Electronic Notes in Theoretical Computer Science
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Typ e Inference for Recursively Constrained
Typ es and its Application to OOP
Jonathan Eifrig Scott Smith Valery Trifonov
Department of Computer Science
The Johns Hopkins University
Baltimore Maryland
feifrig scott trifonovgcsjhuedu
Abstract
We de ne a p owerful typ e inference mechanism with application to ob jectoriented
programming The typ es inferred are recursively constrained typ es typ es that come
with a system of constraints These typ es may b e viewed as generalizations of recur
sivetyp es and Fb ounded p olymorphic typ es the forms of typ e that are necessary
to prop erly enco de ob ject typings The base language we study ISo op incorp o
rates state and records the two features critical to enco de ob jects in a nonob ject
oriented language Soundness and completeness of the typ e inference algorithm are
established by op erational means Our metho d for establishing these prop erties is
somewhat novel We illustrate how the algorithm may b e fruitfully applied to infer
typ es of ob jectoriented programs
Intro duction
This pap er addresses the problem of designing an ob ject oriented program
ming language with an eectivetyp e inference mechanism Recently devel
op ed programming languages including Standard ML and Haskell incorp orate
typ e inference as a core comp onent of the language However typ e infer
ence has yet to achieve practical application to ob ject oriented programming
languages
We strongly feel the core typ e features necessary to mo del ob ject oriented
programming with typ e inference include a notion of subtyping and a no
tion of recursively constrained p olymorphism a generalization of F b ounded
p olymorphism
Recursively constrained typ es are typ es of the form n C with n
reading where C is a set of typ e constraints of the form p ossibly
containing free typ e variables These constraints maybe recursive in that a
Partially supp orted by NSF grants CCR and CCR
Partially supp orted byAFOSR grant F
c
Elsevier Science B V
Eifrig Smith and Trifonov
variable t could o ccur free in b oth and The recursive constraint set
ft Nat t t t Natg expresses t t Nat so recursively constrained
typ es subsume recursivetyp es We will use rctype to abbreviate recursively
constrained typ e
Polymorphic rc typ es are typ es t t n C where constraints
n
in C maycontain typ e variables t t free Polymorphic rc typ es generalize
n
the more well known b ounded typ es t in several ways First
they are recursive so t could o ccur free in this is not allowed in b ounded
typ es Typ es with t o ccurring free in are the so called F b ounded typ es
Polymorphic rc typ es generalize F b ounded typ es byallowing more than
one upp er b ound on a typ e variable as well as allowing multiple lower b ound
constraints t This generalized form of p olymorphic typeisvery useful
in typing ob ject oriented programs that are otherwise untypable irresp ective
of the question of typ e inference An example of such a program is given in
Section b elow
It is not dicult to see how rc p olymorphism is useful in typing classes
and ob jects for it is at least as useful as F b ounded p olymorphism Classes
mayhave so called binary metho ds that refer to the typ e of ob jects of their
own class for instance an ob ject with an equal metho d takes as parameter
another ob ject of its own typ e Thus a self typ e is needed And this self
typ e needs to b e op en ended since a class may b e extended wewishthetyp e
of self to b e an ob ject with all the metho ds currently dened and p ossibly
additional ones Polymorphic rc typ es capture this notion by constraining
the p olymorphic self typ e t to include the current metho ds for instance
t nft equal t Bo ol g
Binary metho ds haveproven very dicult to typ e in a general way it has
even b een suggested that they b e disallowed
One way to understand the usefulness of lower b ounds t in rc typ es
are as generalizations of recursivetyp es It is p ossible to write an rc typ e
t nf t g where lower b ound diers from upp er b ound
it is a recursivetyp e if These generalized forms are useful as
intermediate results pro duced during the typ e inference pro cess as partial
forms of recursivetyp es During the typ e inference pro cess constraints are
