v

and

an

x

X

such

that

x

v

-LRB-

-RRB-

for

each

It

is

no

longer

easy

to

show

that

L

is

complete

in

either

sense

A

special

class

of

weakly

complete

logics

is

of

interest

De?nition

We

say

that

a

logic

L

has

the

nite

model

property

iff

there

is

a

nite

algebra

-LRB-

A

I

-RRB-

such

that

for

each

L

-LRB-

A

I

-RRB-

set

S

-LRB-

P

-RRB-

is

L-consistent

iff

there

is

no

such

that

'n

p

:p

L.

that

k

k

k

k

k

k

and

k

k

XL

k

k.

we

have

that

vL

-LRB-

-RRB-

k

k

for

all

S

-LRB-

P

-RRB-

Note

that

since

each

nite

boolean

algebra

is

also

a

nite

powerset

algebra

we

have

that

the

nite

model

property

implies

weak

completeness

Returning

to

uniform

formulae

again

for

a

moment

we

note

that

FINE

proved

-LSB-

-RSB-

through

a

technique

of

constructing

nite

models

out

of

the

subformulae

of

a

formula

itself

the

following

theorem

Theorem

-LRB-

Fine

-RRB-

All

uniform

normal

mono-modal

logics

have

the

nite

model

property

and

so

if

nitely

axiomatisable

are

decidable

When

FINE

refers

to

the

Finite

Model

Property

he

is

of

course

refering

to

relational

frames

The

conversion

of

a

relational

frame

into

an

S-algebra

is

straightforward

and

is

an

immediate

consequence

of

the

natural

duality

between

normal

modal

algebras

and

relational

frames?note

that

a

nite

relational

frame

gives

rise

to

a

nite

normal

modal

algebra

which

is

an

S-algebra

We

now

return

to

the

notion

of

canonicity

De?nition

For

L

an

intensional

logic

we

de?ne

BL

P

XL

Now

BL

is

a

good

candidate

for

showing

strong

completeness

over

all

L-consistent

sets

but

BL

must

rst

be

out?tted

with

a

JL

agreeable

with

IL

De?nition

An

intensional

logic

L

is

canonical

iff

there

is

a

JL

so

that

AL

IL

is

an

S-subalgebra

of

BL

JL

and

BL

JL

ffl

L

This

de?nition

may

seem

slightly

strange

at

rst

however

the

standard

de?nition

of

canonicity

for

neighborhood

semantics

will

be

examined

in

the

Section

and

readily

seen

to

be

equivalent

to

this

one

Finally

we

note

the

immediate

result

Proposition

If

an

intensional

logic

is

canonical

then

it

is

strongly

complete

Even

Pairs

In

this

section

we

disclose

well

ahead

of

time

the

trick

we

will

use

to

prove

the

canonicity

Remember

that

we

have

to

take

our

power

set

algebra

BL

and

stick

a

JL

onto

it

and

often

it

will

not

be

clear

exactly

how

such

a

JL

could

be

de?ned

Here

we

show

one

technique

of

generating

such

a

sequence

of

functions

in

such

a

way

that

the

even

formulae

that

are

the

subject

of

this

work

are

preserved

Of

course

the

conditions

necessary

for

this

to

go

through

are

relatively

restrictive

De?nition

We

say

that

B

and

A

are

evenly

paired

iff

B

is

a

boolean

algebra

A

is

an

S-algebra

There

is

a

homomorphism

h

B

A

There

is

a

homomorphism

g

A

B

and

For

each

i

I

-LRB-

A

-RRB-

-LSB-

g

-LRB-

Ii

-LRB-

h

ffi

g

-LRB-

a

-RRB-

-RRB-

-RRB-

g

-LRB-

Ii

-LRB-

a

-RRB-

-RRB-

-RSB-

When

this

happens

we

write

h

B

A

g

ffff

Under

these

apparently

restrictive

conditions

we

can

de?ne

our

J

De?nition

The

operations

J

on

B

induced

by

hhh

B

-LRB-

A

I

-RRB-

gii

are

de?ned

as

follows

For

i

I

b

B

Ji

-LRB-

b

-RRB-

g

-LRB-

Ii

-LRB-

h

-LRB-

b

-RRB-

-RRB-

-RRB-

Now

it

is

just

a

matter

of

verifying

at

least

as

far

as

even

formulae

are

concerned

that

this

de?nition

preserves

validity

Theorem

Suppose

that

hhh

B

-LRB-

A

I

-RRB-

gii

and

that

-LRB-

A

I

-RRB-

ffl

L

for

L

an

even

logic

Then

for

J

the

induced

operations

on

B

-LRB-

B

J

-RRB-

ffl

L

Proof

Suppose

that

v

is

a

valuation

on

-LRB-

B

J

-RRB-

De?ne

a

valuation

on

-LRB-

A

I

-RRB-

by

setting

-LRB-

p

-RRB-

h

-LRB-

v

-LRB-

p

-RRB-

-RRB-

Thus

by

h

being

a

boolean

homomorphism

-LRB-

-RRB-

h

-LRB-

v

-LRB-

-RRB-

-RRB-

for

-LRB-

P

-RRB-

Claim

-LRB-

Se

-LRB-

P

-RRB-

-RRB-

-LSB-

v

-LRB-

-RRB-

g

-LRB-

-LRB-

-RRB-

-RRB-

-RSB-

Proof

of

claim

By

induction

on

the

complexity

of

as

an

even

formula

Base

Cases

i

for

-LRB-

P

-RRB-

Thus

v

-LRB-

-RRB-

v

-LRB-

i

-RRB-

Ji

-LRB-

v

-LRB-

-RRB-

-RRB-

g

-LRB-

Ii

-LRB-

h

-LRB-

v

-LRB-

-RRB-

-RRB-

-RRB-

-RRB-

g

-LRB-

Ii

-LRB-

-LRB-

-RRB-

-RRB-

-RRB-

g

-LRB-

-LRB-

i

-RRB-

-RRB-

Inductive

Hypothesis

Assume

that

the

result

holds

for

the

formulae

in

Se

-LRB-

P

-RRB-

Inductive

Step

Suppose

that

is

built

up

from

the

components

of

in

one

step

Case

Then

we

have

the

equalities

v

-LRB-

-RRB-

v

-LRB-

-RRB-

v

-LRB-

-RRB-

v

-LRB-

-RRB-

g

-LRB-

-LRB-

-RRB-

-RRB-

g

-LRB-

-LRB-

-RRB-

-RRB-

g

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

g

-LRB-

-LRB-

-RRB-

-RRB-

g

-LRB-

-LRB-

-RRB-

-RRB-

Case

Then

we

have

the

equalities

v

-LRB-

-RRB-

v

-LRB-

-RRB-

v

-LRB-

-RRB-

g

-LRB-

-LRB-

-RRB-

-RRB-

g

-LRB-

-LRB-

-RRB-

-RRB-

g

-LRB-

-LRB-

-RRB-

-RRB-

g

-LRB-

-LRB-

-RRB-

-RRB-

Case

i

We

get

the

following

sequence

of

equalities

v

-LRB-

-RRB-

v

-LRB-

i

-RRB-

Ji

-LRB-

v

-LRB-

-RRB-

-RRB-

g

-LRB-

Ii

-LRB-

h

-LRB-

v

-LRB-

-RRB-

-RRB-

-RRB-

-RRB-

g

-LRB-

Ii

-LRB-

h

-LRB-

g

-LRB-

-LRB-

-RRB-

-RRB-

-RRB-

-RRB-

-RRB-

g

-LRB-

Ii

-LRB-

-LRB-

-RRB-

-RRB-

-RRB-

g

-LRB-

-LRB-

i

-RRB-

-RRB-

g

-LRB-

-LRB-

-RRB-

-RRB-

Where

the

fth

line

folows

because

of

the

fth

condition

on

B

and

-LRB-

A

I

-RRB-

begin

evenly

paired

End

of

proof

of

claim

Now

assume

that

-LRB-

B

J

-RRB-

L.

Thus

there

is

an

even

axiom

and

a

valuation

v

such

that

-LRB-

B

J

v

-RRB-

Let

be

as

given

above

Hence

g

-LRB-

-LRB-

-RRB-

-RRB-

v

-LRB-

-RRB-

B

but

since

g

is

a

-LRB-

-RRB-

A

and

so

-LRB-

A

I

-RRB-

a

contradiction

to

-LRB-

A

I

-RRB-

ffl

L

This

gives

us

the

criteria

around

which

the

rest

of

this

work

will

be

aimed

Neighborhood

Semantics

In

proving

the

main

result

of

this

paper

we

will

avoid

the

use

of

neighborhood

semantics

However

it

is

important

to

see

how

the

algebraic

constructions

introduced

here

are

equivalent

to

the

neighborhood

semantics

approach

and

we

will

use

neighborhood

semantics

in

our

informal

discussions

as

they

are

more

intuitive

Moreover

the

neighborhood

semantics

approach

does

underline

the

need

for

a

strong

completeness

result

and

it

will

give

the

desired

form

for

our

result

It

should

be

stressed

again

that

we

are

only

considering

the

duality

between

frames

-LRB-

powerset

or

full

algebras

-RRB-

and

full

neighborhood

frames

Of

course

this

duality

can

be

made

precise

for

all

types

of

neighborhood

frames

and

all

algebras

the

reader

is

advised

to

consult

DOSEN

-LSB-

-RSB-

De?nition

A

neighborhood

frame

for

S

is

a

structure

of

the

form

-LRB-

X

N

-RRB-

where

N

and

each

Ni

X

P

-LRB-

P

-LRB-

X

-RRB-

ni

-RRB-

De?nition

A

valuation

v

on

-LRB-

X

N

-RRB-

a

neighborhood

frame

is

a

map

that

assigns

v

-LRB-

p

-RRB-

X

to

each

propositional

letter

p

The

map

v

is

extended

to

v

S

-LRB-

P

-RRB-

P

-LRB-

X

-RRB-

as

follows

v

-LRB-

-RRB-

v

-LRB-

-RRB-

v

-LRB-

-RRB-

v

-LRB-

-RRB-

v

-LRB-

-RRB-

and

For

i

I

v

i

fx

j

v

Ni

-LRB-

x

-RRB-

g

These

neighborhood

models

are

called

minimal

models

by

CHELLAS

in

-LSB-

-RSB-

and

the

semantics

are

sometimes

called

ScottMontague

semantics

Without

loss

of

generality

in

what

follows

take

-LRB-

A

I

-RRB-

to

be

an

algebraic

frame

and

take

A

to

actually

be

P

-LRB-

X

-RRB-

for

some

set

X.

Each

frame

-LRB-

A

I

-RRB-

naturally

de?nes

a

neighborhood

frame

and

vice

versa

De?nition

Given

a

frame

-LRB-

A

I

-RRB-

de?ne

-LRB-

A

I

-RRB-

the

neighborhood

frame

derived

from

-LRB-

A

I

-RRB-

as

follows

Let

-LRB-

A

I

-RRB-

-LRB-

X

N

-RRB-

where

for

i

I

Ni

-LRB-

x

-RRB-

fa

j

ani

X

x

Ii

-LRB-

a

-RRB-

g

If

v

is

a

valuation

on

-LRB-

X

N

-RRB-

-LRB-

A

I

-RRB-

it

is

also

a

valuation

on

-LRB-

A

I

-RRB-

By

noting

that

for

i

I

v

i

x

j

v

Ni

-LRB-

x

-RRB-

x

j

v

fa

j

x

Ii

-LRB-

a

-RRB-

g

x

j

x

Ii

v

we

see

that

the

extnesions

of

v

to

-LRB-

A

I

-RRB-

and

-LRB-

A

I

-RRB-

are

identicle

and

so

precisely

the

same

formulae

hold

over

-LRB-

A

I

-RRB-

as

over

-LRB-

A

I

-RRB-

De?nition

If

-LRB-

X

N

-RRB-

is

a

neighborhood

frame

then

set

-LRB-

X

N

-RRB-

-LRB-

A

I

-RRB-

where

A

P

-LRB-

X

-RRB-

and

Ii

-LRB-

a

-RRB-

fx

j

a

Ni

-LRB-

x

-RRB-

g

In

a

similar

way

we

can

see

that

the

same

formulae

hold

over

-LRB-

X

N

-RRB-

and

-LRB-

X

N

-RRB-

It

is

also

not

hard

to

see

that

-LRB-

A

I

-RRB-

-LRB-

A

I

-RRB-

and

-LRB-

X

N

-RRB-

-LRB-

X

N

-RRB-

The

ultimate

goal

of

this

paper

is

to

look

at

which

logics

L

are

canonical

In

the

sense

of

neighborhood

semantics

this

means

that

we

can

nd

an

NL

such

that

XL

NL

satis?es

XL

NL

ffl

L

and

NLi

-LRB-

x

-RRB-

-LRB-

-RRB-

x

i

for

each

i

I

This

is

the

de?nition

of

canonicity

that

can

be

found

in

CHELLAS

-LSB-

page

-RSB-

modi?ed

to

take

into

account

polyadic

poly-modal

operators

In

the

light

of

what

we

have

just

seen

it

will

be

enough

to

nd

a

JL

so

that

BL

JL

ffl

L

and

AL

IL

is

an

S-subalgebra

of

BL

JL

for

then

setting

XL

NL

BL

JL

will

give

us

XL

NL

ffl

L

and

NLi

-LRB-

x

-RRB-

-LRB-

-RRB-

x

Ji

-LRB-

-RRB-

x

i

This

is

precisely

the

requirement

that

L

is

canonical

Ultrapowers

In

this

section

we

will

introduce

the

notation

associated

with

ultrapowers

an

important

mathematical

construction

which

we

use

to

expand

algebras

so

that

they

encompass

powerset

boolean

algebras

For

this

section

take

A

-LRB-

A

I

-RRB-

be

be

an

S-algebra

take

to

be

a

set

and

take

U

to

be

an

ultra?lter

on

De?nition

The

power

of

A

is

the

algebra

A

A

I

where

A

f

a

Ag

and

the

operations

on

A

are

de?ned

by

-LRB-

a

e

-RRB-

-LRB-

-RRB-

a

-LRB-

-RRB-

e

-LRB-

-RRB-

etc.

