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Top ology Domain Theory and Theoretical

Computer Science

Michael W Mislove

Department of

Tulane University

New Orleans LA

Abstract

In this pap er we survey the use of ordertheoretic top ology in theoretical computer

science with an emphasis on applications of domain theory Our fo cus is on the uses

of ordertheoretic top ology in programming language semantics and on problems of

p otential interest to top ologists that stem from concerns that semantics generates

Keywords Domain theory Scott top ology p ower domains untyp ed lamb da calculus

Subject Classication BFBNQ

Intro duction

Top ology has proved to b e an essential to ol for certain asp ects of theoretical

Conversely the problems that arise in the computational

setting have provided new and interesting stimuli for top ology These prob

lems also have increased the interaction b etween top ology and related areas of

mathematics such as and top ological algebra In this pap er we

outline some of these interactions b etween top ology and theoretical computer

science fo cusing on those asp ects that have b een most useful to one particular

area of theoretical computation

This pap er b egan with the goal of highlighting how the interaction of order

and top ology plays a fundamental role in programming semantics and related

areas It also started with the viewp oint that there are many purely top o

logical notions that are useful in theoretical computer science that could b e

highlighted and which could attract the attention of top ologists to this area

And to b e sure there are many interesting and app ealing applications of

pure top ology certainly in the form of arguments that have

b een made to theoretical computer science But as the work evolved it b e

came clear that the main applications of top ology to the area of programming

1

This work is partially supp orted by the US Oce of Naval Research

Preprint submitted to Elsevier Preprint May

Mislove

semantics involve not just top ology alone but also involve order in an essen

tial way And this approach places domain theory at center stage since it is

the area that has combined order and top ology for application to theoretical

computation most eectively The goal now is to show why this is so and why

it is ordertheoretic top ology that has had such a large impact on theoretical

computer science In particular we highlight those asp ects of domain theory

and its relationship to top ology that have proved to b e of greatest utility and

imp ortance At the same time we do cument the advantages domain theory

enjoys in this area of application and the standard that domain theory has

set in providing solutions to the problems this area of application p oses

Top ology versus Order

Lets b egin with a simple but illustrative example

Example Let N f g denote the natural numb ers and let N

N denote the set of partial functions from N to itself Consider the family of

partial functions f N N dened by

n

n

m if m n

f m

n

undened otherwise

We would like to assert that the functions ff g converge to the

n nN

FAC N N by FACm m m N

To nd a suitable top ology on N N to express this convergence we

rst identify N N with a space of total functions Let b e an element

not in N and dene N N fg We interpret as undened and we

dene an injection

n

f x if f x is dened

f f N N N N by f x

otherwise

Then it is clear that ff j f N N g is precisely the set of selfmaps of

N that are strict ie those that take to itself If we endow N with the

discrete top ology then all the functions in N N are continuous

Prop osition In the compactopen on N N the sequence

ff g converges to FAC

n nN

Pro of Since N is discrete the compactop en top ology on N N is the

same as the top ology of p ointwise convergence so the result is clear 2

We also note that by endowing N with the discrete metric and giving

N N the Frechet metric the Prop osition remains true for the metric

top ology on N N

But even though we have convergence of ff g to FAC after suitable

n nN

identication with ff g something is lost in this assertion Namely the

n nN

functions ff g represent increasing approximations to FAC indeed as n

n nN

increases so do es the amount of information we have ab out the limit function

FAC In fact there is a natural order on N N that makes this idea precise

Mislove

Denition Dene the extensional order on the space N N by

f v g i domf domg g j f

domf

where domf f N

Clearly in this order any increasing family of partial functions has a supre

mum the union Moreover the function FAC is the supremum of the family

ff g Our next goal is to capture this as a convergence in order This

n nN

leads us to the Scott topology

Denition Let P b e a

A subset D P is directed if F D nitex D y v x y F

P is a cpo for short if P has a least element

usually denoted and if every directed subset of P has a least upp er

b ound in P

Note that a must b e nonempty since the empty set is a nite

subset of every set

For example the family N N is a cp o the nowheredened function is

the least element and the supremum of a directed family of functions is just

their union Similarly we can give N the at order whereby x v y if and

only if x or x y for all x y N This corresp onds to the p ointwise

order on the space N N of monotone selfmaps of N and in this order

the supremum of a directed family of functions is the p ointwise supremum

and the constant function with value is the least element

Denition Let P b e a partially ordered set A subset U P is Scott

open if

U U fy P j x U u v y g is an upp er set and

F

D P directed D U D U

Prop osition Let P be a partial ly ordered set

i The family of Scott open sets is a T topology on P

ii If x y P and there is some open set containing x but not containing y

then x y

iii The fol lowing are equivalent

a The Scott topology is T

b P has the discrete order

c The Scott topology is T

d The Scott topology is discrete

Pro of Let P b e a partially ordered set Clearly the union of upp er sets from

F S

U with U op en P is an upp er set And if D P is directed and D

i i

i

F

for each i I then D U for some i I Since U is Scott op en it follows

i i

S

U that D U and so the same is true of D

i i

i

If x v y P then the denition of Scott op en implies that y fz P j

z v y g is Scott closed Since x y we have x P n y which is Scott op en

Hence the Scott top ology is T this proves i

Mislove

For ii if x y P and x U then x y and U U imply y U as

well Thus y U must imply x y

Finally iii follows easily from ii 2

In our example N N it is not hard to show that f is Scott op en if

domf is nite a directed union of functions extends a nite function if and

only if one of the functions in the directed family extends the nite function

and as it happ ens this family of principal upp er sets forms a basis for the

Scott top ology on N N

There are a numb er of imp ortant results that are true of the Scott top ology

Below we summarize some of them they all can b e found for dcp os cp os

without least elements as well as cp os in eg

Prop osition Let P be a cpo and endow P with the Scott topology

F

i If D P is directed then D is a limit point of D and it is the greatest

limit point of D

ii Let I and J be directed sets and let i j x I J P be monotone

ij

Then

F F F F

x x a

ij ij

j J iI iI j J

F F F

x x b If I J then

ii ij

iI iI j J

iii If Q also is a dcpo then f P Q is continuous i f is monotone and

preserves sups of directed sets 2

As we shall see the second part of this result is a very useful to ol in proving

results ab out continuous functions b etween dcp os

One of the most celebrated results ab out cp os is the following We at

tribute it to Tarski who rst proved it for complete lattices However

a numb er of others among them Scott and Knaster have contributed

to this result

Theorem Tarski

If P is a cpo and f P P is monotone then f has a least xed point

F F

n

f f If f is continuous then xf xf

nN Ord

Pro of We conne ourselves to an outline For the rst part the monotonicity

n

of f and the fact that v x x P implies v f and so ff g is

nN

a chain Since P is a cp o this chain has a least upp er b ound which we use to

F

n

f A transnite induction then shows that f dene f

nN

is welldened for all ordinals and that implies f v f

Since this holds for all ordinals there must b e one where the increasing chain

stops growing and the rst place this happ ens is easily seen to b e a xed

p oint of f The fact that v x x P implies f v f y y for all

xed p oints y of f and all ordinals showing that the rst ordinal where

f f f is the least xed p oint of f

If f is continuous then it follows that

G G G

n n n

f f f f f

n n n

2

Mislove

Example Returning to our example we recall that the natural order on

N N makes the empty function the least element this corresp onds to the

p ointwise order on the family N N of monotone selfmaps of N and

the constant function with value is the least element

Now let N N N N b e the family of Scottcontinuous

selfmaps of N N Dene

if n

if n

F N N N N by F f n

n f n otherwise

where we dene n n for all n N Then

F is Scott continuous indeed as the domain of the function f increases

then the domain of F f also increases which implies F is monotone Con

tinuity then follows from the fact that the supremum of a directed set of

functions is just their union

If denotes the function that has value at all p oints on N then

n

F f for n this is a routine induction argument

n

F FAC FAC indeed FAC has maximal domain N and so it cannot

b e extended And it is clear that F FACn n FACn for all n N

FAC x F this follows from the second observation and the fact that

F

FAC f

n

n

Actually this example shows an alternative approach to obtaining a re

cursively dened function from just one functional We shall see a striking

generalization of this result later

Weve already seen that continuous functions can b e characterized purely

ordertheoretically Another fact ab out continuous functions also is imp ortant

to note

Prop osition Let P Q and R be dcpos and let f P Q R Then

f is jointly continuous wrt the Scott topology on P Q if and only if f is

separately continuous wrt the product of the Scott on P and Q

Pro of If D P Q is directed then it is routine to show that tD

t D t D If f P Q R is separately continuous wrt the pro duct

P Q

of the Scott top ologies on P and Q then

0

f tD f t D t D f t d t d

P Q dD P d D Q

0 0

t f d t d t t f d d

dD P d D Q dD d D P Q

t f d d tf D

dD P Q

so f also is continuous wrt the Scott top ology on P Q

Conversely if f P Q R is continuous wrt the Scott top ology of P Q

then f preserves suprema of directed sets in P Q which clearly implies f

preserves suprema of directed subsets of P fy g and fxg Q resp ectively

for all x P and y Q This characterizes separate continuity of f wrt the

pro duct of the Scott top ologies 2

Remark It should b e noted that for dcp os P and Q the pro duct of

Mislove

the Scott top ologies of P and Q is in general weaker than the Scott top ology

of the pro duct However these top ologies do coincide for continuous dcp os

cf Section

Theorem For dcpos P and Q the family P Q of Scottcontinuous

maps is a dcpo in the pointwise order

Pro of Its routine to show that the directed supremum of monotone func

tions is welldened and monotone The fact that order of computing the

supremum of a pro duct of directed sets can b e reversed implies the supremum

function itself also preserves directed suprema hence is continuous 2

Tarskis Theorem guarantees that the the op erator x D D D is

D

welldened and using part ii of Prop osition it is easy to show that this

op erator is continuous The following discussion shows in what sense x is

D

unique

A xed point operator is a family of continuous maps F D D D

D

for each cp o D which satises F f f F f for each f D D

D D

Such a family is called uniform if F g hF f for all continuous maps

E D

f D D and g E E and strict continuous maps h D E satisfying

h f g h

f

-

D D



?

F g hF f

h h

E D

? ?