accumulated on typ es in a b ottom up fashion and so typ es at the leaves
of typing pro ofs have small constraint sets and have fat constraint sets at
the ro ot The lower b ound constrains the output of the typ e what
prop erties ob jects of typ e must have if an ob ject of typ e is used i e
passed to a function of typ e an additional upp er b ound constraint
t will b e placed on the typ e bythetyp e inference mechanism and
this could only b e contradictory if which follows by transitivitywas
contradictory The upp er b ound is the dual of this constraining the input
of the typ e what functions of typ e t must do
The presence of multiple upp er b ound constraints or multiple lower b ound
constraints can b e understo o d as a restricted form of union and intersection
typ e f t tg would b e equivalentto f tg if there were union
Eifrig Smith and Trifonov
typ es in the language a dual relationship exists b etween intersections
and upp er b ounds We b elieve general union and intersection typ es cause to o
many problems to b e worthwhile but this implicit restricted form is quite
natural
In this pap er wedevelop a typ e inference algorithm for the I So op lan
guage Inference Semantics of OOP I So op is not an ob ject oriented lan
guage however it has an expressive enough typ e system so that typ ed OOP
may b e eectively enco ded within I So op We take a translational approach
b ecause we nd the factoring to help clarify ambiguities however there is also
merit in studying languages where ob jects themselves are primitive and
the concepts herein should eventually b e recast as primitiveobjecttypings
I So op s typ e system contains b oth subtyping and p olymorphic rc typ es We
infer shallow p olymorphic rc typ es at let expressions as in the HindleyMilner
algorithm In addition the underlying language includes records and a
notion of state for with these features it is p ossible to obtain an eective
enco ding of ob ject oriented programming Records are needed so record sub
typing can b e used to mo del ob ject subtyping Without state the critical
state holding prop erty of ob jects is lost
Our approach to establishing the soundness of constrained typ e inference
diers from other work in the literature In other approaches eg
a metho d is given that either pro duces a satisfying assignmenttothe
constraints and thus establishes their consistency or establishes that no such
solution exists and the constraints are thus inconsistent In our approach an
rc typ es constraint system is considered consistent if it do es not contain
anyobvious contradictions suchas Nat Bo ol Weshow this view is sound
without ever showing the consistent constraint systems have solutions In
stead we directly establish a sub ject reduction prop ertyover a pro of of typing
with consistent rc typ es at each no de We b elieve the standard
metho d of nding solutions to the constraint sets can b e overly restrictive
for it forces one to havearich enough typ e language or typ e mo del that can
express the solutions as typ es or sets In our language for instance we exp ect
general union and intersection typ es would b e required to express the solution
of constraints as typ es but we do not wish to pay the p enaltyofhaving these
typ es in our language
Wealsotake a more primitive approach to establishing the completeness of
typ e inference ie that all typable programs will successfully have some typ e
inferred by the typ e inference algorithm We rst dene a restricted set of
typing rules the inference rules for whichtyping derivations are deterministic
Then these rules are shown equivalent in strength to the general form of rules
without recourse to a principal typ es prop erty
Related Work
Anumber of typ e inference systems have b een develop ed that b ear on the typ e
inference problem for OOPPap ers of Reynolds Cardelli and Mitchell
are foundational pap ers in the eld that develop the basic concepts of
Eifrig Smith and Trifonov
constraints and subtyping Many pap ers have b een written since we fo cus on
the more recentwork the most relevanttoours
Kaes develops a typ e inference algorithm for a language containing
p olymorphic and recursivetyp es and typ e constraints This work incorp orates
subtyping constraints recursivetyp es and p olymorphism Kaes writes so
called constrained typ es jC in close analogy to our rc typ es n C This