and

Ii

-LRB-

a

-RRB-

-LRB-

-RRB-

Ii

-LRB-

a

-LRB-

-RRB-

-RRB-

Following

our

usual

convention

we

do

not

distinguish

the

boolean

operations

on

A

from

the

boolean

operations

on

A

as

it

will

be

clear

from

context

We

can

use

U

to

de?ne

an

equivalence

relation

sU

on

A

by

declaring

a

sU

e

-LRB-

-RRB-

a

-LRB-

-RRB-

e

-LRB-

-RRB-

U

a.e.

We

can

then

use

this

to

get

the

following

which

can

be

readily

seen

to

be

well-de

ned

De?nition

The

U-ultrapower

of

A

is

the

algebra

A

U

A

U

I

where

A

U

A

sU

and

the

operations

on

A

U

are

de?ned

by

-LSB-

a

-RSB-

sU

-LSB-

e

-RSB-

sU

-LSB-

a

e

-RSB-

sU

etc.

and

Ii

-LRB-

-LSB-

a

-RSB-

sU

-RRB-

-LRB-

-RRB-

-LSB-

Ii

-LRB-

a

-RRB-

-RSB-

sU

Again

the

boolean

operations

are

not

distinguished

We

use

the

accents

and

to

distinguish

elements

of

a

power

and

an

ultrapower

respectively

There

are

particular

members

of

A

that

we

need

to

take

note

of

De?nition

De?ne

a

map

const

A

A

by

setting

const

-LRB-

a

-RRB-

-LRB-

-RRB-

a.

De?nition

De?ne

rep

A

U

A

by

taking

rep

-LRB-

a

-RRB-

to

be

any

element

of

A

such

that

a

-LSB-

rep

-LRB-

a

-RRB-

-RSB-

sU

The

function

const

then

gives

rise

to

an

elementary

embedding

De?nition

The

os

function

l

A

A

U

is

de?ned

by

l

-LRB-

a

-RRB-

-LSB-

const

-LRB-

a

-RRB-

-RSB-

sU

Theorem

-LRB-

os

-RRB-

The

structure

A

U

is

well

de?ned

and

the

map

l

A

A

U

is

an

elementary

embedding

That

is

for

all

rst

order

formulae

and

all

a

A

A

j

-LSB-

a

-RSB-

-LRB-

-RRB-

A

U

j

-LSB-

l

-LRB-

a

-RRB-

-RSB-

Clearly

an

elementary

embedding

of

one

algebra

into

another

is

also

an

injective

homomorphism

We

have

the

following

result

which

could

be

an

immediate

consequence

of

the

above

theorem

had

we

dealt

with

many

sorted

structures

Lemma

Let

u

ult

-LRB-

A

-RRB-

Then

l

-LRB-

u

-RRB-

a

A

U

j

rep

-LRB-

a

-RRB-

-LRB-

-RRB-

u

U

a.e.

ult

A

U

Canonical

Frames

We

saw

in

de?nition

that

to

show

canonicity

we

have

to

nd

a

canonical

frame

which

means

that

we

have

to

stick

a

suitable

JL

onto

BL

In

the

work

on

non-iterative

logics

-LSB-

-RSB-

the

canonical

frame

was

induced

by

a

structure

that

was

an

elementary

extension

of

AL

IL

and

we

adopt

the

same

strategem

here

As

with

-LSB-

-RSB-

we

need

some

preliminary

results

leading

up

to

the

existence

of

such

structures

The

following

Lemma

is

a

mild

extension

of

Lemma

of

-LSB-

-RSB-

and

can

also

be

found

as

Lemma

of

-LSB-

-RSB-

Lemma

Let

D

be

a

nite

boolean

subalgebra

of

BL

Let

F

X

be

nite

Then

there

is

a

boolean

endomorphism

h

D

AL

which

is

the

identity

D

AL

and

-LRB-

F

-RRB-

-LRB-

D

-RRB-

-LSB-

x

d

-LRB-

-RRB-

x

h

-LRB-

d

-RRB-

-RSB-

symbol

j

represents

rst

order

satisfaction

boolean

algebra

D

AL

is

the

boolean

algebra

with

underlying

set

D

AL

Proof

Let

C

D

AL

and

let

Y

fc

C

j

c

is

an

atom

in

Cg

and

Z

fd

D

j

d

is

an

atom

in

Dg

Without

loss

of

generality

we

can

assume

that

F

intersects

each

element

of

Z

at

just

one

point?otherwise

just

increase

D

so

that

it

includes

the

singletons

fxg

for

x

F

For

each

c

Y

let

Zc

fd

Zjd

cg

say

Zc

dcnc

and

so

c

dcnc

Recall

that

each

c

Y

AL

is

really

a

set

of

ultra?lters

and

since

P

is

in?nite

c

is

an

uncountable

set

Thus

there

are

distinct

points

wcnc

c.

By

our

assumption

given

above

we

can

make

our

choice

so

that

all

elements

of

F

are

represented

among

these

ws

By

the

general

theory

of

canonical

models

we

can

nd

ecnc

AL

pairwise

disjoint

sets

such

that

wcj

ecj

and

c

-LSB-

-LSB-

ecnc

Let

D

be

the

boolean

subalgebra

of

AL

generated

by

ecj

j

nc

Now

ecj

is

the

set

of

atoms

of

D

since

each

ecj

does

not

meet

any

distinct

k

and

S

c

j

ecj

S

c

c

A.

Thus

we

can

de?ne

a

map

h

D

D

by

setting

h

dcj

ecj

c

Y

j

nc

This

is

a

onto

map

between

the

atoms

of

nite

boolean

algebras

and

so

h

is

an

isomorphism

Thus

h

D

AL

is

an

endomorphism

To

see

that

h

is

the

identity

on

C

it

is

enough

to

show

that

it

is

the

identity

on

Y

the

set

of

atoms

of

C

Let

c

Y

then

h

-LRB-

c

-RRB-

h

dcnc

h

-LRB-

-RRB-

h

dcnc

ecnc

c

To

nish

off

the

proof

note

that

if

x

F

then

there

is

a

c

Y

and

some

j

such

that

x

wcj

telling

us

that

x

k

-LRB-

-RRB-

c

and

j

k

-LRB-

-RRB-

x

k

-LRB-

-RRB-

x

h

-LRB-

k

-RRB-

Using

the

homomorphic

properties

of

h

we

then

see

that

for

any

x

F

and

d

D

x

d

-LRB-

-RRB-

x

h

-LRB-

d

-RRB-

Now

we

are

in

a

position

to

deduce

that

the

algebra

BL

is

in

some

sense

contained

within

an

ultrapower

of

AL.

Theorem

Let

L

be

an

intensional

logic

Then

there

is

an

index

an

ultra?lter

U

on

and

a

map

h

BL

AL

U

such

that

h

is

a

boolean

endomorphism

into

AL

U

h?AL

l

and

necessarily

even

or

non-iterative

ult

AL

-LRB-

B

-RRB-

-LSB-

uo

b

-LRB-

-RRB-

h

-LRB-

b

-RRB-

l

-LRB-

u

-RRB-

-RSB-

Proof

We

will

suppress

the

superscript

Ls

Set

-LRB-

D

F

-RRB-

j

F

XL

and

D

B

are

both

nite

and

for

-LRB-

-RRB-

set

U

-LRB-

-RRB-

f

-LRB-

D

F

-RRB-

j

D

and

Fg

and

choose

U

to

be

any

ultra?lter

which

extends

fU

-LRB-

D

F

-RRB-

j

-LRB-

D

F

-RRB-

g

This

set

clearly

possesses

the

nite

intersection

property

so

we

can

be

assured

that

it

can

be

extended

to

an

ultra?ler

For

each

-LRB-

D

F

-RRB-

we

can

use

Lemma

to

give

us

a

map

h

D

A

which

is

the

identity

on

D

A

and

satis?es

-LRB-

F

-RRB-

-LRB-

D

-RRB-

-LSB-

x

d

-LRB-

-RRB-

x

h

-LRB-

d

-RRB-

-RSB-

Now

create

a

map

h

B

A

by

setting

h

-LRB-

b

-RRB-

-LRB-

-RRB-

h

-LRB-

b

-RRB-

and

we

are

able

to

complete

our

de?nitions

of

h

B

A

U

by

setting

h

-LRB-

b

-RRB-

-LSB-

h

-LRB-

b

-RRB-

-RSB-

sU

We

then

have

three

conditions

to

verify

h

is

a

boolean

endomorphism

into

A

U

Let

B

and

let

the

boolean

subalgebra

of

B

generated

by

Let

-LRB-

-RRB-

and

-LRB-

D

F

-RRB-

U

-LRB-

-RRB-

Then

h

-LRB-

-RRB-

-LRB-

-RRB-

h

-LRB-

-RRB-

-LRB-

-RRB-

h

-LRB-

-RRB-

h

-LRB-

-RRB-

h

-LRB-

-RRB-

h

-LRB-

-RRB-

-LRB-

-RRB-

where

the

rst

and

last

lines

follow

because

D

Since

this

held

for

each

U

-LRB-

-RRB-

U

we

have

that

h

-LRB-

-RRB-

h

-LRB-

-RRB-

-LSB-

h

-LRB-

-RRB-

-RSB-

sU

-LSB-

h

-LRB-

-RRB-

-RSB-

sU

-LSB-

h

-LRB-

-RRB-

h

-LRB-

-RRB-

-RSB-

sU

-LSB-

h

-LRB-

-RRB-

-RSB-

sU

h

-LRB-

-RRB-

Similarly

h

-LRB-

-RRB-

-LRB-

-RRB-

h

-LRB-

-RRB-

h

-LRB-

-RRB-

h

-LRB-

-RRB-

-LRB-

-RRB-

and

since

this

holds

U

a.e.

we

have

that

h

-LRB-

-RRB-

h

-LRB-

-RRB-

allowing

us

to

conclude

that

h

is

a

boolean

homomorphism

To

see

that

h

is

one-one

note

that

if

then

h

-LRB-

-RRB-

h

-LRB-

-RRB-

since

h

is

one-one

on

D

So

h

-LRB-

-RRB-

-LRB-

-RRB-

h

-LRB-

-RRB-

-LRB-

-RRB-

U

a.e.

Hence

h

-LRB-

-RRB-

-LSB-

h

-LRB-

-RRB-

-RSB-

sU

-LSB-

h

-LRB-

-RRB-

-RSB-

sU

h

-LRB-

-RRB-

h?A

l.

Let

a

A

hfagi

and

-LRB-

-RRB-

We

show

that

h

-LRB-

a

-RRB-

-LRB-

-RRB-

a

U

a.e.

by

showing

that

it

holds

almost

everywhere

on

the

set

U

-LRB-

-RRB-

So

let

-LRB-

D

F

-RRB-

U

-LRB-

-RRB-

Thus

h

is

the

identity

on

D

A

A

fag

and

so

h

-LRB-

a

-RRB-

-LRB-

-RRB-

h

-LRB-

a

-RRB-

a.