g

-

E E

Theorem x is the unique uniform xed point operator dened on the

CPO 2

It is clear that fg is a terminal ob ject for the category CPO of cp os and

continuous maps and that the pro duct of cp os is another such so CPO is

cartesian It also is closed as Theorem shows The fact that P Q

R P Q R also is clear from Prop osition Thus we have

Theorem The category CPO of cpos and Scott continuous maps is

cartesian closed The same also holds of the category DCPO of dcpos and

Scott continuous maps 2

Our aim in this section was to show how order theory together with top ol

ogy provides a richer theory than top ology alone While we have not shown

that top ology alone cannot claim the results we have enumerated it should

b e clear that results we have highlighted are available in a particularly sim

ple way in the cp o setting and that this theory oers some results such as

Tarskis Theorem and Theorem that are not so easily available in other

settings We also shall see that these results are particularly useful in the

Mislove

area of programming language semantics which is at the heart of theoretical

computer science

Domain Theory

While some asp ects of our motivating example in the previous section clearly

are close to computability nothing in the general theory of the category CPO

addresses this directly Domain theory adds this asp ect to the theory we have

outlined

Basic Results

We b egin this development with some standard denitions

Denition Let P b e a dcp o An element k P is compact i k is

Scott op en We let K P fk P j K is compactg and for each x P

K x x K P

For example if we consider N to b e the at natural numb ers then

K N N We already noted that the partial functions with nite do

main are compact in N N from which it follows that K N N

ff j domf is niteg where N N is the space of continuous selfmaps of

N leaving xed

Denition The dcp o P is algebraic if for all x P

K x is directed and

F

x K x

By a domain we mean an algebraic cp o

The following result is basic to the theory

Theorem Let P be a dcpo and let B K P be a family of compact

elements of P If for al l x P

i B x x B is directed and

F

ii x B x

then P is algebraic and B K P

F

Pro of If the conditions hold then x B x can b e used to show that K x

F F

is directed so x B x v K x v x and then P is algebraic If k

F

K P then k B k implies k B k by the denition of compactness 2

We noted earlier that the functions f N N that leave xed and

that have a nite domain are compact elements in N N A corollary of

N ff j domf is niteg Theorem is that K N

Denition Let P b e a partially ordered set An ideal of P is a directed

lower set of P We let IdlP denote the family of ideals of P

Mislove

If P is a p oset then using Theorem it is routine to show that IdlP

is an algebraic dcp o whose compact elements are K IdlP fx j x P g

The denition of algebraicity then implies that a cp o P is algebraic if and

F

only if P IdlP indeed the mapping x K x has I I as its inverse

Moreover IdlP is a cp o if and only if P has a least element

To elevate the ab ove relationship to an equivalence of categories requires

using relations b etween p osets rather than functions Since a continuous map

f P Q b etween domains need not preserve compact elements such a func

tion f do es not restrict to a function from K P to K Q But for each

F

x P f x K f x is completely determined by the ideal K f x of

compact elements of Q This gives rise to the following notion

Denition Let P and Q b e p osets An approximable relation R P Q

is a relation satisfying

x P y y Q xR y w y xR y

x P M Q nite y M xR y z Q xR z M z

x x P y Q x w xR y x R y

These conditions insure that the set fy Q j xR y g is an ideal of Q and

so the relation R is really a monotone function from P to IdlQ this then ex

tends to a unique from IdlP to IdlQ If R P P

and S P P are approximable relations then so is S R P P Hence

there is a category POS of p osets and approximable relations The corre

A

sp ondence taking a continuous mapping f P Q b etween algebraic dcp os

S

to the approximable relation R K P K Q by R ffk g K f x j

f f

k K P g then has as its inverse the assignment taking an approximable re

lation R P Q b etween p osets to the continuous map f IdlP IdlQ

R

F S

by f I R I Thus we have an equivalence of categories POS and

R A

ADCPO b etween p osets and approximable relations and algebraic dcp os and

continuous mappings This equivalence cuts down to an equivalence b etween

the full sub categories POS of p osets with least element and ALG of domains

A

Theorem The category ALG of domains is equivalent to the category

POS of posets with least element and approximable relations 2

A

If by a local ly compact space we mean one in which each p oint has a

neighb orho o d basis of compact sets then the following is obvious from the

denitions

Prop osition If P is a domain then fk j k K P g is a basis for the

Scott topology on P and so P is local ly compact 2

We will see later that domains also are sober for now we leave this issue and

concentrate on bringing computability more to the fore A detailed description

would include an indication of how enumerability can b e captured in this

setting The details are to o many to go through here so we conne ourselves

to the following brief indicator

We showed that the function FAC N N could b e realized as the least

Mislove

xed p oint xF where F N N N N This result has a

striking generalization

Theorem MyhillSheperdson

If g N N is partial recursive then g can be realized as g xG for some

continuous selfmap G N N N N 2

This hints at the close relationship b etween notions of computability and

domain theory We could summarize this relationship with the following slo

gans

Algebraicity captures Computability

k compact if and only if k is computable in nite time

f P P recursive implies F P P P P f x F

Continuous Domains

Many of the basic results outlined in the previous section have an imp ortant

generalization In his seminal pap er Scott comments that the algebraic

lattices he discovered as injective spaces are in some sense zerodimensional

and to close up the class under quotients one needs to consider p ositive

dimensional analogues This was the imp etus for the results in where it

is shown that continuous lattices form the class of ob jects so generated At

the more general level of cp os the corresp onding ob jects are the continuous

cpos Some of their theory was presented in the exercises in but the

nicest presentation we have seen is in Heres a brief outline of the basics

of that theory

Denition Let P b e a dcp o and let x y P

F

We write x y if for all D P directed sets if y v D then D x

If x y then we say x is waybelow y We let y fx P j x y g for

each y P

P is a continuous dcpo if for all y P

i y is directed and

F

ii y y

We let CON denote the category of continuous cp os and Scott continuous

maps

Clearly x P satises x x if and only if x is compact and so each

algebraic cp o is continuous Given a continuous cp o P in analogy with the

p oset K P we can dene the preordered set P In general is not a

partial order x x i x is compact The prop er generalization of K P is

given in the following denition

Denition An abstract basis is a set B together with a transitive rela

tion which satises the interpolation property

INT M B nitex P M x y x M y x

Mislove

Here M x means z x for all z M

Prop osition Let P be a continuous dcpo Then satises INT Hence

P is an abstract basis

x fy P j x y g is Scott open for each x P

Pro of Let P b e continuous and let x P Consider the set A fz P j

w P z w xg It is routine to show that A is directed and clearly

F F F

A v x If A x then x x implies there is some w x with

F

w v A and then the same argument implies there is some z w with

F F F

z v A Then z A so z v A which is a contradiction Hence A x

Next if M P is a nite set with y x for all y M then for each

y M there is some y A with y v y Cho osing z A with y v z for

each y M implies there is some w x with z w But then y v y v z

implies y w x for all y M Hence P satises INT

The rst part of the Prop osition now follows As for the second part if

F F

D x then x D and so INT implies there is some y P with

F

x y D Then d D with y v d and so d y x 2

Corollary If P is a continuous dcpo then P is local ly compact in the

Scott topology

Pro of The fact that x is Scott op en implies x is a Scottcompact neigh

b orho o d of each p oint in x 2

Each abstract basis B has an ideal completion IdlB the set of

directed lower sets of B and this ideal completion is a continuous dcp o

in which x y implies x y in IdlB Moreover given a continuous

dcp o P the mapping x x P IdlP has as its inverse the mapping

F

I I IdlP P

In further analogy to the algebraic case there is a notion of approximable

relations b etween abstract bases and the following theorem holds

Theorem Abramsky Jung

There is an equivalence between the categories ABAS of abstract bases and

approximable relations and COND of continuous dcpos and continuous map

pings This equivalence restricts to an equivalence between the ful l subcate

gories ABAS of abstract bases with minimum elements and CON continuous

cpos 2

What this all says is that there is a uniform approach the algebraic and

continuous cases in which the algebraic structure of continuous cp os can b e

highlighted and used eectively to understand the structure of continuous

cp os It has b een known for some time that certain asp ects of the theory

of domains are more elegantly and simply presented in the continuous case

b ecause of the closure under quotients and the approach of abstract bases

provides a metho d for developing that theory in a way that aords easy access

to the results ab out algebraic cp os that one might wish to highlight

Mislove

Categories of Domains

In the rst section we noted that CPO and DCPO are cartesian closed cate

gories If P and Q are algebraic then it is easy to show that P Q also is

algebraic and that K P Q K P K Q Since the terminal ob ject also

is algebraic ALG the category of domains and continuous maps is carte

sian If we want to know whether ALG also is cartesian closed the following

result shows we dont have to lo ok far for a p otential internal hom

Theorem Smyth Jung

Let C be a ful l subcategory of ALG and let P Q be objects of C

i If C has products then P Q is the product of P and Q in C

ii If C has exponentials then P Q is the exponential of P and Q in C

2

We attribute this theorem jointly to Smyth and Jung Smyth showed

this for algebraic domains ie ones for which K P is countable and

Jung extended the result to the general case

Unfortunately ALG is not cartesian closed Indeed a simple example that

op

hints at the problem is to show that N the natural numb ers in the

op op

dual of the usual order satises N N is not algebraic In

fact K f for any function in this space This example is taken from

So one might ask what cartesian closed categories exist within ALG The

rst one we note is probably the b estknown

Denition A domain P is a if P is closed under the

formation of nonempty inma

Theorem The category SD of Scott domains and continuous maps is

cartesian closed 2

Clearly the pro duct of Scott domains is another such so the pro of of this

result requires only consideration of the Here a little work

is required The p ointwise inmum of a family of continuous maps b etween

Scott domains surely is welldened but it is not necessarily continuous What

one has to take for the inmum is the largest Scott continuous map which is

p ointwise b elow the p ointwise inmum Of course even once it is shown

that this map exists and that it is the inmum it also needs to b e shown

that the family of continuous maps b etween Scott domains is algebraic Here

one explicitly shows that every continuous function is the supremum of step

functions which clearly are compact elements in the function space By taking

nite subsuprema of such stepfunctions one sees that the compact elements

of the function space form a basis

A larger cartesian closed category of domains is obtained in the following

way

Denition An projection pair e p P Q b etween do

Mislove

mains P and Q is a pair of continuous maps e P Q and p Q P satisfying

p e and

P

e p v

Q

Actually an ep pair is a sp ecial case of a Galois adjunction b etween the

domains P and Q e is the lower adjoint and p the upp er adjoint

Denition A domain P is SFP if there is a sequence of nite p osets

and ep pairs fe p P P g such that

nn nn n n nN

P limP p p

n nn mm mnN

n

colimP e e

n nn mm mnN

n

This denition only makes sense once one shows that the indicated limit

and colimit b oth exist and that they coincide This was rst demonstrated

by Plotkin Plotkin constructed the category SFP of SFPob jects and

continuous maps in order to have a cartesian closed category that was closed

under all the op erators he needed to create the sort of semantic mo dels he had

in mind In particular he needed a ccc that was closed under the Plotkin p ower

domain construct and this is something that is not true of Scott domains

Plotkin also conjectured the following result which was proved by Smyth

Theorem Smyth

The category SFP of SFPobjects and continuous maps is the largest cartesian

closed category of algebraic domains 2

In his celebrated thesis Jung greatly extended our knowledge ab out

maximal cartesian closed categories of domains He rst showed that the

category of binite domains those that are simultaneously the limit and

colimit of a directed family of nite p osets under ep pairs is cartesian

closed and in fact is maximal such among those cccs of domains He also

dened the following class of domains

Denition An Ldomain is a domain P in which x is a complete

for each x P

Theorem Jung

There are two maximal cartesian closed ful l subcategories of domains

The category BIFIN of binite domains and continuous maps and

The category LDOM of Ldomains and continuous maps 2

Categorical Generalizations

One of the basic asp ects of the Scott top ology is that directed sets converge

to their suprema Moreover Tarskis Theorem guarantees that continuous

selfmaps on cp os have least xed p oints that can b e computed in a simple

way simply iterate the function starting at the least element Smyth and

Plotkin were the rst to elevate these ideas to the categorical level

In their approach categories of cp os and continuous maps were viewed as

Mislove

large cp os in which colimits of what are called expanding sequences in

corresp ond to suprema in a cp o Furthermore domains satisfying desired

prop erties can b e viewed as xed p oints of asso ciated continuous endofunc

tors of the category and these xed p oints can b e calculated in a way similar

to the calculation of the least xed p oint of a continuous selfmap of a cp o

We now outline this material along with the interesting phenomena that arise

in related categories All of this material is presented in detail in Section of

In order to mimic Tarskis Theorem at the level of a category A of cp os

we rst need to order A This is accomplished by dening not a partial order

on A but rather a preorder a reexive transitive relation on A

Denition Let D and E b e cp os We write D v E if and only if there

is an emb eddingpro jection pair e p D E

Lemma v is a preorder on the class of cpos

Pro of It is clear that the relation is reexive since the identity map forms

an ep pair on any cp o Transitivity follows from the fact that e e p

p D D is an ep pair if e p D D and e p D D are ep

pairs 2

Note however that it is unclear what it means for two dcp os to b e equiv

alent under v

Example Let I denote the unit interval E I I the unit

square in the pro duct order and

D fx x j x g

Clearly D is a subcp o of E and it is easy to see that there is a pro jection

mapping p E D so that the emb edding i D E together with p forms

an emb eddingpro jection pair But likewise E can b e emb edded in D as the

lower square and this also has an asso ciated pro jection p D E Thus D

and E are equivalent under v but they clearly are not isomorphic as cp os

Even though v is not a partial order we can still use it as if it were one

and so our next goal is to show that increasing sequences on cp os in this order

have least upp er b ounds

Denition Let e p D D b e a sequence of ep pairs for each

n n n n

n N We dene

Y

D fx D j p x x g

n nN n n n n

nN

Q

D It is not hard to and we endow D with the order inherited from

n

nN

Q

D since the maps p all are continuous show that D is a subcp o of

n n

nN

We also can dene emb eddingpro jection pairs E P D D by

n n n

P x x and E x f x where f D D by

n n nN n n in iN ij j i

p p if i j

i j

if i j and

f

D

ij

i

e e if i j

i j

Mislove

Theorem If e p D D is a sequence of ep pairs for each

n n n n

n N then E P D D as dened above is a sequence of ep pairs

n n n

satisfying E E e and P p P for each n N

n n n n n n

Moreover if A is a cpo and E P D A is a sequence of ep pairs

n

n n

satisfying E E e and P p P for each n N then there is a

n n

n n n n

unique ep pair E P D A such that E E E and P P P for

n n

n n

each n N

Final ly if E P D A is a cocone over the sequence e p D

n n n n

n n

F

D then the cocone is colimiting if and only if E P 2

n A

n n

Let CPO b e the category of cp os and ep pairs ie the ob jects of the

ep

category are cp os and are pairs of emb eddingpro jection map

pings b etween ob jects The p oint of the previous result is that we can regard

D E P D D as a cocone over the diagram e p D

n n n nN n n n

D in CPO and this result asserts that it is colimiting Viewed as

n nN ep

a colimit D then is the least upp er b ound of the sequence D v v

D v D and so the category CPO of cp os and continuous maps has

n n

least upp er b ounds relative to the order v The construction shows that this

also holds for every full sub category of CPO that is complete Note also that

CPO has a least cp o the onep oint cp o fg since there is an obvious ep

pair from fg to any cp o P

The next p oint is to single out a family of continuous endofunctors for

which we can prove an analogue of Tarskis Theorem The obvious denition

for continuity would b e that a functor preserves least upp er b ounds as dened

in Theorem But to make this precise we rst record a result that shows

CPO is closed under limits and colimits

Theorem If P fe p P P g is a diagram in CPO then

i ij ij i j ij I ep

limP fp g colimP fe g

i ij ij I i ij ij I

2

So if one has a diagram P fe p P P g in CPO then

i ij ij i j ij I ep

the limit of the diagram P fp P P g and the colimit of the dia

i ij i j ij I

gram P fe P P g b oth exist and they coincide This limit can b e

i ij i j ij I

regarded either as a colimit or a limit in the category CPO by taking the appro

priate pro jection from CPO This result allows for a ne analysis of the limit

ep

of such a diagram and this in turn is very useful in applying the techniques

that are needed to construct domains to satisfying certain equations

We already have seen that the colimit of a sequence P fe p P

i ij ij i

P g in CPO can b e regarded as the least upp er b ound of the sequence

j ij I

Moreover the order on CPO ensures that all functors b etween categories of

cp os are monotonic if e p P Q is an ep pair in a category A of cp os

and F A B is a functor then F e F p F P F Q also is an ep

pair So what remains is to nd the appropriate sense in which functors should

b e continuous

Denition Let A and B b e cocomplete categories of cp os and con

tinuous maps The functor F A B is continuous if for every diagram

Mislove

P e p P P in A

i ij ij i j ij I ep

F colimP fe g colimF P fF e g

i j i ij I i j i ij I

While this seems a reasonable denition for continuity alb eit somewhat

opp osite from the usual denition of a continuous functor it can b e a dicult

prop erty to prove The following result shows that there is a simple test that

makes it easy to show certain functors are continuous

Denition The functor F A B b etween full sub categories of CPO is

local ly continuous if for all ob jects P and Q of A

F P Q F P F Q

is continuous

This denition do es make sense indeed b ecause op erations on D E

are dened p ointwise even though D E and F D F E are not

necessarily ob jects of A or B resp ectively nonetheless they are cp os and

F D E F D F E is a welldened function so it makes p erfect

sense that it might b e continuous

Theorem Plotkin Smyth Plotkin

If F A B is a local ly continuous functor between ful l subcategories of CPO

then F A B is continuous 2

Now let F CPO CPO b e an endofunctor and let e p fg F fg

b e the natural ep pair If we let F b e the identity functor then the following

is the analogue to Tarskis Theorem we have b een seeking

Corollary Tarskis Theorem for Categories of Cp os

Let F A A be a continuous endofunctor on a ful l complete subcategory of

n n n n

CPO Then F e F p F fg F fg is a sequence of ep pairs

and

n

I fx j F px x g

n nN n n

n m n

colimF fg F F e

nmN

satises F I I Moreover I is the least such cpo in the sense of Theo

rem 2

Since lo cal continuity implies continuity we can nd a domain satisfying

a desired isomorphism by starting with a continuous endofunctor F CPO

CPO and seeking a cp o P satisfying F P P The technique for nding

such a cp o P is to apply Tarskis Theorem iterate the functor F starting

with the least domain fg using the canonical ep pair from fg to F fg

One should note the analogy to nding xed p oints of continuous selfmaps of

cp os We present p erhaps the simplest example

Example Let L CPO CPO by LP P fg where P and

for f P Q

n

f x if x P

Lf LP LQ by Lf x

otherwise

Mislove

Thus L is the lift functor which adds a new b ottom to the cp o P and which

extends a continuous map b etween cp os by sending the new b ottom in the

domain cp o to the new b ottom in the range cp o Clearly L is an endofunctor

of CPO and the lo cal continuity of L should b e obvious

In seeking a cp o P satisfying LP P we start with the cp o fg and

the emb eddingpro jection pair e p b etween fg and the cp o Lfg which

sends to the least element of Lfg and the pro jection which is the only

map from Lfg to fg This leads to the following diagram in CPO

ep

n

Lp L p

p

n n

Lfg fg L fg L fg L fg

! ! !

e

n

Le L e

Theorem then implies that

n n m

lim L fg fL p L pg

mnN

n

n m n

colim L fg fL e L eg

mnN

n

N

where N is the natural numb ers in the usual order with a top element

added It is imp ortant to note that the reason this is the colimit of the diagram

n m n

L fg fL e L eg is that the colimit is taken in CPO

mnN

where all ob jects must b e directed complete Hence the colimit in POS

N must have a largest element added to make it a cp o

Now since L is lo cally continuous it is continuous Hence

N LN

and this provides a solution to the equation LP P

In analogy to the situation with continuous selfmaps of cp os the solution

LP P we just found is least relative to the preorder we have placed on

CPO There is another way to state this fact which utilizes the notions of

F algebras

Denition Let F A A b e an endofunctor on a category A The ob ject

A of A is an F algebra if there is a F A A in the category

A

A If A and B are F algebras then an F homomorphism from A to B is an

Amorphism f A B such the the following diagram commutes

F f

-

F A F B

A B

? ?

f

-

A B

If F A A is a lo cally continuous endofunctor on the full sub category A

of CPO and if A contains the ob ject fg then we can form the ob ject

n m n

I colimF fg fF e F eg

nN mnN

n

Mislove

where e fg F fg is the emb edding sending to the least element of

F fg and this satises F I I Recall that a map f P Q b etween

cp os is strict if it preserves the least element of P The pro of of the following