approach cannot solve general recursive constraints t generates a non
terminating unication problem in his system if t o ccurs free in while our
approach can handle such constraints without diculty He do es allowa
xing of such a constraintby replacing it with a recursivetyp e t butat
the cost of an imp ortant loss of generalityKaestakes the standard approach
to constraint consistencyby pro ducing a solution to the constraints He also
intends to mo del overloading not record subtyping his system has no record
typ es Sekiguchi and Yonezawa take an approach similar to Kaes but
interpret as subtyping on record typ es making it more directly applicable
to ob ject oriented programming
Palsb erg Schwartzbach et al have written a numb er of pap ers concerning
typ e inference for ob jects The main feature of their work is they
do not take the HindleyMilner approachtotyp e inference Instead their
inference algorithm uses ow analysis to generate a set of constraints ab out a
program and then applies another algorithm to come up with a solution to
these constraints if it exists Their work represents the current state of the art
in having a practical typ e inference algorithm for ob ject oriented programming
languages Other advantages of their approach include asymptotically ecient
inference algorithms and named class typ es Their system however has no
p olymorphism and they take a co de expansion view of inheritance requiring
re typ e checking with each class extension This lack of p olymorphism has
b een partially addressed byPlevyak and Chien
Our work is closest to that of Aiken and Wimmers They develop a typ e
system with subtyping union and intersection typ es and a form of p olymor
phic typ e similar to p olymorphic rc typ es They prove soundness using the
ideal mo del As with the previously mentioned researchers they havean
algorithm that pro duces a satisfying assignment to the top level constraints to
establish consistency of a constraint set The satisfying assignment they pro
duce is an ideal in the ideal mo del Wehave no union intersection or negation
typ es These typ es prove problematic in their system and they are in fact
unnecessary for typ e inference if they are not used in the typ es of atomic
constructs they are not generated by the inference algorithm provided multi
ple upp er and lower b ounds to the same variable are allowed as we do Aiken
and Wimmers have not addressed the problem of using their system for typing
ob ject oriented programs their language lacks imp ortant features necessary
for the enco ding of ob jects In particular their language is a functional lan
guage without records The ideal mo del cannot mo del languages with state
so their approachwould not extend to a language with state Aiken has imple
mented the typ e inference algorithm and this implemented system has an
optimized inference algorithm and an implementation of extensible records
Eifrig Smith and Trifonov
Var x
Num n j j j
Val v x j n j x e j l v j l v j l v
Exp e v j eej let x e in e j l e j el j l e j l e
Fig Syntax of the ISo op language
Enco ding ob ject oriented features within a more basic language is one p os
sible approachtohow ob ject oriented programming should b e done We
could take a similar approachby programming in an ob ject oriented style via
the enco ding of ob jects in I So op that wegive in Section R emygives a
collection of extensions to ML that allow OOP to b e enco ded R emyisthe
only author amongst of those previously discussed who has a pro of of sound
ness of his system in the presence of reference cells His enco ding is missing a
notion of subtyping and thus lacks the core feature of ob ject lifting allowing
sub class ob jects to b e implicitly co erced to b e sup erclass ob jects Instead co
ercion functions must b e explicitly supplied R emys enco ding is more ecient
than the enco ding we use each ob ject creation in our enco ding entails forming
closures for each metho d of the ob ject If our language weretobeusedas
a primitive OOP language some more ecient ob ject representations would
need to b e develop ed R emys system also has a notion of extensible record
whichwe exp ect will b e useful for enco ding delegation style ob ject oriented
programming
Outline
In Section we present I So op and its op erational semantics Section