Hence

h

-LRB-

a

-RRB-

-LSB-

h

-LRB-

a

-RRB-

-RSB-

sU

-LSB-

const

-LRB-

a

-RRB-

-RSB-

sU

l

-LRB-

a

-RRB-

-LRB-

ult

-LRB-

A

-RRB-

-RRB-

-LRB-

B

-RRB-

-LSB-

uo

b

-LRB-

-RRB-

h

-LRB-

b

-RRB-

l

-LRB-

u

-RRB-

-RSB-

Let

u

ult

-LRB-

A

-RRB-

and

set

x

uo

f

S

-LRB-

P

-RRB-

j

k

k

ug

XL

Let

b

B

and

set

hfbgi

and

fxg

We

show

that

uo

b

-LRB-

-RRB-

h

-LRB-

b

-RRB-

u

U

a.e.

by

showing

that

it

holds

on

U

-LRB-

-RRB-

for

if

is

in

this

set

then

uo

b

-LRB-

-RRB-

x

b

-LRB-

-RRB-

x

h

-LRB-

b

-RRB-

-LRB-

-RRB-

h

-LRB-

b

-RRB-

u

Now

we

nish

the

proof

by

noting

that

h

-LRB-

b

-RRB-

l

-LRB-

u

-RRB-

-LRB-

-RRB-

h

-LRB-

b

-RRB-

-LRB-

-RRB-

u

U

a.e.

-LRB-

-RRB-

h

-LRB-

b

-RRB-

u

U

a.e.

on

U

-LRB-

-RRB-

-LRB-

-RRB-

uo

b

Even

Logics

with

the

FMP

In

this

section

we

will

slowly

work

our

way

through

the

main

proof

of

this

paper

namely

that

all

even

logics

that

have

the

fmp

are

neighborhood

canonical

So

x

our

logic

L

which

we

take

to

be

non-trivial

even

and

to

have

the

nite

model

property

Take

U

and

h

to

be

the

result

of

applying

Theorem

to

the

canonical

algebras

for

this

logic

Throughout

this

whole

section

we

suppress

the

superscript

L

so

we

will

take

it

as

given

that

A

represents

AL

J

represents

JL

etc.

Take

C

to

be

the

class

of

nite

S-algebras

that

verify

L

and

take

F

to

be

the

class

of

all

homomorphisms

from

A

to

elements

of

C

Each

homomorphism

in

F

also

gives

rise

to

a

homomorphism

from

A

U

to

C

Proposition

Each

f

A

C

in

F

has

an

associated

homomorphism

f

A

U

C

such

that

f

ffi

l

f

Proof

Let

C

and

f

be

as

in

the

hypotheses

of

this

proposition

Note

that

since

C

is

nite

for

each

a

A

there

is

a

c

-LRB-

a

-RRB-

C

such

that

f

-LRB-

a

-LRB-

-RRB-

-RRB-

c

-LRB-

a

-RRB-

U

a.e.

This

is

because

f

-LRB-

a

-LRB-

-RRB-

-RRB-

has

only

a

nite

number

of

possible

values

hence

one

of

these

values

must

come

up

almost

all

of

the

time

For

a

A

U

set

f

-LRB-

a

-RRB-

c

-LRB-

rep

-LRB-

a

-RRB-

-RRB-

and

then

it

readily

follows

that

f

is

a

homomorphism

and

that

f

ffi

l

f

Now

we

observe

using

the

homomorphisms

in

F

that

every

ultra?lter

in

a

nite

S-algebra

for

L

will

be

in

some

sense

realised

in

an

ultraproduct

of

the

canonical

algebra

Let

C

C

f

F

f

A

C

and

let

z

ult

-LRB-

C

-RRB-

Then

f

-LSB-

z

-RSB-

ult

A

U

De?nition

De?ne

the

set

of

pseudo?nite

points

to

be

W

f

-LSB-

z

-RSB-

j

C

C

z

ult

-LRB-

C

-RRB-

f

A

C

in

F

Note

that

W

is

nonempty

as

the

following

argument

shows

L

is

non-trivial

has

the

nite

model

property

and

so

there

is

a

-LRB-

C

K

-RRB-

C

such

that

-LRB-

C

K

-RRB-

for

some

L.

Thus

there

is

a

valuation

v

on

-LRB-

C

K

-RRB-

such

that

v

-LRB-

-RRB-

De?ne

a

map

f

-LRB-

A

I

-RRB-

-LRB-

C

K

-RRB-

by

taking

f

-LRB-

k

k

-RRB-

v

-LRB-

-RRB-

Then

f

F

Since

A

U

is

an

ultrapower

of

the

canonical

algebra

we

could

think

of

its

ultra?lters

as

being

sets

of

formulae

in

some

non-standard

extended

language

Then

each

element

w

W

would

correspond

to

a

maximal

consistent

set

in

this

extended

language

and

so

W

would

de?ne

in

a

manner

following

GROVE

-LSB-

-RSB-

a

theory

t

-LRB-

W

-RRB-

of

all

non-standard

formulae

which

hold

throughout

W

We

will

show

relative

to

the

appropriate

notions

for

our

algebraic

point

of

view

that

every

element

of

XL

can

be

thought

of

as

being

an

extension

of

t

-LRB-

W

-RRB-

and

inverting

this

procedure

will

give

us

a

map

g

which

will

satisfy

h

B

A

U

g

ffff

The

set

W

can

be

thought

of

as

the

collection

of

points

in

nite

frames

for

L

and

so

we

can

think

of

t

-LRB-

W

-RRB-

to

be

the

theory

of

nite

frames

for

L.

So

what

exactly

is

t

-LRB-

W

-RRB-

De?nition

t

-LRB-

W

-RRB-

a

A

U

j

-LRB-

W

-RRB-

-LSB-

a

w

-RSB-

TW

The

analog

of

a

theory

in

a

language

is

a

lter

in

a

boolean

algebra

Proposition

The

set

t

-LRB-

W

-RRB-

is

a

proper

lter

Proof

This

consists

of

a

number

of

veri?cations

Let

a

e

A

U

a

e

t

-LRB-

W

-RRB-

=-RRB-

a

e

t

-LRB-

W

-RRB-

is

well

de?ned

for

if

k

k

kOk

then

O

L

so

v

-LRB-

-RRB-

v

-LRB-

O

-RRB-

Let

a

e

t

-LRB-

W

-RRB-

and

w

W

Thus

a

e

w

so

a

e

w.

a

t

-LRB-

W

-RRB-

and

a

e

=-RRB-

e

t

-LRB-

W

-RRB-

Let

a

t

-LRB-

W

-RRB-

and

a

e

and

let

w

W

Thus

a

w

and

so

e

w.

t

-LRB-

W

-RRB-

W

so

take

w

W

Thus

w

t

-LRB-

W

-RRB-

Since

t

-LRB-

W

-RRB-

in

some

sense

represents

the

class

of

all

nite

algebras

for

L

it

also

has

some

properties

like

that

of

a

class

of

algebras

Proposition

Let

a

e

A

U

and

i

I

Then

a

e

t

-LRB-

W

-RRB-

=-RRB-

Ii

-LRB-

a

-RRB-

Ii

-LRB-

e

-RRB-

t

-LRB-

W

-RRB-

Proof

Suppose

that

a

e

t

-LRB-

W

-RRB-

let

w

W

and

let

-LRB-

C

K

-RRB-

f

z

give

rise

to

w

Then

a

e

f

-LSB-

-RSB-

for

all

ult

-LRB-

C

-RRB-

Thus

-LRB-

ult

-LRB-

C

-RRB-

-RRB-

f

-LRB-

a

e

-RRB-

=-RRB-

f

-LRB-

a

-RRB-

f

-LRB-

e

-RRB-

C

=-RRB-

Ki

-LRB-

f

-LRB-

a

-RRB-

-RRB-

Ki

-LRB-

f

-LRB-

e

-RRB-

-RRB-

C

=-RRB-

f

-LRB-

Ki

-LRB-

a

-RRB-

Ki

-LRB-

e

-RRB-

-RRB-

C

=-RRB-

f

-LRB-

Ki

-LRB-

a

-RRB-

Ki

-LRB-

e

-RRB-

-RRB-

z

=-RRB-

Ki

-LRB-

a

-RRB-

Ki

-LRB-

e

-RRB-

w

f

-LSB-

z

-RSB-

The

rst

of

these

clearly

holds

so

we

left

with

the

last

and

desired

result

The

next

two

propositions

show

in

effect

that

each

x

XL

can

be

made

into

a

maximal

consistent

extension

of

t

-LRB-

W

-RRB-

which

could

be

thought

of

as

meaning

that

each

x

XL

extends

to

a

point

in

a

nite

frame

Proposition

For

all

u

ult

-LRB-

A

-RRB-

the

set

l

-LSB-

u

-RSB-

has

the

nite

meet

property

Proof

Let

l

-LRB-

a

-RRB-

l

-LRB-

e

-RRB-

l

-LSB-

u

-RSB-

for

a

e

u.

Thus

a

e

u

and

so

l

-LRB-

a

-RRB-

l

-LRB-

e

-RRB-

l

-LRB-

a

e

-RRB-

l

-LSB-

u

-RSB-

So

all

we

have

to

show

now

is

that

each

element

of

l

-LSB-

u

-RSB-

is

non

Assume

that

l

-LRB-

a

-RRB-

for

a

u.

Thus

by

l

being

an

elementary

embedding

-LRB-

and

so

one-one

-RRB-

and

by

l

-LRB-

-RRB-

l

-LRB-

a

-RRB-

we

have

that

a

so

a

u

a

contradiction

So

l

-LSB-

u

-RSB-

Proposition

For

all

u

ult

-LRB-

A

-RRB-

the

set

l

-LSB-

u

-RSB-

-LSB-

t

-LRB-

W

-RRB-

has

the

nite

meet

property

Proof

Assume

not

Individually

l

-LSB-

u

-RSB-

and

t

-LRB-

W

-RRB-

have

the

nite

meet

property

so

there

must

be

an

a

u

and

a

e

t

-LRB-

W

-RRB-

such

that

l

-LRB-

a

-RRB-

e

Thus

e

l

-LRB-

a

-RRB-

so

l

-LRB-

a

-RRB-

t

-LRB-

W

-RRB-

But

a

since

a

u

so

a

We

can

take

a

k

k

so

L

and

thus

by

L

having

the

nite

model

property

there

is

an

algebra

C

C

for

L

which

denies

Hence

there

is

a

valuation

v

on

C

such

that

v

-LRB-

:'-RRB-

C

As

before

v

induces

a

homomorphism

f

A

C

such

that

f

-LRB-

a

-RRB-

f

-LRB-

k

k

-RRB-

v

-LRB-

:'-RRB-

C.

Take

z

ult

-LRB-

C

-RRB-

such

that

f

-LRB-

l

-LRB-

a

-RRB-

-RRB-

f

-LRB-

a

-RRB-

z.

Set

w

f

-LSB-

z

-RSB-

and

we

see

that

l

-LRB-

a

-RRB-

w.

Thus

l

-LRB-

a

-RRB-

w

a

contradiction

Let

us

now

pick

a

function

which

witnesses

the

extendibility

of

x

XL

to

maximal

consistent

extensions

of

t

-LRB-

W

-RRB-

De?nition

De?ne

a

function

ult

-LRB-

A

-RRB-

ult

A

U

as

follows

-LRB-

u

-RRB-

-LRB-

l

-LRB-

u

-RRB-

if

l

-LRB-

u

-RRB-

W

any

ultra?lter

extending

l

-LSB-

u

-RSB-

-LSB-

t

-LRB-

W

-RRB-

otherwise

We

now

observe

that

W

is

completely

covered

by

Proposition

range

-LRB-

-RRB-

W

Proof

Let

w

W

and

suppose

that

this

is

witnessed

by

C

f

F

and

z

ult

-LRB-

C

-RRB-

That

is

w

f

-LSB-

z

-RSB-

Set

u

f

-LSB-

z

-RSB-

and

we

are

done

with

the

following

claim

-LRB-

since

then

-LRB-

u

-RRB-

l

-LRB-

u

-RRB-

w

-RRB-

Claim

l

-LRB-

u

-RRB-

w

Proof

of

claim

Let

a

A

U

and

take

a

rep

-LRB-

a

-RRB-

a

l

-LRB-

u

-RRB-

-LRB-

-RRB-

a

-LRB-

-RRB-

u

f

-LSB-

z

-RSB-

U

a.e.

