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
UnitedKingdom
lvIarch 1992
Partial Metric
Spaces
Steve Matthews
Dept. of Computer Science
University of Warwick
Coventry, CV4 7 AL
email sgm@uk.ac.warwick.dcs
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
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.
defined
d(Nil ,1)
VreL
=
V s,s'€ S V 1, l'e L
d(s:I,S':1') = l2x
=l
if 1+Nil
1
d( 1 ,1')
if s=s'
ifs*s'
Unfortunately Godel's decidability results force us to include partial objects such as
undefined data object alongside the complete
ones. This leads to Scott's
l-
the totally
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.
1 -
BCTCS8'.gz
Definition I
A Partial Metric ( Pmetric )
(P1)
(PZ)
(P3)
(P4)
V
V
V
V
is
afunction p
AX
A + B,
such that"
x=Y (+ P(x,x) = p(x,y)=p(y,y)
x,YeA
p(x,x)
x,y€A
p(x,Y) = P(Y,x)
x,y € A
p(x,z)
X,y,ze A
A metric is precisely a pmeric p
V xeA
:
AX
A -+ R,
p
:
AX
A -+
lR
,
:+
p(x,y) = 0
V x,Y€A
this being "half' of the metric reflexive axiom
a data
in which,
p(x,x) = Q
Also note that for each pmeric
definition of when
:
Ml .
x=Y
We can use pmetrics to make the following
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
Example
partial ::= p(x,x) > 0
I
AFlatPmetricisapmetric pt
for a set S, and l- + S , ild,
V x,Y€St
: StX S-l+ {0,1}
where, 51
::=SU{I}
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.
2 -
BCTCS8'.92
Example 2
A KahnPmetric
isapmetric pS :
KaS
{2-n l n € (, }U
X KaS -t
{0}
where
KaS is defined to be the set of all finite & infinite sequences over the set S and,
V yeKas
Ps(<>,Y) =
V x,y e (dS V n,m > 0 .
1
ps( (xgr...,\-l)
= In x ps(<xl ,...,\-l),(yl
= 1
V x e cds ps(x,x) =
, (Y0,...,Ym-t >
,...,ys1-1
)
)) if *o=yo
if*O+yO
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
Br(x)
foreach
.
€>0
p : A X A -+ IR
,,= { y.A
and
are the sets of the form,
I p(x,y)<€
}
xeA.
Theorem I
Theserofallopenballsofapmetric
A.
Proof:
Proof usingDefinition
p: AXA +
IR wrth
A
formthebasisofaopologyon
A4 andTheorem Al
p: AXA -r IR isaPmetric.
Then, A = U*"A Bil*,*)+t (x)
Suppose
and,
) 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)}
for any balls B,( x
tr
3 -
BCTCS8',92
Theorem 2
Foreachpmetric
p: AXA -+ IR, openball Br(a), and xe A,
xcB6(x)EBr(a)
xeBr(a) :+ 3 6>0
Proof:
Suppose
;
e B,(a)
Then p(x,a) < €
Let 6 ::= € - p(x,a) + (x,x)
Then 6>0 as €tp(x,a)
as €tp(x,a)
Also, p(x,x)<6
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-+
Suppose x+y e A
Thenfrom
IR
isapmetric.
Pl & P2 p(x,x) < p(x,y) or p(y,y) < p(x,y)
p(x,x)<p(x,y) then,
leBr(x) n y* Br(x) where, €::= (p(x,x)+p(x,y))/2
Wlog suppose
tr
Note that the open balls (with A) of the form,
Br(x)
form
a basis
r,= { y.A
I p(x,y)<t
}
for the same topoplogy as the balls (with A) of the form,
B'r(x)
rt= { y.A
I p(x,y)
B',(x) : Br+X*,*;(x)
V €>0 V xeA
Br(x) = B'r_p1r,*;(x)
andas, V €tp(x,x)
Br(x) = g
and, V0<€<p(x,x)
xs,
4 -
BCTCS8',gz
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
I eA iff :l li-n__* p(\,I)
= p(I,tr)
convergesto
Proof:
X converges to I
ThenV€>0 3k>0 Vn>k
ThenV€>0 3k>0 Vn>k
ThusbyP2 3 t-o_*" p(4,,I)
Suppose
.
