Research Report 222
The Topology of Partial Metric Spaces
Matthews
SG
RR222
t
world of Scott's topological models used in the denotational semantics of programming
languages may at frnt sight appear to have nothing whatever in common with the Hausdorff
world of metric space theory. Can this be true though when the notion of "distance" is so
important in the application of inductive proof theory to recursive definitions? This paper shows
that existing work on the application of quasi metrics to denotational semantics can be taken much
further than just describing Scott topologies. Using our "partial metric" we introduce a new
approach by constructing each semantic domain as an Alexandrcv topology "sandwiched"
The
between two metric topologies.
to be presented at the Eighth Summer Conference on General Topology and Applications, June
18-20 1992, Queens College, New York City.
Deparrnent of Computer Science
Univenity of Warwick
Coventry CV4 7AL
United Kingdom
June 1992
The Topology of
Partial Metric Spaces
Steve Matthews
Dept. of Computer Science
University of Warwick
Coventry, CV47AL
email
1.
sgm@dcs.warwick.ac.uk
Introduction
In the study of the denotational semantics of programming languages topological models
are created for programming languages represented by systems of logic. More often than not this
means a T o model for the lamMa calculus in the spirit of Scott. However, the inherent
assumption in this approach that all suitable models must be Tp appears to remove any possibility
that the theory of metric spaces ( which are all T2) can be applied in any way to domain theory in
Computer Science. Rare exceptions to this rule are the use of quasi metrics
in [La87].
in
[Sm87] and metrics
If metrics are to be used at all then the more conventional wisdom in Computer Science
would dismiss Scott's T6 approach in favour of a purely metric approach ldB&7421. Unfortuantely
the latter T2 approach pays the price of loosing any notion of partial order, a concept of
fundamental importance in any Tarskian approach to fixed point semantics in denotational semantics.
The distinct advantage of using quasi metrics is that such generalised metrics can be used to define T9
topologies with partial orders, and so allowing Tarskian semantics. Quasi metrics are not without
their problems though. Being non-symmetric a quasi metric is arguably an "unnatural" notion of
distance.
Perhaps a more important crticism is that they shed linle light on how to develop proof
theory for programs based upon metric reasoning.
In [Sm87]
the
phrase Reconciling Domains
with Metric Spaces is used to suggest that conventional wisdom from the theory of menic spaces can
be applied to programming language semantics.
[Ko88] contains a proof that All Topologies
from Generalised Metrics. This may
or may not be of interest to topologists in general as many of the more pleasant T2 properties usually
come
associated with metric spaces may be lost in a process of generalisation. However, the point that
metrics
can, if only in principle,
be generalised to explore non - T2 topologies is established.
Combining this with Smyth's work on quasi metrics we are naturally lead to the following question.
To wlnt extent can
the T2 topological metlnds
applied to progrwntning langtage sennntics
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?
of metric space tlwory be
to this question can be found by showing that
This report aims to demonstrate that a positive answer
suggested'
taken much further than has previously been
the existing work on quasi metrics can be
but
metric be used to describe Scott Tp topologies
Not onry can a generalised merric such as a quasi
here the w'eighted metric' a version of a metric
we can also use metrics as well. we introduce
equivalent to a restricted form of quasi meric'
algebraically
method of constructing domains
Defined using partial metics we propose here a
asAlexandrovsub-topologiesofweightedmetrictopologies.
of metric spaces ever closer to Scott's theory of
The purpose of this work is to bring the theory
with metric Spaces a reality'
domains, and so make the reconciliation of domains
2.
Backgound Definitions
Definition
A
2.L
Topotogy over a set
&
Results
U is a set tr g 2u
such that '
) @eT
(T2) U e f,
(T1
(T3) vsgtr user
ono'eT
(T4) vo,o'er
Members
of T
are
called open sets
and their complements closed sets '
Definition 2.2
for a topology
The T0, T1 , and T2 (i.e. Hausdorff ) separation Axioms
To
=
V x*x'eU
TI
=
V x*x'e
T2
=
V x*x'e
t
are,
SOeT
(x eO A x'* o)
v( x' e o A x * o)
U
(3O'O'ef
x e O-O'
A x' e O'-O )
U
(3O'O'ef
x e O A x'e
A OnO'= g )
O'
If f is T2 then f is Tt, andif f is Tt then f is T0'
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2
Definition
A
2.3
Basis for a topology
f
is a set
E c tr
such that,
r = { us I ss03 }
Lemma
i3 c 2u
2.1
X over U il
is a basis for a topology
and vo,o'eE 3 sga3 0no' = us
u = uE
Lemma 2.2
E , E' g 2u
are each a basis for the same topology
o = US'
O'= US
V oefJ 3 S'gtrJ'
v o'e?3' 3 Sctr3
Definition
A
and,
2.4
Partial Order (poset)
)
(POz)
(PO3)
(POl
Definition
on U if ,
isapair < U,
Vxe(J
Vx,yeU
V x,l,ze(J
<<
c
U
XU
x<<x
x<<y Ay<<x + x=!
