Syracuse University SURFACE
College of Engineering and Computer Science - Former Departments, Centers, Institutes and College of Engineering and Computer Science Projects
1995
Objects, Interference, and the Yoneda Embedding
Peter W. O'Hearn Syracuse University
Uday S. Reddy University of Illinois at Urbana-Champaign
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Recommended Citation O'Hearn, Peter W. and Reddy, Uday S., "Objects, Interference, and the Yoneda Embedding" (1995). College of Engineering and Computer Science - Former Departments, Centers, Institutes and Projects. 14. https://surface.syr.edu/lcsmith_other/14
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Electronic Notes in Theoretical Computer Science to app ear
Ob jects Interference and the Yoneda
Emb edding
Peter W O Hearn
Syracuse University
Uday S Reddy
University of Il linois at Urbana Champaign
Dedicated to John C Reynolds in honor of his th birthday
Abstract
We present a new semantics for Algol like languages that combines metho ds from
two prior lines of development
the ob ject based approach of where the meaning of an imp erative program
is describ ed in terms of sequences of observable actions and
the functor category approach initiated by Reynolds where the varying na
ture of the run time stack is explained using functors from a category of store
shap es to a category of cp os
The semantics gives an account of b oth the phemomena of lo cal state and irre
versibility of state change As an indication of the accuracy obtained we present a
full abstraction result for closed terms of second order typ e in a language containing
active expressions i e value returning commands
Intro duction
In his in uential Turing award lecture John Backus criticized imp erative
programming languages for promoting a view of programming as word at a
time pro cessing John Reynolds expressed his resp onse to this criticism in
a meeting of IFIP working group in around which he rep eated to
several p eople privately including the second author The view put forward
by Backus Reynolds said is that imp erative programming is like working with
1
Supp orted by NSF grant CCR
2
Supp orted by NSF grant CCR
c
Elsevier Science B V
O Hearn and Reddy
pigeon holes All that one do es is to take a pigeon out from a hole or to
put a new pigeon in a hole But with ob ject oriented programming he said
one works with turkey holes rather than pigeon holes Instead of taking out
a pigeon or putting in a pigeon one do es more sophisticated manipulations
such as rotate a turkey or tilt a turkey
The turkey holes that Reynolds sp oke of are what programmers call
ob jects They incorp orate some physical resources such as memory and
provide op erations for the manipulation of these resources Programs are built
by putting such ob jects together and letting them invoke each other s op era
tions The state of an ob ject cannot b e manipulated by other ob jects without
the co op eration of the ob ject itself in terms of the op erations it provides
Such an ob ject based view we nd is implicit throughout Reynolds s work
on imp erative programming
In his seminal pap er on Algol like languages Reynolds treats pro ce
dures not as actions on the global state but as actions on the state at the
p oint of their de nitions To elab orate a pro cedure can only access the ob
jects that have b een allo cated when the pro cedure is de ned Other ob jects
created after the p oint of de nition but b efore the pro cedure is called do not
a ect its b ehavior Every pro cedure lives in its own turkey hole so to sp eak
Reynolds also shows how to treat variables pigeon holes as a sp ecial case
of turkey holes ob jects with op erations for setting and reading values stored
in them This essentially frees imp erative programming from the limitations
suggested by Backus and sets up a truly ob ject based paradigm for thinking
ab out imp erative programs
Reynolds s program for the semantics of imp erative languages was further
develop ed by Oles and Tennent and continued and expanded in
a numb er of works In a separate line of development
a mo del based more explicitly on a notion of ob jects has b een formulated
in Reynolds s conception of imp erative programming expressed ab ove
formed an imp ortant pre theoretic motivation for this work though its theo
retical development also draws inspiration from linear logic syntactic control
of interference and the relation b etween them In this pap er we obtain a new
semantics for Algol like languages via a synthesis of these two lines
the ob ject based approach of where the meaning of an imp erative
program is describ ed in terms of sequences of observable actions and
the functor category approach initiated by Reynolds where the varying
nature of the run time stack is explained using functors from a category of
store shap es to a category of cp os
In the remainder of this intro ductory section we give an informal overview of
the construction and discuss the sp eci c semantic issues addressed by it
Semantic issues Locality and irreversibility
In imp erative computation there is an idea of destroying information by over
writing parts of computer memory This is clearly imp ortant for implementa
O Hearn and Reddy
tion But supplying direct access to assignment in the programming language
also results in p ositive information that programmers make use of Consider
a parameterless pro cedure gensym that returns a di erent integer each time
it is called In reasoning ab out a program using gensym for instance gen
erating fresh names when implementing substitution in calculus we would
use the prop erty that any call to gensym returns an integer that was not
returned by it previously
This prop erty exempli es one of the most basic intuitions ab out state
the general irreversibility of state change By this we mean not only that
p ortions of the store are destructively up dated during the course of a compu
tation but that in the presence of abstraction or lo cal state this irreversibility
manifests itself in observable prop erties of programs
A typical implementation of gensym would use a lo cal integer variable
that is incremented on each call When we say that gensym returns a di erent
integer each time it is called it is crucial that other pro cedures or ob jects
do not access the lo cal state of gensym directly and reset the value to a
previously encountered one This statement ab out the gensym pro cedure
implicitly involves interactions b etween the pro cedure and any other pieces
of a program The following co de illustrates the kind of prop erty of such
interactions we have in mind
b egin
integer x
integer pro cedure gensym f x x return x g
x
P gensym
if gensym then diverge
end
A client pro cedure P is passed a parameterless pro cedure gensym for
generating new names P can use its argument a numb er of times we are
assuming call by name though the e ect can obviously b e simulated in call
by value and if it uses its argument at least once then we exp ect that the
whole blo ck will diverge Since the non lo cal pro cedure P cannot access the
lo cal variable x if x is up dated by calling gensym then pro cedure P has
no way of resetting its value to zero It follows by intuitive reasoning that
this blo ck should have termination non termination b ehavior equivalent to
P diverge
This co de is not a realistic program but it is interesting for the reasoning
principle it illustrates Generally when we have an ob ject consisting of some
internal state and observable op erations it is not p ossible for a client program
to cause the internal state of the ob ject to backtrack to previous states This is
b ecause the only changes to the internal state that the client can p ossibly e ect
come ab out by using the provided op erations The observable rami cations
of irreversibility of state change are inextricably b ound up with lo cality
Irreversibility has proven di cult to capture in semantics b ecause most
mo dels allow for snapback op erations These op erations work by accepting a
O Hearn and Reddy
pro cedure as an argument running the pro cedure and then restoring the state
to the value it had b efore the argument was executed this would contradict