Supposenowthat
Also,
suppose
Xn.Be*4r,fy(I)
p(Xn,I)-p(I,I)
= p(I,tr)
3 hro_r- p(Xn,I) = p(I,I).
that )r e B,( a ) .
X is eventually in B,( a ).
As [m"-* p(Xn,tr) = p(],])
wecanchoose k>0 suchthat,
p(Xn,I)-p(tr,I)
Vn>k
p(4,,a) < p(Xn,I) - p(I,I) + p(I,a)
Vn>k
We have to show
Vn>k
that
=€
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
V x,y€A
Theorem 5
Foreachpmetric
x .p y
p: AXA+ IR
isthebinaryrelationdefinedby,
<+ p(x,x)=p(x,y)
.p
isapanialorder.
Proof:
VxeA.
Vx,y€A
5 - BCTCS8',92
x.px
as p(x,x)=p(x,x)
x.py n y.px
+ p(x,x)=p(x,y)= p(y,y) (bVP3)
(bypl)
=e x=y
n Y'Pz
Vx,y,zcA.x'PY
+ 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)
(byP2)
p(x,z) = p(x,*)
( by definition of .p )
x .p z
tr
ps
p
.p1
domain, while for each Kahn pmetric
.pS istheusual"initials€gment"orderingonsequences. Forametric d: AXA + lR
For each flat pmetric
is the usual ordering on a flat
.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
sequences have unique
h
TO spaces as not all
limits, even if chains do.
Theorem 6
Suppose XeoA convergesto IeA inapmetricspacewithpmetric p:
and that I' e A is such that I' 'p I . Then X converges to I' as well.
AXA-+
IR
Proof:
By Theorem 4 it is sufficent to show that
3 lhq,->0,
P( Xn
then as
Suppose €
we can choose k >
V n>k
Vn>k
, I' )
X
0
,
=
convergesto
such that,
p( Xn, I )
. p( Xn, I')
P(
I', I')
I
(byTheorem4)
- P(I,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.
6 -
BCTCSS'92
Definition 4
Asequence
Converges
Xe @A inapmetricspacewithpmetric p : AXA +
to I e A if X convergesto tr and,
B, Properly
3 [rnn-- p(4,,)q) = p(],tr)
Inotherwords
Xe oA
properlyconvergesto
3 lirnnr- p( 4, ,4 )
Ie A if 3 limo-- p(4t,I )
and'
an4
limo-- p(Xn,\)
= p(]'])
= [-r,-- P(4t,])
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
X e oA properly converges to both tr and I'
p: AXA + lR, then I' 'P tr.
Suppose
Proof:
in
a
pmetric space with pmetric
X e oA properly converges to both I and I'
Choose € > 0 , thenwecanchoose k > 0 suchthat,
p(I,I)
p( Xn,I )
Suppose
.
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€
tr
Thus p( l' , I' ) =
And so I' .p )
p(
I , I')
as
€
was an arbitrary choice.
The implication of the Theorem 7 is that limits to properly convergent sequences are unique. This is
for non-Hausdorff Tg spaces.
Other standard metric constructions also generalise to pmetric
an interesting result
considerable & surprising ease.
7 -
BCTCS8'.92
spaces
with
both
Definition 5
A sequence X c oA inapmetricspacewithpmetric
V
Definition
€
>0 3 k>0 V n,m>k
p: AXA -+ lR is Cauchy f'
p(4,,Xn') - P(Xn,,Xm)<€
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:
f : A + A is a contraction in_a complete puqal. meqc space with
IR, andthat 0Sc<lissuchthat,
pmetric p: AXA+
V x,y e A . p( f(y), f(x) ) p( f(x), f(x) )
Suppose
Let a e A, andlet X e @A besuchthat V n > 0
We will first show that
X
\ = fn(a).
