Partial metric spaces

(1)

http://wrap.warwick.ac.uk/

Original citation:

Matthews, S. G. (1992) Partial metric spaces. University of Warwick. Department of Computer Science. (Department of Computer Science Research Report). (Unpublished) CS-RR-212

Permanent WRAP url:

http://wrap.warwick.ac.uk/60901 Copyright and reuse:

The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.

Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.

A note on versions:

The version presented in WRAP is the published version or, version of record, and may be cited as it appears here.For more information, please contact the WRAP Team at:

(2)

Research

R.port

272 Partial

Metric

Spaces

S

G

Matthews

RR212

Scott models are-topolggt"4 models of complete partial orders used for Tarskian fixed point

semantics of the lamMa calculus. As of yet there are no methods for deriving Scon m6dels from

specificationg of the "complete" objects beyond an arbitrary choice. This paper introduces

"partial metrics" for generalising a theory of complete objects into a Scott model including partiat

objects.

Deparrnent of Comp,uter Science

University of Warwick Coventry CV4 7AL

(3)

Partial

Metric

Spaces

Steve

Matthews

Dept.

of

Computer Science

University

of

Warwick

Coventry,

CV4

7

AL

email

[email protected]

Introduction

Metric

space

topology provides

an

excellent framework

for

studying

the

behaviour

of

continuous functions

in

many

Tz

topologies.

For example,

Banach's

contraction

mapping

theorem

provides

a foundation

for

much inductive

proof

theory

for

continuous

functions.

Metric

spaces are of much interest to programming language designers as they provide the domain of totally defined

or

complete

programmable data

objects.

For example,

for

the set

of all flat finite &

infinite

lists LS

over a

set

S

we can define a "Baire" style metric

on LS

as follows.

VreL

V

s,s'€

d(Nil

,1)

=

1

if 1+Nil

d(

1

,1')

Unfortunately

Godel's decidability results force us to include partial objects such

as

l-

the

totally

undefined data object alongside the complete

ones.

This

leads to Scott's partial order topological

models as used in denotational

semantics. However,

by necessity these models

are T6

and so not describable by any metric as all metric spaces

areT2

(i.e.

Hausdorff).

This would seem to infer that

metric space topology is not appropriate

for

denotational semantics. deBakker

_{&Zucker [dB&232]}

have used

metric

spaces but

without the

usual Scott

partial

order

topology.

_{Smyth [Sm87]}

has

generalised the metric axioms by dropping the symmetry axiom (see _{M2 in}

_{Definition A5 or}

_[Su75])

in

order

to

define

paftial

orders.

Also,

Kopperman

_[Ko88]

has shown that any

topology

can be

generated

by

an appropriate generalised

metric.

This

raises the

possibility

that there

may

be an

appropriate notion

of

a generalised

metric

suitable

for

Scott

topologies.

In

this paper we provide

such a generalisation as an alternative to the approach taken by

Smyth.

In our approach the complete objects

form

a metric subspace

of

a

pantial

metric space.

Our generalisation keeps the symmeny

axiom

MZ

while

using a

new generalised

reflexive

axiom to distinguish

between

partial

and

complete

objects.

This

new approach promises an approach

to

denotational semantics which combines the elegance

of

Tarskian semantics and Scott topologies

with

conventiional metric space

technology.

if

s=s'

ifs*s'

S V 1, l'e

L

d(s:I,S':1')

_{= l2x}

=l

(4)

Definition

I

A

Partial

Metric

(

Pmetric

₎

is

afunction

p

:

A

X

A

+

B,

such that"

(P1) V x,YeA

x=Y

(+

_P(x,x)

= p(x,y)=p(y,y)

(PZ) V x,y€A

p(x,x)

(P3) V

x,y

€

_A

p(x,Y) =

_P(Y,x)

(P4) V

X,y,ze

A

p(x,z)

A

metric is precisely a

pmeric p

:

A

X

A

-+

R,

in which,

V

x

e

A

p(x,x)

=

Q

Also

note that for each

pmeric p

:

A

X

A

-+

lR

,

V x,Y€A

_{p(x,y) = 0 :+ x=Y}

this

being

"half' of

the metric reflexive

axiom

Ml

.