x<<y A yKz =t x<<z
2.5
An Alexandrov
Topology overaposet
<U,<<> isatopology T over U
such
that,
VOetr
xeO Ax<<y -
V x,yeU
yeO
T = { { ", n+l , ... , @ } I n e u J over
( t t U { * }, S > is an example of an Alexandrov topology. The Full
Topology tr [ <</ overaposet < U, << > is the topology
Thetopology,
x<<y -
{ OsU I VxeO,y€(J
of all
upward closures [Vi89l
ft <l
cuNY
,
'92
however
,
}
The topology
,oo} I neuU{@}
.'.'= { {n,n+l
is the Full Alexandrov Topology
is always T6
.
yeO
Abxandrov
over (
<o u
l
{* }
not every Alexandrov Topology
is Tp
as the indiscrete topolog;'
3
T = { A , {0,1} }
shows.
Definition 2.6
A Scott Topology is an Alexandrov Topology f
< U , << ) such that for each chain X e @U ,
VOef
The
only T2
fubXeO
over a chain complete poset (cpo)
= 3k>0
Vn>k
XneO
Scott topologies are the "flat" trivial ones in which,
V x,y e U
x
<<
y € x=J
The least element J- (bottom) is not included here in our definition of a Scott topology as it is not
essential to the development of the work presented in this
repon. later work using our methods
for frxed point semantics and reflexive domains will need to include -L.
Definition
A Metric
2.7
[Su75] is a function
(Ml )
(M2)
(M3)
d :
U
Vx,yeU
V x,y e(l
V x,y,z e (J
Definition 2.8
An Open Ball forametric d :
U
Bdr(x) ,,= t y.U
X U -+ 9I
such that,
x=! e d( x,y)*0
d(x,y) = d(y,x)
d(x,z)
XU -+ fr
isasetof rheform,
I d(x,y)
forany x eU and t>0.
Lemma
2.3
Thesetofallopenballsofametric
on U.
d : UXU I 9t isabasisfora T2 topology ftdl
Definition 2.9
A Quasi Metric isafunction q : UXU -) n
(Ql
)
(Q2)
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Vx,y€(l
Vx,y,ze(J
x--y €
q(
suchrhar,
q(x,y)=q(y,x)=0
x,z)
4
Lemma
2.4
q : UXU -) fr
Foreachquasimetric
xKo! e
Vx,yeU
therelation <q
I UXU
definedby,
q(x,y)=0
is a partial ordering.
Lemma
2.5
For each quasi
metric q :
Bs€(x)
U
X
U -+ fr
{ yrU
"=
the set of
all
open
balls of the form,
I q(x,y)
forany x e(J and €>0 isthebasisforanAlexandrov T0 topology ftql
<
u,
over
>.
<<q
Example
2.1
d : U X U -) fr and for each collection of closed sets
Uc c 2u the function q : Uc X Uc -+ fi where,
For each complete
metric
V X,Y e 2u
q(X,Y)
isaquasimetricsuchthat,
Lemma
2.6
Foreachquasimetric
VX,YeW
q : UXU q fr
V x,y eIJ
isametricsuchthat
.'.'= sup{ inf{ d(x,y)l(1+d(x,y))
XgY
thefunction
a X
lyeY } I xeX l
<<qY
q^ : UXU + fr
where,
q^(x,y) ::= q(x,y) + q(Y,x)
ft qI gft q^l
Lemma 2.7 ( Quasi Metric Contraction Mapping Theorem )
Foreachquasimetric q : UXU -+ fr suchthat q^ iscomplete,
function f :U-+U suchthat,
3 0Sc<l
thereexistsaunique a e
V x,y€(l
andforeach
q(f(x),f(y))
U suchthat a = f( a )
.