result can b e found in

Theorem

i If A is an F algebra for which F A A is an isomorphism then

A

there is a least F homomorphism f A B for any F algebra B

B A

ii I is a subcpo of every xed point B F B of F

iii I is an initial F algebra in the category A of Aobjects and strict contin

uous maps from A 2

So if we take the case F L the lift functor then this says that the

lift algebra N is a lift algebra in CPO that is the least xed p oint in

the category in the sense that there is a least homomorphism from it to any

other lift algebra Moreover if we force the least element of N to b e

mapp ed to the least element of a target lift algebra B then there is a unique

lift algebra homomorphism from N to B

Lastly in Section we will see how the assignment P P P can b e

made functorial and how the techniques outlined here allow one to construct

a nondegenerate xed p oint for the asso ciated functor This result provides

us with a mo del of the untyp ed lamb da calculus of Church and Curry

Further Results

The results we have outlined b egin to make a case that domain theory has

a numb er of interesting results to oer From the start there have b een

several attempts to duplicate the results we describ e in other settings For

example a numb er of authors have examined the p ossibility of developing

analogous results in categories of metric spaces Most notable among these is

the seminal result of America and Rutten where it was shown that one

of the most imp ortant techniques solving recursive domain equations like

our lift algebra equation can b e carried out in the metric setting Analogous

results also have b een obtained by Flagg and Kopperman who use the

dierent setting of quantales Perhaps the most p enetrating results so far have

b een obtained by Wagner who has shown that the domaintheoretic and

metric space approaches can b e understo o d as instances of a common theme

This theme is to regard the categories CPO and MET as enriched categories

For CPO the enrichment is over the twop oint lattice while for MET it is over

op

the quantale R of real numb ers in the opp osite order equipp ed with

as the tensor pro duct

Another p oint that is worth making is that there are concerted attempts

to understand just what p ortion of the prop erties of CPO are fundamental to

a basis for theoretical computation In this regard we mention two research

eorts

i The work of Freyd on algebraical ly compact categories If T is an

Mislove

endofunctor of a category C then T Inv denotes the category of triples

hA f g i where hA f i is a T algebra hA g i is a T coalgebra and f g

and g f are the identity maps T is algebraical ly bounded if T Inv has

an ob ject that is b oth terminal and initial If C is bicomplete ie every

op

covariant endofunctor of C or of C has an initial algebra then C is

algebraical ly compact if every endofunctor is algebraically b ounded For

example the category of countable sets is algebraically complete These

notions app ear to characterize what is necessary for each endofunctor to

have a least xed p oint in the category

ii The work of Plotkin Fiore et al on a system of axioms for domain

theory The axiom system p ostulates a pair of categories in which there

is a forgetful functor from one to the other whose left adjoint is analogous

to the lift functor The relation b etween CPO and CPO is the prime

example here CPO is the category of cp os and strict continuous maps

The forgetful functor from CPO to CPO has lift as its left adjoint The

motivation is that N is an initial lift algebra and this is exactly

what is needed to develop a theory of cp os and continuous maps

partial orders with least element where countable chains have least upp er

b ounds and in which maps preserve the suprema of such chains

Domains as Top ological Spaces

The traditional approach to domain theory emphasizes realizing domains

as dcp os P which are isomorphic to the family IdlK P of order ideals

of the set of compact elements The results in extend this approach to

continuous domains by utilizing the notion of an abstract basis In our opinion

this approach suers from the drawback of having to deal with approximable

relations which we view as much less intuitive than continuous functions

In this section we outline an alternative approach that emphasizes top ology

p erhaps to the detriment of not highlighting the algebraic character of domains

that the traditional approach oers Nonetheless we b elieve this approach has

some intuitive advantages

Ordertheoretic Topology

To b egin we recall the wellworn connection b etween top ology and algebra

that has b een extensively studied under the rubric ordertheoretic top ology

A basic reference for this approach is the b o ok However we prefer to

fo cus on the closed sets of a top ological space rather than the op en sets

Let TOP b e the category of top ological spaces and continuous maps

Denition If X is a top ological space then we dene the family X

C g of closed subsets of X If f X Y is continuous we fC X j C

dene f Y X by f C f C

Denition A Brouwerian lattice is a L for which x

V V

x y for all x L and all families C L A morphism of C

y C

Mislove

Brouwerian lattices is a mapping f L M that preserves all inma and all

nite suprema

Theorem If CBL denotes the category of Brouwerian lattices and Brouw

op

erian lattice maps then TOP CBL is a contravariant functor 2

To go back the other way we rst need some terminology

Denition For a complete lattice L an element p L is coprime if for

all F L nite if p v F then F p We denote by Sp ec L the

family of coprimes of L

For example given a top ological space X the set fxg is coprime in X

for each x X Note that the least element of L cannot b e coprime since

F is a p ossibility

We want to top ologize Sp ec L so we make the following denition

Denition If L is a lattice we dene C Sp ec L to b e closed if

C x Sp ec L for some x L The hul lkernel topology on Sp ec L has

these sets as its family of closed sets

Of course for this denition to make sense it must b e shown that the

family of closed sets we have dened is closed under all intersections and

T

all nite unions The former is true since f x Sp ec L j i I g

i

V

x Sp ec L while the latter is an easy exercise using the fact that all

i

i

elements of Sp ec L are coprime

Prop osition Let L M be a morphism of Brouwerian lattices We

V

dene the lower adjoint of by M L by x x Then

i and ie is a Galois adjunction between

M L

L and M

ii preserves al l suprema

iii Sp ec M Sp ec L

iv j Sp ec M Sp ec L is hul lkernel continuous

Sp ec M

_

Pro of It is clear that is welldened since M is a complete lattice and

part i is then a routine exercise Since preserves all inma part ii follows

from the general theory of adjunctions Part iii follows from the fact that

preserves nite suprema and part iv again is easy 2

Using this Prop osition we can prove the following result

op

Corollary There is a functor Sp ec CBL TOP given by Sp ecL

Sp ec L and for L M Spec j 2

Sp ec L

_

Our aim is to use the functors Sp ec and to establish as equivalence of

categories but this is not true in the generality we are in For example not

every top ological space is of the form Sp ec L for some complete Brouwerian

fxg and lattice L Indeed the unit X Sp ec X satises x

X X

so this map is injective if and only if X is T On the other side the counit

W

L Sp ec L given by x X Sp ec L certainly is onto

L L

Mislove

but it is onetoone if and only if every x L is the supremum of the set

x Sp ec L We make these sp ecial spaces and lattices the sub ject of our

next denitions

Denition A closed subset C X of the top ological space X is irre

ducible if C is a coprime in X ie if C is not the union of two prop er

closed subsets The space X is sober if every irreducible closed subset C sat

fxg for a unique p oint x X We let SOB denote the category of ises C

sob er spaces and continuous maps

The following prop osition is routine

Prop osition

i If L is a complete Brouwerian lattice then Spec L is a sober space in

the hul lkernel topology

ii If X is a topological space then X Sp ec X is a continuous

X

and open mapping onto its image 2

Corollary The functor Sp ec TOP SOB is left adjoint to the

inclusion functor 2

For a top ological space X the space Sp ec X is called the sobrication

of X it is the largest space having the same top ology as X

On the lattice side we have the following

Denition A complete Brouwerian lattice L has enough coprimes if

W

x x Sp ec L for all x L We let SCBL denote the category of such

lattices and maps L M that are upp er adjoints to CBLmaps from M

to L

Prop osition

i If X is a topological space then X has enough coprimes

ii If L is a complete Brouwerian lattice then the mapping L Sp ec L

L

is a monomorphism of complete Brouwerian lattices 2

Corollary The functor Sp ec CBL SCBL is left adjoint to the

inclusion functor 2

A complete Brouwerian lattice also is called spatial if L has enough co

primes All of this culminates in the following result

op op

op

Theorem The functors j SOB SCBL and Sp ecj SCBL

SOB SCBL

SOB form a dual equivalence 2

Continuous Posets

We know by now that we can endow each dcp o with its Scott top ology and

obviously this would b e a way to take advantage of the equivalence of cate

gories we have just outlined Unfortunately in this generality it is not clear

whether every dcp o can b e retrieved from its Scott top ology But we will b e

Mislove

able to do this for continuous dcp os and it is convenient to generalize from

the setting of dcp os just a bit

Denition Let P b e a p oset If x y P then we write x y if for

F F

all directed sets D P if D exists in P and y v D then D x

We say P is continuous if for all y P

y fx P j x y g is directed and

F

y y

Likewise x P is compact i x x and P is an algebraic poset if K x

F

is directed and satises x K x for all x P We let CPOS denote the

category of continuous p osets and Scott continuous maps and APOS denote

the full sub category of algebraic p osets

The only dierence b etween the denitions we just made and the earlier

notions of continuity and algebraicity is that we no longer assume the under

lying p oset P is directed complete Algebraic and continuous p osets also have

b een studied in and in resp ectively We shall see that the equiva

lence just outlined for sob er spaces and spatial Brouwerian lattices yields a

very satisfying theory for the categories CPOS and APOS We b egin our study

with the following result whose pro of is the same as that for Prop osition

Lemma If P is a continuous poset then satises the property

INT x y z P x z y

2

Hence x is Scott open for each x P

Lemma If P is a continuous poset and C Sp ec P then fx

F

C j x C g is directed and C fx jx C g

S F

Pro of Any closed set C fx j x C g and each x P satises x x

S

since P is continuous Thus C fx j x C g Supp ose that x y C

satisfy x C y C If x y C then C C n x C n y

is the disjoint union of prop er closed sets which means C Sp ec P This

S

fx j x C g shows fx C j x C g is directed And since C

F

it follows that C fx j x C g 2

Prop osition If P is a continuous poset then

i Sp ec P is a dcpo

ii C D Sp ec P i x y P C x z D

iii Sp ec P is continuous

iv The mapping P Sp ec P is a homeomorphism onto its image

P

and the topology P inherits from the Scott topology on Sp ec P

P

is the hul lkernel topology of P

P

Pro of If L is a complete lattice and D L is a directed family of coprimes

F

then it is easy to show that D also is coprime This shows i

For ii supp ose that x y P and that D Sp ec P is a directed

S

D Since family of closed sets whose supremum dominates y Then y

Mislove

S

x y it follows that x is a Scott op en set containing y and so D x

Since closed sets are lower sets there is some set C in the family D with x C

and this means x v C Thus x y in Sp ec P It then follows that

C D for any sets C and D with C x and y D

Conversely if C D in Sp ec P then the preceding lemma implies

F

D f x j x D g and this supremum is directed Hence x D

x D C x Since x D Lemma implies there is some

y D x with y D Then the rst part of the pro of implies x y

and so C x y D which proves part ii

Part iii follows from part ii Lemma and the continuity of P The

rst part of iv follows from the fact that P is T in the Scott top ology Since

directed sets in Sp ec P converge to the same p oint in the Scott top ology

of Sp ec P as they do in the hullkernel top ology since Sp ec P

is closed under directed suprema in P the identity map is continuous

from the Scott top ology to the hullkernel top ology Conversely if C U

Sp ec P and U is Scott op en then there is some x P with C fD j

x D g U Then

Sp ec P n fD j x D g fD Sp ec P j D x g

fD Sp ec P j D P n xg

which clearly is hullkernel closed and so the top ologies are the same 2

Since Sp ec P is sob er for any continuous p oset P the following result

is clear

Corollary The functor Sp ec CPOS CON is left adjoint to the

forgetful functor Hence the continuous poset P is sober if and only if P is a

dcpo 2

Thus the sobrication of a continuous p oset is a continuous dcp o with

the same wayb elow relation Of course we can restrict our attention to

the algebraic case to obtain the following

Corollary If P is an algebraic poset then Sp ec P is an algebraic

dcpo with K Sp ec P fk j k P g Hence P is a dcpo if and only P

is sober in the Scott topology 2

One might ask which algebraic p osets P satisfy the prop erty that IdlP

Sp ec P the answer is the following

Prop osition An algebraic poset P satises IdlP Sp ec P if

and only if P K P 2

All of the material we have presented has b een for continuous p osets and

the resulting directed complete partial orders are continuous dcp os Clearly

a similar development can b e made for continuous p osets with least element

and then the resulting directed complete partial order would b e cp os

Our stated motivation was to present a theory that avoided the use of

approximable relations This theory do es that but it do es not have the purely

algebraic avor that using approximable relations aords On the other hand

Mislove

this theory provides a nice example of how the sobrication functor can yield

pleasing results relating categories of incomplete partial orders to ones that are

complete This highlights that fact that sob er spaces might b est b e thought

of in terms of completeness rather than separation

Finally one might ask whether the theory we have presented can b e ex

tended to a larger class of p osets endowed with the Scott top ology While

this may b e true such a theory cannot include all dcp os as the target of the

sobrication functor as Johnstones example of a dcp o whose Scott

top ology is not sob er shows

Duality Theories

One of the app ealing asp ects of domain theory is the fact that rich duality

theories can b e devised for it These theories rely on b oth asp ects of domains

their intrinsic top ological structure as represented by the Scott top ology and

their intrinsic algebraic structure represented by the role that compact ele

ments play in the structure of domains The basic theory relies on analyzing

the use of sp ectral theory of the previous section somewhat more carefully

In applying the sobrication functor we passed through the family P

for P a continuous p oset Since P and Sp ec P have the same closed sets

we can investigate the complete Brouwerian lattice P assuming that P is

a continuous p oset or a continuous dcp o

Denition A complete lattice L is continuous if L is a continuous cp o

Likewise L is algebraic if L is algebraic as a cp o The lattice L is bicontinuous

op

resp bialgebraic if b oth L and L are continuous resp algebraic

An examination of the pro of of part ii of Prop osition shows that

x y P implies x y in P for a continuous p oset P It then follows

that if x y for each i n then x y in P It then is

i i i i i i

routine to show that any closed subset C of a continuous p oset P satises

G

C fF j F C nite F C g

That is P is a continuous cp o

Dually it can b e shown that P n y is wayb elow P n x in P

if x y P again for P a continuous p oset If C P is closed then

T

C fP n F j F P n C niteg and P n F C in P for each

such F Since this family is easily seen to b e directed under reverse inclusion

it follows that P also is continuous Hence P is bicontinuous if P is

a continuous p oset Since P is Brouwerian Prop osition VI I then

implies that P also is completely distributive

Theorem If P is a continuous poset then P is a completely dis

tributive bicontinuous lattice Moreover C D in P if and only if

there are nite subsets F G P such that C F G D Dual ly

C D in P if and only if there are nite subsets F G P such that

C P n F P n G D

Moreover P is algebraic if and only if P is algebraic in which case

Mislove

op

K P f F j F K P niteg and K P fP n F j F

K P niteg 2

Lawson duality also asserts that the converse of this Theorem holds Namely

if L is a completely distributive bicontinuous lattice then Sp ec L is a con

tinuous dcp o and L Sp ec L cf

Theorem can b e raised to the level of a duality theory by applying

the general equivalence b etween sob er spaces and spatial complete Brouwerian

lattices Indeed the continuous functions b etween continuous dcp os are the

Scott continuous maps and these corresp ond precisely to the maps b etween

the resp ective closedset lattices that preserve all suprema and coprimes To

state this precisely we let CDCPO denote the category of continuous dcp os

and Scott continuous maps and BDL denote the category of bicontinuous

completely distributive lattices and maps preserving all suprema and all co

primes

Theorem The functors j CDCOP BDL and Sp ecj BDL

CDCPO BDL

CDCOP form an equivalence

These functors further cut down to an equivalence between the ful l subcat

egories ALG of algebraic dcpos and BAL of bialgebraic completely distributive

lattices 2

Power Domains

One of the most imp ortant constructs for semantics is that of power domains

The idea is to have a mo del for nondeterminism There are three traditional

p ower domains and these constructs can b e dened purely algebraically ie

ordertheoretically We b egin with the denitions and then pro ceed to recast

them top ologically

Nondeterministic choice is meant to b e a binary op eration which satises

some simple algebraic rules associativity commutativity and idempotency

Thus a mo del of nondeterministic choice is simply a The tradi

tional path to building a mo del for nondeterminism is to start with a mo del

for sequential comp osition and p erhaps some additional op erations as well

and then to construct a mo del for nondeterminism on top of the existing

mo del Thus one usually b egins with a continuous algebra relative to some

signature ie a algebra whose underlying set is a continuous cp o such

that the interpretation of all of the op erations is continuous and seeks to

add a semilattice op eration to the mo del The following development mo d

ularizes this by rst constructing free ordered over p osets and

then extending them naturally to b e continuous algebras

Denition Let P b e a p oset We dene the family

a L P fF j F P niteg with

n

F v G i F G and F G F G

L

b U P fF j F P niteg with

n

F v G i G F and F G F G

U

Mislove

c C P fhF i F F j F P niteg with

n

hF i v hGi i F v G F v G and hF i hGi hF Gi

C L U

Prop osition Let POS be the category of posets and monotone maps

i If SUP is the category of supsemilattices and supsemilattice maps then

the functor L POS SUP by LP L P and Lf F f F is

n

left adjoint to the forgetful functor from SUP to POS

ii If INF is the category of infsemilattices and infsemilattice maps then

the functor U POS INF by U P U P and U f F f F is

n

left adjoint to the forgetful functor from INF to POS

iii If OSEM is the category of ordered semilattices and orderedsemilattice

maps then the functor C POS OSEM by C P C P and C f hF i

n

hf F i is left adjoint to the forgetful functor from OSEM to POS

Pro of We outline the pro of for iii the others are similar Let P b e a p oset

and S an ordered semilattice ie a semilattice with a partial order relative

to which the semilattice op eration is monotone and supp ose f P S is a

monotone map

The family C P C P is a semilattice under the op eration hF i hGi

n

hF Gi Moreover if hF i v hF i and hG i v hG i then

C C

hF G i F G F G F G hF G i

and similarly

hF G i F G F G F G hF G i

Thus the semilattice op eration is monotone on C P

n

Now dene C f C P S by C f hF i x x where F

n n

fx x g and is the semilattice op eration on S It is routine to show C f

n

is welldened and that C f is a semilattice map Finally C f fxg f x

is clear and this is the unique semilattice map from C P to S satisfying

n

this prop erty since C P is generated by the image of P under the map

n

x fxg hfxgi 2

In Section we p ointed out that the category of algebraic dcp os and Scott

continuous maps is equivalent to the category of p osets and approximable

relations There is another relationship b etween the category of DCPO dcp os

and Scott continuous maps and the category POS of p osets and monotone

maps that is worth p ointing out

Prop osition The functor Idl POS DCPO dened by IdlP fI

P j I I directedg and Idlf I f I is left adjoint to the forgetful

functor

Pro of Certainly IdlP is a dcp o for any p oset P And if f P Q is

a monotone map from the p oset P to the dcp o Q then we can dene the

F

mapping f IdlP Q by f I f I Since f is monotone and I is an

ideal it follows that f I is directed so this supremum is welldened And

Mislove

if C Q is a Scottclosed set then

G

f C fI IdlP j f I C g fI IdlP j f I C g

It is clear that this is a lower set in IdlP and it is routine to show that this

family is closed under directed suprema which are just increasing unions

Moreover if we dene P IdlP by x x then this map is

P P

continuous and f f and this is the unique continuous map g IdlP

P

Q satisfying g f since P is Scottdense in IdlP 2

P P

Next we note the following result

Prop osition If S is an ordered semilattice then IdlS fI P j

I I is directedg is an algebraic dcpo which also is a semilattice and the

semilattice operation on IdlS induced from that of S is continuous

Pro of Let S S S b e the semilattice op eration on S We dene

IdlS IdlS IdlS by I J fx y j x I y J g Since I and J are

directed and S S S is monotone it follows that fx y j x I y J g

also is directed and so I J is an ideal of S Also using Prop osition

and the fact that IdlP Q IdlP IdlQ for all p osets P and Q the

fact that S S S is monotone implies that IdlS IdlS IdlS

is continuous 2

Note that a corollary of this last result is that if S is a sup resp ectively

an inf semilattice then so is IdlS under the op eration induced from the

semilattice op eration from S

Corollary The restriction of the functor Id l to each of the categories

SUP INF and OSEM respectively is a left adjoint to the inclusion functor from

the associated category of continuous semilattices and continuous semilattice

maps The composition of this restriction with each of the left adjoints L

U and C gives a left adjoint to the inclusion of the associated category of

continuous semilattices into POS respectively 2

This last result can b e used along with the following purely categorical

result to lift the free ordered semilattices just constructed to free algebraic

semilattices thus building of universal algebraic semilattices over algebraic

cp os

Theorem Let A B and C be categories and suppose F A B is

AB

left adjoint to U B A and that F A C is left adjoint to U C A

AB AC AC

Also suppose there is a functor U C B satisfying U U U

BC AB BC AC

and nal ly suppose that for each object b in B there is an object Gb in A such

that F Gb b Then there is a left adjoint F B C to U given by

AB BC BC

F b F Gb and F F F

BC AC BC AB AC

2

By we mean natural isomorphism

Mislove

F

AB

-



A B

U

AB

6



F

BC

F U

AC AC

U

BC

?