We can use pmetrics to make the

following

definition of

when a data object is to be regarded as complete.

x

iscomplete

::=

p(x,x)

₌₀

Any

object which is not complete is called

partial,

thus,

x

is

partial ::=

p(x,x)

>

0 Example

I

AFlatPmetricisapmetric

pt : StX S-l+ {0,1}

where,

51 ::=SU{I}

for

a set

S,

and

l-

+

S

,

ild,

V x,Y€St

_{Pr(x,Y)=o c+ x=Y€S}

I-ater we

will

show how flat pmetrics relate to the usual partial order notion of a flat domain.

(5)

Example

2

A

KahnPmetric

isapmetric

pS

:

KaS

X

KaS

-t

_{2-n

l

n

€

(, }U

{0}

where

KaS

is defined to be the set

of all finite

&

infinite

sequences over the set

S

and,

V yeKas

Ps(<>,Y) =

1

V

x,y e

(dS

V

n,m

>

0 .

ps(

(xgr...,\-l)

_,

(Y0,...,Ym-t >

)

= In x

ps(<xl

_{,...,\-l),(yl}

,...,ys1-1

))

if

*o=yo

= 1

if*O+yO

V

x

e

cds

ps(x,x)

=

0

Kahn

pmetrics are used

for

describing the partial

order domain used

by Kahn

lKa74l to

give

a

denotational semantics to

pipeline

data

flow

networks.

Later on we

will

show

how

Kahn pmetrics can be used to describe Kahn's partial ordering

on

KaS .

Definition

2

The

Open

Balls

for

a

pmetric

p

:

A

X

A -+

IR

are the sets

of

the form,

Br(x) ,,= { y.A

I p(x,y)<€

}

foreach

€>0

and

xeA.

Theorem

I

Theserofallopenballsofapmetric

p:

AXA

+

IR wrth

A

formthebasisofaopologyon

A.

Proof:

Proof

usingDefinition

A4

andTheorem

Al

Suppose

p: AXA -r

IR

isaPmetric.

Then,

A = U*"A

Bil,)+t

(x)

and,

for

any

balls

B,(

x

)

and

86(

y

)

,

Br(x)

n B6(y)

-Ut Bn(z) I ze

B.(x)

n

B6(y)

where,

_I

::= p(z,z)+min{

e-p(x,z),6-p(y,z)}

tr

(6)

Theorem

2

Foreachpmetric

p:

AXA

-+

IR,

openball

Br(a),

and

xe

A,

xeBr(a)

:+ 3 6>0

xcB6(x)EBr(a)

Proof:

Suppose

;

e B,(a)

Then

p(x,a)

<

€

Let 6

::=

€

_-

p(x,a)

+

(x,x)

Then

6>0

as

€tp(x,a)

Also,

p(x,x)<6

as

€tp(x,a)

Thus

x

e

B6(x)

Suppose now

that

y

e

86(

x ₎

P(Y,x)

<

6

p(y,x) <

€

_{-p(x,a)+p(x,x)}

p(y,x)

+

p(x,a)

_-

p(x,x)

p(y,a)

< €

(byPa)

y

e

B6(a)

Thus

86(x)

c

B€(a).

tr

Theorem

3

Pmetric topologies

re

To.

Proof:

Suppose

p: AXA-+

IR

isapmetric.