Note that the unique fixed point of a quasi metric contraction mapping need not be maximal as the
following simple example shows.
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5
q : {a,b}2 -, {0,1}
Q(a,b)=0 , q(b,a)- I
f : {a,b} -+ {a,b}
f(a) = f(b) = a
Thus this theorem cannot be used to prove that recursive definitions deftne total objects
Section 5 we present a second theorem specifically designed for this purpose.
.
In
3. Partial Metrics
Definition
A
3.1
PartialMetric (Pmetric) isafunction
The pmetric axioms
Ml
suchthat,
Q p(x,x)-p(x,y)=p(y,y)
x=!
Vx,yeU
p( x,x )
V x,y e U
Vx,YeU
P(x,Y) = P(Y,x)
p(x,z)
V x,y,z eU
(P1)
(P2)
(P3)
(P4)
axioms
p: UXU -+91
thru
Pl
M3
thru P4
are intended to be
a "minimal" generalisation of the metric
such that objects do not necessarily have to have zero distance from
themselves. In this generalisation we manage to preserve the symmetry axiom M2 to get P3 , but
have to "massage" the transitivity axiom M3 to produce the generalisation P4 . Consequently a
metric is precisely a pmetric p : U X U -+ 9t in which,
VxeU
For each
p(x,x) -
0
pmetric p : U X U -> fi
we preserve
"half'
of the metric reflexive axiom Ml
giving,
V x,y eU
Example
p(x,y)
3.1
- {0}, < > ofpositiverealnumberscanbedefinedbythepmetric
p(x,y) .'.'= mer{ Ilx,lly
p : (9t - {0})2 --+91 where, V x,y e U
as V x,y e (J x <y e p(x,x) = p(x,y)
Theposet < 9t
Example
3.2
Thewellknown Baire
S
is defined by
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}
,
Metric
dB
:
a'S
X ars + X
of
@-sequencesoveraset
e a'S
Vx+y
dB(x,y)
= 2-^in{neul{n)+y(n)}
Thisdefinitioncanbeextendedtotheset ( U{ {0""'n-l}S
finite and c^r - sequences over S by the Baire Pmetric ,
f
I new } ) u "tS
ofall
-+ fr
: ((U{{0,...'n-1 }S lneco })u.S)2
Inthisexample
where, V neu), x € {0,-.-,n-1}S f( x, x ) ::= 2-n
\f,.e can use the condition p( x, x ) = 0 to disringuish between those x which are in the Baire
Space and those which are not.
Example 3.3
A FlatPmetric isapmetric pt j S_rXSr+ {0,IJ
for a set S, and L + S, and,
pt(x,y)=0
Vx,)€S_r
Here an object
xe
Sl is
totally defined if
€
where, 51 .'.'= SU{l}
x=yeS
and only
if pt( x ,x ) = 0
Definition 3.2
The OpenBalls forapmetric
Bp€(x)
',=
p: UXU+fr
{ y.A
arethesetsoftheform,
I p(x,y)<€
}
foreach €>0 and xeA
Theorem
3.1
Thesetofallopenballsof apmetricp;
Proof
UXU +fr
isthebasisofatopology
TIp] on U
:
Proof using Lemma 2.1
Suppose
Then, A
p: UXU -+fr
=
isapmetric.
UaeA BPp1xil+l (x/
and,
foranyballs BP€(x) and Bn6(l),
BPt(x) n Bo5(t) =
U{ Ben(z) I zeBn6(x)nBr5(t)
where, q .'.'= p( z,z) + min{ t-p( x,z ), 6-p(y,z ) }
D
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7
Theorem
3.2
Foreachpmetric
p: UXU+91 ,
xeBP6(a)
Proof
I
openbalt BP€(a), and
3 6>0
.r € Bn6(x) I
xeU,
BPe(a)
:
Suppose p: UXU ->fr
Suppose I e BPs(a)
isapmetric.