2

C

Our particular application of this Theorem is to the case that A POS

is the category of p osets and monotone maps B CPO is the category of

algebraic cp os and F Idl is the ideal functor The function G on ob jects

AB

of B asso ciates to each algebraic cp o P the set K P of compact elements of

P By cho osing C to b e an appropriate category of algebras over POS in the

case of p ower domains these will b e categories of continuous semilattices we

nd that having a universal Calgebra over a p oset K P naturally leads to a

universal Calgebra over the algebraic cp o P We see that each of the p ower

domains arises in exactly this fashion so the construction of these ob jects

has b een broken down into two steps rst form the appropriate free ordered

semilattice over the family of compact elements of an algebraic cp o and then

apply Corollary to obtain the free continuous algebra over the cp o

We let

ADCPO denote the category of algebraic dcp os and continuous maps

ASUP denote the category of algebraic dcp os having a continuous supsemi

lattice op eration and continuous maps preserving nite suprema

AINF denote the category of algebraic dcp os which also have a continuous

infsemilattice op eration and continuous maps preserving nite inma and

ASEM denote the category of algebraic dcp os having a continuous semilattice

op eration and continuous maps preserving nite pro ducts

Theorem Hennessy Plotkin

i The functor P ADCPO ASUP dened by

L

P P Idl L K P and P f P P P Q by

L n L L L

G

P f I ff F j F K P F I g

L

is left adjoint to the forgetful functor from ASUP to ADCPO

ii The functor P ADCPO AINF dened by

U

P P Idl U K P and P f P P P Q by

U n U U U

G

P f I ff F j F K P F I g

U

is left adjoint to the forgetful functor from AINF to ADCPO

iii The functor P ADCPO ASEM dened by

C

P P Idl C K P and P f P P P Q by

C n C C C

G

P f I fhf F i j F K P hF i I g

C

is left adjoint to the forgetful functor from ASEM to ADCPO

Mislove

Pro of Again we conne ourselves to an outline of the last assertion the oth

ers b eing similar We know from Prop osition that the functor C from POS

to OSEM is left adjoint to the forgetful functor and Corollary implies the

restriction of the functor Idl is left adjoint to the forgetful functor from ASEM

to OSEM Corollary further implies that Idlj C POS ASEM

OSEM

is left adjoint to the inclusion and Theorem then implies it induces

P ADCPO ASEM which is left adjoint to the inclusion Given the def

C

initions of Idl and of C it is routine to show that P acts on ob jects and

C

morphisms as indicated 2

If we apply the construction for the free supsemilattice to K P for an

algebraic p oset P and then take the ideal completion to obtain P P then

L

we have the lower power domain or the Hoare power domain as it sometimes

is called Similarly P P is the upper power domain or Smyth power domain

U

using the free infsemilattice over K P and P P is the convex power do

C

main or Plotkin power domain over the algebraic dcp o P It was rst p ointed

out in that each of these yields a left adjoint to the forgetful functor from

a category of ordered semilattice dcp os into ALG

Theorem serves to dene the three traditional p ower domains for all

algebraic dcp os The rst of these also has a simple top ological representation

Prop osition If P is an algebraic dcpo then P P P where

L

P denotes the family of nonempty closed subsets of P

Pro of For nonempty nite subsets F G of K P F v G i F G i

L

F G and this implies the mapping F F L K P P is an

n

isomorphism of L P onto K P Clearly this mapping preserves the

n

semilattice op eration and since each of IdlL K P v and P is an

n L

algebraic dcp o the isomorphism extends to one of the structures themselves2

This leads to a denition of an analogue to the lower p ower domain for

continuous dcp os We let SUPCON denote the category of continuous dcp os

endowed with a Scott continuous supsemilattice op eration and Scott contin

uous maps preserving the supsemilattice op eration

Prop osition The functor CON SUPCON which sends each con

tinuous dcpo P to P endowed with the union operation and each mapping

f C is left adjoint to f P Q to the supsemilattice mapping f C

the forgetful functor

Pro of The functor clearly is welldened on ob jects And the unit of the

adjunction is the mapping x x P P Supp ose that S is a sup

semilattice continuous dcp o and that f P S is continuous Then we

can dene f x f x for each x P Given C P we have

F

C f F j F C F niteg and this sup is directed So we can extend

F

f by dening f C f F This mapping is welldened and

S

continuous 2

For the other p ower domains the upp er and convex we require some

more development and a restriction of the class of dcp os considered for a

Mislove

top ological analogue for them to b e derived We b egin with the following

denition

Denition A subset S X of the top ological space X is saturated if

it is the intersection of the op en sets that contain it

It is routine to show that a subset A X of a top ological space X is

T

compact if and only if SatA fU j A U op eng is saturated Moreover

the saturated subsets of a partially ordered set endowed with the Scott top ol

ogy are precisely the upp er sets this follows since P n x is Scott op en for all