Suppose

x+y e

A

Thenfrom

Pl

&

P2

p(x,x)

<

p(x,y) or

p(y,y)

<

p(x,y)

Wlog

suppose

p(x,x)<p(x,y)

then,

leBr(x) n y*

Br(x)

where,

€::= (p(x,x)+p(x,y))/2

tr

Note that the open balls (with

A)

of the

form,

Br(x) r,= { y.A

I p(x,y)<t

}

form

a basis for the same topoplogy as the balls

(with

A)

of the form,

B'r(x) rt= { y.A

I

p(x,y)

xs,

V €>0 V xeA

B',(x) :

Br+X,;(x)

andas,

V €tp(x,x)

Br(x) =

B'r_p1r,*;(x)

and, V0<€<p(x,x)

Br(x) =

g

(7)

The next theorem gives us a

pmetric

analogue to the

familiar

menic condition

for

convcrgence. A

sequence

X

e

@A inametricspacewithmetric

d: AXA -r

R,

convergesto

I

c

A

lff,

3 h-n--r-

d(Xn,I)

=

0

Theorem

4

Asequence

X

e

@A inapmetricspacewithpmetric

p:

AXA

_{-r lR}

convergesto

I eA iff :l

li-n__*

p(\,I)

_{= p(I,tr)}

Proof:

Suppose

X

converges

to

I

.

ThenV€>0 3k>0 Vn>k

Xn.Be*4r,fy(I)

ThenV€>0 3k>0 Vn>k

p(Xn,I)-p(I,I)

ThusbyP2

3 t-o_*"

p(4,,I)

_{= p(I,tr)}

Supposenowthat

3 hro_r-

p(Xn,I) = p(I,I).

Also,

suppose

that

)r

e

B,(

a ₎.

We have to show

that

X

is eventually

in

B,(

a

).

As

[m"-* p(Xn,tr) = p(],])

wecanchoose

k>0

suchthat,

Vn>k

p(Xn,I)-p(tr,I)

V

n >

k

p(4,,a)

<

p(Xn,I)

_{- p(I,I)}

+

p(I,a)

=€

Vn>k

Xn

€

Br(a).

tr

A

primary motivation behind the development of generalising metrics to get pmetrics was that there

should be a natural way of defining a partial order on a pmetric space, and so open up such spaces to

applications in denoational semantics.

Definition

3

Foreachpmerric

p: AXA-+ IR .pg AXA

isthebinaryrelationdefinedby,

V x,y€A

x .p y <+ p(x,x)=p(x,y)

Theorem

5

Foreachpmetric

p: AXA+ IR

.p

isapanialorder.

Proof:

VxeA.

x.px

as

p(x,x)=p(x,x)

Vx,y€A

x.py n y.px

+ p(x,x)=p(x,y)= p(y,y)

(bVP3)

=e x=y

(bypl)

(8)

Vx,y,zcA.x'PY

n Y'Pz

+

p(x,x)

=

p(x,y) n

P(Y,Y) = P(Y,z)

but,

p(x,z) < p(x,y)+p(y,z) -p(y,y)

(byPa)

p(x,z)

S p(x,

x)

p(x,z)

=

p(x,*)

(byP2)

x

.p

_z

( by

definition

of

.p

₎

tr

For each flat pmetric

p

.p1

is the usual ordering on a flat

domain,

while

for

each Kahn pmetric

ps

.pS

_{istheusual"initials€gment"orderingonsequences.}

Forametric

d: AXA +

lR

.d

is the equality

relation.

As we regard each metric space as a partial metric subspace

of

complete objects

it

is appropriate that this should be so 4s ne fstally defined object should be comparable with

another distinct totally defined objecr

The next theorem provides a warning

of

the dangers

of

working

h

TO

spaces as not all

sequences have unique

limits,

even

if

chains do.

Theorem

6

Suppose

XeoA

convergesto

IeA

inapmetricspacewithpmetric

p: AXA-+

IR

and

that

I'

e

A

is

such

that

I'

'p

I .

Then

X

converges

to

I'

as

well.

Proof:

By Theorem 4

it

is sufficent to show that ,

3 lhq,--

P(

Xn

,

I' ) =

P(

I', I')

Suppose €

>0,

then as

X

convergesto

I

(byTheorem4)

we can

choose

k >

0

such

that,

V

n>k

p(

Xn,

_{I ) - P(I,I)}

Vn>k

.

p(

Xn,

I') -

_P(

I', I')

= p(4,,I)-p(I,I)

(as)t'.pI)

tr

In

the above

proof

I'

is a "phoney"

limit

in

the sense that it would not correspond to a chain

limit

if

the sequence were a chain. The intention

of

the next definition is to overcome the problem

of

having sequences

with

more than one

limit

by introducing

a restricted notion

of

convergence. This

will

ensure that the topological

limit

of a chain is also the least upper bound.