presents the I So op typ e system sketches of the pro ofs of sub ject reduction
and typ e inference app ear in Section Then to showhow OOP can b e faith
fully enco ded an extended example is worked in Section This example also
serves to illustrate the p ower of the typ e inference system Wedraw some
nal conclusions in Section
The ISo op Language
We b egin by dening the I So op language which is roughly call by value PCF
with records variants reference cells and let expressions see Figure
l v is shorthand for l v l v for some The vector notation
k k
k l v is shorthand for the same and indicates that i will range over the
i i
elements of the vector The set B fsucc pred is zero ref setg Var
contains the names of built in primitive functions on numb ers and reference
cells Variants are dual to records the injection l e tags the value of e with
l e similar to the Standard ML fn construct can lab el l and the match
b e applied to a tagged value to extract it The b o oleans and conditional are
Eifrig Smith and Trifonov
derived from variants true and false are dened as true and false
resp ectively and if e then e else e stands for true e false e e
weuse e to denote x e for some x not free in e
A store ranged over by s is a nite mapping from variables to values
A conguration hs ei is a pair of a store and an expression Computation
is dened via a single step relation between congurations A reduction
context R is an expression with a hole in it into which one mayputa
sub expression via Re Reduction contexts serve to isolate the next step of
computation to b e p erformed it is always in the hole
De nition A reduction context is dened inductively in Figure
R jRej vRj let x R in e
j l v l v l R l e l e j Rl
i i i i i k k
j l v l v l R l e l e j l R
i i i i i k k
Fig ISo op reduction contexts
De nition is the least relation on congurations satisfying the con
ditions shown in Figure where
ee x is the capture free substitution of e for x in e
x v is the map dened only on x with result v
f jjg is the functional extension of f by g
hs Rx e v i hs Revxi
hs Rlet x v in ei hs Revxi
hs Rsucc ni hs Rn i if n n
hs Rpred ni hs Rn i if n n
hs Ris zero i hs Rtruei
hs Ris zero ni hs Rfalsei if n
hs R l v l i hs Rv i
hs R l v l v i hs Rvvi
hs Rref v i hsjj x v Rai x Dom s B
hs Rxi hs Rsxi x Dom s
hs Rset cell x val v i hsjj x v Rv i x Dom s
Fig The singlestep computation relation
Eifrig Smith and Trifonov
Here is a sample execution
h xsucc xeld eld ref i
h y x succ xeld eld y i
h y succ eld y eldi
h y succ y i
h y succ i
h y i
Lemma
ei and hs ei hs e i then there i is deterministic if hs ei hs
is a uniform renaming of variables in s and e to those in s and e
respectively
ii is compositional if hs ei hs eithen hs Rei hs Re i for
every reduction context R
ISo op Typ es
The monomorphic typ es of the language are
TyVar t j u
l j l j ref Typ j Nat j j
def
where t ranges over the applicative typ e variables AppTyVar ft t g
def
and u ranges over the imperative ones ImpTyVar fu u g This
division of variables into two classes is similar to that of Standard ML The
set of free typ e variables in a typ e is FTV is imperative if FTV
ImpTyVar
A typeconstraint is a subtyping assertion b etween two monomorphic
typ es written We will require all sets of constraints used in typ es
and rules to b e implicitly closed under obvious laws
De nition Constraint System A set of typ e constraints C is closed
i
i If C and C then C
ii If C then f g C
l l l g fl g then f C and f g C iii If
i i j i j j
j j
iv If l l l g fl g then f C and f g C
i i j i j i
j i
v If ref ref C then f g C
A closed set of constraints is a constraint system
WeletC range over implicitly closed constraint systems and thus will
b e careful to make sure any new set of constraints we form is closed The
Eifrig Smith and Trifonov
closed union of sets of constraints is denoted by C C an op eration that by
insp ection can b e seen to b e asso ciative
De nition Constraint Consistency A constraint is consis
tent if
i TyVar or TyVar
ii Nator ref and refor and
for some
l l and flgfl g or iii
iv l l and flg fl g
Otherwise a constraintisinconsistent
For example Nat tNat and tref m Nat are inconsistent con
straints while t tNat t uandu Nat are each consistent A con
straint system is consistent if all the constraints in the system are consistent