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
= cn x (1-"k+l;
1-c
B
BCTCS
8
'.92
, Xn ) - f( X1' , Xn )
x (f( Xl ,& ) - f( X0,& ))
Thus
X
is seen to be a Cauchy sequence.
Thus as our prnetric space is complere
We now show that
I
is a fixed point
Choose € >
0,
Vn>k
p( I ,
X
properly converges
to I e A
say.
of f .
X properlyconvergesto tr wecanfind k 2 0 suchthat,
Xn ) - p( x' Xo ) < e/(l+c)
thenas
n p(\,l)
Thus Vn>k
- P(I,l)
I ) - P(I'I
p( f(I),
)
+p(\+r,I)-p(l,I)
+ p( Xn+l , I ) - p( I , I
)
=€
Thus,as€isarbitrary,
Similarly,
Vn>k
) - p( f(I),
.p( f(I),I
(*)
- p(I,tr)
p(f(I),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 ))
p()',I))
E cx(p(),Xn)
+E/(l+c)
=€
Thus, as €
isarbitrary, p( f(I),
Thusfrom
(*) andPl l=f(l),
have a fixed
9
BCTCS
8
'92
l ) = p( f(I),
f(I)
andsof hasbeenshownto
point. It just reamins to show that
I
is unique.
)
Suppose
P(
I' c A and l' = f( )') ,
I' *''=
I
then
,
l,';,^','nt', ) - p( f(I'), r(I')
=0
p(I,I')-P(I',I')
I',pI
Similarly we can show,
I 'p I'
as
and
)
0Sc<1
so tr =
tr'
tr
Weighted Metrics
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
So far we have explained partial metrics in terms of a generalisation
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, ||
andaWeightFunction ll:A+
d:AXA+lR
V x,y e A
Theorem
IR
where,
d(x,Y)
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,
P(x,y) = (d(x,Y) +lxl + lYl) /
Vx,y € A
We will first show that
(P1=+)Trivially
10 -
p
BCTCS8',92
- P4
x=y + p(x,x) = p(x,y) = p(y,y)
is a pmetric by proving Pl
V *,y € A
2
.
(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
(byMl)
+ d(x,y) = 0 + x=Y
(byM2)
p(x,y) = p(y,x)
(P2) V x,y e A
(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,y ) +lx | + ly
d(x, z) + lx | + | z I
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
V x e A lxl ::= p(x,x)
d(x,Y) := 2rP(x,Y)
V x,Y€A
is a weighted metric by proving
(M1+) V x,YeA
( d, ll>
lxl
11
BCTCS
V x,y € A
8
'.92
lYl
Ml - M3.
x=Y + d(x,y)=0(bydefrnitionof <d,ll>)
d(x,y) =Q
O{1e) V x,y€A
lxl
lyl = 0
=t 2*p(x,y)
+ (p(x,y)-p(x,x)) + (p(y,x)-p(y,y))
+ p(x,x) = p(x,y) = p(y,y)
:+ x=y
(N42)
definedby,
d(x,y) = d(y,x)
= 0 (byP3)
(byP3)
(byP2)
(byPl)
(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,
x <Y
V x,Y€A
(+
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,
spaces. The remainder of this paper
demonstrates that such constructions do exist. First we need to apply to pmetrics a standard
sum, and function
space constructions to build useful
construction used for turning an unbounded metric into a bounded metric.
Definition 9
Foreachpmetric p : A X A -) IR , P^ : A X A -+ [0rl)
suchthat
V x,y€A
Using Theorem
Theorem
43
p
to check P4 it can easily be verified that
p"
is indeed a pmetric
.
the topology induced
by p"
is the same as p
.