(9)

Definition

4

Asequence

Xe @A

inapmetricspacewithpmetric

p : AXA +

B,

Properly

Converges

to

I

e

A

if

X

convergesto

tr

and,

3 [rnn--

p(4,,)q) =

p(],tr)

Inotherwords

Xe

oA

properlyconvergesto

Ie A if 3

limo--

p(4t,I )

and'

3 lirnnr-

p(

_4,

_{,4 )}

an4

limo--

p(Xn,\)

= [-r,-- P(4t,])

= p(]'])

The next Theorem shows that properconvergence captures the

limits

we really want although notice, we have not used chains here to obtain unique

limia

in a

T6

space.

Theorem

7

Suppose

X

e

oA

properly converges to

both

tr

and

I'

in a pmetric space

with

pmetric

p:

AXA

+

lR,

then

I'

_'P

tr.

Proof:

Suppose

X

e

oA

properly converges

to

both

I

and

I'

.

Choose

€ >

0 _,

thenwecanchoose

k >

0

suchthat,

p(

Xn,I )

p(I,I)

n p(Xn,I')

P(tr',I')

n lp()q,,Xn) p(I,I)l

p(

I

_,

I')

_P(

I', I')

+ (p(X",I')

p()',I'))

s€

Thus

p(

l'

_,

I' ) =

p(

I

,

I')

as

€

was an arbitrary choice.

And

so

I'

.p

)

tr

The implication of the Theorem

7

is that limits to properly convergent sequences are

unique.

This is an interesting result

for

non-Hausdorff

Tg

spaces.

Other

standard

metric

constructions

also

generalise

to

pmetric

spaces

with

both considerable

&

surprising ease.

(10)

Definition

5

A

sequence

X

c

oA

inapmetricspacewithpmetric

p:

AXA

-+

lR

is

Cauchy

f'

V

€

>0 3 k>0 V n,m>k

p(4,,Xn') -

P(Xn,,Xm)<€

Definition

6

A

pmetric space

is

Complete if

every Cauchy sequence properly converges.

Definition

7

A

Contraction inapmetricspacewithpmetric

p:

AXA

+

IR

isafunction

f

:

A

-r

A

such that,

3 0 < c

<

I V

x'Y

€

A

p(

f(y)

_,

f(x)

) -

P(

f(x) '

f(x)

)

Theorem

8

Each contraction in a complete partial metric space has a unique fixed

poinr

Proof:

Suppose

f

:

A

+ A

is a contraction in_a complete puqal.

meqc

space

with

pmetric

p: AXA+ IR,

andthat

0Sc<lissuchthat,

V x,y

e

A

.

p(

f(y), f(x) )

p(

f(x), f(x)

)

Let a e

A,

andlet

X

e

@A

besuchthat

V

n

>

0 \ =

fn(a).

We

will

first

show

that

X

is a Cauchy sequence.

V n>0 . f( 4r+2,\+r ) -

f( 4r+r,4r+r

)

V n>0 . f(

_4,+z

,

Xn+l

_{) - f( Xi+r ,}

Xn+l

)

V n,k)0

.f(

Xn+k+l

_,

Xn

_{) - f( \,}

Xn

)

+ f(

Xn+k,

Xr,

_{) - f( Xn,}

Xn

)

+f(Xn+k,Xn)-f(Xn,4,)

V n,k ) 0 . f(

Xn+k+l

_,

Xn

_{) - f(}

X1'

_,

Xn

)

=

cn

x (1-"k+l;

B

BCTCS

8 '.92

(11)

Thus

X

is seen to be a Cauchy sequence.

Thus as our prnetric space is complere

X

properly converges

to

I

e

A

say.

We now show

that

I

is a fixed point

of

f

.