the reasoning ab out gensym ab ove The snapback e ect requires restoration
of even lo cal state
The phenomenon of irreversibility is not so clear cut in languages that
violate the abstractness of lo cal state such as C or when programming on a
system level where one might want access to the entire computer memory
One could in some instances achieve the e ect of snapback by a series of
incremental state changes But on the level of programmable ob jects where
abstraction is central irreversibility is a familiar phenomenon one that arises
in Scheme ML Algol and most ob ject oriented languages
These intertwined notions of irreversibility and lo cality are fundamental
and should b e accounted for by a satisfactory theory of state
Overview of Approach Objects plus Yoneda
The mo del presented here builds up on the work rep orted in where a
semantics is presented based on identifying an imp erative computation with
a stream of observations For example commands are mo delled not as state
to state functions but as sequences of signals indicating a message to a
command ob ject More accurately a command in context C comm
translates demands for output into requests of typ ed entities Similarly
active integers are mo delled using streams of integers where we read a
stream h i as indicating an ob ject with a single op eration that returns the
rst time it is used and the second
There is a view of an active integer as an ob ject p ossessing an internal state
that may change and a metho d for accessing this state But the representation
of the state is nessed in the mathematical description of ob jects given in
state is regarded as implicit in a history of events One b ene t of such
a stateless account of state is that it forces lo cality to b e resp ected when
comp osing meanings Since the internal state of an ob ject is not part of
the mathematical description the ways of combining these entities do es not
tamp er with the internal state in the way that early denotational mo dels do
Also there is no explicit state to b e sub ject to a snapback e ect though
care is needed to comp ose meanings in a way that resp ects some temp oral
ordering
The work rep orted in formalizes these ideas and results in a mo del
that accounts for lo cality and irreversibility quite well But there is one di
culty in the treatment of state as an implicit attribute it is not easy to give a
satisfactory account of shared state Put another way the ob jects of are
non interfering and it is not obvious how to deal smo othly with interference
A notion of function typ e is de ned but it forms a monoidal closed structure
obtained as the adjoint of a non interfering and non cartesian pro duct
whose comp onents do not interfere As a result the semantics is de ned only
for syntactic control of interference a restricted form of calculus
The constraints in this framework disallow interference b etween pro cedure and
O Hearn and Reddy
argument or client and ob ject
In order to treat the full typ ed calculus a semantics is called for based
on a cartesian closed category See e g The approach that we use
here is mathematically straightforward We b egin with a category C of ob ject
spaces suitable for the semantics from and simply apply a Yoneda em
op
C
that maps this ob ject based semantics into a cartesian b edding C Cp o
closed category of certain functors where Cp o is the category of complete
p ointed p osets and continuous functions So for instance where the typ e of
commands is interpreted as an ob ject comm of category C in the functor
category it is interpreted as the functor C comm using the order struc
ture of C Interpretations of rst order constants are obtained immediately
using the morphism part of the emb edding functor and the fact that Yoneda
preserves pro ducts This is the mathematical side of the story
The computational intuition b ehind this construction is as follows The
op
typ es of the language are mo delled as functors F C Cp o The ob ject part
of such a functor sp eci es a cp o F X of computations for each store shap e
X Recalling that C is the category of ob ject spaces this means that our
store shap es are ob ject spaces whose elements represent ob jects with internal
state Thus a store in our sense is an arbitrary ob ject which might
b e comp osed with various comp onent ob jects such as variables input output
streams windows etc The morphism part of F maps morphisms f Y X
of C to continuous functions F f F X F Y The morphism f Y X
represents a metho d for simulating a store of shap e X using a store of shap e
Y Thus Y may b e thought of as a larger store shap e than X The function
F f then sp eci es the asso ciated conversion of a computation of typ e F X
on the small store to one over the large store F Y These intuitions are
exactly as in Reynolds s functor category semantics for Idealized Algol
The only di erence is that instead of mo delling stores by their state sets we
are mo delling them as ob jects with implicit internal state and observable
b ehavior
To interpret lo cal variable declarations we follow Reynolds s ideas The
meaning of a command new x C at store shap e X is given in terms of the
meaning of C in an enlarged store shap e X var where var is a space interpret
ing a storage variable Note that this interpretation of new do es not follow
from Yoneda emb edding b ecause it semantically corresp onds to a second
order op eration
How do es this approach account for interference Generally sp eaking a
store shap e parameter X provides for a common p oint of interaction for ob jects
in F X This idea represented in the interpretation of higher typ es in the
following way A pro cedure of typ e t t for a store shap e X is a natural
When such a pro cedure is called from transformation from C X F to F
a larger store of shap e Y obtained from allo cation of additional variables we
provide a map f C Y X which serves to pro ject the small store X from
Y and an argument a F Y The pro cedure s own action on the store is
carried out via the map f which pro jects back to its turkey hole On the
other hand the argument a can act on Y directly The provision of the map
O Hearn and Reddy
f which intuitively connects the store shap e at p oint of de nition with the
p oint of call is the key ingredient added to the ob ject based semantics of
It enables a pro cedure and argument to work with the same store shap e and
thus allows interfere by interacting with a common store ob ject A concrete
instance of this phenomenon is shown in Example
It is natural to ask whether we could obtain a similar treatment without
passing to a functor category by expressing the ideas of directly in a
cartesian closed category obtained p erhaps by leaving the framework of co
herent spaces This might b e p ossible if we were to take a concurrent view
of ob jects and accept non determinism but the details of such a treatment
are by no means obvious As we explain in section the Yoneda interpreta
tion accounts for interference via a determinate use of interleaving in which
interfering ob jects are interpreted in a shared context of evaluation
Overview of the pap er
After xing the syntax of a sample programming language Sec and
reviewing the relevant details of ob ject spaces Sec we present the mo del
obtained by the Yoneda emb edding in Sec In Sections and we analyze
the accuracy of the mo del arriving at the following technical results
We give explicit representations of rst order typ es and show that all natu
ral transformations b etween pro ducts of base typ es are least upp er b ounds
of de nable elements The language used for de nability is an Algol like
language containing active expressions i e value returning commands
We give a full abstraction result for closed terms of second order typ e
In App endix A we also include a short discussion of ob jects with internal
state which motivates the technical de nitions concerning ob ject spaces This
discussion is not used in the main b o dy of the pap er except for motivational
purp oses
We assume the reader s familiarity with the theory of complete partial
orders The text also has a discussion of dI domains and stable
functions which we mention in passing Coherent spaces form a particularly