-LRB-

-RRB-

f

-LRB-

a

-LRB-

-RRB-

-RRB-

z

U

a.e.

-LRB-

-RRB-

-LRB-

C

-RRB-

-LSB-

f

-LRB-

a

-LRB-

-RRB-

-RRB-

c

U

a.e.

and

c

z

-RSB-

-LRB-

-RRB-

-LRB-

C

-RRB-

-LSB-

f

-LRB-

a

-RRB-

c

U

a.e.

and

c

z

-RSB-

-LRB-

-RRB-

f

-LRB-

a

-RRB-

z

U

a.e.

-LRB-

-RRB-

a

f

-LSB-

z

-RSB-

w

End

of

proof

of

claim

As

the

t

function

is

anti-monotonic

and

since

each

-LRB-

u

-RRB-

extends

t

-LRB-

W

-RRB-

we

can

conclude

Corollary

t

-LRB-

range

-LRB-

-RRB-

-RRB-

t

-LRB-

W

-RRB-

We

already

have

the

h

which

was

given

by

Theorem

so

in

order

to

apply

Theorem

we

need

to

produce

a

g

De?nition

Take

g

A

U

B

to

be

de?ned

by

g

-LRB-

a

-RRB-

x

XL

j

a

-LRB-

x

-RRB-

Since

B

is

a

powerset

boolean

algebra

it

is

immediate

that

Proposition

The

map

g

is

a

boolean

homomorphism

Our

next

result

will

show

that

g

respects

t

-LRB-

W

-RRB-

Proposition

a

A

U

-LSB-

a

t

-LRB-

W

-RRB-

=-RRB-

g

-LRB-

a

-RRB-

B

-RSB-

Proof

Let

a

A

U

and

suppose

that

a

t

-LRB-

W

-RRB-

Thus

a

T

range

-LRB-

-RRB-

so

g

-LRB-

a

-RRB-

x

XL

j

a

-LRB-

x

-RRB-

XL

B

To

show

that

we

are

in

a

position

to

apply

Theorem

we

only

need

prove

Lemma

h

B

A

U

g

ffff

Proof

We

just

need

to

verify

the

last

condition

of

De?nition

which

in

our

case

is

For

each

i

I

a

A

U

g

-LRB-

Ii

-LRB-

h

ffi

g

-LRB-

a

-RRB-

-RRB-

-RRB-

g

-LRB-

Ii

-LRB-

a

-RRB-

-RRB-

We

do

not

need

to

verify

something

so

complicated

since

by

Proposition

together

with

Proposition

we

need

only

show

a

A

U

-LSB-

h

ffi

g

-LRB-

a

-RRB-

a

t

-LRB-

W

-RRB-

-RSB-

So

let

a

A

U

w

W

and

let

x

XL

be

such

that

x

-LRB-

w

-RRB-

so

l

-LRB-

x

-RRB-

w.

Thus

by

the

last

condition

on

h

given

by

Theorem

a

w

-LRB-

-RRB-

a

-LRB-

x

-RRB-

-LRB-

-RRB-

x

g

-LRB-

a

-RRB-

-LRB-

-RRB-

x?o

g

-LRB-

a

-RRB-

-LRB-

-RRB-

h

-LRB-

g

-LRB-

a

-RRB-

-RRB-

l

-LRB-

x

-RRB-

w

Thus

a

h

-LRB-

g

-LRB-

a

-RRB-

-RRB-

w

So

applying

Theorem

we

get

a

J

such

that

-LRB-

B

J

-RRB-

ffl

L

and

canonicity

can

quickly

follow

Theorem

If

L

is

an

even

logic

with

the

nite

model

property

then

L

is

neighborhood

canonical

Proof

Since

we

have

that

-LRB-

B

J

-RRB-

ffl

L

we

need

only

verify

that

-LRB-

A

I

-RRB-

-LRB-

B

J

-RRB-

and

for

this

we

need

only

show

-LRB-

I

-RRB-

-LRB-

A

-RRB-

-LSB-

Ji

-LRB-

a

-RRB-

Ii

-LRB-

a

-RRB-

-RSB-

Let

i

I

and

a

A

Then

Ji

-LRB-

a

-RRB-

g

-LRB-

Ii

-LRB-

h

-LRB-

a

-RRB-

-RRB-

-RRB-

g

-LRB-

Ii

-LRB-

l

-LRB-

a

-RRB-

-RRB-

-RRB-

g

-LRB-

l

-LRB-

Ii

-LRB-

a

-RRB-

-RRB-

-RRB-

x

XL

j

l

-LRB-

Ii

-LRB-

a

-RRB-

-RRB-

-LRB-

x

-RRB-

x

XL

j

l

-LRB-

Ii

-LRB-

a

-RRB-

-RRB-

l

-LSB-

x

-RSB-

x

XL

j

Ii

-LRB-

a

-RRB-

x

Ii

-LRB-

a

-RRB-

where

the

rst

line

follows

by

de?nition

which

gave

us

our

J

the

fth

line

by

-LRB-

x

-RRB-

l

-LSB-

A

-RSB-

l

-LSB-

x

-RSB-

and

the

most

of

the

other

lines

by

l

being

an

elementary

embedding

As

with

our

result

on

non-iterative

logics

we

have

also

essentially

shown

that

each

even

logic

that

also

has

the

nite

model

property

is

complex

Another

simple

extension

of

this

theorem

is

that

if

we

had

another

powerset

boolean

algebra

P

-LRB-

-RRB-

where

l

-LSB-

W

-RSB-

ae

ae

XL

then

we

can

also

nd

a

suitable

J

such

that

-LRB-

J

-RRB-

ffl

L

In

some

sense

the

non

nite

points

were

irrelevant

to

neighborhood

canonicity

Conclusion

Our

main

conclusion

forshadowed

in

our

introduction

is

Corollary

All

uniform

normal

modal

logics

are

neighborhood

canonical

Proof

As

FINE

showed

in

-LSB-

-RSB-

each

logic

in

this

class

has

the

nite

model

property

Since

they

are

uniform

they

are

also

even

and

the

result

follows

Corollary

The

logic

KMcK

is

neighborhood

canonical

This

answers

a

question

posed

in

-LSB-

-RSB-

This

paper

has

enlarged

the

class

of

logics

which

we

know

are

neighborhood

canonical

This

still

leaves

many

gaps

since

there

are

many

logics

which

are

neither

uniform

nor

have

the

nite

model

property

Thus

it

is

natural

to

ask

if

there

are

any

other

syntactically

de?ned

classes

like

the

class

of

even

logics

for

which

similar

tricks

can

be

carried

out

Also

how

important

was

the

assumption

that

the

logics

have

the

nite

model

property

We

needed

it

because

niteness

was

essential

in

taking

a

homomorphism

f

A

C

and

ex

tending

it

to

a

homomorphism

f

A

U

C

If

we

had

another

class

of

even

logics

and

associated

algebras

for

which

we

can

do

this

we

would

get

a

similar

result

Perhaps

this

technique

could

be

used

to

demonstrate

that

neigborhood

canonicity

in

one

cardinal

implies

neighborhood

canonicity

in

higher

cardinals?a

neighborhood

analog

of

a

question

posed

by

GOLDBLATT

-LSB-

-RSB-

Finally

we

can

ask

how

our

neighborhood

constructions

relate

to

the

canonical

relational

constructions

Are

there

lessons

to

be

learnt

here

for

relational

canonicity

set

-LRB-

u

-RRB-

extends

l

-LSB-

u

-RSB-

which

exhaustively

bisects

l

-LSB-

A

-RSB-

Acknowledgements

This

work

was

undertaken

while

the

author

was

a

recipient

of

a

CSFP

award

This

program

is

administered

in

conjunction

with

the

Department

of

Employment

Education

Training

and

Youth

Affairds

representing

the

Commonwealth

of

Australia

References

-LSB-

-RSB-

BENTON

ROY

A.

Strong

modal

completeness

with

respect

to

neighborhood

semantics

Unpublished

manuscript

Department

of

Philosophy

The

University

of

Michigan

-LSB-

-RSB-

CHELLAS

BRIAN

F.

Modal

Logic

An

Introduction

Cambridge

University

Press

Cambridge

-LSB-

-RSB-

DOSEN

KOSTA

Duality

between

modal

algebras

and

neighbourhood

frames

Studia

Logica

vol

-LRB-

-RRB-

no.

pp.

-LSB-

-RSB-

FINE

KIT

Normal

Forms

in

Modal

Logic

Notre

Dame

Journal

of

Formal

Logic

vol

-LRB-

-RRB-

no.

pp.

-LSB-

-RSB-

GROVE

ADAM

Two

modellings

for

theory

change

Journal

of

Philosophical

Logic

vol

-LRB-

-RRB-

pp.

-LSB-

-RSB-

GOLDBLATT

ROBERT

The

Mathematics

of

Modality

CSLI

Lecture

Notes

Stanford

California

-LSB-

-RSB-

GOLDBLATT

ROBERT

The

McKinsey

axiom

is

not

canonical

The

Journal

of

Symbolic

Logic

vol

-LRB-

-RRB-

no.

pp.

-LSB-

-RSB-

SURENDONK

TIMOTHY

JOHN

Canonicity

for

intensional

logics

without

iterative

axioms

Journal

of

Philosophical

Logic

-LRB-

To

appear

-RRB-

-LSB-

-RSB-

SURENDONK

TIMOTHY

JOHN

Neighborhoods

Ultra?lters

and

Canonicity

Technical

Report

Automated

Reasoning

Project

Australian

National

University

Canberra

-LSB-

-RSB-

WANG

XIAOPING

The

McKinsey

axiom

is

not

compact

The

Journal

of

Symbolic

Logic

vol

-LRB-

-RRB-

no.

pp.

Automated

Reasoning

Project

Australian

National

University

Canberra

ACT

Australia

Email

Timothy.Surendonk@anu.edu.au

WWW

canonicity
strong completeness
uniform logics

neighborhood frame semantics
non-normal modal logics


Degree-Constrained

Spanners

for

Multi-dimensional

Grids

Arthur

L.

Liestman

Thomas

C.

Shermer

Christopher

R.

Stolte

School

of

Computing

Science

Simon

Fraser

University

Burnaby

B.C.

Canada

CMPT

TR

August

Abstract

A

spanning

subgraph

S

-LRB-

V

-RRB-

of

a

connected

simple

graph

G

-LRB-

V

E

-RRB-

is

a

f

-LRB-

x

-RRB-

spanner

if

for

any

pair

of

nodes

u

and

v

dS

-LRB-

u

v

-RRB-

f

-LRB-

dG

-LRB-

u

v

-RRB-

-RRB-

where

dG

and

dS

are

the

usual

distance

functions

in

graphs

G

and

S

respectively

The

delay

of

the

f

-LRB-

x

-RRB-

spanner

is

f

-LRB-

x

-RRB-

x

We

construct

two

spanners

with

maximum

degree

for

d-dimensional

grids

one

with

delay

and

one

with

delay

-LRB-

e

dd

e

-RRB-

and

discuss

generalizations

of

these

spanners

We

also

construct

a

-LRB-

x

-RRB-

spanner

with

maximum

degree

for

d-dimensional

grids

For

the

particular

cases

of

and

grids

we

improve

upon

this

construction

giving

-LRB-

x

-RRB-

spanners

and

-LRB-

x

-RRB-

spanners

respectively

This

research

was

supported

by

the

Natural

Sciences

and

Engineering

Research

Council

of

Canada

under

Grants

No

and

Introduction

and

Definitions

There

are

several

popular

topologies

used

for

constructing

parallel

computers

Our

goal

is

to

determine

substructures

of

such

topologies

with

smaller

maximum

degree

We

require

also

that

these

substructures

called

spanners

have

the

property

that

the

distance

between

two

vertices

in

the

substructure

is

not

significantly

larger

than

the

corresponding

distance

in

the

original

structure

A

network

is

represented

by

a

connected

simple

graph

G

-LRB-

V

E

-RRB-

We

use

the

notation

dG

-LRB-

u

v

-RRB-

to

denote

the

distance

between

vertices

u

and

v

in

G

The

maximum

degree

of

G

is

denoted

G

and

the

average

degree

is

denoted

ffiG

The

subscript

G

may

be

omitted

if

it

is

clear

from

context

We

introduced

a

general

definition

of

graph

spanner

in

-LSB-

-RSB-

A

spanning

subgraph

S

of

a

connected

simple

graph

G

is

an

f

-LRB-

x

-RRB-

spanner

if

for

any

pair

of

nodes

u

and

v

dS

-LRB-

u

v

-RRB-

f

-LRB-

dG

-LRB-

u

v

-RRB-

-RRB-

We

call

dS

-LRB-

u

v

-RRB-

dG

-LRB-

u

v

-RRB-

the

delay

between

vertices

u

and

v

in

S

denoted

-LRB-

u

v

-RRB-

For

an

f

-LRB-

x

-RRB-

spanner

S

we

let

f

-LRB-

x

-RRB-

f

-LRB-

x

-RRB-

x

and

refer

to

f

-LRB-

x

-RRB-

as

the

delay

of

the

spanner

Note

that

f

-LRB-

x

-RRB-

is

an

upper

bound

-LRB-

but

not

necessarily

a

tight

bound

-RRB-

on

the

maximum

delay

in

S

between

any

pair

of

vertices

at

distance

x

in

G

It

may

be

possible

to

express

the

delay

f

-LRB-

x

-RRB-

in

several

ways

Although

any

spanner

S

of

a

finite

graph

G

is

an

-LRB-

x

c

-RRB-

spanner

where

c

is

the

maximum

delay

between

any

pair

of

vertices

in

S

a

more

careful

analysis

of

S

may

reveal

a

closer

relationship

between

the

distance

in

G

and

the

delay

in

S.