Choose

€

>

0,

thenas

X

properlyconvergesto

tr

wecanfind

k

2 0

suchthat,

Vn>k

p(

_{I ,}

Xn

_{) -}

p(

x'

Xo

_{) < e/(l+c)}

n p(\,l)

_{- P(I,l)}

Thus Vn>k

p(

_{f(I), I ) - P(I'I}

)

+p(\+r,I)-p(l,I)

+ p(

Xn+l

_{, I ) -}

p(

I

_,

I

)

=€

Thus,as€isarbitrary,

p(f(I),I)

- p(I,tr)

(*)

Similarly,

Vn>k

.p(

f(I),I

) -

p(

f(I), f(I)

)

+

_{p( 4r+r}

_{, I ) -}

p(

_{f(I) , f(I)}

)

= (p( f(I) ,Xn+l ) - p( f(I) , f(I)

))

+

(P(

_4r+r

_{, I ) -}

_P(

ll+r ,

_4r+r

₎₎

E cx(p(),Xn)

p()',I))

+E/(l+c)

=€

Thus, as

€

isarbitrary,

p(

f(I), l ) =

p(

f(I), f(I)

)

Thusfrom

(*)

andPl

l=f(l),

andsof

hasbeenshownto

have a fixed

point.

It just reamins to show

that

I

is unique.

(12)

Suppose

I'

c

A

and

l'

=

f(

)')

,

then ,

P(

I'

*''=

I

l,';,^','nt', )

-

p(

f(I'),

r(I')

)

p(I,I')-P(I',I')

=

0 as

0Sc<1

I',pI

Similarly

we can

show,

I 'p I'

and

so tr

=

tr'

tr

Weighted Metrics

So far we have explained partial metrics in terms

of

a generalisation

of the metric

axioms

Ml

-

M3

However,

there

is

another method

for

introducing

partial

metrics.

This

second approach sheds more

light

on the relationship bet'ween metrics and

partial

metrics.

As

has been

clearly shown already partial metrics do allow discussion

of

Scott style partial

objecs in

the spirit of

metric

spaces

by introducing

the idea that an object need not necessarily have zero distance from

itself,

i.e.

V xeA .p(x,x)

>

0 insteadof

V xeA

.

d(x,x)

=

0.

Byconcentrating

on

the idea that each object has

a

weight

which

in

general

is

a non-negative

real

gives us an

alternative way

to

define

partial

metrics.

The result

of

this

is

the

conclusion that

the notion

of

pmetric

is precisley the combination

of

the ideas

of metric

and

weight.

Definition

8

A

WeightedMetric

overaset

A

isapair

<

d,

||

d:AXA+lR

andaWeightFunction

ll:A+

IR

where,

V x,y

e

A

d(x,Y)

Theorem

9

Partial metrics and weighted metrics can be defined in terms of each other.

Proof :

Suppose

<

d, ll >

isaweightedmetricovertheset

A. Lrt

p

:

A

X

A

-+

lR

be the function such that,

Vx,y

€

A

_{P(x,y) = (d(x,Y) +lxl}

+

lYl) /

2

We

will

first

show

that

p

is a pmetric by

proving

Pl

_-

P4 .

(P1=+)Trivially

V *,y

€

A

x=y + p(x,x)

=

p(x,y)

=

p(y,y)

(13)

(Ple)V

x,y

e

A

_P(x,x)

=

P(x,Y)

= P(Y,Y)

d(x,x)+lxl+lxl

d(x,y)+lxl

+lyl

d(y,y)+lyl+lyl

=) 2xlxl

= d(x,y)+lxl +lyl = 2 x

lyl

+

d(x,Y) = lxl-lYl

= lYl-lxl

+

d(x,y) = 0 + x=Y

(byMl)

(P2)

V x,y

e

_A

p(x,y) = p(y,x)

(byM2)

(P3)

V

x,Y

€

_{A p(x,x) =}

lxl

(P4)

V x,y,ze A

d(x,z)

+

d(x,z)+lxl+lzl

d(x,

z)

+

lx

| + | z

I

d(x,y

)

+lx

|

+

ly

I

2

d(y,z)+lyl+lzl

lvl

2

:+ p(x,z) < p(x,y)+p(y,z) -p(y,y)

Thus

p

has been shown to be a pmetric.