simple class of dI domains which we make use of in our techincal results ex
tensively Discussion of coherent spaces may b e found in as well as the
pap ers where they are applied to semantics of linear logic and ob ject
spaces resp ectively The semantics texts ab ove also contain an intro ductory
treatment of cartesian closed categories and basic de nitions concerning func
tors and adjunctions Additional discussion of categorical concepts can b e
found in and standard category theory texts such as
Syntax
We consider a language with the following base typ es
comm the typ e of commands and
aint the typ e of active integer expressions active integers for short
O Hearn and Reddy
By active expressions we mean computations that p otentially cause state
changes and return values We form other typ es using binary pro duct and
function space We follow Reynolds and regard a typ e var of storage vari
ables as sugar for aint comm aint De referencing is second pro jection
and assignment is accomplished with the rst pro jection and pro cedure call
For instance x x desugars as x succ x
1 2
The typ e system is that of simply typ ed calculus with binary pro ducts
The constants are as follows
succ pred aint aint Y t t t
t
ifz aint b b b aint
new var comm comm skip comm
letval b b b b
where b b range over base typ es and t over typ es We can also consider a
version of new that taken an integer returning pro cedure and pro duces an
integer
The arithmetic constants are just those of sequential PCF For commands
we have constants for lo cal creation and a form of sequential comp osition
letval The phrase letval M y N evaluates M binds the value obtained
to y and then executes N In case M is a command y is b ound to skip
after the execution of M The key p oint here is that the execution of M
can change the state but subsequent uses of y do not Also the side e ect
of M is p ersistent and not a snapback We use notation C C as sugar for
letval C x C where x not free in C or C This is for any combination of
base typ es for C and C When C is a command and C an integer this gives
us a side e ecting or active integer
In our very bare sample language there is no input output or global vari
ables for programs to act up on Storage variables are created using new as
in new x C This creates a lo cal variable x initialized to that may b e
up dated within C recall the sugaring of assignment ab ove but this storage
variable is de allo cated on blo ck exit As a result a closed term of typ e comm
do es not change the state at all it must b e equivalent to Y x x or skip
But even for this bare language there are many interesting examples that
illustrate principles of imp erative computation
Example The gensym example from the Intro duction is represented as
O Hearn and Reddy
the following term
new x
gensym
letval x dd
1
letval P gensym dd
ifzhpred gensym skip Y x x i
letval x succ x y x
1 2 2
where are pro jections For obvious reasons we will use a sugared syntax
i
when the desugaring is clear
Remark Since expressions in this language are active typical prop erties
such as commutativity of addition are lost It is p ossible to add a typ e int of
passive side e ect free expressions and our semantic approach can handle
these quite well But we have not obtained de nability and full abstraction
results in the presence of passivity Among other things the old problems with
sequential functions reapp ear
Remark Active expressions are not necessary to raise the problem of
irreversibility For example we can just use the command typ e as in the
blo ck
b egin
integer x
x
P x x
if x then diverge
end
with P comm comm This blo ck is equivalent in our language to
P diverge i e has the same termination nontermination b ehavior in all
contexts In a language with I O or jumps these terms would b e inequivalent
Then irreversibility would b e exempli ed not by a pure equivalence but as a
more complex prop erty such as equivalence of termination b ehaviour under
the precondition that P do es not p erform a jump
A Category of Ob ject Spaces
In this section we will de ne the category of p ossible worlds based on the
free ob ject spaces of
De nition Let A jAj be a coherent space i e a re exive and
A
symmetric binary relation on a countable set jAj The free ob ject
A
space yA associated with A is the coherent space where jyAj jAj is the set
of nite sequences of tokens in jAj and a a b b i
1 n 1 m
yA
i f min n m g a a b b a b
1 i 1 1 i 1 i i
O Hearn and Reddy
The intuition in this de nition is that tokens in yA are sequentialized
One may think of a sequence a a as representing a series of observations
1 n
made on an ob ject The coherence relation indicates when it is consis
yA
tent to regard two traces as arising from the same computational ob ject see
Example b elow Further motivation for the de nition based on a discus
sion relating to ob jects and automata may b e found in App endix A A fuller
treatment is in An intro ductory account of coherent spaces and linear
maps can also b e found in that pap er as well as in the original sources
Letters X Y W will b e used to range over the free spaces yA We will
often consider X jyAj as a monoid with unit empty sequence and mul
X
tiplication concatenation written simply by juxtap osition x x x x will
1 2 1 n
typically denote a multiplication where each x is a sequence while a a
i 1 n
denotes a sequence of tokens a We write singleton sequences as hai when
i
necessary for disambiguation
A regular map f X Y of ob ject spaces constructs a Y ob ject from an
X ob ject by simulating the op erations of the Y ob ject on the given X ob ject
Generally f will b e given by a relation f jX j jY j with elements written
as x y Note that x and y are themselves sequences here A pair x y
signi es that the Y op eration y is simulated by carrying out the op eration x
on an X ob ject Now we think of x y x y x y as indicating
that x and y p ossess the same input information cf App endix A and we
require that the input part of y determine the input part of x i e
y y x x
Secondly the output part of x together with the input part of y must deter
mine the output part of y i e
x x y y
These are standard conditions for linear functions To these we add conditions
concerning the preservation of monoid structure
De nition A regular map f X Y is a relation f jX j jY j such
that for al l x y x y f
1 1 2 2
i x x y y and
1 2 1 2
ii y y x x
1 2 1 2
satisfying
iii f
X Y
iv x y x y f x x y y f and
1 1 2 2 1 2 1 2
v x y y x x x x x x y x y f
1 2 1 2 1 2 1 1 2 2
The condition ii can also b e written as x x y y x x
1 2 1 2 1 2
The conditions iii v in the de nition state that regular maps are state
indep endent or history free For example the condition iv means that
if x y f signifying that an action y is simulated by x then this
2 2 2 2
simulation can always b e tacked on later on top of another simulation
Even though our programming language is imp erative a form of history
freeness is appropriate in global maps b ecause these corresp ond to denotations
O Hearn and Reddy
of closed terms In a language ob eying the stack discipline state is securely
encapsulated in lo cal declarations new x C so the closed terms themselves
are e ectively stateless This viewp oint on global maps is also found in the
p ossible world mo dels
De nition The category Ob of free ob ject spaces has as objects the
spaces y A The morphisms are regular maps with relational composition
We can order the hom sets of this category using the inclusion order of rela
tions this order corresp onds to the stable order
Commands are mo delled using the space y where where is the one
token coherent space The idea is that a command corresp onds to an ob ject
with one op eration which when invoked simply runs the command We write
comm for y
Active integers are mo delled using yint where int is the discrete coherent
space of non negative integers with the equality relation as Since any
int
two integer tokens are inconsistent all the tokens have the same input part
So We write aint for y int
The opp osite of int plays an input role in this category The coherent
space int has the same tokens as int but all the tokens are considered con
sistent We regard the information of a token as purely input Intuitively an