Then p(x,a) < €
Let 5 ::= € -p(x,a) + p{ r,x)
Then 6>0 as trp(x,a)
Also, p(x,x) < 6 as €rp(x,a)
Thus x e Be5(x)
Supposenowthat
y
e BP5(x)
p(y,x) < 6
p(y,x) < € -p(x,a) + p(x.x)
p(y,x) + p(x,a) - p(x,x1
p(y,a) < 6
y
(byPa)
e Bn6(a)
Thus Br5(x) I BP€(a).
u
Theorem
3.3
Pmetric topologies are To
Proof
.
:
p: UXU +fr
Suppose x+y e U.
Suppose
isapmeric.
Pl & P2 p(x,x) < p(x,)" ) or p(y,y) < p(x,y)
Thenfrom
p(x,x)<p(x,y) then,
xeBp6(x) AylBe6(x) where, f .'.'= (p( x,x)+p(x,y))/2
Wlog suppose
U
Definition
3.3
Foreachpmetric
p: UXU +X,
Vx,yeA
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x
Kp
)
(o S UXU
e
isthebinaryrelationdefinedby,
p(x,x)=p(r,yl
Theorem
3.4
Foreachpmetric
Proof
p: UXU -rfr,
<p isapartialorder.
:
Suppose p: UXU -+fr isapmetric.
We prove POI thru PO3
(POl
x((px as p(x,x)=p(x,x)
Vxe(J
)
(PO2) Vx,y eU xKr! AyKpx
+ p(x,x)=p(x,y)= p(y,y)
+ x=!
(byP3)
(bypl)
(PO3) V x,J,zeA. xKo! A y Koz
+ p(x,x) = p(x,y) A p(y,y) = p(y,z)
(byPa)
but, p(x,z) S p(x,y)+p(y,z)-p(y,y)
p(x,7) < p(x, x)
(byP2)
p(x,z) = p(x,x/
( by definition of ((o )
x ((o z
tr
Theorem
3.5
Every pmetric topology is an Alexandrov Topology.
Proof:
p: UXU +fr
then we have to show that f
Suppose
isapmetric,
tp
I c tr[ <p ]
.
It is sufficient to show that,
Vxe(1
Suppose
, €>0
< p(x,y) + p(y,z) - p(y,y)
= p(x,y)
Thus, z e BPt(x)
lyeBpt(x)}
and €>0 aresuchthat yeBtr(x)
x,J,zeU
Then, p(x,z)
Bp€(x) = U{{zly<<rz}
andy<<pz.
(byPa)
as !
Kpz
.
tr
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I
Theorem
3.6
Foreachpmetric
<p] rfandonlyif,
BPr(t) = {ylx<<pY}
3€>0
Vxe(J
Proof
{[ p]=TI
p: UXU+X,
:
Supposefirstthat,
Then,
V x e (J 3 € > 0
BP€(x) = { y I x <<ol }
= urco{ylxKof
voetr[ <<r] o
}
e rt pl
TI <p] c Tt Pl
{t p I : TI <, ]
( b;" Theorem 3.5 )
, TI p ] - tr[ <p ]
Then, V x e(l
{ y I x Kol } e ft pl
Thus , by Theorem 3.2 ,
x e BP€(x) c { y lxKr!
V xeU 3 €>0
Suppose now that
}
But, if x e Bpt(x) then t y I x <<ol ) .g BP€(x)
BPt(x) = {ylxKol}
3€>0
Thus, VxeU
tr
4. Partial &
Theorem
Quasi Metrics
4.1
Foreachpmetric
p: UXU ->fr
Proof
q: UXU +fr
where,
q(x,Y) .'.'= P(x,Y) - P(x,x)
V x,y e U
isaquasimetricsuchthat
thefunction
ftpl =
Tt
ql and Ko= ((n.
:
Suppose
p: UXU -+fr
Let q: UXU -+9l
Vx,yeU
We show first that
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q
isapnetric.
besuchtha:,
q(x,)'t = P(x,Y)-P(x,x)
is a quasi metric by proving
Ql
and Q2.
10
V x,! e (J
(Ql+)
(Ql
q(x,y) =q(y,x)=0
Vx,yeIJ
- p(x,y)-p(x,x) = p(y,x)-p(y,y) =
:+ p(x,x) - p(x,y) = p(y,y) (byP3)
(bypl)
- x =y
e)
(Q2)
Thus
x = ! - q(x,y) = 0 (bydefinirion ofq
)
0
p( x,z) S p( x,y)+p(y,z)-p(y,y)
V x,y,ze(J
= q(x,z) S q(x,y) + q(y,z)
q
is a quasi metric
Now,
Tt
.
pl = ft ql
V xe(J,
&s,
E>p(x,x)
Bpt(x) =
V xeU,0<€<p(x,x)
V xe(.1 , €>0
and by
\*mma 2.2
Finally
,
((p
Be6_o1ra1
(x)
BP€(x) = O
Bc€(x) =
BPr*4rr. (x)
.