Supposenowthat

p

isapmetric.

Wewillshowthatthepair

( d, ll>

definedby,

V

x

e

_A

_lxl

::= p(x,x)

V x,Y€A

d(x,Y) := 2rP(x,Y)

lxl

lYl

is a weighted metric by

proving

Ml

-

M3.

(M1+)

V x,YeA

x=Y +

d(x,y)=0(bydefrnitionof

<d,ll>)

O{1e) V x,y€A

d(x,y)

=Q

=t 2*p(x,y)

lxl

lyl =

0

+ (p(x,y)-p(x,x)) + (p(y,x)-p(y,y)) =

0 (byP3)

+ p(x,x)

= p(x,y)

=

p(y,y)

(byP2)

:+ x=y

(byPl)

(N42)

V x,y

€

A

d(x,y) = d(y,x)

(byP3)

(14)

(M3) V x,y,z. A

P(x,z)

d(x,

z)+lx

l+

lzl

d(x,Y)+

lx l+lY

I

2

d(y,z)+lyl+lzl

lvl

+

d(x,z)

tr

Using the one to one relationship between paftial and weighted metrics used

in

the last proof we can

define the equivalent of the pmetric ordering on weighted metrics by,

V x,Y€A

x <Y (+ d(x,Y) = lxl-lYl

Now we move on to the problems of how to build larger pmetric spaces from smaller pmetric

spaces.

For pmetrics to be

of

much use

in

denotational semantics we must have (at least) product,

sum,

and

function

space

constructions

to build

useful

spaces.

The remainder

of

this

paper

demonstrates

that

such

constructions do

exist.

First

we

need

to apply to

pmetrics a

standard construction used

for

turning an unbounded metric into a bounded metric.

Definition

9

Foreachpmetric

p : A X A -)

IR

_{, P^ : A X A -+ [0rl)}

isthepmetric

suchthat

V x,y€A

p^(x,y) = p(x,y) /( 1*p(x,y))

Using Theorem

43

to

check

P4

it

can easily be verified

that

p"

is indeed a pmetric .

Theorem

10

For each

pmetric

p

the topology induced

by

p"

is the same as p .

Proof

:

Suppose

p : A X A ->

IR

isapmetric,

then,

VxeA

V€>0

B,(x)= B^6111a6;(x)

and,

VxeA

V0<e <1

B"r(x) = Br4r-ry(*)

and,

V xeA V €>1

B"€(x) = U{

Bp(*,r)+r(y)

ly€B^r(x)

}

tr

(15)

Also note that

for

each

pmetric

p

,

.p

=

.

(

p"

₎

.

Definition

f0

TheCountableProductof

thepmetrics

pn: \XA1, -+ IR

(n20)isthefunction

p' : (Xn>oAn)2 + IR

where,

V X,y € Xn>oA,, p'(x,y) = In>o (pn)"(xn,yn) x

2-n-l

Theorem

l0

The countable product of pmetrics is a pmetric.

Proof:

Suppose

p' : (Xn>04,)2 +

lR

isthecountableproductofthepmetrics

Pn: AnX\ +

IR

We

will

show the countable product

o

be a pmetric by

proving

Pl

-

P4.

(Pl=+)

trivial.

(Ple) V X,y .

Xn>04r

Pt(x,x)

=

Pt(x,Y

) =

Pt(Y,Y)

=i

In>O

(pn)"(\,\)

x

2-n'r

= In>o

(pn)"(

xn

_,

yn

_{) x}

Z-n-l

=

In>o

(Pn)"(

yn

_,

yn

_{) x}

2-n-l

+ In>o (

(pn)^(

_\,

yn

₎

(pn)n(

_*n,

_{\ ) ) x}

z'n'r

=In>0 ( (po)"(\,yn)

(pn)"(yn,yn)

) x

2-n-l

=Q

+

V n)0

(Pn)^(\,\)

_=(Pn)n(\,yn)

=

(pn)"(

yn,

yn

₎

:+

_{V n20 .}

xn

_{= yn}

_{=+ x=y}

(P2)

by

Y2

for each pn.