ob ject for yint is an integer acceptor that accepts an integer and uses it
to p otentially alter its internal state We write acc for y int
Remark Given an ob ject space yA let x y denote the pre x relation
z xz y It follows from De nition that x y whenever x y In
particular x for all x An object behavior is a subset L jX j that is
pre x closed x y y L x L and pairwise consistent For example
the ob ject b ehavior for gensym is the set of initial sequences n jaintj
Thus ob ject spaces form an instance of Winskel s event structures Ob ject
b ehaviors form a dI domain under the inclusion ordering A regular map
f X Y determines a function f from ob ject b ehaviors of X to ob ject
b ehaviors of Y
f L f y jY j x L x y f g
Such a function is stable continuous and preserves consistent glb s and linear
preserves all the lubs that exist
The free ob ject space construction extends to a functor y CohL Ob
from the category of coherent spaces and linear maps to the category of ob ject
spaces We use the standard convention of writing linear maps as f A B
The morphism part of y sends a linear map f A B to
yf f a a b b a b f i n g
1 n 1 n i i
This functor is the right adjoint to the forgetful functor U Ob CohL
CohL U X A Ob X yA
If f UX A is a linear map the corresp onding regular map f X yA is
f f x x a a x a f i n g
1 n 1 n i i
O Hearn and Reddy
We often use this formula to de ne regular maps f simply by giving the
corresp onding linear maps f Note that the adjunction gives a comonad
U y on coherent spaces which we write as y or simply y
L
The category Ob has nite pro ducts Recall rst the de nition of categor
ical pro duct in the category CohL of coherent spaces and linear maps
A B jAj jB j with
A&B
a a a a
A&B A
b b b b
A&B B
a b always
A&B
The pro duct y A y A of ob ject spaces is y A A This is immediate from
1 2 1 2
the fact that y is a right adjoint The pro jections are
f i a i a a a a jA j k n g
i 1 n 1 n k i
For pairing if f yC y A then hf f i y C y A A is
i i 1 2 1 2
hf f i f x x i a i a x a f i f g k n g
1 2 1 n 1 1 n n k k i k
k
The terminal ob ject in Ob is I yemp where emp is the empty coherent
space The unique map X I is f g
X X I
To mo del storage variables we use var acc aint y int int In
tuitively an ob ject for this space has an op eration of typ e acc for setting its
value and an op eration of typ e aint for reading the value We regard the
tokens of var as strings over f put i i jint j g f get i i jint j g for
mnemonic value
The ob ject b ehavior cf Remark cel l jvarj consists of those se
quences t satisfying
t get i get i i i
t put i get i i i
t get i i
This ob ject b ehavior mo dels a declared storage variable with initial value
Example The coherence relation is meant to indicate consistency
yA
of observed b ehaviors To illustrate this consider the case A var where
we regard put i tokens as input and get i tokens as output Two sequences
a a a a and a a a a are coherent i a and a
1 n n+1 n+2 1 n n+1
n+1 n+2 n+1
are coherent The interesting case is when a a If these are output to
n+1
n+1
kens get i and get j then the sequences are incoherent b ecause i and j indicate
di erent or inconsistent output observations notice the implicit determinacy
assumption For coherence if a a then one must b e a put j token
n+1
n+1
There is no inconsistency b etween an input action put j and any other action
b ecause we do not immediately observe the internal result of the input
action Notice that there is no relationship b etween a and a
n+2
n+2
For example consider two sequences put get and put get The
sequences di er coherently in the rst p osition and so are deemed coherent
O Hearn and Reddy
n
zero I aint f h i n g
I
succ aint aint f i i i i
1 n 1 n
n i jint j g
k
pred aint aint f i i i i
1 n 1 n
n i g
k
n
skip I comm f h i n g
I
n n
seq comm comm comm f h i n g
deref var aint f get i get i i i n i jint j g
1 n 1 n
k
n
assign var aint comm f i put i i put i h i n i jint j g
1 1 n n
k
a i B cond aint yA yA yA f i B a a a
1 n i X 1 i n 1 n
n
1
n i jint j a jAj g
k k
Table Examples of regular maps
even though they are incoherent in the second p osition This is reasonable
b ecause we could certainly conceive of the following ob ject when given a
put it changes its state to when given a put it changes its state to and
when a get request is issued it simply returns the value of its internal state
So it is logically consistent to regard the two sequences as arising from the
same ob ject This is why is de ned so that sequences must b e coherent
yA
only at the rst place they di er With di erent changes of state such as
in put and put there is no inconsistency in having completely unrelated
subsequent observations
Example Some examples of regular maps are given in Table The
notation B a in the de nition of cond means a if i and a otherwise
i X
Each of these maps may b e understo o d as a simulation of the op erations of
one typ e on ob jects of another typ e For example the map seq simulates
the unique op eration of a command ob ject comm on an ob ject with two
command op erations
By virtue of isomorphism many of the maps in Example are
uniquely determined by linear maps of coherent spaces for instance
seq y f g
0
We sum up some of this discussion for the record
Lemma The category Ob has nite products The forgetful functor U
Ob CohL has a right adjoint y
O Hearn and Reddy
Remark The induced comonad y on CohL do es not satisfy the isomor
L
phism y A y B y A B characteristic of in linear logic The
L L L
reason is that y interleaves tokens from A B and the order of interleaving
L
is imp ortant
Remark The category Ob is the category of free coalgebras for y which
L
is equivalent to the Kleisli category of y The de nition of ob ject spaces in
L
is more general b ecause it uses coalgebras other than the free ones This is
needed for closure under tensor pro ducts and for the treatment of passivity
But for the example programming language considered here the free coalgebras
su ce
Finally we note an imp ortant prop erty of the space aint of active integers
it is a generator for the category Ob in the following ordered sense
Lemma For maps f g X Y in Ob
f v g e aint X e f v e g
Pro of The direction is trivial Conversely supp ose x y is a pair in f
that is not in g where x a a We want to nd a map e aint X
1 n
such that e f v e g Treat a as a function f ng jX j If i i
1 k
f ng is a string write a for a a Let e yint X b e the regular
i i
1
k
map f a f ng g To see that this is indeed a regular map
note that two strings and are consistent in yint i one of them say is a
pre x of the other In that case a a and we have a a If in
addition a a then and must b e p ermutations of each other Since
is a pre x of this means The other conditions of regular maps can
b e veri ed easily Now n y is a pair in e f but not in e g 2
This prop erty will play a key role in connecting the mo del to the programming
language with the typ e aint used to generate distinguishing contexts
Interference via Yoneda
The category Ob has a categorical pro duct for mo delling in our program
ming language But it do es not have exp onentials with a natural isomorphism
Ob X Y Z Intuitively the problem is that a regular Ob X Y Z
map f X Y Z is a simulation using an X Y ob ject i e an ob ject
with X and Y op erations on some shared state The currying transformation
would require us to separate the X and Y parts of the X Y ob ject But
they are not separable as they act on shared state
To obtain the required interpretation we emb ed this semantics together
with its treatment of rst order maps in Table into a Cartesian closed cat
egory of functors using a Yoneda emb edding Thus we interpret comm as
Ob comm and aint as Ob aint and the function typ e using the functor
category exp onent The computational intuition underlying this reinterpre
tation is the following We now regard an Algol command as a regular map
W comm where the role of W is something like that of the store parameter
in traditional denotational semantics A map W comm is the simulation
O Hearn and Reddy
of a command in a W typ ed store All Algol typ es are similarly parameter
ized by W s and this allows interference or sharing to b e accounted for by
considering meanings dep endent on the same parameter W