x P

Theorem Hofmann and Mislove

Let X be a sober space and let F be a lter basis of compact saturated subsets

of X Then

T

F is compact and i

T

ii if F U with U open then there is some C F such that C U 2

This theorem can b e proved by rst showing that the family of Scott

compact saturated subsets of a sob er space X is isomorphic to the semilattice

of Scottop en lters in the lattice of Scottop en subsets of X A simple analysis

of this structure then yields the result Alternatively as in a direct pro of

can b e given

It is imp ortant to note that the second part of the theorem implies that if

the intersection of a lter basis of compact saturated sets is empty then one

of them is empty

Finally recall from Corollary that continuous dcp os are sob er in the

Scott top ology so the ab ove result applies to them

We now are ready to give a top ological representation of the upp er p ower

domain This result was rst discovered by Smyth for the case of do

mains Given the to ols at our disp osal however we can extend the denitions

to all continuous dcp os To b egin let INFCON denote the category of con

tinuous dcp os which also have a continuous infsemilattice op eration and

continuous infpreserving maps and recall CON denotes the category of con

tinuous dcp os and continuous maps

Prop osition Let P be a continuous dcpo and let C P denote the family

of nonempty Scott compact saturated subsets of P Then

i C P is a continuous dcpo infsemilattice

ii The functor C CON INFCON which associates C P to the contin

uous dcpo P and to the continuous mapping f P Q the mapping

C f C P C Q by C f C f C is left adjoint to the forgetful

functor

iii If P is algebraic then C P P P

U

Pro of Supp ose P is continuous Then each compact saturated subset C of P

can b e written as the ltered intersection of sets of the form F where C F

and F P is nite Conversely Theorem implies that each ltered

Mislove

intersection of sets of the form F for F P nite is Scott compact and so

this accounts for all compact saturated subsets of P Moreover Theorem

also implies that C P is a dcp o under reverse containment

Now supp ose C is saturated and compact and C F for F P nite

T

If F is a lter basis of compact saturated subsets of P such that C F

then since F is Scott op en and C F Theorem again implies that

X F for some X F It follows that F C in C P in the opp osite

order The continuity of P then implies C P is a continuous dcp o and

it is clear that union is a continuous infsemilattice op eration on C P

This proves i

For part ii it is clear that C CON INFCON is welldened Supp ose S

is a continuous dcp o with a continuous infsemilattice op eration and supp ose

T

f P S is continuous If C C P then C fF j C F F niteg

F

Then f F f F is welldened and f C f f F j C

S S

F T

F F niteg denes f C If F C P is directed then F F and

F T

clearly ff C j C F g S is ltered Now f F v f F holds just

S

b ecause f is monotone

T T F

Conversely let f F s for some s S Then f F f f F j

S

T T

F F F niteg and so there is some nite subset F P with F F

T

and s v f F Since F is a neighb orho o d of F Theorem implies

S S

C f U for some C F and the monotonicity of f then implies s v

S

T T F

fs S j s f F g we conclude f F v f C Since f F

S S

S

T F

f F f F and this implies f C P S is continuous This that

S

map clearly preserves nite infs and it is completely determined by f This

proves ii

For part iii we note that the pro of of the second part shows that C

F

fF j C F F niteg characterizes the wayb elow relation in C P

If P is algebraic then the nite sets F P such that C F can b e taken

to consist of compact elements in which case F F It then follows that

K P P fF j F K P niteg is a basis for C P and so P P

U U

and C P are isomorphic 2

This aords the desired generalization of the upp er p ower domain to con

tinuous dcp os Obtaining a generalization of the convex p ower domain re

quires more work still To derive the result we seek we restrict ourselves to

an interesting sub class of continuous dcp os

Denition A domain P is coherent if it is Scott compact and the in

tersection of Scottcompact saturated subsets of P again is Scott compact

Prop osition A compact domain P is coherent if and only if

F G K P niteH K P nite F G H

More general ly a continuous compact dcpo P is coherent i

F G P nite F G is compact

3

By F we mean the pro duct in S of the elements of the nite set F

S

Mislove

Pro of First supp ose P is a domain If F G K P are nite then clearly

F and G are Scott compact and so coherence implies the same is true of

F G But this set also is Scott op en and so it is the union of sets of the

form k for k K P Then compactness implies we can nd nitely many

such compact elements whose union is the set F G and this is the nite

set H we seek

To show the converse rst recall the sets of the form k for k K P are

a basis for the Scott top ology and any compact upp er set A is the intersection

of a lter basis of sets of the form F for F K P nite So given compact

upp er sets A and B we conclude that

A B fF j A F g fG j B Gg

fF G j A F B Gg

where the sets F and G all are nite subsets if K P and the intersections are

ltered Theorem then shows that A B is compact

More generally if P is a coherent dcp o and F G P are nite then F

and G are compact and so the same is true of F G Conversely if this

condition holds and C D P are compact then we can write each of these

sets as a ltered intersection of sets of the form F where F is nite and the

set in question is within F The same argument as in the algebraic case then

implies C D is compact 2

So far we have fo cused on the Scott top ology on dcp os We now intro duce

a renement of that top ology

Denition Let P b e a continuous dcp o We dene the Lawson topology

on P to b e the smallest top ology for which U n F is op en for all U P Scott

op en and all F P nite

It is routine to show that the collection fxn F j x P F P niteg

is a basis for the Lawson top ology on a continuous dcp o P in particular if P

is algebraic then fxg F can b e taken to b e consist of compact elements

Prop osition Let P be a continuous dcpo Then

i The Lawson topology on P is Hausdor and the order v is closed in P P

in the product of the Lawson topologies

ii The Lawson topology on an algebraic dcpo P is total ly disconnected

iii P is coherent if and only if the Lawson topology is compact

Pro of For i let x y P with x v y Since P is continuous there is some

z P with z x and z v y Then x z y P n z and these are disjoint

Lawsonop en sets This same argument shows that P P n v is op en in the

pro duct of the Lawson top ologies

For ii we simply note that the basis k n F where k K P and

F K P is nite consists of clop en subsets of P since k is clop en in the

Lawson top ology if k is compact

Finally for iii we rst assume that P is coherent and we employ the

Mislove

Alexander subbasis theorem to show P is Lawson compact Note that a sub

basis for the Lawson top ology consists of sets of the form x and P n x for

x P Assume that

P fz j z Ag fP n z j z B g

Since P is coherent the sets fz j z F B niteg are saturated and com

pact and this family is ltered Thus Theorem implies the intersection

T

fz j z B g is compact and saturated and it do es not intersect P n z for

T S

any z B It follows that f z j z B g f z j z Ag and so there

T

is a nite sub cover fz j z B g z where F is a nite subset of

z F

T

A Now since z is op en and contains f z j z B g Theorem

z F

implies there is a nite subset G B with z z It then follows

z G z F

that f z j z F g fP n z j z Gg is a nite sub cover of P and so P is

Lawson compact

The converse that P Lawson compact implies P coherent follows from

the fact that F is a Lawsonclosed hence compact upp er set for any nite

set F and the fact that Lawson compact upp er sets are Scott compact in any

continuous dcp o which is easy to show 2

Note that the second part of this result gives substance to Scotts intuition

that algebraic dcp os are zerodimensional Also note that the last part of

the pro of shows that the Scottcompact saturated sets ie the memb ers of

the Smyth p ower domain C P all are Lawsonclosed sets in P

Given a subset X P of a dcp o P we dene hX i X X the convex

hul l of X

Prop osition Let P be a coherent continuous dcpo Then P

fX P j X is closedg is a continuous lattice with respect to reverse

inclusion In particular P is compact and Hausdor in its Lawson topol

ogy

Pro of Since P is coherent P is compact and Hausdor in the Lawson top ol

ogy It is then routine to show that each compact subset X P is the ltered

intersection of the family X fY X j X Y Y closedg Moreover

T

if the intersection of a lter basis F of closed sets satises F X then

for any Y X there is some F F with F Y This means Y X for

each Y X so this set deserves its name 2

Prop osition Let P be a coherent continuous dcpo and suppose F

P is a lter basis of compact subsets of P Then

h F i hF i

F F

Thus the family of Lawson closed orderconvex subsets of P is a continuous

lattice under reverse inclusion and X Y for such sets if and only if Y

X

Pro of If X is a closed subset of P then X is compact and so X and

X also are closed hence compact all b ecause P is coherent Thus X

Mislove

hX i P P is welldened and it clearly is monotone with resp ect

T T

hF i to reverse inclusion Thus h F i

F F

T

hF i Then for each F F For the converse we supp ose that x

F F

there is a pair of elements a b F with a v x v b Since P is compact

F F F F

in the Lawson top ology the nets fa g fb g must have subnets which

F F F F F F

converge to p oints a b P and wlog we assume that the nets fa g and

F F F

fb g already converge to these p oints resp ectively Since each of the sets

F F F

in F is closed and F is a lter basis we must have a b F for each F F

T

and so a b F And since the order v on P is closed in the pro duct

Lawson top ology and a v x v b for all F it follows that a v x v b Hence

F F

T

x h F i so the two sets are equal

Now since the mapping X hX i P P preserves ltered

intersections it is a continuous kernel op erator and so its image which is

precisely the family of Lawson closed orderconvex subsets of P also is a

continuous lattice under reverse inclusion The characterizing prop erty of the

way b elow relation in this lattice follows from the same prop erty in P

cf Corollary IV 2

Denition For a coherent continuous dcp o P let

D P fC P j C C C C P C C P g

and dene C v C i C v C and C v C

D L U

Prop osition Let P be a coherent continuous dcpo Then

C g is the family of nonempty i D P fC P j C C C

Lawson closed orderconvex subsets of P

ii If C D D P and D C the interior in the Lawson topology then

there is a nite subset F C such that F D and D F

Pro of The forward inclusion of part i follows from the fact that Scott

closed sets are Lawson closed as are Scottcompact saturated sets The reverse

inclusion follows from the fact that coherent continuous dcp os are Lawson

compact

For part ii since P is coherent D and hence D are Lawsoncompact

T

Then the continuity of P implies D fF j F nite F D g and this

intersection is ltered Since D C it follows that D C and so

Theorem implies the result 2

Note that Prop osition shows that D P is a continuous cp o in

which X Y if and only if Y X We now investigate the other order on

D P

Prop osition If P is a coherent continuous dcpo then D P v is a

D

continuous dcpo in which

C D i F P nite C v hF i v D D F

D D

and for which the operation

C D hC D i C D C D D P D P D P

Mislove

is continuous

S

fC j C F g Pro of Assume that F D P is v directed and let A

D

b e the Scott closure of the union of the lower sets of the memb ers of F Then A

is Scott closed and hence also Lawson closed Now the sets fA C j C F g

form a ltered intersection and each is nonempty and Lawson compact since

P is coherent so their intersection also is nonempty and Lawson compact

eg by Theorem Let B b e that intersection We claim that B

F

F Indeed it is obvious that B C for all C F For the other

D P

direction C B for each C F given x C F the fact that F is

v directed implies that x A C is nonempty and compact for each

D

C F so Theorem shows the same is true of the x B Thus B is an

upp er b ound for F and a similar argument shows that B is the least upp er

b ound of F in the order v so D P v is a dcp o

D D

Next if C D P then C is a convex Lawsonclosed subset of P and C

is Lawson compact hence also Scott compact So we can write C as the

ltered intersection of sets F where C F and F is nite Clearly we can

arrange it so that F C for each such F Then hF i F F D P and

hF i v C We now show that hF i C

D D

F

First since C F if F D P is directed and C v F then

D

F F T

F C F The rst part of the pro of implies F fA C j

S F

C F g where A fC j C F g Since this expresses F as a ltered

intersection Theorem implies there is some C F with A C F

Thus C A C F hF i

F F

On the other hand C v F also implies that C v F and so F

D

S F F F T

F C that F C F F Using the facts that F

0

C F

and that F is nite we conclude there is some C F with F C Since

F is directed we can cho ose a C F such that C C v C and it then

D

follows that hF i v C This all go es to show that hF i C in D P as

D D

claimed

It is easy to show that the family fhF i j F nite hF i v C C

D

F g is v directed so we conclude that D P is continuous and that fhF i j

D

nite hF i v C C F g is a basis for the wayb elow set of each C in

D

D P Hence D P v is continuous and if C C in D P then there

D D

is some F P nite with C v hF i v C and C F

D D

The pro of that C D hC D i D P D P D P is continuous is

straightforward 2

Thus D P is a coherent continuous dcp o by Prop osition while

D P v continuous dcp o by Prop osition We now investigate the

D

relationship b etween these distinct orders on D P We b egin with a technical

lemma

Lemma Let F P be a nite subset of the continuous dcpo P If F is

an antichain then

hF i fX D P j X F X F n fxg x F g

D

Pro of If hF i v X then X F and F X and it is easy to show that

D

Mislove

X is in the set on the right side of the claimed equality using the fact that F

is an antichain Conversely if X is in the set on the right side of the claimed

equality then X F and since X F n fxg for any x F it must b e

that x X for each x F But then for each x X y X x v y

which means x y X Hence F X so F v X 2

D

Prop osition Let P be a coherent continuous dcpo Then

i The identity mapping on D P is continuous from the topology on

D P to the Scott topology on D P v

D

ii The identity mapping on D P is continuous from the topology on

D P to the lower topology on D P v

D

Pro of For i we note that given C U D P v with U Scott op en

D

then there is a nite subset F P with C hF i hF i U By

D v

D

substituting the minimal elements of F for F itself we may assume that F

is an antichain and so hF i F Now if F is a singleton set then

D

hF i fD D P j D F g which is Scott op en in D P Thus

C hF i U

D

In case F has more than one element we also can assume that C has

more than one minimal element for otherwise we are back in the case of one

element in F Then we also can assume that C x for any x F if x F

with C x then the fact that C is closed in the compact space P implies

that there is a nite set G P with C G and x G We also can

cho ose the elements of G so that C y for any y G since C has more than

one minimal element and so we can substitute the nite set G for F in our

argument Now we consider the op en subset of D P given by

hF in F n fxg

xF

which consists of the sets in D P that are subsets of the Scottop en set F

but are not subsets of the set F n fxg Clearly C is in this set which

xF

is op en since hF i is the Scott interior in D P of the set hF i

Moreover any set X which lies in this set satises hF i v X by the previous

D

lemma This shows part i

For part ii if C D P n D for some D D P then C D or

D

D C In the rst case D D P as P is coherent and C D P n D

Moreover if X D P n D then X D and so X D P n D

D

Finally in the second case D C implies x C for some x D

Then x C and since C is compact we can cho ose y x in P with

y C It follows that P n y D P and C P n y Thus

P n y C and if X D P with P n y X then

D P D P

X y and so D v X 2