(P3)

by

P3 foreach

pn.

(16)

(P4) V x,y,z e Xn>o\

Pt(x,z)

p'(x,z)

=

In>'

(pn)"(

_{\, h ) x}

z-n-r

andas

p"(x,y)

+

p'(y,z)

_{- p'(y,y)}

₌

In>o (

(pn)^(

xn

_{,yn )}

+

(pn)"(

yn

,

zn)

-

(Pn)"(

_Yo

_,

_Yn

_{) )}

x

z'n'r

tr

The countable product has the "pointwis€"

ordering,

i.e.

V X,y € Xn>04,. x'(p")y

ce

_{V n20. \ r(}

(pn)n)

yn

Definition

11

TheDisjointSumofafamilyofpmetrics

pi:

A'1

X,A.1

-)

IR

(ieI)

isthe

pmetric

p+: Ui.l{<i,x>lxeAi}

+ [0,1]

where,

V <i,x> , (j,Y> . Ui.t

{

<i,x ) lx

e

Ai

}

p+(

<i,x> , (j,y)

₎

=

(pl)"(x,y,

ii;i

The topology and partial ordering for a disjoint sums are the expected ones.

We now come to the more involved problem of how to constnrct a pmetric function space.

As

with

function space constructions of othen we are forced to make certain assumptions on the type

of functions allowed

in

such a function space.

Definition

12

Foreachpmetric

p: AXA+

IR aset A*cA

isProperlyDenseinAifeach

member

x

e

A

is the

limit

of a sequence

in

A*

properly converging

to

x

.

Definition

13

Aset A

withpmetric

p: AXA -r

IRis

Sufferable

iftherethereexistsacountableproperly

denseset

A*cA

andfunction

p*: A-+R-{0}

suchthat

forany

x,Y.

4*10,21

,

IaeA*

p*(a)

t

Xa

_{= Iu"6*}

p*(a) x

Y"

(+

_X

₌

Y

Sufferability

is not an umeasonable assumption as any space

of

interest to a programming language

(17)

Erratum

(RR212

Partial Metric

Spaces)

Definition

13 is unnecessarily strong and should be replaced

by,

Definition

13

A

set

A

with

pmetric

p

: A

X

A

+

IR

is

Sufferable

if

,

3 r20

e

_{IR V x,y e A}

p(x,y)

andthereexistsacountableproperly

dense

set

A*c

A

with

function

p*:

A*

_{-r R-t0)}

such that

IaeA*

P*(a)

In

Definition

14 read,

The

set

of

all

such

properly

continuous functions over

(18)

closure

of

a recursively enumerable

set.

Although not proved here we can show that the countable

product

of

sufferable spaces is sufferable.

Definition

14

A

continuous

function

f

: A

-)

A'

over

pmetric

spaces

is

Properly

Continuous

if

for each

sequence

{

e

orfi

properlyconvergingto

x

e

A

thesequence

Y

e

A'

where,

V n>0

Yn

=

f(Xn)

properly converges

to

f(

x

₎

.

The

set

of

all

such properly

continuous functions is denoted by

A.+

A'

Definition

15

For

pmetrics

p:AXA+lR

and

p':A'XA'-rlR,

d):(A-A')X

(A'+A') +

IR

and

ll : (A'+A') -)

IR

arethe functions such that,

V f

e

A"+

A'

lfl = I3ctr*

p*(a)

x

lf(a)l'

V f,g € A'+ A'

dt(f,g) = Ia.A*

p*(a)

x

d'(f(a),

g(a)

)

where

<

d'

,

I

l'

>

is

the weighted metric equivalent

for

(p')"

as constructed

in

the proof

of

Theorem

9

.

Theorem 1l

<

_{d) ,}

ll>

isaweightedmetric.