Domains Functors and the Yoneda Embedding
We will b e working with an enriched version of the Yoneda emb edding see
for enriched notions We use Cp o to denote the category of complete
p ointed p osets and continuous functions and Cp o for the sub category of
strict functions We refer to the ob jects simply as cp os
Supp ose C is a small Cp o enriched category This means that each hom
set C X Y comes equipp ed with a cp o structure and that comp osition is
continuous with resp ect to this structure Cp o itself has the obvious enriched
op op
structure We can then lo ok at enriched functors C Cp o where C
uses the same ordering as C In this case enriched functors are simply ordi
op
nary functors whose action on the hom sets C X Y Cp o F X FY is
continuous
De nition Given a smal l Cp o enriched category C the category M is
C
de ned as fol lows
op
Objects Cp o enriched functors F C Cp o that factor through the
inclusion functor Cp o Cp o
Morphisms Al l natural transformations of such functors
The factoring condition is from
The category M contains a copy of C For any ob ject X of C there is
C
op
a functor C X C Cp o whose action is to send
an ob ject Y to the cp o C Y X and
an arrow f Y Y to the continuous function g f g C Y X
C Y X
The functor C X lives in M and represents the ob ject X in M
C C
Lemma C X is an object of M
C
Pro of C X g f g is clearly continuous in f Thus C X is an
enriched functor Since C f X is strict in f it factors through the inclusion
Cp o Cp o 2
op
Lemma Yoneda Lemma For any functor F C Cp o Nat C X F
F X where the set of natural transformations on the left is ordered pointwise
Pro of This is a standard Yoneda Lemma argument Given C X F
id is an element of F X and this element uniquely determines by the
X X
naturality condition Moreover v implies id v id Hence we
X X X
X
have an order isomorphism 2
It follows that the functor Yon C M given by Yon X C X is a full
C
and faithful emb edding taking F C X in the Yoneda Lemma we obtain
Nat C X C X C X X It is also noteworthy that the emb edding
O Hearn and Reddy
x t u u x
X
x M s t u Y f a M f u x a
X Y
MN t u M u X id N u
X X X X
M N t t u M u N u
1 2 X X X
M t u M u
i i X i X
Table Interpretation of a typ ed lamb da calculus
Yon X Yon Y preserves all the existing pro ducts in C Yon X Y
This follows from a basic prop erty of categorical pro ducts viz C X Y
C X C Y
A second fact of interest is that we can interpret typ ed calculus and
recursion in M
C
Lemma M is Cartesian closed I It has a least xed point
C
combinator Y A A A for each functor A in M
A C
Pro ducts in M are de ned p ointwise as is usual in functor categories The
C
exp onent is de ned with the help of the Yoneda lemma On C ob jects
F G X Nat C X F G ordered p ointwise
and on morphisms when f Y X
F G f p Y g Y Y a p Y g f a
Fixed p oints are given by de ning Y X p to b e the least xed p oint of
A
a p X id a A X A X
X
Y satis es typical uniformity criteria for xed p oints such as dinaturality
See e g for the uniformity criterion
To indicate how the interpretation go es we show in Table the semantics
of a typ ed lamb da calculus in M The meaning of a term in typing context
C
M t is a natural family in X of continuous functions X
t X We also use the abbreviated notation M when the typing judgement
is clear from context If is the typing context x s x s then we
1 1 n n
regard X as the cp o of environments fx a x a g where
1 1 n n
each a s X
i i
Remark The role of the functor C X in F G X is just as in
standard functor category semantics except that its order structure is also
taken into account This will allow certain of these hom functors to play a
double role used for quanti cation over contexts and for interpreting base
typ es in the programming language See Lemma for where this is used
Remark Oles used the strictness condition on functors in order to obtain
Cartesian closure With hindsight we can see this condition arising in another
op
C
way The category Cp o is symmetric monoidal closed and there is an
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op op
C C
endofunctor Cp o Cp o obtained by comp osing on the
right with lifting that has a comonad structure M is equivalent to the
C
Kleisli category of Thus Oles s strictness condition arises naturally if we
take Cp o together with the lifting comonad as fundamental and lo ok for a
mo del of intuitionistic linear logic based on functors into Cp o rather than
lo oking directly for a mo del of intuitionistic logic cf
Semantic Model
The semantics is given in M For the typ es de ne
Ob
comm Ob comm
aint Ob aint
s t s t
s t s t
The de ned typ e var gets the interpretation
var aint comm aint
Variables of this kind Algol variables can b e more complicated than variable
ob jects in store contexts but note that the latter can b e easily turned into Al
gol variables Sp eci cally there is a natural injection Ob var var
de ned by X v a r where
a Y f e hf v ei assign
r v deref
The data assign and deref may b e found in Table
The interpretations of rst order constants are obtained from the maps
in Table by the Yoneda emb edding For instance ifz is interpreted by the
comp osite map iso Ob cond where iso is the appropriate isomorphism
b
iso
-
Ob aint Ob b Ob b Ob aint b b
All that is left is to interpret new and letval
To interpret letval we de ne a map letval b b b b which is
determined uniquely by the following prop erty x hai letval X p q i
x x jX j x x x
1 2 1 2
n jbj x hni p x hai q X id k
1 2 X n
By fo cusing on a single output token hai we are essentially using the Kleisli
representation of regular maps The idea is that we evaluate the argument p
consuming x from the state context and then we consume x while pro ducing
1 2
a k Ob X aint is the evident constantly n active integer the unique
n
map containing e n in the case that b aint and it is the constantly
X
command skip if b comm Sending k as an argument to q shows how
n
further evaluations of this argument always yield the same integer or command
action
O Hearn and Reddy
For the semantics of new we need a map
new var comm comm
For every pro cedure p var comm X new X p must b e a regular
map X comm There are two main parts to obtaining such a amap new X
must call p in an enlarged store typ e X where there is an additional variable
v var X This gives a command p X comm Second new X must
convert p to a command X comm by supplying it with an appropriate
enlarged store of typ e X
The rst part is done by taking the space X X var and calling p with
the variable obtained from the second pro jection X var var
2
p p X var X var X var comm
1 2
where Ob var var is the emb edding de ned earlier For the second
part of converting p to X comm de ne new X p as the unique regular map
including the following pairs
f x x h i s s cel l x s x s x h i p g
0 k 1 k 0 1 1 k k
Again this is the Kleisli representation which by the adjunction U a y
determines the map completely We are using the monoid multiplication jux
tap osition here so for example some of the sequences x may well b e empty
i
The idea of this de nition is that the uses s of the lo cal variable are simply
i
ignored at the non lo cal level Note that while we can convert commands
X var comm to X comm we do not have a corresp onding regular map
X X var Indeed since regular maps are history free they cannot create
new ob jects
This completes the de nition of the mo del
Example Consider the application map app aint aint aint
aint On the level of functor categories the de nition is app X p a
p X id a On the level of ob ject spaces the e ect is as follows Applying
X
the Yoneda lemma a numb er of times we nd that this application map
determines a continuous function
aint aint aint X aint X
Ob X aint aint Ob X aint Ob X aint
The induced function takes a pair p f of maps and pro duces a regular map
X aint
hid f i
p
- -
X X aint aint
This comp osite is the unique regular map containing the following pairs
x y x y x hai whenever there is k k jaintj such that
0 1 1 n n 1 n
y hk i f x k x k x hai p
i i 0 1 1 n n
This is the form of sharing or interference that we obtain by placing ob jects
into the same context the common context here b eing X The x and y in
i j
x y x y x represent interleaved uses of X by p and f Thus the Yoneda