D

Corollary If P is a coherent continuous dcpo then so is D P v

D

Pro of Since D P is coherent its top ology is compact and the

top ology of D P v is Hausdor The previous Prop osition then implies

D

these top ologies are the same so the top ology of D P v also is compact

D

making D P v coherent 2

D

Mislove

Lemma If S is a coherent continuous dcpo with a continuous semilattice

operation S S S then there is a continuous mapping f D S S

S

such that f hC D i f C f D and f fxg x for al l x S

S S S S

Pro of If x x v y y if F fx x g and G fy y g

m S n m n

are nite subsets of S with hF i v hGi This implies that hfx x gi

D m

x x is welldened and monotone on sets of the form hfx x gi

m m

Since each set C D S is the directed supremum of sets of the form hF i for

F nite we have a welldened mapping f D S S To see that f is a

S S

continuous assume f C U S is Scott op en Then there is some nite

S

subset F S with C F U and clearly we can assume F v C Hence

D

C hF i f U so f U D S is op en It also is routine to check

D S

S S

that this mapping preserves the semilattice op eration on nitely generated

subsets of S and so it must on all of D S by continuity Finally if x S

then clearly f fxg x 2

S

Theorem Let SCCON be the category of coherent continuous semilattice

dcpos and Scott continuous semilattice maps Then

i The functor D CCON SCCON which assigns to each coherent con

tinuous dcpo P the coherent continuous semilattice D P and to each

morphism f P Q the mapping D f C hf C i is left adjoint to the

forgetful functor from SCCON to CCON

ii If P is algebraic then D P v P P v

D C C

Pro of If P is a coherent continuous dcp o then the previous result implies

D P is a coherent continuous semilattice Supp ose S is a coherent continuous

semilattice and f P S is continuous Then the previous results ab out

the lower and upp er p ower domains imply there are continuous semilattice

maps f P S and f P P P S that uniquely extend the

L U U U

map f It is routine to show that the mappings C C D P P

and C C D P P P are continuous semilattice maps and then the

U

mapping C f C f C D P D S also is a continuous semilattice

L U

mapping If we comp ose this map with f D S S we have the desired

S

map This proves part i

Part ii follows from part ii of the previous result where we showed that

the sets of the form hF i with F nite form a a basis at C for each C D P

If P is algebraic then we can assume the nite sets F consist solely of compact

elements and so this basis is isomorphic the the basis K P P fhF i j

C

F K P niteg 2

These results end our presentation of the p ower domain constructions For

each of the three standard p ower domains we have constructed an analogous

p ower domain over continuous coherent domains and shown these top ological

constructions agree with the algebraic ones in the case the domain P on which

they are built is algebraic

Mislove

Abramskys Program

Domains have demonstrated utility in devising mo dels for highlevel program

ming languages Part of this mo deling pro cess includes devising logics that

allow one to reason ab out the programming language under study One of the

most impressive accomplishments in the area of domain theory has b een the

work of Abramsky in which a tight connection b etween domaintheoretic

mo dels and program logics is detailed We very briey outline these results

here

Abramskys starting p oint is the realization that for domains the compact

elements completely determine the domain and for logics the Lindenbaum

algebra provides a denitive mo del from which the logic can b e recovered

So if there were a canonical way to create a domain from the Lindenbaum

algebra of a logic and a canonical way to create a Lindenbaum algebra from

the compact elements of a domain then this would lead to a canonical metho d

for asso ciating domains to logics and viceversa

Now a Lindenbaum algebra is a sp ecial sort of distributive lattice If one

wants a classical logic then this algebra should b e Bo olean On the other

hand if one seeks an intuitionistic logic then the lattice should b e a Heyting

algebra If one starts with a domain D then K D can b e identied with

certain Scott op en sets the basis fk j k K D g In fact we can take the

distributive lattice K D fC D j C is compact and Scott op eng

Lemma For a domain D K D fF j F K D niteg 2

Now if D also is coherent then the family K D is a lattice and so we

can view it as the Lindenbaum algebra of some logic And this logic will b e

intuitionistic since its Lindenbaum algebra is a Heyting algebra

We can carry this a bit further The Scottop en sets are ones that are in

accessible by directed suprema Viewing the compact elements as represent

ing nite amounts of information the set k then represents any information

that sup ersedes that of k But the p oint is that we can observe in nite time

whether the supremum of a directed set gets in k

On the other hand F for F K D nite is Scottcompact This

means it is determined by a nite amount of nite information Thus the

sets in K D form the basis of the top ology of the nitely observable as

Abramsky likes to phrase it

Going back the other way given a Lindenbaum algebra L what domain

could it represent If we lo ok at the algebra K P for P a coherent domain

we see it is the sets k for k K P that we need to retrieve In the algebra

K P these are distinguished by the fact that f k j k K P g is the set

the coprimes So it is Stone Duality that we need to apply to retrieve the

domain P from the lattice K P

Of course this all is rather fatuous since it is not so easy to take any

domain even any coherent domain P and gure out what logic has Lin

denbaum algebra K P So what is needed is a metho d for going back and

forth b etween domains and logics in a way that allows one to identify precisely

Mislove

the logic that a given domain generates This is where Abramskys program

fo cuses He sets out a numb er of basic constructors of domain theory and

shows how each corresp onds to a constructor for the pro of system in the Lin

denbaum algebra of the desired logic In the end his theory can b e applied

to domains freely generated by these constructors

Abramskys theory applies to the category SFP of SFP ob jects and contin

uous maps This is reasonable since the domains should b e countably based

to reect computational reality and some of the constructors are wellb ehaved

only if the domains in question are coherent The fact that SFP is the largest

cartesian closed category of countablybased domains that are coherent makes

it the right target for Abramskys theory

The domain constructors Abramsky incorp orates are the following

Lift LP P

Coalesced sum P Q the disjoint union of P and Q with the least elements

identied

Products P Q

Function space P Q

Power domains P C P and D P and

Recursion

Winskel was the rst to observe that each of the p ower domains can

b e characterized logically by suitable mo dal op erators the lower p ower do

main corresp onds to the sometimes op erator the upp er p ower domain to the

eventual ly op erator and the convex p ower domain to the combination of the

two

The inclusion of recursion in this list is fundamental It is based on the

notion of an admissible predicate on a cp o a nonempty Scottclosed subset

of the cp o and the Principle of Fixed Point Induction cf

If f P P is a monotone selfmap of a cp o and if f x satises a given

admissible prop erty whenever x do es then x f also satises the prop erty

An imp ortant outstanding question is how to extend Abramskys theory to

domains that are not freely generated The problem is reected by the fact

that quotients of domains need not b e algebraic they can b e continuous

But even if the quotients in question again are domains there still is no clear

way to extend Abramskys theory to handle them More concretely the mo del

for the language CSP is intrinsically a Scott domain and it would b e very

nice to have an extension of Abramskys theory that provided a logic for this

mo del

The Lamb da Calculus

Top ology is at the center of the only known approach to giving mo dels of

the untyped This system originally was devised by Alonzo

Church in the s in an attempt to nd a foundation for mathematics and

Mislove

logic that placed functions at the forefront While Churchs original formu

lation had inconsistencies Curry put forth subsystems that are consistent

Because the theory fo cuses on the computational asp ects of functions in the

spirit of say the First Recursion Theorem it attracted the attention of com

puter scientists But the lack of mathematical mo dels for the theory made it

little more than a formal and unmotivated notation in the words of Dana

Scott cf the Foreword to Not long after he made this observation

however Scott found the rst of a whole family of mo dels for the system and

thus b egan the presentday study of the calculus in earnest

Syntax

We b egin our discussion of the lamb da calculus by describing the calculus and

what it means to have a mo del for it Let C denote a set of constants V a

set of identiers or variables and then consider the following set of BNFlike

pro duction rules

M c j x j MM j xM

where c C and x V One way to think of the rules given ab ove is as

the signature of a singlesorted universal algebra In this case the algebra

has nullary op erators ie constants c C and x V a family of unary

op erators fx j x V g and one binary op erator M N MN which

is given as the third clause of the set This op erator is called application and

it is meant to b e an abstract mo del for the application of a function the rst

M in the clause to its argument the second M in the clause The op erator

in the last clause is called abstraction and it is meant to mo del the way we

can take say the p olynomial x and make it into a function x x R R

here dened on the reals So in lamb da notation the function f R R by

f x x would b e denoted x R x

The term x R x is from a typed universe where functions are sp ecied

as having sp ecic domains and co domains The typ e of this term is given

by the syntax x R which denotes that the variable x is restricted to the set

R of real numb ers But the calculus whose syntax is given in equation

has no such restriction all terms are assumed to have the same domain the

family of all ob jects dened by the syntax In this sense the untyp ed lamb da

calculus could also b e viewed as a unityped system where there is just one

typ e Since terms either can b e arguments for functions the second M in

the application clause of the BNF or functions the rst M in the clause

the abstraction clause provides a way of taking a term and making it into a

function of the variable x the term xM makes the term M into a function

of the variable x

With no further rules then the lamb da calculus whose abstract syntax

is given ab ove is simply the initial universal algebra with the signature just

describ ed This algebra consists of all terms we can create by rep eatedly

applying the op erators starting with the constant terms C X Such algebras

often are called term algebras

But the lamb da calculus is not meant to b e just an abstract anomic uni

Mislove

versal algebra Indeed it is supposed to b e an abstract mo del of functions and

how they op erate This is why Church chose to fo cus on the two op erations we

have included application and abstraction In order to mo del functions more

accurately we imp ose three conversion rules these simply are equations we

want to hold in the algebra and so the calculus is really the universal algebra

we get as ab ove mo dulo the algebra congruence the following rules generate

xM y M y x for y neither free nor b ound in M

xM N M N x

xM x M for x not free in M

We let C denote the set of terms of the lamb da calculus with constants

from S mo dulo the equations and These rules presupp ose the

notions of when a variable is bound in a term ie within the scop e of a

abstraction and when it is free not within such a scop e In essence the

rule says that b eing a function of a variable has nothing to do with the

name of the particular variable that is used The rule provides the vital link

b etween the two op erators application and abstraction via the usual notion

of substitution And nally the rule says that all terms can b e regard as

functions Certainly all functions we encounter satisfy these laws and so it

seems reasonable to include them in the calculus

The Notion of a Lambda Model

One question to consider is what sort of models the calculus might have Since

the calculus is meant to b e an abstract version of functions we exp ect to nd

some mathematical model D in which

the constants c C are interpreted as constants in c D

the variables x X also are constants in x D

the term xM is interpreted as a function xM D D

application MN is interpreted as the application of the term M interpreted

as a function M D D to the term N D

In all this says we exp ect to nd a mathematical ob ject D which is a universal

algebra with the same signature as the calculus and a homomorphism of

universal algebras C D from the terms of the calculus to D Of

course the calculus and the identity map form one such mo del but it is

dicult to think of any other mo del

This diculty is reected in certain asp ects of the calculus For example

the idea that ob jects can b e functions and arguments for functions at the same

time seems a bit o dd The fact that all terms live at the same level seems

unintuitive This clash with intuition is brought home when we consider the

term

xxx

4

It is traditional to denote the semantic meaning of a term x in a mo del by x

Mislove

This certainly is a valid term of our calculus And for any term M of the

calculus it expresses the fact that we can apply M to itself xxxM MM

by the rule Since xxx can b e applied to any term of the calculus each

selfmap of the domain arising as the interpretation of an element of the calculus

must also b e an element of the domain Transp orting this to our hop edfor

mo del D we see that there must some way to interpret each element of D as

a selfmap of D This means that at least there must b e maps

p D D D and e D D D such that p e

D D

Here D D denotes the family of selfmaps of D In general then we seek

a mathematical ob ject D satisfying the equations in whatever universe

D resides in What seems to b e required minimally is a cartesian closed

category A and an ob ject D from A for which AD D is a retract of D in A

Such ob jects are called reexive so we seek a reexive ob ject is some cartesian

closed category as a mo del for the calculus

We note that C also has an op eration of composition intrinsically de

ned on it

Prop osition The operation M N M N xM N x C C

C denes a monoid structure on C whose identity is the term I

xxx

Pro of This is a routine verication using the conversion rules 2

Lemma If X is an object of a ccc A then X X has a monoid struc

ture

Pro of This also is a standard result ab out cartesian closed categories A

pro of can b e found in the app endix of 2

Using these observations and denoting by MON the category of monoids

and monoid homomorphisms we now can formulate precisely what we mean

by a lambda model

Denition An ob ject X in a ccc A is a lambda model if there is a mapping

p X X X and maps

C X in SET and

C X X in MON

such that p and M N MN

app

A

-

X X X

X

*



6 





app

C





C C

From this denition it is clear that we are interested only in mo dels that

are SETbased The following result claries the situation further a more

complete presentation ab out this connection can b e found in Chapter of

Mislove

Theorem There is a onetoone correspondence between models of the

untyped lambda calculus and reexive objects in cccs 2

We remark that there are other notions of what a lambda model should

b e Of particular note are the results of which elab orate the relationships

b etween a numb er of approaches to dening this concept as well as the de

tailed discussion in These involve concepts such as combinatory algebras

and combinatory mo dels

For us the question is how to nd an example of a nondegenerate reexive

ob ject in a ccc The place to start a search for a lamb da mo del is the category