:
Suppose
p : A X A ->
VxeA
V€>0
VxeA
V0<e
V xeA
V €>1
tr
12
p^(x,y) = p(x,y) /( 1*p(x,y))
10
For each pmetric
Proof
isthepmetric
BCTCS
I
'.92
IR isapmetric, then,
B,(x)= B^6111a6;(x)
<1
B"r(x) = Br4r-ry(*)
and,
and,
B"€(x) = U{ Bp(*,r)+r(y) ly€B^r(x)
}
Also note that for each pmetric p
, .p = . ( p" )
.
Definition f0
TheCountableProductof thepmetrics
p' : (Xn>oAn)2 + IR
pn: \XA1, -+ IR (n20)isthefunction
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
+
Pn: AnX\
We
isthecountableproductofthepmetrics
IR
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)
x 2-n'r
=i In>O (pn)"(\,\)
= In>o (pn)"( xn , yn ) x
= In>o (Pn)"( yn , yn ) x
+ In>o ( (pn)^( \, yn )
=In>0 ( (po)"(\,yn)
Z-n-l
2-n-l
\ )) x
(pn)"(yn,yn) ) x
(pn)n( *n,
=Q
+
:+
V n)0
V n20 . xn = yn
(P2) by Y2 for each pn.
(P3) by P3 foreach pn.
13 BCTCS 8
'92
(Pn)^(\,\)
= (pn)"( yn,
=(Pn)n(\,yn)
yn
)
=+ x=y
z'n'r
2-n-l
(P4) V x,y,z e Xn>o\
Pt(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
p'(x,z)
tr
The countable product has the "pointwis€" ordering, i.e.
V X,y € Xn>04,. x'(p")y
Definition
ce V n20.
\ r(
(pn)n)
yn
11
TheDisjointSumofafamilyofpmetrics
pmetric p+: Ui.l{<i,x>lxeAi}
pi:
A'1
X,A.1
+ [0,1]
-) IR (ieI)
isthe
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
Definition
a
function space.
12
p: AXA+
IR aset A*cA isProperlyDenseinAifeach
member x e A is the limit of a sequence in A* properly converging to x
Foreachpmetric
.
Definition 13
Aset A withpmetric p: AXA -r IRis Sufferable iftherethereexistsacountableproperly
andfunction p*: A-+R-{0}
denseset A*cA
x,Y.
4*10,21
suchthat forany
,
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
designer
14
is likely to
BCTCS
B
have some
'92
kind of countable
dense subset as the universe
will probably
be the
Erratum (RR212 Partial Metric Spaces)
Definition 13 is unnecessarily strong and should be replaced by,
Definition
A set
A
13
with pmetric
p:AXA
+ IR is Sufferable if ,
3 r20 e IR V x,y e A
andthereexistsacountableproperly dense set
p(x,y)
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
sufferable
A
is denoted bY A
'+
A'
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
sequence { e orfi properlyconvergingto x e A thesequence Y e A' where,
Yn = f(Xn)
V n>0
properly converges
A.+
to
f( x
).
each
The set of all such properly continuous functions is denoted by
A'
Definition 15
For pmetrics p:AXA+lR
and p':A'XA'-rlR,
(A'+A') + IR and ll : (A'+A') -) IR
d):(A-A')X
arethe
functions such that,
V f e A"+ A'
V f,g € A'+ A'
where < d'
Theorem 9
, I l' >
lfl = I3ctr* p*(a) x lf(a)l'
dt(f,g) = Ia.A* p*(a) x d'(f(a), g(a) )
is the weighted metric equivalent
for
(p')"
as constructed
in the proof of
.
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
d)(f,g) -0
(M2e) V f,g€ A*r A'
d'(f(a),g(a)) = 0
+ VaeA*
f(a) = g(a)
+ V a e A*
+ flA* = glA*
+ f =g asf &garecontinuousandA*isdenseinA.
(M2) d) is symmetric as d' is symmetric
15
BCTCS
8
',92
.
dt(f,h)
V f,B,hc A.+ A'
= IarA* P*(a) x d'( f(a) ' h(a)
(M3)
)
= 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
v'''
.=^;"1.