0 1 1 n n
emb edding leads not only to a treatment of function typ es that is technically
O Hearn and Reddy
correct but an implementation of sharing that is intuitively reasonable and
which has proven di cult to come by otherwise
op
Remark It is p erhaps surprising that a category C can b e used as
the category of worlds where C is a category of functions In previous work
the categories of worlds typically involved morphisms that were
more than even opp osites of functions they were pairs of functions one
for de allo cation of storage variables and one for overwriting small pieces
of large states In an explicit state setup when mo delling commands as
state to state functions b oth the co and contravariant roles of state need
to b e accounted for in order to get a functor of command meanings The
op
completely contravariant account given here via C using only de allo cations
Weakenings to interpret declarations is p ossible b ecause of the demand
driven nature of the treatment of commands in
First order de nability
We know that the spaces Nat s t of natural transformations are cp os but
to study de nability in the mo del we need more information on their structure
In this section we use the Yoneda lemma to calculate the structure precisely
by showing that for base typ es s and t these cp os are algebraic In fact we
show much more each of these cp os is isomorphic to the set of p oints of a
coherent space
Given this characterization we move on to show that every nite element
in these domains is de nable by a closed term in the programming language
By algebraicity every element is then the lub of de nable ones Standard
ccc manipulations allow us to obtain an analogous result for all global el
ements I t where t is an arbitrary rst order typ e The order of a
typ e is de ned inductively order aint order comm order s t
max order s order t and order s t max order s order t
Lemma Suppose b b b are base types Then
1 n
Nat b b b
1 n
with pointwise order is isomorphic to a coherent space
In the statement of the lemma and throughout we confuse a coherent space
with the cp o of its p oints ordered by inclusion
Pro of Let A and B b e the coherence spaces used in the interpretations of
i
b and b int in the case of aint in the case of comm Let us calculate
i
Nat b b b
1 n
Nat Ob yA Ob yA Ob yB de nition
1 n
Nat Ob yA yA Ob yB Yoneda preserves
1 n
Ob yA yA yB Yoneda lemma
1 n
Ob y A A yB de nition of
1 n
CohL y A A B U a y y U y
L 1 n L
O Hearn and Reddy
2
Prop osition Given base types b b and b any nite element in
1 n
Nat b b b is de nable by a term in context
1 n
x b x b Q b
1 1 n n
Pro of We use the representation in terms of y A A B and con
1 n
sider tokens of A A as of the form i a for i n the i indicating
1 n
the comp onent Let f b e a nite linear map De ne the size of f to b e the
numb er of tokens of A A and B in its trace The pro of is by induction
1 n
on the size of f There are three cases
f Then Q some divergent term of typ e b
a f Coherence of f implies that f f ag If b is comm
a
let Q skip If b is aint then let Q succ
i a s b f Coherence of f means that if i a s b f then
i i Supp ose that b is aint Let z b e the collection of those a where
i
i a s b f for some s b Since f is nite z is nite For each a z
0 0
is de nable by a fs b i a s b xg By induction f let f
a a
0
Let k k b e an enumeration of z Note that z is not empty term M
1 n a
Then f is de nable by the following term using evident notation for if where
x aint is the identi er corresp onding to b
i i
letval x m aint
i
if m k then M
1 k
1
else if m k then M
2 k
2
else if m k then M
n k
n
else
If b is comm the pro of is simpler 2
i
Notice that there is a form of sequentiality at work in case of the pro of
Coherence of a nite element f means that if i a s b f and i a s
b f then i i This corresp onds to the intuition that the i th comp onent
is queried rst by f which is why we are accounting prop erly for sequential
facilities at rst order The active nature of the arguments is crucial here as
this kind of account of sequentiality do esn t adapt to PCF style computation
Corollary For any rst order type t Nat I t is isomorphic to a coher
ent space each of whose nite elements is de nable by a closed term Q t
Pro of From the prop osition using standard syntactic versions of ccc ma
nipulations involving currying pairing and the cartesian isomorphism A
A B A C 2 B C
Example Closed terms of typ e comm comm are interpreted as ele
ments of Nat I comm comm Let us calculate this hom set using the
O Hearn and Reddy
argument in the pro of of lemma
Nat I comm comm
Nat comm comm enriched ccc isomorphism
Nat Ob y Ob yB de nition
Ob y y Yoneda lemma
CohL y U a y y U y
L L
N calculation
where N is the at natural numb ers see Each n N corresp onds
n 0
to a Church numeral c c of typ e comm comm where c skip and
i+1 i
c c c The least element this typ e is the divergent command Y c c
Thus every element in the hom set Nat I comm comm is de nable
This representation of Nat I comm comm should b e compared
op
to where the corresp onding representation yields N Vnat with
op
Vnat the upside down vertical natural numb ers and the smash pro duct
op
The Vnat comp onent has entirely to do with snapback op erations which in
this case lead to a more complex domain
A Full Abstraction Result
In reasoning ab out second order terms we need to consider the denotations
of rst order typ es at various p ossible worlds and not only global elements
I t for rst order t Syntactically this corresp onds to the fact that the
context lemma do es not hold in our example language one needs more
than closed applicative contexts to distinguish closed terms of functional typ e
Semantically it corresp onds to the fact that the category is not well p ointed
to distinguish parallel maps f g s t it is not enough to comp ose on the
left with maps I s out of the terminal ob ject So the de nability result of
the previous section do es not immediately give us full abstraction for closed
terms of second order
To get full abstraction at second order we rst show that for the ap
propriate typ es di erent natural transformations can b e distinguished at the
p ossible world aint This then enables us to use the programming language
typ e aint together with new to build distinguishing contexts It su ces to
consider applicative contexts with a single free identi er of typ e aint wrapp ed
in the scop e of a new variable declaration used to generate an active integer
to bind to this free identi er
Lemma v s t t aint v aint for order types
t s t
Pro of The direction is trivial Supp ose v Then for some X
0
using a representation of X v X Ob X A A Ob X A
s t t
the typ es calculated as in Lemma Consider f Ob X A A where
s t
X f v X f By Lemma there exists a map e aint X such that
O Hearn and Reddy
e X f v e X f Naturality of and with resp ect to e then implies that
aint e id f v aint e id f and we are done 2
Next we want a de nability result ab out rst order typ es instantiated at
world aint
Lemma For order types s and t s t X is isomorphic to a coherent
space Further each nite element of s t aint is de nable in an evident
sense by a term in context y aint M s t
The term M determines an element of s t aint using the isomorphism
Nat aint s t s t aint
This is where Ob aint is playing a double role used in the de nition of
s t aint and as the interpretation of aint in Nat aint s t
Pro of We can calculate the domain explicitly using a Yoneda lemma argu
ment again as in Lemma
s t X
Nat Ob X s t de nition
Ob X yA yA as b efore
CohL U X yA A by
This gives the rst part of the lemma
In the case that X aint yint we use the de nition of pro duct in Ob
to obtain the representation
CohL y int A A
L
Once again A and A are the coherent spaces used in the interpretation of s
and t p ossibly applying pro duct preservation of y
To de ne the nite elements of this domain recall that we have seen that it
is isomorphic as a cp o to the space of natural transformations Nat aint s
t and we have already shown that these nite elements are de nable by
terms in context
y aint x s Q t
The desired term in context
y aint Q s t
de nes the corresp onding nite element of s t aint 2
Theorem Inequational Full Abstraction