For

example

the

-LRB-

x

c

-RRB-

spanner

mentioned

above

may

also

be

determined

to

be

a

In

general

we

prefer

to

express

f

-LRB-

x

-RRB-

in

a

manner

that

bounds

the

delay

as

clearly

as

possible

Spanners

were

introduced

by

Peleg

and

Ullman

-LSB-

-RSB-

who

used

these

structures

in

order

to

efficiently

simulate

synchronous

networks

on

asynchronous

networks

Various

aspects

of

spanners

have

been

investigated

in

recent

papers

-LSB-

-RSB-

Our

particular

interest

here

is

in

spanners

of

multi-dimensional

grids

Liestman

and

Shermer

-LSB-

-RSB-

described

how

to

construct

low

average

degree

-LRB-

t

x

-RRB-

spanners

for

grids

In

-LSB-

-RSB-

they

proved

that

maximum

degree

spanners

of

two-dimensional

grids

require

-LRB-

x

-RRB-

delay

and

showed

how

to

construct

such

spanners

with

O

-LRB-

x

-RRB-

delay

In

this

paper

we

show

how

to

construct

low

degree

spanners

of

d-dimensional

grids

for

d

It

will

be

convenient

to

describe

our

constructions

on

infinite

grids

and

then

to

consider

only

a

finite

part

of

the

construction

An

infinite

d-dimensional

grid

denoted

G

-LRB-

d

-RRB-

has

the

vertex

set

V

Zd

Its

edges

are

between

pairs

of

vertices

whose

labels

differ

by

in

exactly

one

position

that

is

vertex

-LRB-

uk

ud

-RRB-

is

connected

to

exactly

those

vertices

-LRB-

uk

ud

-RRB-

and

-LRB-

uk

ud

-RRB-

where

k

d

A

finite

d-dimensional

grid

denoted

nd

is

the

induced

subgraph

of

G

-LRB-

d

-RRB-

on

vertices

-LRB-

ud

-RRB-

with

uk

nk

for

k

d.

To

avoid

confusion

we

will

insist

that

nk

for

each

k.

Construction

of

Grid

Spanners

with

Maximum

Degree

In

this

section

we

describe

maximum

degree

spanners

of

G

-LRB-

d

-RRB-

called

the

layered

and

double

layered

spanners

Both

of

these

spanners

can

easily

be

modified

to

obtain

spanners

for

finite

grids

We

also

discuss

several

generalizations

of

these

spanners

To

construct

a

maximum

degree

layered

spanner

S

of

an

infinite

d-dimensional

grid

G

-LRB-

d

-RRB-

we

connect

every

vertex

-LRB-

ud

-RRB-

to

-LRB-

ud

-RRB-

and

also

to

-LRB-

uk

uk

uk

ud

-RRB-

where

-LRB-

k

-RRB-

-LRB-

mod

d

-RRB-

For

any

integer

j

the

jth

layer

denotes

the

collection

of

vertices

of

the

grid

having

first

coordinate

j

In

the

layered

spanner

each

layer

contains

all

of

its

dimension

k

edges

for

some

k

d.

Figure

shows

a

dimensional

slice

-LRB-

dimensions

and

-RRB-

of

a

layered

spanner

Theorem

The

layered

spanner

S

is

a

-LRB-

x

-RRB-

spanner

of

G

-LRB-

d

-RRB-

with

S

and

ffiS

PROOF

Given

two

vertices

u

-LRB-

ud

-RRB-

and

v

-LRB-

vd

-RRB-

dG

-LRB-

u

v

-RRB-

Pdi

jui

vij

We

wish

to

bound

the

delay

between

u

and

v

in

S

Since

S

contains

all

edges

of

dimension

there

is

a

path

from

u

to

v

consisting

of

a

path

in

dimension

from

u

to

-LRB-

ud

-RRB-

followed

by

a

path

from

to

v.

Thus

dS

-LRB-

u

v

-RRB-

dS

-LRB-

v

-RRB-

Since

dG

-LRB-

u

v

-RRB-

dG

-LRB-

v

-RRB-

-LRB-

u

v

-RRB-

-LRB-

v

-RRB-

From

vertex

-LRB-

ud

-RRB-

we

can

move

in

dimension

i

-LRB-

mod

d

-RRB-

without

delay

In

particular

we

can

move

from

-LRB-

ui

ud

-RRB-

to

-LRB-

ui

vi

ui

ud

-RRB-

using

exactly

jui

vij

edges

From

this

vertex

we

move

to

-LRB-

ui

vi

ui

ud

-RRB-

using

edge

From

this

vertex

we

can

move

in

dimension

j

-LRB-

mod

d

-RRB-

without

delay

Note

that

j

will

be

either

i

or

-LRB-

The

latter

occurs

when

i

d.

-RRB-

d

d

d

d

Figure

A

slice

of

a

layered

spanner

showing

dimension

vertically

and

dimension

horizontally

Label

i

on

a

vertex

indicates

that

the

vertex

is

connected

to

its

neighbors

in

dimension

i

As

before

we

move

until

the

coordinate

in

dimension

j

is

vj

By

performing

d

iterations

of

this

process

of

stepping

positively

one

edge

in

dimension

and

then

ju

v

j

edges

in

dimension

we

eventually

arrive

at

-LRB-

-LRB-

d

-RRB-

vd

-RRB-

From

we

follow

d

edges

in

dimension

to

arrive

at

v

The

length

of

this

path

from

to

v

is

jui?vij

P

k

d

k

i

juk

vk

j

-LRB-

d

-RRB-

-LRB-

d

-RRB-

dG

-LRB-

v

-RRB-

Thus

-LRB-

v

-RRB-

-LRB-

u

v

-RRB-

and

thus

for

any

x

f

-LRB-

x

-RRB-

Let

nd

be

a

finite

d-dimensional

grid

and

be

the

subspanner

of

the

d-dimensional

layered

spanner

S

induced

by

the

vertices

of

If

then

the

analysis

in

the

above

proof

holds

for

except

when

-LRB-

-RRB-

-LRB-

d

-RRB-

In

this

case

one

can

not

step

positively

d

times

in

dimension

as

required

by

the

description

of

the

path

However

by

substituting

negative

steps

in

dimension

for

positive

ones

-LRB-

and

vice

versa

-RRB-

we

stay

within

the

grid

and

the

same

analysis

holds

We

can

view

the

constructed

path

as

a

cycle

of

length

in

dimension

interspersed

with

one

single

dimensional

path

Pk

in

each

other

dimension

k

It

is

not

necessary

that

all

of

the

Pk

's

are

grouped

at

the

beginning

of

the

cycle

In

fact

any

cycle

in

dimension

that

encounters

d

distinct

dimension

coordinates

can

be

used

in

the

construction

inserting

the

Pk

appropriately

In

particular

we

can

use

any

cycle

that

consists

of

l

dimension

edges

in

the

positive

direction

followed

by

d

dimension

edges

in

the

negative

direction

followed

by

d

l

dimension

edges

in

the

positive

direction

Thus

we

can

relax

our

requirement

of

in

our

analysis

of

the

delay

of

to

d

In

fact

as

long

as

ni

d

for

some

i

d

we

can

construct

a

layered

spanner

with

the

same

delay

bound

by

exchanging

dimensions

i

and

Corollary

a

For

any

finite

d-dimensional

grid

nd

with

ni

d

for

some

i

d

there

exists

a

-LRB-

x

-RRB-

spanner

of

with

and

ffi

To

construct

a

maximum

degree

double

layered

spanner

S

of

an

infinite

d-dimensional

grid

G

-LRB-

d

-RRB-

we

begin

by

connecting

every

vertex

-LRB-

ud

-RRB-

to

-LRB-

ud

-RRB-

When

is

even

we

connect

-LRB-

ud

-RRB-

to

-LRB-

uk

uk

uk

ud

-RRB-

where

k

-LRB-

mod

e

-RRB-

When

is

odd

we

consider

three

cases

If

-LRB-

mod

e

-RRB-

then

we

connect

-LRB-

ud

-RRB-

to

-LRB-

ud

-RRB-

If

e

-LRB-

mod

e

-RRB-

and

d

is

odd

we

add

no

further

connections

to

-LRB-

ud

-RRB-

Otherwise

we

connect

-LRB-

ud

-RRB-

to

-LRB-

uk

uk

uk

ud

-RRB-

where

k

-LRB-

mod

e

-RRB-

e

Figure

shows

a

dimensional

slice

-LRB-

dimensions

and

-RRB-

of

a

double

layered

spanner

Theorem

The

double

layered

spanner

S

is

a

-LRB-

-LRB-

dd

e

-RRB-

x

-RRB-

spanner

of

G

-LRB-

d

-RRB-

with

S

and

ffiS

PROOF

Given

two

vertices

u

-LRB-

ud

-RRB-

and

v

-LRB-

vd

-RRB-

dG

-LRB-

u

v

-RRB-

Pdi

jui

vij

We

wish

to

bound

the

delay

between

u

and

v

in

S

Since

S

contains

all

edges

of

dimension

there

is

a

path

from

u

to

v

consisting

of

a

path

in

dimension

from

u

to

-LRB-

ud

-RRB-

followed

by

a

path

from

to

v.

Thus

dS

-LRB-

u

v

-RRB-

dS

-LRB-

v

-RRB-

Since

dG

-LRB-

u

v

-RRB-

dG

-LRB-

v

-RRB-

-LRB-

u

v

-RRB-

-LRB-

v

-RRB-

From

vertex

we

move

to

-LRB-

ud

-RRB-

by

taking

b

b

dd

c

dd

e

dimension

edges

to

the

nearest

vertex

with

neighbors

in

dimension

followed

by

a

path

of

edges

in

dimension

and

b

more

edges

to

-LRB-

a

-RRB-

d

d

-LRB-

b

-RRB-

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

Figure

A

slice

of

a

double

layered

spanner

showing

dimension

vertically

and

dimension

horizontally

Label

i

on

a

vertex

indicates

that

the

vertex

is

connected

to

its

neighbors

in

dimension

i

We

will

construct

a

path

from

to

v

by

interspersing

single

dimensional

paths

in

dimensions

through

d

into

a

short

cycle

The

cycle

consists

of

paths

of

e

edges

in

dimension

joined

on

both

ends

by

a

single

dimension

edge

Note

that

such

a

cycle

will

encounter

a

vertex

with

edges

in

dimension

k

for

all

k

d

At

each

such

vertex

we

include

the

appropriate

path

of

length

juk

vkj

in

dimension

k

The

total

path

length

from

u

to

v

is

-LRB-

-RRB-

e

Pdk

juk

vkj

dG

-LRB-

u

v

-RRB-

e

dG

-LRB-

u

v

-RRB-

e

e

Note

that

the

maximum

delay

in

the

double

layered

spanner

is

d

which

is

better

than

the

delay

of

the

layered

spanner

by

about

d.