= Kq
V x,y e (l
&s ,
p(x,x) = p( x,y) e
u
q(x,y) =
0
Definition 4.1
A Weighted Quasi Metric overaset U isapair ( q ,ll ) consistingof aquasimetric
q: UXU --+9t anda Weight Function ll: (l -+fr where,
(WQ)
Aquasimetric
Vr,!e(J
q is Weightable
q(x,y) +lxl
ifthereexistsaweightfunction
=
ll
q(y,x)+ lyl
suchthat
(q,ll)
is a weighted quasi metric.
Theorem
4.2
Foreachweightedquasimetric
<q,ll>
overaser
u
thefunction
p: uXu -+fr
where.
V x,y eU
isapmericsuchthat
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p(x,y) .'.'= q(x,y) + lxl
{t pl =
TI
ql and ($=
((0.
11
Proof
:
Suppose
( q ,ll )
isaweightedquasimetricoveraset U.
I"rt p: UXU --rfrbesuchthat,
p(x,y) .'.'= q(x,y) + lrl
V x,y eU
We show first that
p
proving
is a pmetric by
Pl
thru P4
.
(Pl =)
Trivial
(Ple)
p(x,x) = p(x,y) = p(y,y)
V x,! e(J
+ lxl = q(x,y)+lrl = lyl
(byQl)
= lxl = q(x,y) + lxl = q(y,x) + ly | = lyl
(bywQ)
:+ q(x,y) -- q(y,x) =
= x=)
(P2)
Vx,yeU
V x,y e U
V x,y e U
(P3)
Vx,ye(I
(P4)
Vx,y,ze(J
Thus
p
Now,
(byel
lrl
p(x,x)
q(x,y) +lxl = q(y,xi +lyl
p(x,y) = p(y,x)
V x,Y e(l
is a pmetric
Tt
)
0
(byWQ)
q(x,z)
V x,y,z e (J
q(x,z) + lxl
V x,y,z e (I
p(x,z)
.
pI = ft qI
asbyLemma2.2,
V xe(J,6>lxl
BP€(x) = Bq€_*t(x)
V xe(J ,0< 6-<lrl
V xe(J, €>0
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0
Bp€(x) = O
Bc€(x) = Bpe*w(x)
12
Finally
= Kq
((p
,
&S ,
p(x,x) =p(x,Y) e
Vx,yeU
q(x,Y)=0
tr
Note that Theorems 4.1
& 4.2 establish an algebraic equivalence
restricted form of the quasi metric
Theorem
between the
partial metric
and a
.
4.3
Not every quasi metric is weightable
Proof
:
I-et q : {a,b,c}2 I {0,1,2,3}
q(a,b) = 0
Q(a,s) = I
Q(b,c) = 3
Q(b,a) = 2
Q(c,a) = I
Q(c,b) = 0
Supposethatthereexistsaweightfunction
for q,
betheuniquequasimetricsuchthat,
ll : { a,b,c} + {0, 1,2,3 }
then,
'b'
.
',
:*' "'j .i,;,"?ri,',:
= lcl + q(c,a)
( lcl + q(c,b) ) + I
= ( lbl + q(b,c) ) + I
(bywQ)
(bywQ)
(bywQ)
But this is impossible.
U
Unfortunately Theorem 4.3 does not answer the question of whether or not every quasi metric
topology can be defined using a partial metric. Theroem 4.3 only shows that the method of
defining a pmetric for a quasi metric topology using a weight function will not always work. The
next result does answer the question for finite quasi metric topologies.
Theorem
4.4
q: UXU -rS overafiniteset U thereexissapmetric
suchthat Tt pl = Tt ql and Kp--<q.
Foreachquasimetric
p:UXU-+9t
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13
Proof Outline
:
Lrt P .'.'= < U , <<q > be the partial order induced by a quasi metric
q: UXU -+9t overafiniteset U.