SET of sets and functions But there Cantors Lemma says no nondegenerate

set can admit its space of selfmaps as a quotient let alone a retract It turns

out that with one exception it is relatively dicult to nd cartesian closed

categories that have any nondegenerate reexive ob jects Well say more

ab out the search for reexive ob jects in other categories in a moment but

rst we want to present the construction of one such mo del

Finding a Lambda Model

While the rst mathematical mo del of the untyp ed lamb da calculus was found

by Scott in the category of algebraic lattices and Scottcontinuous functions

there is an underlying construction technique which is applicable much

more broadly and which highlights the nature of the mo del much b etter than

simply reiterating Scotts construction verbatim

We b egin with the observation that what makes SET fail to have a non

degenerate mo del is that there are to o many functions in SET Namely the

set of selfmaps of a nondegenerate set has larger cardinality than the set

itself and this is what is getting in the way But if we restrict our attention

to top ological spaces and continuous maps then this no longer is a problem

For example the space of continuous selfmaps of a space X is of the same

cardinality as X providing X has a dense subspace of cardinality smaller

than the cardinality of X For example jR R j jR j for precisely this

reason But it turns out that nding a mo del in Hausdor spaces is not a

simple matter in fact it remains an unsolved problem So we turn our

attention to spaces not satisfying such a strong separation condition

One place to nd such spaces is within the area of partially ordered sets

As we have seen the categories DCPO and CPO are cartesian closed and the

Scott top ology is T but rarely T Moreover we already have encountered

a technique for nding domains with desired prop erties solving recursive

op op

domain equations Our example of the domain N LN is a case

in p oint So one way to nd a cp o satisfying our needs would b e to seek a

solution of a similar domain equation

The problem is to nd a nondegenerate cp o that is isomorphic to its cp o of

continuous selfmaps We cannot exp ect an actual setinclusion P P P

since this is forbidden within ZermeloFrankel by the Foundation

Mislove

Axiom so the the nearest we can come is an isomorphism In this approach

the likely equation is P P P and this gives rise to the op erator

F P P P on cp os However this op erator has P in two places and

it is not at all clear how to turn F into a functor We now recall a denition

that allows us to overcome this problem

If P and Q are dcp os then the pair of Scottcontinuous maps e P Q

and p Q P is an embeddingprojection pair if pe and pe Given

P Q

an emb eddingpro jection pair e p P Q we can dene related functions

F e P P Q Q by F ef e f p

and

F p Q Q P P by F pf p f e

Moreover these mappings F e and F p are Scottcontinuous since they

are dened via comp osition The following result captures an imp ortant fact

ab out emb eddingpro jection pairs

Lemma If e p P Q is an embeddingprojection pair then so is the

pair of mappings F e F p P P Q Q

Pro of If f P P then F ef e f p Q Q and so

F p F ef p e f p e f

since p e Similarly for g Q Q

P

F e F pg e p g e p g

since e p 2

Q

Both DCPO and CPO are complete categories the limit of diagram G

DCPO dened on the directed graph G N E is the usual family

Y

lim n e fx n j ex x e n n g

neN E n n n i j

i j

nN

Both categories also are cocomplete the colimit of the diagram G

DCPO is the ideal completion under directed suprema of the colimit in POS

Theorem shows that for emb eddingpro jection pairs the limit and colimit

that arise naturally are in fact the same If G DCPO is a diagram in

ep

the category of dcp os and emb eddingpro jection pairs For each edge e in G

let e e p where e n n Then

ij ij i j

lim fn p j n N e n n g lim fn e j n N e n n g

ij i j ij i j

If we start with the canonical emb eddingpro jection pair

ep f g F f g where exy x and pf f

then we can consider the ob ject

n n m

I colimF f g fF e F eg

mnN

n

5

This follows by a simple argument using the von Neumann denition of ordered pair

x y ffxg fx y gg and the identication of functions as sets of ordered pairs

Mislove

We would like to claim that F I I but to do this we need to know that

F is continuous Since F is not the same sort of functor as b efore we need

the following denitions and result

op

Denition Let F A B C b e a functor dened on categories of

dcp os which is contravariant in its rst argument and covariant in its second

such a functor is said to b e of mixed variance We say F is continuous if for

all diagrams G A and all diagrams G B we have

ep ep

0 0 0

F limn e colim n e

neG n e G

A B

0 0 0

colimF n n F e e

nen e GG

C

Also G is local ly continuous if for all ob jects P P of A and Q Q of B

op

G A BP Q P Q CGP Q GP Q

is continuous

In analogy to Theorem we have the following

op

Prop osition If G A B C is mixed variance local ly continuous

functor then G is a continuous functor 2

op

Applying this to our functor F CPO CPO CPO we see that lo cal

continuity of F is sucient to prove the desired isomorphism I F I

And indeed the pro of that F is lo cally continuous is straightforward The

ob ject thus dened I F I is the D mo del rst discovered by Scott

This rather general treatment of how to construct such a mo del is taken

from Section of where more details of pro ofs can b e found and where

further results ab out the canonicity of solutions to domain equations for mixed

variance lo cally continuous functors also can b e found

There is one subtle p oint we have skipp ed over here Namely in generating

the xed p oint I we started with the domain f g instead of the least

domain fg Of course the reason is that were we to apply the functor

F P P P to fg we would never get anywhere and our solution to

P P P would b e fg But there is a way to bring this construction

into the realm of generality considered in Section Namely we redene the

functor F Instead of using F P P P we can take instead the functor

F P A P P where A is the at domain dened on the set A

of constants we wish to include in the syntax of the calculus and denotes

coalesced sum Since this set A can b e assumed to include the variables it is

nonempty and so the resulting domain P F P must b e nondegenerate

For example taking A to b e a singleton yields the original D mo del of Scott

The Search for Other Models

We now have constructed a nondegenerate mo del of the untyp ed lamb da

calculus in CPO An interesting question is whether mo dels exist in other

categories We already commented that this is not p ossible in SET The next

place one might lo ok is the category POS of p osets and monotone maps

Mislove

Models in POS

It is wellknown that POS is cartesian closed the terminal ob ject is the one

p oint p oset pro ducts are ordered in the pro duct order and the internal hom

is the space of monotone maps

Denition If P is a p oset we let LP fY P j Y Y g denote the

family of lower sets of P

The following result of Gleason and Dilworth is crucial to our devel

opment

Theorem Gleason Dilworth

If P is a poset then there is no monotone surjection of a subposet of P onto

LP 2

We also need the following result whose pro of is straightforward

op

Lemma If f P Q is an injection then x P Q

Qnf x

is also an injection where denotes the twopoint lattice 2

Theorem POS has no nondegenerate lambda models

Pro of Abramsky

Supp ose that P is a p oset and that P P is a retract of P in POS If P is

P

an antichain ie if P is a set with the discrete order then P P P

and the result follows from Cantors Lemma

So we assume P is not an antichain Then there are a b P and so

P retracts P onto Then P is a retract of P P and

P na

hence also of P Applying the same reasoning again we see that P P

also is a retract of P But the mapping

op

I LP P

I

is an orderisomorphism and this implies that LP is isomorphic to a sub

p oset of P P Since the latter is a retract of P there is some

subp oset Q P which maps under the retraction onto LP and this con

tradicts the GleasonDilworth Theorem 2

We should note that an alternative pro of of this is contained in

Models in Complete Ultrametric Spaces

Another wellknown ccc is the category CU of complete ultrametric spaces

and nonexpansive mappings In this category the terminal ob ject is the one

p oint space pro ducts are given the pro duct metric and the internal hom is

the space of nonexpansive mappings b etween the spaces We pro ceed to show

that CU has no nondegenerate lamb da mo dels

Denition A metric d X X R is an ultrametric if x y z

X dx z maxdx y dy z

Lemma If X d is an ultrametric space and x X and then

B x is closed as wel l as open

Mislove

Pro of Supp ose that y B x Then dx y Now consider B y We

claim B x B y Indeed if z B x B y then dx z

and dy z Hence dx y maxdx z dy z max

which is a contradiction 2

Finally we recall the paradoxical combinator Y C dened by

Y M f xf xxf xx

A routine calculations show that for all terms M in C M Y M Y M

ie YM is a xed p oint of M This means that in any lamb da mo del each

selfmap must have a xed p oint

Theorem CU has no nondegenerate lambda models

Pro of Plotkin

If X is a nondegenerate complete ultrametric space then there are distinct

p oint a b X If da b then B a is a clop en ball in X which do es

not contain b The mapping f X X by

a if x B a and

f x

b otherwise

clearly has no xed p oints Moreover it is nonexpansive If x y B a or

x y X n B a then f x f y so this is clear On the other hand if

x B a and y X n B a then the argument in the pro of of the lemma

shows that dx y and so df x f y dx y Thus X cannot b e

a lamb da mo del 2

It is interesting that this pro of is quite dierent from the ones for SET and

POS b oth of which relied on a cardinality principle Of course knowing that

all selfmaps in a lamb da mo del must have xed p oints already says top ological

spaces that are lamb da mo dels must b e connected

Hausdor Lambda Models

A last category we consider in our search for lamb da mo dels is a ccc of Haus

dor spaces If one starts with the category of compact spaces and continuous

maps then the natural ccc one comes to is the category K of k spaces and

continuous maps A K space is a top ological space in which a subset is op en

if and only if its intersection with each compact subset of the space is op en

in the subspace If we are given a top ological space X we can kify it by

taking the top ology generated by the sets satisfying this prop erty The basic

results ab out the category K are contained in They include the fact that

the terminal ob ject is the onep oint space that the pro duct is obtained by k

ifying the pro duct top ology and the internal hom is the space of continuous

maps endowed with the kication of the compactop en top ology

In the last subsection we intro duced the combinator Y which assigns to

each term of the lamb da calculus a canonical xed p oint Along with Y

there is another combinator K This combinator is dened by

K M N M

Mislove

and it gives a way of recognizing constant functions in the lamb da calculus In

fact it is not hard to show that KM N M using reduction Moreover

Y K I is the identity op erator

Our rst result ab out nondegenerate lamb da mo dels in K eliminates com

pact spaces To obtain this result we rst derive a simple result from the

theory of compact semigroups This result is not new it can b e found eg

in

Prop osition Let S be a compact monoid If x y S satisfy x y

then y x

S S

Pro of First we show that any compact semigroup T has a smallest closed

ideal M satisfying M I for any closed ideal I T Indeed the semi

T T

group T itself is an ideal and if I and J are closed ideals then so is I J the

setpro duct of I with J Moreover I J I J so I J Thus the family

of nonempty closed ideals is ltered and so it has a nonempty intersection

This intersection also is an ideal and so it must b e the minimal ideal we seek

Next we note that if T is commutative then M is a group Indeed if

T

x M then x M M is compact b eing a translate of a compact set

T T T

and T x M T M x T M x M x x M Dually

T T T T T

x M T x M T x M Thus x M is an ideal and so it must

T T T T

b e equal to M as M is minimal Similarly M M x so M is indeed a