The rules will require all constraint systems to implicitly b e consistent
The typ e system assigns I So op expressions rctypes of the form
n C
to indicate an expression of typ e which is constrained by the constraints in
C Since the rules implicitly require C to b e consistent it makes sense to view
as a typ e and to write C on the right side of the turnstile as part of the
typ e
We dene the following notion of subtypingonrctyp es
De nition Subtyping rc Typ es n C n C provided that C is
consistentand C f g C
Stronger notions of subtyping could b e dened but for our purp oses this
denition suces The typ e schemes are as follows
j
Note that since n C can contain an arbitrary collection of constraints C
shallow F b ounded p olymorphic typ es are a sp ecial case of these p olymorphic
rc typ es
ISoop Typing Rules
Before giving the rules we describ e notation used in the rules Notation used in
sequent judgements includes the following A type environment A is a mapping
from variables to typ e schemes we use the more intuitive notation x in
stead of x Given a typ e environment A the pro of system assigns to an
expression e arctyp e n C written as the type judgement A e n C un
der the condition that C is consistent as mentioned previously all constraint
sets C app earing in the rules implicitly must b e consistent we o ccasionally
Eifrig Smith and Trifonov
A e
Sub Num
A e A n Nat n
A e n C e n C Ajj x e n C
App Abs
A x e n C A e e n C C
is a substitution on fg Ax Ax
PVar Var
A x n A x
A A e l n C e n C
i i i
Record Sel
U
C A e l n C A l e l n
i i i i i
i
A A e n C e n C
i i i
Match Inj
U
C A l e l n C A l e l n
i i i i i
i
A e n C Ajj x n C e n C
Let
A let x e in e n C C
if e is expansive then AppClos n C A
g where f
else Clos n C A
Fig Typing rules of ISo op
may write A e n C e n C to indicate several typ e judge
ments provable in the same environment Programs are typ e checked in the
initial environment A assigning the following typ e schemes to the built ins
zero NatBo ol A succ NatNat pred NatNat is
ref u uu ref t treft set t cell tref val t t
where Bo ol stands for the typ e true false Asubstitution on f g is
amap TyVar Typ whichistheidentityon TyVar nf g and maps
ImpTyVar to imp erativetyp es a renaming of f g is a substitution on fg
with co dom TyVar An expression is expansive if and only if it is not a
value following Tofte weformtyp e schemes by making the sets of typ e
variables we generalize over dep endent on the expansiveness of the expression
The denitions of these sets are
Clos n C A FTV FTV C n FTV A
AppClos n C A Clos n C A AppTyVar
where the functionalityof FTV is extended as usual to constraint systems rc
typ es typ e schemes and typ e environments
The typing rules for I So op are given in Figure Most of the rules have
obvious relation to those of standard systems with subtyping and records as
Eifrig Smith and Trifonov
in Toftes system the typing of ref intro duces imp erativetyp es The main
dierence is the addition of constraints as part of typ es the asso ciated sub
sumption rule on these typ es and the way consistent constraints accumulate
from the leaves to the ro ot of a typing pro of It is imp ortant to observethat
consistency of constraints is implicitly enforced byeach rule Other presenta
tions of constrained typ e systems do not require lo cal consistencyso
the constraints in the rules havebotha hyp othetical and assertional comp o
nent They are hyp othetical in that they may b e inconsistent and they are
assertional in that they assert prop erties of the typ e if they are consistent
For this reason they write C on the left of the turnstile and p erform some
top level consistency check b efore a proved typing is true Since constraints
are never inconsistent in our rules wehavenohyp othetical comp onentand
constraints are thus written on the right hand side of the turnstile
Some justication is required for the Let rule in which the constraint
system of the let expression contains not only the constraints in C necessary
for typing its b o dy but also those in C accumulated for the typ e of the
bound variable Leaving the latter constraints out as do but corrected
in results in a system unsound with resp ect to the standard call by value
semantics of the let expression C may contain constraints on typ e variables