Let P" c 2u
subsets of U \.
be the set of all chains
Lrt $ : U -+ t't
VxeU
be the function where
$(x)
Then it can be shown that the
V x,y e U
)tut+t
in P
p(
( i.e. all non - empty totally ordered
,
.'.'= )nax{ lcl I ceP* A lubc <<qx }
function p : U X U -) o)
x,y )
where
,
.'.'=
met{ Lrr"$(x) f cep* A lubcKox A
isapmetricsuchthat
tubc<<,
l}
Ttpl = ft ql and ((p= <c.
tr
An importanr implication of Theorem 4.4 is that any finite partial order can be defined by a parrial
metric .
5.
Metrics
Definition
&
Partial Metncs
5.1
A Weighted Metrtc overaset U isapair < d ,ll > consistingof ametric
d: UXU -tfr anda Weight Function ll: -rfr where,
(WM)
Ametric
d
Vx,JeU
d(x,y)
isWeightable ifthereexistsaweightfunction
ll
suchthat
<d,ll>
isa
weighted metric.
The next two results show the algebraic equivalence between the partial metric and the weighted
metric
.
Theorem
5.1
Foreachpmetricp:UXU+sthepair(p^:UXU+9|,||:U+fr>
where,
cuNY
'92
1
4
Vx,ye(Jp^(x,y)::=2'p(x'y)P(x'x)-p(y'y)
lxl .'.'= P(x,x)
V x eU
is a weighted metric such that
,
TtPl I tr[ P^] , and'
V x,y e (J p(x,y) ::=
Proof
:
(
p^(x'y) + lrl + lyl ) /
P : UX[] 1S isaPmetric'
Then, by Theorem4'l , thefunction q :
SuPPose
V x,y eU
is a quasi metric such
Tt
Thus, byLemma2'6, thefunction
isametricsuchthat,
XU -) fr
where'
q(x,Y) ""= P(x'Y) - P(x'x)
that tt p I =
V x,y e (l
U
2
q
I
'
{ : UXU + fr
where'
f (x'y) ::= 2'p(x'y) - p(x'x) - p(y'y)
V x'y e (J
{(x'y)
= q(x'y) + q(y'x)
Thus, tt Pl I ft{l
Finally, wMholdsasbyP2
Vx'y€u
f(x'y)
> lrl
lyl
tr
Theorem
5.2
Foreachweightedmetric
where
overaset
U thefunction p: UXLI _, fr
,
V x,y e (J
suchthat
Proof
<d,||>
d=p^
p(x,y) ""= ( lxl + lyl + d(x'y) ) /
lrl=P(x'x)
and VxeU
2
:
as Ml = Pl and,
WM + PZ and,
MZ + P3 md,
Ml&M3 + P4
tr
CUNY '92
15
We now turn to constructing a contraction mapping theorem for partial metrics. The beauty of the
result in Theorem 5.3 is that its formulation is virtually the same as Banach's original theorem for
metric spaces.
Definition
5.2
Foreachpmetric
Sequence if ,
p : uXu -) N,
V€>0
3ke(t
andforeach
Vn,m>k
x e u(J, x isa
p( Xn,X^)
cauchv
< t
Definition 5.3
A pmetric p : UXU -) X is Complete ifforeachCauchysequence X e uIJ
exists aeU suchthat, 3 limn-os p( Xn, a ) - A
Note that these definitions
for Cauchy sequence and complete pmetic
are consistent
there
with the
usual metric definitions.
Theorem 5.3 ( Partial Metric Contraction Mapping Theorem )
Foreachcompletepmetric
that,
3 0Sc<I
fintly,
Proof
p:uXu
-r N,
f :U+U
such
p(f(x),f(y))
Vx,yeU
thereexists a unique
andforeachfunction
a e (J
such
that a
= f( a ),
and secondly p(
a,
a
)=0
:
Suppose
p,f , & c
Suppose
are as
in the statement of the Theorem
u e U.
Webeginbyshowingthat A n e
CUNY '92
.
u .f"(u)
isaCauchysequence.
16
Thusas, Vne(t
p(f"(u),f"(u))
S ,"*p(u,u)
weseethat ,l n e u) . f"(u) isaCauchysequence.