Corollary

a

For

any

finite

d-dimensional

grid

nd

with

ni

e

for

some

i

d

there

exists

a

-LRB-

-LRB-

dd

e

-RRB-

x

-RRB-

spanner

of

with

and

ffi

PROOF

We

assume

that

the

dimensions

have

been

permuted

so

that

e

We

could

construct

a

finite

double

layered

spanner

by

simply

taking

an

induced

subspanner

of

the

infinite

double

layered

spanner

However

if

-LRB-

mod

e

-RRB-

is

large

then

vertices

with

the

largest

coordinate

in

dimension

would

incur

a

large

delay

because

they

are

far

from

all

of

the

short

cycles

used

in

the

paths

from

to

v

in

the

proof

of

Theorem

Consequently

to

construct

a

double

layered

spanner

for

nd

we

first

shift

the

infinite

double

layered

spanner

-LRB-

mod

d

e

-RRB-

c

units

positively

in

dimension

and

then

take

the

induced

subspanner

The

analysis

is

then

the

same

as

in

Theorem

except

that

may

not

be

on

a

short

cycle

If

this

is

the

case

then

the

path

from

to

contains

a

vertex

with

edges

in

both

dimensions

and

This

vertex

is

on

a

short

cycle

and

thus

the

short

cycle

-LRB-

interspersed

with

single

dimensional

paths

-RRB-

can

be

spliced

into

the

path

at

this

point

-LRB-

rather

than

occuring

after

it

-RRB-

Thus

we

arrive

at

the

same

delay

bound

of

-LRB-

e

dd

e

-RRB-

There

are

several

possible

generalizations

of

the

layered

and

double

layered

spanners

Note

that

paths

in

the

layered

spanner

were

described

in

terms

of

a

cycle

of

dimension

edges

interspersed

with

single

dimensional

paths

in

higher

dimensions

Paths

in

the

double

layered

spanner

were

described

in

terms

of

a

cycle

of

two

paths

of

roughly

dimension

edges

joined

by

two

dimension

edges

An

obvious

generalization

would

be

to

construct

a

cycle

of

four

paths

of

roughly

dimension

edges

joined

by

two

dimension

edges

and

two

dimension

edges

A

crucial

point

in

the

analysis

of

this

generalized

spanner

is

to

note

that

vertices

with

edges

in

dimension

on

one

short

cycle

are

adjacent

by

dimension

edges

to

vertices

with

edges

in

dimension

on

another

short

cycle

This

helps

us

to

obtain

a

short

path

from

u

-LRB-

ud

-RRB-

-LRB-

ud

-RRB-

This

generalization

yields

a

-LRB-

e

e

x

-RRB-

spanner

of

G

-LRB-

d

-RRB-

with

and

ffi

To

describe

our

next

generalization

we

consider

the

layers

of

a

double

layered

spanner

Starting

from

a

layer

with

dimension

edges

by

increasing

the

dimension

coordinate

we

encounter

dd

e

layers

that

each

contain

edges

in

two

different

dimensions

This

pattern

of

a

dimension

layer

followed

by

dd

e

other

layers

is

cyclically

repeated

across

the

entirety

of

dimension

When

d

is

large

one

may

improve

the

delay

of

the

spanner

by

interleaving

k

additional

dimension

layers

equally

spaced

in

each

set

of

dd

e

layers

This

increases

the

length

of

the

cycle

by

but

decreases

the

delay

from

to

in

our

analysis

This

results

in

a

spanner

with

delay

approximately

d

with

and

ffi

We

can

also

generalize

these

constructions

by

allowing

the

maximum

degree

to

increase

For

instance

we

may

create

a

spanner

like

the

layered

spanner

but

where

each

layer

contains

all

of

the

edges

in

k

dimensions

instead

of

just

a

single

dimension

This

gives

Theorem

There

exists

a

-LRB-

e

x

-RRB-

spanner

S

of

G

-LRB-

d

-RRB-

with

S

and

ffiS

By

performing

a

similar

generalization

starting

with

the

double

layered

spanner

we

obtain

a

spanner

of

G

-LRB-

d

-RRB-

with

S

and

ffiS

with

delay

approximately

dk

Construction

of

Grid

Spanners

with

Maximum

Degree

In

this

section

we

construct

a

good

spanner

of

G

-LRB-

d

-RRB-

with

maximum

degree

We

begin

by

discussing

the

general

method

of

construction

and

then

present

a

more

detailed

specific

construction

We

first

decompose

G

-LRB-

d

-RRB-

into

hypercubes

connected

by

edges

Each

vertex

-LRB-

ud

-RRB-

belongs

to

a

hypercube

H

ffd

where

i

c

for

i

d

The

edges

of

H

ffd

are

those

induced

by

the

vertices

of

H

ffd

in

G

-LRB-

d

-RRB-

The

remaining

edges

of

G

-LRB-

d

-RRB-

connect

adjacent

hypercubes

A

highway

is

an

infinite

line

of

edges

in

some

dimension

i

that

is

the

subgraph

induced

in

G

-LRB-

d

-RRB-

by

the

set

of

vertices

-LRB-

ui

x

ui

ud

-RRB-

where

the

u

's

are

constant

and

the

x

varies

over

the

integers

A

d-dimensional

general

bypass

spanner

is

a

spanner

of

G

-LRB-

d

-RRB-

such

that

every

hypercube

H

ffd

is

intersected

by

at

least

one

highway

in

each

dimension

and

each

hypercube

H

ffd

induces

a

connected

subgraph

of

the

spanner

Consider

the

two

vertices

u

-LRB-

ud

-RRB-

in

H

ffd

and

v

-LRB-

vd

-RRB-

in

H

fid

To

find

a

path

from

u

to

v

in

a

general

bypass

spanner

S

of

the

grid

G

G

-LRB-

d

-RRB-

we

examine

the

indices

d

and

d.

Suppose

that

i

i

for

some

i

d.

Then

by

moving

from

u

to

a

vertex

in

H

ffd

from

which

a

highway

in

dimension

i

exits

the

hypercube

in

the

direction

of

H

ffi

fii

ffi

ffd

and

following

the

dimension

i

highway

from

H

ffd

to

the

first

vertex

of

H

ffi

fii

ffi

ffd

encountered

we

have

reduced

the

number

of

indices

in

which

our

source

and

destination

hypercubes

differ

By

repeating

this

process

we

eventually

arrive

at

a

vertex

of

H

fid

-LRB-

which

could

be

u

if

i

i

for

i

d

-RRB-

To

get

from

to

v

we

follow

any

path

between

and

v

in

H

fid

Because

each

hypercube

of

the

construction

is

connected

any

such

path

between

two

vertices

of

the

same

hypercube

is

of

length

no

more

than

Let

denote

the

number

of

dimensions

i

such

that

i

i

The

constructed

path

from

u

to

v

consists

of

subpaths

each

in

a

single

hypercube

alternating

with

sections

of

highway

edges

Thus

the

total

path

length

is

at

most

-LRB-

-RRB-

-LRB-

-RRB-

plus

the

length

of

the

highway

sections

As

the

highway

sections

used

in

the

path

construction

are

shortest

paths

between

the

hypercubes

involved

the

length

of

the

section

of

dimension

i

highway

is

jui

vij

Thus

the

total

length

of

all

highway

sections

is

Pdi

jui

vij

dG

-LRB-

u

v

-RRB-

This

implies

that

the

total

path

length

in

S

is

at

most

-LRB-

-RRB-

-LRB-

-RRB-

dG

-LRB-

u

v

-RRB-

and

thus

the

delay

is

at

most

-LRB-

-RRB-

-LRB-

-RRB-

As

d

this

gives

delay

at

most

-LRB-

d

-RRB-

-LRB-

-RRB-

Thus

any

general

bypass

spanner

of

the

infinite

d-dimensional

grid

is

a

-LRB-

c

x

-RRB-

spanner

for

some

constant

c

If

a

general

bypass

spanner

is

such

that

the

highways

are

regularly

spaced

and

each

hypercube

contains

the

same

substructure

we

call

it

a

regular

bypass

spanner

To

describe

a

regular

bypass

spanner

S

we

need

only

describe

which

edges

of

are

included

in

S

and

designate

at

least

one

such

edge

in

each

dimension

as

a

highway

edge

The

structure

included

in

is

included

in

each

hypercube

H

ffd

that

is

if

the

edge

from

-LRB-

ui

ui

ud

-RRB-

to

-LRB-

ui

ui

ud

-RRB-

is

included

in

S

then

all

edges

-LRB-

ui

i

i

ui

i

ud

d

-RRB-

to

-LRB-

ui

i

i

ui

i

ud

d

-RRB-

are

also

included

in

S

The

endpoints

of

a

dimension

i

highway

edge

also

have

edges

leading

to

their

other

neighbor

in

dimension

i.

Consider

two

adjacent

hypercubes

H

ffd

and

H

ffi

ffi

ffi

ffd

These

two

hypercubes

each

contain

a

designated

dimension

i

highway

edge

Furthermore

there

is

a

dimension

i

edge

connecting

these

two

designated

highway

edges

To

complete

S

we

include

all

such

connecting

highway

edges

We

now

construct

a

maximum

degree

regular

bypass

spanner

Sd

of

G

-LRB-

d

-RRB-

for

d

The

constant

delay

of

this

spanner

is

O

-LRB-

d

-RRB-

which

is

significantly

better

than

the

O

-LRB-

-RRB-

delay

of

a

general

bypass

spanner

We

label

the

nodes

of

a

d-dimensional

hypercube

with

d-bit

binary

strings

We

follow

the

convention

that

the

leftmost

bit

corresponds

to

the

vertex

's

coordinate

in

dimension

and

the

rightmost

bit

corresponds

to

the

vertex

's

coordinate

in

dimension

d

A

double

-LCB-

rooted

binary

tree

is

a

full

binary

tree

in

which

the

root

node

has

been

split

into

two

nodes

-LRB-

one

with

the

left

subtree

hanging

off

of

it

and

the

other

with

the

right

subtree

-RRB-

that

are

connected

by

an

edge

Each

of

the

vertices

created

by

the

split

is

known

as

a

root

of

the

double

-LCB-

rooted

binary

tree

and

the

edge

between

them

is

called

the

root

edge

A

double

-LCB-

rooted

binary

tree

is

shown

in

Figure

Following

-LSB-

-RSB-

we

construct

a

double

-LCB-

rooted

binary

tree

DRB

-LRB-

d

-RRB-

as

a

spanning

subgraph

of

a

d-cube

as

follows

First

we

let

DRB

-LRB-

-RRB-

DRB

-LRB-

-RRB-

and

DRB

-LRB-

-RRB-

be

the

trees

shown

in

Figure

b

and

c.

Note

that

for

DRB

-LRB-

-RRB-

the

root

edge

of

the

tree

is

in

dimension

d

and

is

incident

on

the

origin

The

two

edges

adjacent

to

the

root

edge

are

in

dimension

d

-LRB-

from

-RRB-

and

in

dimension

d

-LRB-

from

-RRB-

We

wish

for

our

general

construction

to

have

the

above

properties

i.e.