;.ijl ; l:lr"r,',s(a),'
|
|
is a weight.
)
is a weighted metric
= d)(f,g)
Thus
I
I
is a weight
Thus < d), I I >
for
A !t
A'
is aweightedmetricfor
A l+ A' .
tr
As in the proof of Theorem 9 we can construct a pmetric
a potential
A'
.
An important result for
function space is the following.
Theorem
12
f .p)g
Vf,geAD+A'
Proof
p) for A '+
<+ VaeA
f(a).p'g(a)
:
V f ,g e A l+ A'
(+ d(f,g)
f 'P)
= lfl
g
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*
<+ V aeA
n
16 BCTCS 8 ',92
f(a).p'g(a)
f(a) .p'g(a) asA*isproperlydenseinAand
f & g are properly continuous and Theorem A4
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
n xe€l-
x.py
X,Y € A
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.
References
tdB&2321
Derntaional Senawics of Concunency ,
J. W. de Bakker & J. I. Zucker , Dept. of Computer Science report
Processes andthe
NV 209182
lKa74l
,
Stichting Mathematisch Centrum.
Proc. IFIP Conf. 1974
17 -
for Parallel Progranvning
pp. 471 - 475 .
The Semantics of a l-anguage
BCTCSB'.92
,
,
Gilles Kahn
,
[Ko88]
All Topologies comc from Generalised Metrics, Ralph Kopperman,
American Mathematical Monthly, Vol.95 , No.2, February 1988.
[Sm87]
Quui - Untformitics : Reconciling Dornains withMetric Spaces ,
M. B. Smyth, Mathematical Foundations of Prognmming Language
Semantics
,
3rd Workshop
,
Tulane 1987
,
in
LNCS 298
,
eds. M. Main et. al.
[Su75]
Introduction
n Metric andTopological Spaces, W. A. Sutherland,
Clarendon Press. Oxford 1975
.
Appendix
Definition A1
A TopologJr on a set A
is a set
t c 2A such that,
(Tl) A e 0(T2) Aef,
(T3) vscr
USe€r-
(T4) V Sclf
( Members
of f
Definition
A topology
are
lSl
<""
called open sets )
on a set
A is Tg if,
V x+y e A 3 Oef,
( xeO n y+O ) v
A topology
f,
on A is
T,
Definition
A4
basis for a set
)
(i.e. Hausdorff) if,
A
is a set
E c 2A
(Bl) A = Uf3
(F.2) V 81 ,BrefJ
18 -
( y.O n x*O
A3
V x+y e A 3 O,O'ef
xeO n y€O' n
A
nS € t
A2
f
Definition
:+
BCTCSB',92
OnO'=A
such that,
3 A c tr3
BlfiBz =
U,S
Theorem Al
Foreachbasis
E fqanon-empty
Definition A5
AMetricisafunction
set
A,
d: AXA+ n
(Ml) Vx,y€A
.
(l{12) Vx,I€A
(M3) V X,y,zeA
Definition A6
An Open Ball forametric d
:
AX
,r= { y.A
Br(x)
U
G
isaopologyonA.
suchthat,
(+ d(x,y) =
x=y
d(x,y) = d(y,x)
0
d(x,z)
A + IR
is a setoftheform,
I d(x,y)<€
}
forany x€ A and €>0.
Theorem A2
Theopenballsofametric
topology on A.
d: AXA + IR with A
formabasefora
Tz
Theorem A3
V &,b,c,d ) 0
a
abcd
1+a
l+b
l+c
l+d
Theorem A4
p : AX A -+ IR isapmetric, and X,Y e oA, &Dd x,y €
aresuchthat X properlyconvergesto x and Y properlyconvergesto y, and,
Suppose
V
Then
n>
0
Xn .p Yn.
x .p y.
19 -
BCTCS8',92
A