Thus
as
p is complete we can choose a e U
limn-* p( f"(u) , a ) - 0
But, p(f(a),a)
V n e ut
= 0 BS,
p( f(a) , a )
)
p( f"+1(u)
such that,
+ p( f"*I(u),
yn-t(u) )
a)
S c xp( a , f"(u) ) + p( f"rI(u) , a )
Thus
a
=
f( a)
eU is ,r.h thut b
p(a,b) = p(f(a),f( b)
Suppose
b
Thus, p(a,b)=0,
Thus the fixed point
of
/
Finally, p(a,a)=Q
tr
6.
Conclusions
&
andso
is unique
f(
b),
rhen,
a=b
.
as,p(a,a) = p(f( a),f(a))
Further Work
The principle conclusion from the research in this r€port is that generalised metric topology
has a largely unexplored potential in the field of partial sder denorational semantics. This conclusion
is justified for the following reasons. Firstly , this research supports the quasi metric approach
used
in [Sm87] and the metric approach in [La8fl
to model panal order topologies. Secondly
,
distance function. Thirdly ,
we have
shown that such work can be conducted using a symmetric
we have
shown that such work can be conducted within metric topology itself by adding Ore concept of a weight
to points in a metric space. Finally , we have shown that Banach's Contraction Mapping Theorem
can be generalised for proof rheory applications in Scon topologies.
Whether or not an arbitrary quasi metric topology can be defined by a weighted merric is an
open question which needs an answer in order to decide exactly which topologies can be defined using
our approach. However
, it is unlikely
that there exists any simple algebraic method for converting an
arbitrary quasi metric into a weighted quasi
metric.
Similarly
,
it is not yet known exacrly which
Scott topologies can be defined using this approach. If a favourable answer to this question can be
established then perhaps weighted metric models for the lamMa ialculus could be feasible.
CUNY '92
17
The Wadge CycleswnTest [WaS1] for proving Kahn data flow networks free of deadlock
within Kahn's fixed point semantics. By formulating
the domain of all finite & infinite streams as a partial metric space we plan to use the Partial Metic
Contraction Mapping Theorem to prove Wadge's test. Beyond this we hope to use partial metrics
to provide a denotational semantics for intensional programming languages such as LUCID
[Va&A85] which have been inspired by Kahn's work [Ka74]. For more results on partial metrics
such as partial metric products , partial metric sums , and partial metric function
has yet to receive a proper denotational proof
sryces
7.
see
[Ma92]
.
References
[dB&282)
Processes and the Denotational Semantics of Concunency
J. W. de Bakker & J. I. Zucker , Dept. of Computer Science report
NV 249
lKa74l
,
/82
[Ko88]
Stichting Mathematisch Centrum.
The Semantics of a Simple l-anguage for Parallel
Gilles Kahn
All
,
,
Proc. IFIP Conf. 1974
,
Processing
,
pp. 471- 475 .
from Generalised Metrics, Ralph Kopperman
American Mathematical Monthly , Vol. 95 , No. 2, February 1988
Topologies come
,
.
tk87l
The Versatile Continuous Order,
J. D. Lawson
,
Mathematical Foundations of Programming Language Semantics
Tulane 1987
Ma92l
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LNCS 298
Partial Metic Spaces,
Theoretical
in Research Report 212,
of Warwick
al.,
Springer 1988
.
An
Spaces
,
W. A. Sutherland
,
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o Metic and Topological Spaces,
hess . Oxford 1975
Introduction t
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of Dataflow Deadlock, W. W. Wadge ,
Computer Science 13 , pp. 3 - 15
Extensional Treatment
Theoretical
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Springer 1988.
- Unifurmities : Reconciling Domains with Metric
M. B. Smyth , Mathematical Foundations of Programming Language
Semantics , 3rd Workshop , Tulane 1987 , in LNCS 298 ,
Clarendon
[Wa81]
3rd Workshop,
8th British Colloquium for
S. G. Matthews
, University
,
Quasi
eds. M. Main et.
[Su75]
eds. M. Main et. al.
,
Computer Science, March 1992 ,
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[Sm87]
,
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tWa85l
Lucid , the Dataflow Programming I'anguage
W. W. Wadge & E. A. Ashcroft,
,
APIC Studies in Data Processing No. 22 , Academic Press 1985
tvi89l
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Topology via
l-ogic,
S . Vickers
,
.
Cambridge University Press
.
19