the

root

edge

is

in

dimension

d

from

and

the

two

edges

adjacent

to

the

root

are

in

dimension

d

from

and

in

dimension

d

from

When

constructing

DRB

-LRB-

d

-RRB-

we

assume

that

this

is

true

for

all

of

the

trees

DRB

-LRB-

-RRB-

where

d

We

now

construct

another

embedding

of

the

double

-LCB-

rooted

binary

tree

symmetric

to

and

de

Figure

A

double

-LCB-

rooted

binary

tree

-LRB-

c

-RRB-

-LRB-

a

-RRB-

-LRB-

b

-RRB-

Figure

The

embedded

double

-LCB-

rooted

binary

trees

DRB

-LRB-

-RRB-

DRB

-LRB-

-RRB-

and

DRB

-LRB-

-RRB-

DRB

-LRB-

d

-RRB-

TWIST

-LRB-

d

-RRB-

DRB

-LRB-

d

-RRB-

Figure

Making

a

double

-LCB-

rooted

binary

tree

from

two

smaller

ones

rived

from

DRB

-LRB-

d

-RRB-

We

call

this

embedding

TWIST

-LRB-

d

-RRB-

Each

vertex

X

-LRB-

xd

xd

xd

xd

-RRB-

of

DRB

-LRB-

d

-RRB-

is

mapped

to

a

vertex

OE

-LRB-

X

-RRB-

-LRB-

xd

-RRB-

of

TWIST

-LRB-

d

-RRB-

where

xd

xd

and

xd

Each

edge

-LRB-

X

Y

-RRB-

of

DRB

-LRB-

d

-RRB-

gives

rise

to

an

edge

-LRB-

OE

-LRB-

X

-RRB-

OE

-LRB-

Y

-RRB-

-RRB-

of

TWIST

-LRB-

d

-RRB-

In

TWIST

-LRB-

d

-RRB-

the

root

edge

is

in

dimension

d

from

and

the

two

edges

adjacent

to

the

root

are

in

dimension

d

from

and

in

dimension

d

from

To

construct

DRB

-LRB-

d

-RRB-

we

start

by

placing

DRB

-LRB-

d

-RRB-

in

the

d-cube

with

-LRB-

d

-RRB-

coordinate

and

placing

TWIST

-LRB-

d

-RRB-

in

the

d-cube

with

-LRB-

d

-RRB-

coordinate

We

then

add

three

edges

in

dimension

d

from

the

vertices

and

to

the

vertices

and

respectively

We

then

remove

the

edge

from

to

and

the

edge

from

to

Conceptually

we

have

spliced

together

two

double

-LCB-

rooted

binary

trees

to

obtain

a

larger

one

as

in

Figure

The

resulting

tree

is

DRB

-LRB-

d

-RRB-

and

indeed

it

has

the

root

edge

in

dimension

d

from

and

the

two

edges

adjacent

to

the

root

edge

in

dimension

d

from

and

in

dimension

d

from

DRB

-LRB-

-RRB-

is

shown

in

Figure

1111 1001 1011

1100 0110 0010

-LRB-

c

-RRB-

-LRB-

a

-RRB-

-LRB-

b

-RRB-

Figure

The

tree

DRB

-LRB-

-RRB-

Lemma

For

d

a

vertex

is

a

leaf

of

DRB

-LRB-

d

-RRB-

iff

it

is

a

leaf

of

either

the

DRB

-LRB-

d

-RRB-

or

the

TWIST

-LRB-

d

-RRB-

used

in

the

construction

of

DRB

-LRB-

d

-RRB-

PROOF

For

d

DRB

-LRB-

d

-RRB-

was

constructed

from

the

DRB

-LRB-

d

-RRB-

and

TWIST

-LRB-

d

-RRB-

by

adding

and

deleting

some

edges

incident

on

the

roots

of

these

trees

or

incident

on

vertices

adjacent

to

such

roots

As

d

the

depths

of

the

underlying

double

-LCB-

rooted

binary

trees

for

both

DRB

-LRB-

d

-RRB-

and

TWIST

-LRB-

d

-RRB-

are

at

least

two

Thus

no

leaf

is

a

root

and

no

leaf

is

adjacent

to

a

root

Therefore

the

degree

of

any

leaf

in

either

the

DRB

-LRB-

d

-RRB-

or

TWIST

-LRB-

d

-RRB-

is

unchanged

when

the

two

copies

are

put

together

Furthermore

no

new

leaves

are

created

Corollary

a

For

d

the

vertices

and

are

leaves

of

DRB

-LRB-

d

-RRB-

We

now

describe

which

edges

of

are

included

in

Sd

The

structure

in

is

composed

of

a

short

cycle

containing

highway

edges

in

each

dimension

and

a

large

tree

connected

to

the

cycle

at

a

single

vertex

The

large

tree

is

essentially

composed

of

several

instances

of

DRB

-LRB-

i

-RRB-

of

different

sizes

i

all

hanging

off

of

a

short

path

For

d

the

vertices

of

the

d-cube

can

be

decomposed

into

the

vertices

of

a

-LRB-

d

-RRB-

cube

Qd

with

xd

a

-LRB-

d

-RRB-

cube

Qd

with

xd

and

xd

a

-LRB-

d

-RRB-

cube

Qd

with

xd

and

xd

xd

and

so

on

until

we

have

one

with

and

xd

and

another

Q

with

xd

For

i

d

we

form

a

tree

T

-LRB-

i

-RRB-

in

Qi

by

first

placing

DRB

-LRB-

i

-RRB-

in

Qi

and

then

deleting

the

three

vertices

-LRB-

i

-RRB-

-LRB-

i

-RRB-

and

-LRB-

i

-RRB-

that

have

the

first

i

coordinates

and

respectively

Note

that

these

vertices

were

leaves

of

DRB

-LRB-

i

-RRB-

by

Corollary

a

and

thus

T

-LRB-

i

-RRB-

is

a

tree

The

cycle

of

the

spanner

of

will

contain

the

vertices

of

Q

and

all

of

the

vertices

-LRB-

i

-RRB-

and

-LRB-

i

-RRB-

The

vertices

-LRB-

i

-RRB-

will

be

used

to

connect

together

the

trees

T

-LRB-

i

-RRB-

to

form

a

large

tree

that

will

be

connected

to

the

cycle

in

Qd

The

order

of

vertices

on

the

cycle

is

-LRB-

d

-RRB-

-LRB-

d

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

d

-RRB-

The

high

-LCB-

dimensional

part

of

this

cycle

is

shown

in

Figure

The

highway

edges

are

in

dimension

d

between

-LRB-

d

-RRB-

and

-LRB-

d

-RRB-

in

dimension

d

between

-LRB-

d

-RRB-

and

-LRB-

d

-RRB-

in

dimension

d

between

-LRB-

d

-RRB-

and

-LRB-

d

-RRB-

and

so

on

alternating

between

vertices

and

vertices

dimension

-LRB-

-RRB-

edges

v

-LRB-

-RRB-

v

-LRB-

-RRB-

Q

Q

QQ

v

-LRB-

-RRB-

v

-LRB-

-RRB-

-LRB-

-RRB-

v

-LRB-

-RRB-

d

dimension

edges

Figure

Upper

part

of

cycle

with

highway

edges

shown

as

thick

edges

-LRB-

a

-RRB-

-LRB-

b

-RRB-

Figure

Lower

part

of

cycle

with

highway

edges

shown

as

thick

edges

If

d

is

odd

this

alternation

ends

at

a

dimension

edge

between

-LRB-

-RRB-

and

-LRB-

-RRB-

The

remaining

highway

edges

are

in

dimension

between

-LRB-

-RRB-

and

in

dimension

between

and

in

dimension

between

and

and

in

dimension

between

and

-LRB-

see

Figure

-RRB-

If

d

is

even

the

alternation

ends

at

a

dimension

edge

between

-LRB-

-RRB-

and

-LRB-

-RRB-

The

remaining

highway

edges

are

in

dimension

between

-LRB-

-RRB-

and

in

dimension

between

and

in

dimension

between

and

and

in

dimension

between

and

-LRB-

see

Figure

-RRB-

Recall

that

vertices

on

highway

edges

are

connected

outside

of

the

hypercube

in

the

direction

of

the

highway

edge

Thus

each

highway

edge

effectively

contributes

two

to

the

degree

of

its

endpoints

in

the

final

bypass

spanner

construction

Note

that

in

the

cycle

as

constructed

no

vertex

has

more

than

one

highway

edge

incident

on

it

and

thus

these

vertices

will

have

degree

at

most

three

in

the

bypass

spanner

Furthermore

the

vertex

-LRB-

d

-RRB-

has

no

highway

edges

incident

on

it

and

thus

has

only

degree

two

so

far

To

complete

our

construction

-LRB-

forming

the

large

tree

referred

to

above

-RRB-

we

extend

the

path

-LRB-

d

-RRB-

-LRB-

d

-RRB-

-LRB-

d

-RRB-

-LRB-

-RRB-

from

the

vertex

-LRB-

d

-RRB-

on

the

cycle

This

makes

-LRB-

d

-RRB-

degree

-LRB-

-RRB-

degree

and

the

remaining

vertices

-LRB-

i

-RRB-

degree

In

each

Qi

we

attach

T

-LRB-

i

-RRB-

to

this

path

by

the

edge

in

dimension

between

the

vertex

r

-LRB-

i

-RRB-

with

the

first

i

coordinates

and

-LRB-

i

-RRB-

which

has

the

first

i

coordinates

Adding

these

trees

makes

the

degree

of

each

of

the

vertices

-LRB-

i

-RRB-

be

at

most

Note

that

the

vertex

r

-LRB-

i

-RRB-

being

the

origin

of

Qi

is

a

root

of

DRB

-LRB-

i

-RRB-

and

hence

has

degree

two

in

T

-LRB-

i

-RRB-

Adding

the

edge

to

-LRB-

i

-RRB-

makes

each

r

-LRB-

i

-RRB-

have

degree

This

completes

the

construction

of

the

substructure

of

and

together

with

the

inclusion

of

the

appropriate

highway

edges

between

adjacent

hypercubes

as

described

above

it

also

completes

the

construction

of

the

bypass

spanner

Sd

of

G

-LRB-

d

-RRB-

which

has

maximum

degree

three

Since

the

construction

of

the

bypass

spanner

Sd

partitions

the

d-grid

into

d-cubes

and

then

replicates

the

same

structure

on

each

d-cube

the

average

degree

for

the

spanner

is

the

same

as

the

average

degree

for

a

single

hypercube

In

any

such

d-cube

there

are

vertices

Recall

that

the

structure

of

Sd

in

a

d-cube

is

a

cycle

with

an

attached

tree

and

therefore

has

edges

Each

of

these

edges

contributes

to

the

total

degree

in

the

d-cube

and

each

of

the

highway

edges

leading

to

other

d-cubes

contributes

to

the

total

degree

Thus

the

average

degree

in

Sd

is

d

Given

two

vertices

u

-LRB-

ud

-RRB-

in

H

ffd

and

v

-LRB-

vd

-RRB-

in

H

fid

we

construct

a

path

from

u

to

v

in

Sd

consisting

of

three

sections

section

A

from

u

to

vertex

-LRB-

d

-RRB-

of

H

ffd

section

B

from

vertex

-LRB-

d

-RRB-

of

H

ffd

to

vertex

-LRB-

d

-RRB-

of

H

fid

and

section

C

from

vertex

-LRB-

d

-RRB-

of

H

fid

to

v

If

u

is

a

vertex

on

the

cycle

in

H

ffd

then

section

A

is

the

shortest

path

on

the

cycle

from

u

to

-LRB-

d

-RRB-

consisting

of

at

most

d

edges

Otherwise

section

A

is

the

path

from

u

to

-LRB-

d

-RRB-

in

the

large

tree

of

H

ffd

which

is

of

length

at

most

d

edges

Thus

in

either

case

section

A

contains

at

most

d

edges

Similarly

section

C

contains

at

most

d

edges

Section

B

can

be

thought

of

as

one

traversal

of

the

cycle

from

-LRB-

d

-RRB-

to

itself

interspersed

with

single

dimensional

paths

in

each

dimension

i

in

which

i

i

The

length

of

this

section

is

Pdi

i

ij

edges

By

the

way

the

grid

has

been

subdivided

into

hypercubes

we

have

that

jui

vij

i

ij

for

all

i

The

number

of

edges

in

the

path

from

u

to

v

may

be

rewritten

as

Pdi

-LRB-

vij

-RRB-

Pdi

Pdi

-LRB-

vij

-RRB-

d

dG

-LRB-

d

-RRB-

-LRB-

u

v

-RRB-

Thus

the

delay

in

Sd

is

at

most

giving

Theorem

The

bypass

spanner

Sd

is

a

-LRB-

x

-RRB-

spanner

of

G

-LRB-

d

-RRB-

with

and

ffi

d

We

may

also

obtain

good

spanners

of

finite

grids

by

taking

induced

subspanners

of

of

Sd

Corollary

a

For

any

finite

d-dimensional

grid

nd

with

ni

even

i

d

there

exists

a

-LRB-

x

-RRB-

spanner

of

with

and

ffi

d

Construction

of

Some

Specific

Spanners

We

have

constructed

other

bypass

spanners

which

improve

the

above

delay

bound

for

and

dimensions

We

now

construct

a

maximum

degree

spanner

S

of

the

infinite

grid

This

spanner

is

much

like

a

regular

bypass

spanner

except

that

each

hypercube

does

not

induce

a

connected

subgraph

of

the

spanner

We

begin

by

creating

the

highways

We

create

the

dimension

highways

by

connecting

-LRB-

-RRB-

to

-LRB-

-RRB-

when

is

even

and

is

odd

We

then