If M t t t t and N t t t t are closed
1 n 1 n
terms of second order type then
M v N C C M v C N
Here C ranges over ground contexts
Pro of Only the direction needs to b e proven Supp ose M v N We
will construct a command typ ed context C where C M v C N
O Hearn and Reddy
Since M and N are closed terms they determine natural transformations
t t t t Using Lemma Lemma algebraicity of
1 n
t t t aint and continuity we may calculate
1 n
M v N
M aint v N aint
nite d t t t aint M aint d v N aint d
1 n
By Lemma there is y aint Q t t t that de nes d
1 n
We do the rest of the pro of for the case of t aint comm b eing easier
Given such a d we know that the trace sets of M aint d and N aint d in
comm aint CohL y int int are such that M aint d N aint d
say s m M aint d and s m N aint d We construct a term in
context x var c aint as follows If s k k then c is the term
1 n
x x if x then k
1
else if x then k
2
else if x n then k
n
else
Here recall that we are using a sequencing combinator C E as sugar for
letval C z E where z is not free in C or E If s then c is it
do esn t matter what c is in this case
With this c and Q a context distinguishing M and N is
new x x
if y aint Q c m then skip else
where we test for equality explicitly in if which can b e programmed using ifz
and other constants 2
We have formulated the full abstraction result for second order functions
that take a single rst order function as an argument It should b e clear from
the form of the pro of that the argument works for all second order typ es We
don t know if the result can b e extended to higher typ es
Example We illustrate the semantics for the example from the Intro
duction First we have a regular map gensym var aint that builds the
b ehavior of gensym by simulating its output in terms of var typ ed actions
gensym is given by
f get i put i get j get i put i get j j j
1 1 1 n n n 1 n
i j jint j g
k k
As mentioned in Remark the regular map determines a function from
ob ject b ehaviors of var to those of aint In particular when applied to the
b ehavior cel l jvarj the function gives an ob ject b ehavior f h ni n
jintj g of typ e aint This corresp onds to how gensym is de ned in terms of a
declared variable
O Hearn and Reddy
The meaning of the blo ck
b egin
integer x
integer pro cedure gensym f x x return x g
x
P gensym
if gensym then diverge
end
is a natural transformation of typ e aint comm comm Its
action at a p ossible world W is a continuous function
Nat Ob W aint comm Ob W comm
which using a Yoneda lemma calculation reduces to a continuous function
f Ob W aint comm Ob W comm The action of f on a regular map
p W aint comm may b e calculated as the following map
f w w t cel l t jvarj i jint j s s jaintj
1 n 1 n
t hput it hget i put i get i i i
t s s gensym
1 n
w s w s p g
1 1 n n
For clarity we have shown a linear map of typ e W The corresp onding
regular map W comm is obtained by iterating this b ehavior The se
quence t denotes the op erations p erformed on the variable x Given that the
nal value of x must b e no greater than the sequences t and s s can
1 n
only b e empty Thus the linear map is equal to
f w w p g
It is clear that the meaning of P diverge maps p to precisely the same regular
map
We must admit that the reasoning in this example is rather technical
Nevertheless it illustrates an interesting feature of the ob ject based semantics
After applying a Yoneda lemma argument we see that the denotation of
pro cedure P is a regular function W aint comm with W corresp onding
to the context of evaluation and aint to the argument The semantics in this
case works by communication b etween the pro cedure P and the lo cal blo ck
Where P exp ects an argument of typ e aint the blo ck simulates the argument
in terms of the var typ ed b ehavior cel l The interesting p oint is that the
domain W aint for P do es not mention the space var corresp onding to lo cal
variable x at all or for that matter any other typ e that may b e used in a
simulation of the aint argument This corresp onds to the intuition that any
meaning for pro cedure P is de ned without reference to the lo cal variable
O Hearn and Reddy
Related Work
Although there has b een a go o d deal of theoretical work on the foundations of
ob ject oriented programming most of it has concentrated on typing issues in
a purely functional context see for example and so b ears little relation
to our work For us the initial conception of ob ject involves at least a hidden
lo cal state together with op erations acting up on it
Much closer to our concerns is work on translating ob jects into pro cess
calculi e g In this approach an ob ject is treated as a pro cess of
a certain form with the state implicit in the history of events this asp ect
is clearly related in pre theoretic conception to the approach of But
the results and details are di cult to compare Here the fo cus has b een on
denotational metho ds and examining the connection full abstraction with
an example programming language In comparison the pro cess approach can
b e thought of as b eing broader handling more features but as far as we are
aware no analysis indicating the accuracy of the resultant enco dings has yet
b een given
Closer still to our concerns is a variety of applications and extensions of
functor category semantics One of these is the work of Pitts and Stark on
dynamic allo cation where a language is considered in which mere equality
of names is the basic op eration b esides lo cal allo cation they obtain a full
abstraction result for rst order typ es Equality of names or lo cations do es
not t so easily into the ob ject based mo dels which follow Reynolds s lead
in taking a lo cation free view of state It do es not app ear that the
phenomenon of irreversibility arises in this very bare setting of lo cal names
but neither is it certain that actual storable values are necessary for mild
cases of irreversibility to arise For instance something similar app ears to b e
present in a simple form in the language SPCF of though we are unsure
of the exact relationship
Sieb er has built a mo del for an Algol like language in which functors are
equipp ed with logical relations that are used to constrain function typ es
and has obtained a full abstraction result for the closed terms of second order
The pro of is subtle and original making use of nitely determined natural
transformations it is not obvious whether the cp o s in Sieb er s semantics are
even algebraic The pro of given here is much less sophisticated using the
usual metho d of de nability of nite elements
There are imp ortant di erences b etween our language and the one in
First and foremost is that Sieb er s results are for a language with a snapback
combinator so in comparison to the work rep orted here we may say that his
mo del accounts for lo cality to a go o d degree but not for irreversibility An
other di erence is that Sieb er s language has a form of side e ect free integer
expression whereas we have used active integers Our mo del can easily b e
extended to deal with passive integers but in that case we have not obtained
a full abstraction result the old problems with sequential functions crop up
again
But we should emphasize that though it do es not have passive integers
O Hearn and Reddy
the language considered here is sequential it is one where the order of evalua
tion of at least base typ e arguments can b e recorded using storage variables
cf It would seem to make sense to try to push this explanation of
active sequentiality utilizing coherent spaces and the stable order on reg
ular functions as far as p ossible b efore abandoning coherent spaces And of
course full abstraction is not the ultimate aim of the semantics though in the
course of proving the result we did nd legitimate structure asso ciated with
imp erative typ es this structure is of more interest than the result itself which
is only a technical indicator
But it is interesting that b oth Sieb er s results as well as ours stop at full
abstraction for second order typ es The reason is essentially that we are inter
preting Algol typ es as parameterized typ es functors and our present under
standing of parameterized typ es at higher orders is sketchy at b est While we