free in the environment and their omission may lead to accepting programs
which get stuck while evaluating the expression assigned to the b ound variable
As an example consider the expression
x let y x in succ x
By rules PVarVar Sub and App the constraint system C of the rc typ e
of x contains ref for some typ e where is the typ e asso ciated with
x by the rule Abs This constraint will lead to inconsistency when combined
with the constraint Nat at the outermost rule of the typing pro of App
If it were omitted from the constraint system of the let the other constraint
on namely Nat from the b o dy succ xwould not cause an inconsistency
and the program would typ e check however its execution obviously leads to
the stuck state h let y in succ i
While the typ e language do es not have recursivetyp es x x x can b e given
the rc typ e t t nft t t gWedonothave a b ottom typ e but its
p ositive o ccurrences maybesimulated by an unconstrained typ e variable eg
x x x x x x has the rc typ e
t nft t t t t t g
An unconstrained variable can also b e used instead of a top typ e in negative
p ositions Positive o ccurrences of top maybesimulated byoverconstraining
from b elow
A if true then else t nfNat t tg
This constraint system is consistent Note that not all typable programs are
of this particular top typ e but they are provably of typ e t nfNat t
Eifrig Smith and Trifonov
tgC for some C and fresh t by a single use of Sub Similarly overconstraining
from ab oveachieves the eect of b ottom in negative p ositions
Sub ject Reduction Soundness and Typ e Inference
We prove soundness of the typ e system by demonstrating a sub ject reduction
prop erty First we strengthen the Let rule of the system to
n C e n C is a renaming of fg A e n C Ajj x
Let
A let x e in e n C C
if e is expansivethen AppClos n C A
g where f
else Clos n C A
Obviouslyany use of the original Let rule can b e trivially transformed
into a use of the stronger rule bycho osing to b e the identity renaming
This renaming do es not add anypower to the typing system any program
that is typable with the stronger Let rule is also typable with the original it
is intro duced only to avoid certain technical complications which arise during
reductions within a let expression
Next we extend the notion of typing to congurations
De nition A hs ei n C if and only if
A e n C
Dom A Dom A Dom s Dom A Dom s andA agrees
with A on Dom A
for each x Dom swehave Ax ref and A sx n C for
x x x
some and C C
x x
Theorem Sub ject Reduction If A hs ei then either e Val
or else hs ei hs ei and there exists an environment A such that A
hs ei
Wepresent only a sketch of the pro of in this abbreviated version The
pro of pro ceeds in the standard fashion given a conguration and a pro of of
its typability p erform one step of computation and transform the original
typing pro of into a pro of for the new conguration The interaction b etween
let p olymorphism and reference cells is known to cause signicant diculty
our approach to this problem derives from avoiding Toftes complex
greatest xed p oint construction
The dierences b etween our pro of and that of result from the con
straint systems of rc typ es and p olymorphic rc typ es Each step of computa
tion is accompanied by a pro of transformation that pushes constraints present
near the top of the pro of tree towards the leaves The complications of the
pro of arise when these constraints are pushed through uses of the Let rule
Eifrig Smith and Trifonov
demonstrating that the typ e generalizations p erformed in the initial applica
tion of the rule remain valid is non trivial
This pushing of constraints from the ro ot of the typing pro of towards the
leaves during reduction can b e considered a lazy approach to pro of canoni
calization An alternative approachwould b e to regularize the initial typing
pro of of a program to canonical form by pushing all of the constraints present
at the ro ot to the leaves b efore p erforming any computation This would re
sult in a more straightforward sub ject reduction pro of at the exp ense of a
more complicated pro of canonicalization lemma
The soundness of the typ e system is a corollary of the Sub ject Reduction
theorem
Theorem Soundness If A e then either e diverges or e com
putes to a value
Pro of By induction on the length of computation using Theorem