create

the

dimension

and

dimension

highways

by

connecting

-LRB-

-RRB-

to

-LRB-

-RRB-

when

is

even

and

is

odd

and

-LRB-

-RRB-

to

-LRB-

-RRB-

when

is

even

and

is

odd

To

complete

the

spanner

we

connect

all

vertices

-LRB-

-RRB-

that

are

not

on

any

highways

-LRB-

which

we

shall

refer

to

as

connector

vertices

-RRB-

to

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

Theorem

The

spanner

S

described

above

is

a

-LRB-

x

-RRB-

spanner

of

G

-LRB-

-RRB-

with

S

and

ffiS

PROOF

Given

two

vertices

u

-LRB-

-RRB-

and

v

-LRB-

-RRB-

dG

-LRB-

u

v

-RRB-

jui

vij

We

wish

to

bound

the

delay

between

u

and

v

in

S

If

one

of

the

vertices

say

u

is

a

vertex

on

a

highway

of

dimension

i

let

be

the

vertex

on

that

highway

with

ith

coordinate

equal

to

vi

One

path

from

u

to

v

in

S

consists

of

a

section

along

the

ith

dimensional

highway

from

u

to

followed

by

a

path

from

to

v

The

delay

along

this

path

is

equal

to

the

delay

from

to

v

and

-LRB-

u

v

-RRB-

-LRB-

v

-RRB-

Otherwise

both

u

and

v

are

connector

vertices

If

u

and

v

differ

in

two

or

fewer

coordinates

let

u

clearly

-LRB-

u

v

-RRB-

-LRB-

v

-RRB-

If

u

and

v

differ

in

all

three

coordinates

then

one

of

them

say

u

has

lower

first

coordinate

From

this

vertex

we

can

step

one

edge

upward

in

the

first

dimension

to

a

highway

in

one

of

the

other

dimensions

i

We

follow

this

highway

in

the

appropriate

direction

until

we

reach

a

vertex

with

ith

coordinate

equal

to

vi

As

the

first

step

and

the

highway

section

are

both

in

the

direction

of

v

in

G

-LRB-

u

v

-RRB-

-LRB-

v

-RRB-

In

any

case

differs

from

v

in

at

most

two

coordinates

is

either

a

connector

vertex

or

is

on

a

highway

orthogonal

to

a

plane

containing

and

v

and

-LRB-

u

v

-RRB-

-LRB-

v

-RRB-

CHCHCHCH

HCHCHCHC

CH

HC

CHCH

HCHCHC

CH

Figure

A

two

dimensional

slice

of

spanner

S

Consider

the

plane

containing

and

v

if

there

is

more

than

one

such

plane

choose

one

orthogonal

to

the

highway

used

to

get

from

u

to

if

any

By

symmetry

we

can

orient

this

plane

to

look

like

Figure

where

H

denotes

vertices

on

highways

orthogonal

to

the

plane

and

C

denotes

connector

vertices

Also

by

symmetry

we

can

assume

that

the

connector

vertices

are

each

incident

on

an

edge

orthogonal

to

this

plane

which

connects

to

a

vertex

on

a

horizontal

highway

For

convenience

we

will

call

the

horizontal

direction

dimension

the

vertical

direction

dimension

and

the

orthogonal

direction

dimension

Note

that

is

labeled

either

C

or

H

If

v

is

on

a

highway

in

dimension

then

let

be

the

vertex

on

that

highway

with

the

same

dimension

coordinate

as

One

path

from

to

v

in

S

goes

from

to

and

then

along

the

highway

to

v

thus

-LRB-

v

-RRB-

-LRB-

-RRB-

If

is

a

vertex

of

type

H

then

a

path

from

to

can

be

formed

by

stepping

down

in

dimension

orthogonally

in

dimension

to

a

highway

in

dimension

following

the

highway

stepping

from

the

highway

to

a

connector

vertex

via

a

dimension

edge

moving

right

on

a

dimension

edge

and

finally

stepping

up

on

a

dimension

edge

The

two

possible

shapes

of

this

path

are

shown

in

Figure

As

we

can

see

from

the

figure

this

path

has

delay

at

most

If

is

a

type

C

vertex

then

a

similar

path

from

to

can

be

found

by

omitting

u

v

u

v

Figure

Possible

path

shapes

the

first

and

last

edges

of

the

path

described

above

Such

a

path

has

delay

at

most

It

remains

to

consider

the

case

where

v

is

either

a

type

C

or

a

type

H

vertex

Let

be

v

if

v

is

a

type

C

vertex

or

the

type

C

vertex

adjacent

to

v

by

a

dimension

edge

otherwise

Similarly

let

be

if

is

a

type

C

vertex

or

the

type

C

vertex

adjacent

to

by

a

dimension

edge

otherwise

Note

that

-LRB-

v

-RRB-

-LRB-

-RRB-

since

either

dG

-LRB-

v

-RRB-

dG

-LRB-

-RRB-

and

dS

-LRB-

v

-RRB-

dS

-LRB-

-RRB-

or

dG

-LRB-

v

-RRB-

dG

-LRB-

-RRB-

and

dS

-LRB-

v

-RRB-

dS

-LRB-

-RRB-

Since

and

are

both

type

C

vertices

we

can

construct

a

path

from

to

by

stepping

orthogonally

in

dimension

to

a

highway

in

dimension

following

the

highway

stepping

from

the

highway

to

a

connector

vertex

via

a

dimension

edge

stepping

along

a

dimension

edge

to

a

highway

of

dimension

following

the

highway

and

then

stepping

from

the

highway

to

along

a

dimension

edge

as

shown

in

Figure

As

the

distance

traveled

along

the

highway

sections

is

the

distance

between

and

this

section

has

delay

and

hence

-LRB-

v

-RRB-

Thus

in

all

cases

we

have

shown

that

-LRB-

v

-RRB-

and

therefore

-LRB-

u

v

-RRB-

We

now

construct

a

maximum

degree

regular

bypass

spanner

S

of

the

infinite

v

v

u

u

Figure

Path

from

to

grid

which

has

two

highways

of

each

dimension

passing

through

each

hypercube

The

edges

we

include

in

form

the

Hamiltonian

cycle

The

highway

edges

within

are

alternate

edges

of

this

cycle

starting

with

the

edge

between

and

This

structure

is

shown

in

Figure

with

thick

edges

indicating

highway

edges

Theorem

The

spanner

S

described

above

is

a

-LRB-

x

-RRB-

spanner

of

G

-LRB-

-RRB-

with

S

and

ffiS

PROOF

Consider

any

pair

of

vertices

u

in

H

and

v

in

H

If

i

i

i

then

both

u

and

v

are

on

the

cycle

of

H

and

there

exists

a

path

along

that

cycle

between

the

two

vertices

of

length

at

most

and

therefore

-LRB-

u

v

-RRB-

At

this

point

we

could

proceed

with

an

analysis

similar

to

that

in

our

previous

proof

using

a

path

consisting

of

a

full

cycle

traversal

interspersed

with

single

dimensional

paths

However

since

our

cycle

includes

two

highway

edges

in

each

dimension

we

can

improve

on

this

obvious

bound

Note

that

all

vertices

of

H

are

symmetric

under

a

cyclic

permutation

of

dimensions

and

possibly

a

reflection

and

we

may

thus

without

loss

of

generality

assume

that

u

is

the

vertex

1110 10010001

Figure

Cycle

in

of

H

that

has

all

coordinates

even

The

vertex

u

is

therefore

on

a

highway

in

dimension

and

incident

on

a

non-highway

edge

to

a

vertex

on

a

highway

in

dimension

as

shown

in

Figure

We

consider

two

paths

from

u

to

v

in

S.

Let

be

the

vertex

in

H

with

dimension

coordinate

odd

and

the

other

coordinates

even

One

path

from

u

to

v

consists

of

a

path

from

u

to

corresponding

to

a

walk

along

the

cycle

-LRB-

counterclockwise

in

Figure

-RRB-

to

interspersed

with

single

dimensional

paths

followed

by

the

shortest

path

along

the

cycle

from

to

v

in

H

Similarly

let

be

the

vertex

in

H

with

dimension

and

coordinates

odd

and

the

other

coordinates

even

Another

path

from

u

to

v

consists

of

a

path

from

u

to

corresponding

to

a

walk

along

the

cycle

-LRB-

clockwise

in

Figure

-RRB-

to

interspersed

with

single

dimensional

paths

followed

by

the

shortest

path

along

the

cycle

from

to

v

in

H

Consider

a

highway

section

in

dimension

i

contained

in

either

of

these

paths

not

including

the

highway

edge

-LRB-

if

any

-RRB-

of

the

conceptual

cycle

that

we

are

traversing

This

section

goes

between

two

vertices

whose

ith

coordinates

have

the

same

parity

and

whose

other

coordinates

are

equal

This

section

is

of

length

i

ij

which

is

at

most

jui

vij

Thus

in

the

path

from

u

to

we

have

at

most

dG

-LRB-

u

v

-RRB-

edges

with

four

of

the

extra

edges

coming

from

the

four

highway

sections

and

u

u

u

Figure

Redrawn

cycle

in

the

other

five

coming

from

the

traversal

of

our

conceptual

cycle

Similarly

the

number

of

edges

in

the

path

from

u

to

is

at

most

dG

-LRB-

u

v

-RRB-

Regarless

of

the

location

of

v

on

the

cycle

the

shorter

of

the

two

resulting

paths

from

u

to

v

is

of

length

at

most

dG

-LRB-

u

v

-RRB-

In

fact

the

length

is

at

most

dG

-LRB-

u

v

-RRB-

unless

v

has

first

coordinate

odd

and

all

other

coordinates

even

-LRB-

the

vertex

clockwise

from

u

in

the

figure

-RRB-

In

this

latter

case

we

will

use

the

path

through

with

the

modification

that

we

replace

the

highway

section

and

cycle

edge

in

dimension

with

a

highway

section

in

dimension

consisting

of

exactly

edges

This

gives

a

path

length

of

at

most

dG

-LRB-

u

v

-RRB-

Conclusions

We

have

constructed

spanners

with

maximum

degrees

and

of

d-dimensional

grids

having

constant

delay

-LRB-

for

a

fixed

dimension

-RRB-

In

particular

we

constructed

a

-LRB-

x

-RRB-

spanner

and

a

-LRB-

-LRB-

dd

e

-RRB-

x

-RRB-

spanner

of

G

-LRB-

d

-RRB-

with

and

ffi

and

a

-LRB-

x

-RRB-

spanner

of

G

-LRB-

d

-RRB-

with

and

ffi

d

We

discussed

generalizations

of

these

spanners

and

how

to

adapt

these

spanners

to

finite

grids

For

grids

we

obtained

a

-LRB-

x

-RRB-

spanner

with

and

for

grids

we

obtained

a

-LRB-

x

-RRB-

spanner

with

It

is

likely

that

the

idea

used

in

the

construction

-LRB-

using

more

than

one

highway

in

each

dimension

through

each

hypercube

-RRB-

can

also

be

used

to

obtain

good

grid

spanners

in

higher

dimensions

References

-LSB-

-RSB-

Cai

L.

Tree

Simon

Fraser

University

Tech

Report

No

-LRB-

-RRB-

-LSB-

-RSB-

Liestman

A.

L.

and

T.

C.

Shermer

Additive

Graph

Spanners

Networks

-LRB-

-RRB-

-LSB-

-RSB-

Liestman

A.

L.

and

T.

C.

Shermer

Additive

Spanners

for

Hypercubes

Par

Proc

Lett

-LRB-

-RRB-

-LSB-

-RSB-

Liestman

A.

L.

and

T.

C.

Shermer

Degree-Constrained

Network

Spanners

With

Nonconstant

Delay

Simon

Fraser

University

Tech

Report

No.

-LRB-

-RRB-

-LSB-

-RSB-

Liestman

A.

L.

and

T.

C.

Shermer

Grid

Spanners

Networks

-LRB-

-RRB-

-LSB-

-RSB-

Monien

B.

and

I.

H.

Sudborough

Embedding

one

Interconnection

Network

in

Another

Computing

Suppl

-LRB-

-RRB-

pp.

-LSB-

-RSB-

Peleg

D.

and

A.

A.

Schaffer

Graph

Spanners

J.

Graph

Theory

-LRB-

-RRB-

-LSB-

-RSB-

Peleg

D.

and

J.

D.

Ullman

An

Optimal

Synchronizer

for

the

Hypercube

Proc

ACM

Symp

on

Princ

Distr

Comp

-LRB-

-RRB-

-LSB-

-RSB-

Richards

D.

and

A.

L.

Liestman

Degree-Constrained

Pyramid

Spanners

J.

Par

Distr

Comp.

to

appear