T T T T T

group

Finally let x y S with S a compact monoid and supp ose x y Let

S

n

fx j n N g b e the closed subsemigroup of S that x generates Since the S

x

semigroup of p owers of x is commutative and multiplication is continuous on

S it follows that S is a commutative and so this is a compact commutative

x

semigroup Its minimal ideal M then is a group and we let e e b e the

S

x

identity of this group Furthermore since M is a group x e M has an

S S

x x

inverse x in this group

n

We claim that e Indeed since e S there is a net fx g

S x

n n

fx j n N g such that e lim x Now S is compact and so the net

n

fy g has a cluster p oint and by picking subnets if necessary we can assume

n n n

lim y z S Then x y for all n N and so

S

n n

e z lim x y

S

and this means

e z e e z e e z e e

S S

Now x e x x and so x is a memb er of the group of units of S and

S

x is the inverse of x in S But since x y it follows that y x and

S

so y x as well 2

S

Theorem Hofmann Mislove There are no nondegenerate

compact Hausdor lambda models in K

Pro of Supp ose that X is a compact Hausdor space that is a reexive ob ject

in K Then X X is a retract of X and so it to o is compact and Hausdor

X X also is a top ological semigroup under the op eration of comp osition

Mislove

and the identity map is the identity of this semigroup Now the combinators

Y and K can b e recognized in X X and so there are functions Y K

X X satisfying the prop erty that Y K It is routine to verify

X

that the combinator K gives rise to the constant picker K x X X by

K xy x Hence the image of X under K consists of the constant maps

Now the previous Prop osition shows that in the compact monoid X

X if s t then t s as well It follows that K Y in

X X X

X X and this in turn implies that K X is a constant map Hence

X

X is degenerate 2

Of course this result do esnt eliminate nondegenerate lamb da mo dels

from K it merely says they cannot b e compact The class of p ossible mo dels

can b e shrunk further by the following result It is taken from

Prop osition Let X be a lambda model in a ccc A and suppose that Z

is a retract of X in A Then there is a morphism Y in AZ Z Z such

Z

that f Y f Y f for al l f Z Z

Z Z

Pro of Since X is a lamb da mo del X X is a retract of X in A and

the interpretation Y of the paradoxical combinator Y in X X implies

X

each morphism f X X satises f Y f Y f Now if Z X

X X

expresses Z as a retract of X in A then it is routine to show that the mappings

f f Z Z X X and g g X X Z Z

express Z Z as a retract of X X Another calculation shows that

f Y f Z Z Z pro duces a xed p oint combinator for

X

Z Z and clearly Y Y Z Z is the desired morphism2

Z X

Recall that the unit interval in the usual top ology is an absolute

neighb orho o d retract

Theorem Hofmann Mislove

If X is a normal space that contains a homeomorphic image of then X

cannot be a lambda model

Pro of Indeed if X is normal and contains a copy of then there is

a retraction of X onto Then the previous result implies has a

continuous xedp oint picker But this is not true Indeed consider the

mapping

tx t if t

H by H tx

tx if t

The xed p oints of the mappings H t are

fg if t

FixH t if t

fg if t

Clearly Fix cannot b e made to b e a continuous function on the arc of functions

fH t j t g 2

Mislove

Rice has an alternative approach to proving these results His meth

o ds rely more on the syntactical structure of the lamb da calculus and less on

the structure of the ob jects of K

Models of the Icalculus

The calculus we have fo cused on has included the constant combinator K

xy x This calculus is sometimes called the callbyname calculus since

it do es not require evaluating terms b efore other terms can b e applied to them

Another calculus is the Icalculus which corresp onds to the callbyvalue

calculus We b egin by giving its syntax

The Icalculus includes

the constants c C and the variables x X

the application MN of any term M in I to any term N in I and

the abstraction xM only if x FV M the set of free variables of M

Clearly this eliminates the constant functions from the calculus In fact any

term of the calculus can b e dened from terms in the Icalculus and the

combinator K xy x cf But our argument showing compact Hausdor

mo dels of the calculus are degenerate uses the combinator K crucially To

prove a similar result for the Icalculus we rst dene three combinators

B xy z xy z

C xy y x and

hIi xxI xxy y the socalled list of I

B is the combinator that instantiates comp osition in the calculus The follow

ing result is a routine computation

Prop osition In the Icalculus we have

i BChIi z ttz I and

ii BhIiC I 2

Corollary There are no nondegenerate compact Hausdor models of

the Icalculus

Pro of According to Prop osition if there is a compact Hausdor mo del

then xx I z ttz I If we take any element a in the mo del then this

implies a xxa z ttz Ia ttaI So for any b in the mo del

ab ttaIb baI Taking a I we see that b Ib bI I bI for all

b in the mo del and so ab baI ba

Now by a basis for the calculus we mean a family K of terms such that

every term in the calculus can b e realized as the application of terms in K

ie every term of the calculus is in the set of terms that K generates under

the op eration of application A result of Rosser cf and Prop osition

of states that the terms

I xx and

Mislove

J w xy z w xw z y

form a basis for the Icalculus But according to our result ab ove J IJ

JI and so the only terms that I and J generate are p owers of J But further

J IJ JI xy z xz y J

and so further

J J IJ J I y z z y J

and nally

J J IJ J I z z I

Thus the mo del is degenerate 2

Furio Honsell is to b e thanked for p ointing out to the author that

the combinators B C and hIi can b e used as describ ed ab ove to show that

compact Hausdor mo dels of the Icalculus also are degenerate in the same

way as K and Y can b e used to show compact Hausdor mo dels of the

calculus are degenerate

Programming Languages and Other Applications

Domain theory b egan in an attempt by to avoid the untyp ed

lamb da calculus and nd a more intuitive mathematical structure for pro

viding mo dels for programming languages As it happ ened the search also

pro duced the rst mathematical mo dels of the untyp ed lamb da calculus But

we have not said much ab out the metho ds used to build programming lan

guage mo dels themselves We close this pap er with some comments along this

line they are more hints at places to lo ok for details than they are precise

descriptions of such mo dels themselves

The classical languages which domain theory has proven most useful for

mo deling are functional languages this is understandable since the lamb da

calculus itself is a prototypical such language An excellent intro duction here

is the b o ok by Gunter A great deal of research has gone into this area

and Gunters b o ok also is a go o d resource for nding further applications along

this line

An area which the author has b een motivated by is process algebra and

languages supp orting concurrent computation Among the most prominent

of these are the languages CSP studied by Hoare Brookes Roscoe

Reed citebhrrr and others at Oxford and CCS invented by Mil

ner and studied by many p eople In either case the approach is to fo cus

on the communication events that take place b etween machines running in a

concurrent environment rather than on the actual computations each machine

makes Domain theory has proved a fruitful to ol in this area For example the

seminal pap er of Hennessy and Plotkin showed how p ower domains

corresp ond to forms of nondeterminism The pap er carries this theme

further by showing for a simple parallel programming language how each of

the p ower domains corresp onds to a distinct form of nondeterminism

Mislove

Pro cess algebra also provides an interesting contrast to two opp osing ap

proaches for giving denotational mo dels As we have tried to demonstrate

from the outset the role of domain theory is to give meanings to recursive

constructs and the prop erty of domains that facilitates this is the least xed

p oint prop erty that continuous selfmaps on domains enjoy Indeed these xed

p oints not only exist they are canonical The alternative approach here is to

use complete metric spaces and the Banach Fixed Point Theorem This ap

proach has b een championed by de Bakker and his colleagues This ap

proach has a shortcoming however in that it requires a restriction to socalled

guarded terms in order to guarantee the asso ciated selfmaps are contractive

A synthesis b etween the domaintheoretic and metricspace approaches

also has b egun to emerge A seminal result is the pap er of America and

Rutten in which it is shown how to solve recursive domain equations

within the metric space world Further work along this line has b een done

by Flagg and Kopperman as well as Alessi Baldan Belle and

Rutten But the most extensive results along the lines of synthesizing

domain theory and metric spaces are due to Wagner

Our discussion of p ower domains fo cused on presenting them rst in their

original algebraic formulation and then from a top ological view There are

top ological analogues to these constructs which have evolved from the original

Vietoris hyp erspaces These constructs have received new interest b ecause of

the work of Edalat In these works Edalat has found new and exciting

applications of domain theory to the areas of fractals neural networks and

p erhaps most notably to the theory of integration Indeed Edalat has used

domain theory to provide a very simple derivation of the Riemann integral

and at the same time he has found solutions to problems that do not seem

available from the more traditional metho ds

Lastly we mention set theory as an area of application One of the mo

tivations for Church in devising the lamb da calculus was to provide a new

foundation for mathematics Through work on set theory and pro cess alge

bra Aczel devised a new formulation for set theory in which he replaces

the traditional Foundation Axiom by a more general axiom that allows a

much wider family of sets including ones that contain themselves as mem

b ers This set theory has sp ecial app eal for theoretical computation since

it provides simple mo dels for pro cesses that want to call themselves An

obvious topic is then the relation b etween this new set theory and the more

traditional domain theory One asp ect of this relation is presented in

where it is shown how to present a domaintheoretic mo del for the hereditar

ily nite p ortion of Aczels theory A direct application of Aczels theory to

providing programming mo dels also can b e found in

Acknowledgements

This pap er arose from a series of lectures the author delivered at the Summer

Top ology Conference held in Portland Maine in August The author

wants to express his thanks to the organizers of that meeting including Bob

Mislove

Flagg Southern Maine and Ralph Kopperman Queens College as well

as to Jerry Vaughan UNC Greensb oro for inviting him to present those

lectures and for urging him also to prepare this article Thanks also to the

National Science Foundation which sp onsored the conference

Thanks go also to numerous p eople who have commented on some of the

work rep orted here First and foremost the authors longstanding collab ora

tion with Frank Oles IBM has b een a continued source of knowledge and

insight ab out theoretical computation Thanks go also to Karl Hofmann

TH Darmstadt with whom the author collab orated on work on mo dels of

the untyp ed lamb da calculus and many other topics b efore that one Finally

thanks go to Furio Honsell Udine whose close attention to the lectures

at Maine led to his comments on mo dels of the Icalculus rep orted in this

pap er

Thanks go to the PhD students at Tulane who have listened patiently to

initial presentations of some of the results in this pap er and who have read

over drafts of this pap er p ointing out errors Thanks also are due to the

referee for p ointing out errors that had gone unnoticed and for recommending

improvements in the presentation

Finally thanks also to the US Oce of Naval Research and to Dr Ralph

Wachter the Program Ocer for the Software Research Program for their

generous supp ort of this research

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