Note wehavethus proved soundness of the constrained typ e system with
out ever having shown the systems of constraints have a solution
Type Inference
Wenow dene the typ e inference algorithm and prove it is complete ie if a
program has a typ e derivation the inference algorithm will infer a typ e for it
The strategy wetake to reach this desired outcome is the following
Dene a new set of rules the inference rules for whichtyping derivations
are deterministic
Prove the inference rules are equivalent in strength to the general rules
we had b een using previously
The inference rules app ear in Figure
Theorem For al l terms e and environments Aitisdecidable whether
there exists a such that A e
inf
Pro of Sketch By insp ection of the rules there is only one rule for typing
each expression construct By further insp ection the only nondeterminism
that maybeintro duced in rule application is the choice of typ e variables used
in rules AbsandPVar Wethus cho ose normal pro ofs that use fresh
variables in every place p ossible If a pro of exists there clearly must then
b e a corresp onding normal pro of For expression e the normal pro of is
unique mo dulo conversion Thus a decision pro cedure may b e dened for
constructing such a canonical pro of The algorithm fails when an inconsistent
constraint system is obtained when combining the constraint systems inferred
for subterms and detection of such inconsistencies is trivially decidable
Wenow relate the inference rules to the general rules
Eifrig Smith and Trifonov
Ajj x t e n C Ax
inf
Abs Var
A x e t n C A x n
inf inf
A e n C A e n C
inf inf
Inj Sel
A e l t n C f l t g A l e l n C
inf inf
e n C A
i i i inf
Num Record
U
C A l e l n A n Nat n
i inf i i i i inf
i
A e n C
inf i i i
Match
U
C f t tg A l e l t t n
i i i inf i i i i
i
A e n C e n C
inf
App
A e e t n C C f tg
inf
Ax is a renaming of fg
PVar
A x
inf
A e n C Ajj x n C e n C
inf inf
Let
A let x e in e n C C
inf
if e is expansivethenAppClos n C A
g where f
else Clos n C A
Fig Typ e inference rules of ISo op
Theorem Completeness of Typ e Inference Given an environment
A and an expression e the typing judgement A e is provable for some
if and only if A e is provable for some
inf
Pro of Sketch If A e is provable A e is obviously provable
inf
as well each inference rule is a sp ecial case of a combination of Sub and a
general rule
Converselytyping pro ofs in the general set of rules may b e transformed
into ones using only the inference ones in a two step pro cess First the pro of is
transformed into preinference form in which each rule used one of the infer
ence rules or p ossibly Sub In the pro cess certain typ es used in the pro of
such as in the conclusion of rules Var Sel App and the like are replaced
byfreshtyp e variables t the corresp onding typ e constraints f t t g
are added to the constraint system and bubbled to the top Similarlyeach
assumption x is replaced with an assumption of the form x t together with
the constraints f t t g for some fresh t The result is a larger set of
constraints mentioning these new typ e variables Demonstrating the consis
tency of these richer constraint systems as these new constraints propagate to
the ro ot of the pro of is non trivial
Second the pro of is reworked again eliminating uses of rule Sub induc
Eifrig Smith and Trifonov
tively This transformation takes a pre inference pro of of A e n C and
pro duces an inference pro of of A e n C whereC C and either
or f g C This is p ossible b ecause the antecedents of each inference
rule are simply of the form A e n C the typ e need not b e in any sp ecial
form for the rule to b e applicable Essentially this means that a use of Sub
followed by another rule can b e exchanged thus moving the subsumptions to
the ro ot of the pro of where they can b e eliminated
Thus from Theorems and wemay conclude that every program
typable under the general rules has a typ e inferred bythetyp e inference algo
rithm Note we establish no principal typing prop ertyThetyping pro duced
by the inference algorithm is indeed minimal in an intuitive sense but it is
not formally minimal since our denition of is weak tNat nft Natg
is not a subtyp e of NatNat n even though any term that can b e given the
former typ e can also b e given the latter Weleave the question of principal