were able to show that all rst order Algol typ es hace he structure of coherent
spaces we do not even know if the cp o s arising at second order are algebraic
A more thorough understanding of parameterized typ es at higher orders is
needed to make further progress on this front
The parametricity mo dels based on PERs and logical relations presented
in do not account for irreversibility either However we may under
stand the main message of that work as applying more broadly than to the
sp eci c mo dels The prop osal there was that the abstractness of lo cal state
could b e understo o d in terms of Strachey s concept of parametric uniform
p olymorphism This leads to quite a convincing explanation of lo cal
ity Furthermore it has recently b ecome clear that a slight variation on the
parametricity semantics based on a strict function mo del of linear even rel
evant p olymorphism rules out the snapback and other unwanted op erations
There should b e close connections b etween the parametric and ob ject based
semantics
We exp ect that some readers will feel with us that the mo del here works
in a slightly mysterious fashion without providing an explanation of lo cality
and irreversibility The metho ds of building up computational entities in the
mo del do not mention any conditions related to these prop erties The prop er
ties to the extent we know what they are arise as a consequence of the way
ob jects are constructed It may simply b e that an axiomatic approach to these
issues fo cusing more on properties characteristic of lo cality and irreversibility
is b est carried out within the context of an explicit state semantics though
this is by no means certain In any event we have shown that the mo del is
quite accurate and so we exp ect that such an explanation should also b e
consistent with the ob ject based semantics
Ultimately we do not b elieve that there should b e a con ict b etween the
explicit state view as exempli ed by the the parametricity mo dels and the
view of state as implicit in histories of events Very often it is most e cient
to conceive of ob jects as computational entities with pieces of state and op
erations though at other times it can b e more e cient to work directly in
terms of traces or similar representations For instance here we were able to
calculate the domain theoretic structure of typ es with great ease while the
O Hearn and Reddy
principles explicitly adopted in the parametricity mo dels often but not al
ways lead to smo other reasoning ab out sp eci c examples Ideally one would
hop e to have precise means of linking these two forms of description enabling
passage back and forth b etween one and the other These connections await
further development
Acknowledgement
We are grateful to Bob Tennent for comments and discussions
App endix
A Ob jects Coherent Spaces and Automata
We think of an ob ject as a computational entity with a mutable internal store
and a collection of observable op erations that can read and alter the store In
this resp ect ob jects are much like automata Elementary notions of ob jects
can receive some illumination by comparing to concepts of automata theory
Let M M e b e a monoid i e a set with an asso ciative op eration
M
and a unit e for this op eration We often write a pro duct x y as simply
M
xy An automaton for M is a pair Q Q M Q where Q is a
set of states and is a partial function the transition function satisfying
q e q
M
A
q xy q x y
These identities are understo o d in the context of partial functions if either
side is de ned the other side is de ned and equal Such automata are also
called monoid actions or M sets with partial functions
Often one takes M to b e a free monoid the set of strings over an
alphab et In this case is uniquely determined by giving a one step
transition function Q Q If is a one element set f g then
0 0
reduces to a function Q Q An automaton of this form can b e regarded as
a command ob ject an ob ject with a single op eration that alters the store
Another example is an active integer that returns the rst time it is used
the second and continues doubling its value thereafter As an automaton one
representation is obtained by taking Q f g f g and
n
setting then n n This active integer illustrates the irreversibility
0
of state changes mentioned earlier we never return to the state after an
initial use of the ob ject assuming as the initial state The representation
of active integers as automata is not entirely satisfactory in that we would like
to think of the integer as an output of the automaton but nothing in the
de nition suggests this This is remedied b elow in the de nition of ob jects
with reference to coherent spaces
If Q M Q is an automaton its behavior at a state q Q is de ned
O Hearn and Reddy
as
L q f x M q x is de ned g
L q is left closed i e xy L q x L q By the second identity
of A if q xy is de ned q x must b e de ned Conversely given any
left closed subset X M we can recover an automaton from it though not
uniquely A canonical choice is to take Q X and de ne X M X by
x y xy
Then L e X This is the initial automaton with b ehavior X The nal
M
automaton is obtained by identifying all the right congruent elements in X
These two automata sandwich all the other automata with b ehavior X at
designated start states
This discussion illustrates how we might regard automata as intensions and
their b ehaviors as extensions We can obtain technical economy by identifying
automata with their b ehaviors
Objects de nable in Algol are similar to automata but with one di erence
The op erations of an ob ject have b oth input and output information This
is in contrast to the instructions of an automaton the elements of M which
are to b e regarded as having only input information The input and output
parts of an ob ject op eration can b e causally interlinked in a complex fashion
So stream lined constructions like Mealy machines will not do
We use coherent spaces to treat the complex input output breakdown of
the ob ject op erations We equip a monoid with a consistency relation that
we conceptualize in intuitive terms as follows For elements x y M we say
that x and y are consistent and write x y if x and y have di ering input
information or have the same output information The complement relation
x y x y signi es the opp osite while the inconsistency relation
x y x y x y signi es that x and y have the same input
information Supp ose Q M Q is the transition function of an ob ject
Whenever q x is de ned we exp ect that the output part of x as well the
nal state q x is uniquely determined by q and the input part of x In
other words
q x and q y are b oth de ned x y A
For example for an active integer ob ject we de ne that two distinct integers
are always inconsistent This ensures that q i is de ned for at most one i
which is then regarded as the output of the ob ject in the state q Supp ose
x x x and y y y in A ab ove Condition A shows that q x
1 2 1 2 1
and q x are b oth de ned So we exp ect x y Secondly if x y then
2 1 1 1 1
q x x and q x y are b oth de ned So we exp ect y y
1 2 1 2 1 2
This motivates the basic de nition of an ob ject space
De nition A An ob ject space is a pair X jX j where jX j
X
jX j e is a monoid and is a re exive symmetric binary relation on
X
X
jX j such that
x x y y x y x y x y
1 2 1 2 1 1 1 1 2 2
X X X
Then yA creates the free ob ject space asso ciated with a coherent space A
O Hearn and Reddy
Finally we can regard an object for an ob ject space X as a pair Q
Q jX j Q satisfying the condition A The b ehavior L q for any
state q Q is a left closed pairwise consistent set
Regular maps f A B determine functions from A ob jects Q
to B ob jects Q The transition map Q jB j Q is given by
q y q x x y f q x q This is the formaliza
tion of simulation mentioned in Section Conversely all functions from
A ob jects to B ob jects that are uniform in state sets Q in an appropriate
sense arise from regular maps in this fashion
While ob jects as considered here su ce for the treatment of Algol like
languages one would want additional structure to treat other features of
ob ject oriented languages such as references comparison op erations and the
notion of self
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