Proof

:

(M1+)

Vf eA'+A'

d)(f,f)

= Ia.A*

P*(a)

x

d'(

f(a)

_'

f(a)

)

= Isctr*

p*(a)

x

0

=0

(M2e)

V f,g€ A*r A'

d)(f,g) -0

+ VaeA*

d'(f(a),g(a)) =

0

+ V

a

e

A*

f(a) =

g(a)

+ flA*

=

glA*

+

f

=g

asf

&garecontinuousandA*isdenseinA.

(M2)

d)

is symmetric as

d'

is symmetric .

(19)

(M3) V f,B,hc A.+ A'

dt(f,h)

= IarA* P*(a)

x

d'(

f(a) '

h(a)

)

= Ise tr. P*(a)

x

d'(

f(a) '

g(a))

I Ia.A* P*(a) x

d'(

g(a) ,

h(a)

)

= dt(f,g) + dt(g,h)

Thus

d?

is proven to be a

metric.

It justreamains to show

that

|

is a

weight.

v'''

.=^;"1.

;.ijl

;

l:lr"r,',s(a),'

)

is a weighted metric

= d)(f,g)

Thus

I

is a weight

for

A !t

A'

Thus

<

d),

I

>

is

aweightedmetricfor

A l+

A'

.

tr

As

in

the

proof of

Theorem 9 we can construct a

pmetric

p)

for

A '+

A'

.

An important result for

a potential

function

space is the following.

Theorem

12

Vf,geAD+A'

f .p)g

<+ VaeA

f(a).p'g(a)

Proof :

V

f

_,g

e

A l+ A'

f 'P)

g

(+ d(f,g) = lfl

lgl

Gt Iu.6*

P*(a) x

d'(

f(a) '

g(a)

)

= I".4* p*(a) x ( lf(a)l' - lg(a)l'

)

c+ V aeA*

d'(f(a),g(a))

= lf(a)l'-

lg(a)l'

( by the definition of

p*

₎

<+ V aeA*

f(a).p'g(a)

<+ V aeA

f(a) .p'g(a)

asA*isproperlydenseinAand

f

&

g

are properly continuous and Theorem A4

n

(20)

Conclusions

Pmetrics

_{( = weighted metrics}

)

allow the application

of

metric Hausdorff methods to the

non

Hausdorff

Tg

topologies required

for

denotational semantics based upon partial

orders.

For Computer Scientists this approach promises a fresh approach to denotational semantics using

well

understood metric mathematics. For Mathematicians this approach suggests

thu

the standard theory

of

metric

spaces can be generalised

to non-Hausdorff

spa@s

without losing too

many Hausdorff properties such as

limits of

sequences being

unique.

Perhaps more importantly there is a lesson to be learnt

by

both Computer Scientists

&

Mathematicians

here.

Too often the former have had to

invent

thek

own mathematics because the latter have not found conoputing problems mathematically

interesting.

The coincidence between the late David Park's

work

on bisimulation for process calculi and Peter

Aczel's

theory on non

well

founded sets was perhaps an

earlier

example

of

the same

lesson.

The challenge is to extend

familiar

mathemaical methods

for

reasoning about

"total"

well founded objects to include "partial" non

well

founded ones and so apply these methods for reasoning about programs.

Thetopology

tl-

ofapmetricspacewithpmetric

p

: AXA +

IR

alwayshasthe

first of

the t'wo definitive properties,

V

X,Y €

A

x.py n

xe€l-

y€€r

which characterise a Scott

topology.

The second definitive property is that the least upper bound

of

a chain must be a topological

limil6f

ttrat

chain.

In the context of pmetric spaces this is equivalent to

saying that

all

chains must be properly

convergent.

Thus

if

a

Scott

pmetric

is defined to be one

in

which

every chain is properly convergent and

for

which

there exists a special

element

l-

e

A

such that

P(I,a)

=

sup{

_{P(x,x) | xeA}

}

then the

topology

of

a Scott pmetric

is

always a Scott

topology.

The conclusion

from

this

is

that

pmetrics can be used to define Scott topologies, and

so

must be relevant t o denotational semantics.

The open question is how many Scott topologies cannot be defined using pmetrics.