typings for future study since completeness is ultimately all the programmer
desires
Applications to OOP
Wenow illustrate howthistyp e inference algorithm is useful for typing ob ject
oriented programs the main motivation for our work We show its utilityin
class based OOP we exp ect it also applies to delegation style OOP but that
topic is b eyond the scop e of this pap er The basic OOP concepts we wish
to incorp orate include standard notions of ob ject metho d instance variable
class inheritance metho dinstance hiding and ob ject lifting The more
advanced notions we wish to account for include p olymorphism multiple in
heritance and binary metho ds Without binary metho ds in general metho ds
that take ob jects as parameters or return ob jects as values the ob ject typ
ing problem is not overly dicult ob jects maybeinterpreted as records of
functions metho ds and cells instance variables inheritance is subtyping
and ob ject lifting is accomplished by a subsumption rule As weshow typing
b ecomes considerably more dicult in the presence of binary metho ds
The ideal waytoshow applicability to OOP would b e to dene a complete
OOP language typ es and inference algorithm this is b eyond the scop e of
this pap er however Instead we will showhow a collection of simple macros
allow OOP to b e emb edded into I So op
The basic idea of the representation is to interpret classes as functions on
records s where s is the self new then takes the xed p oint of a class
to pro duce an ob ject in the form of a record see We cannot quite use
this enco ding First it is dicult to create xed p oints which are records in a
call by value language Second when taking a xed p oint via a Y combinator
the semantics entails re evaluating the record with each recursive access and
thus erroneously re initialize any instance variables In previous work we
avoided these problems by using a memory based xed p oint Unfortunately
Also called implicit ob ject co ercion or ob ject subsumption
Eifrig Smith and Trifonov
this enco ding will not work here as the use of reference cells to form the xed
point will infer imp erative p olymorphic typ es for ob jects Wethus opt for
an enco ding using a Y combinator with an initial instance variable allo cation
phase In a more complete treatment of this topic a limited form of memory
based xed p oint such as the single assignment reference SAR of could
b e used We ignore the issue of information hiding in this presentation though
it is not dicult to incorp orate
De nition The ob ject syntax is dened by the macros given in Figure
def
where Y y x x x x z y x x z is a call by value Y combinator and
def
! Yx x
and those Note that the class macro binds o ccurrences of s free in the e
k
and e of u free in e
i
k j
class
class s sup er meth u of e inst x e m e
i i j k
j k
e in let x !in u u u let
i i
i i
let y x ref e u u in slet sin
j i
j i
inst y meth m e
k
k
new new x Yx
message send e m e methm
instance read ex e instx
instance write e x e set cell e instx val e
Fig Macros for ob ject syntax
We illustrate the typing problems involved with binary metho ds through
an example of a GcdNum class that has a binary metho d gcd that takes another
GcdNum and recursively computes the GCD of itself and the other GcdNum
In order to keep the example very simple we assume the instance variable
containing the actual number val is publicly accessible and that GcdNum
denes no other metho ds ZGcdNum is a sub class of GcdNum with an addi
tional unimp ortant metho d zero Here mo d is taken to b e a function that
computes the mo dulus of twonumbers
let GcdNum class s sup er
inst
val
meth
zero sval then s gcd num if is
else if is zero mo d numval sval then s
else sval mo d sval numval num gcd s
Eifrig Smith and Trifonov
The gcd metho d takes another GcdNum ob ject num as argument Because
num is of the same typ e as the typ e of ob jects of the class we are cur
rently dening expressing the typ e of the gcd metho d will require some self
referentiality
We rst consider appropriate typ es for the inheritance is subtyping paradigm
This is known to have serious limitations but is nonetheless frequently
found in commercial OOP languages In this paradigm wegive GcdNum the
typ e
GcdNum GcdTyp eGcdTyp e where
GcdTyp e t val Nat ref gcd tt
Note that is the usual recursivetyp e constructor We use it instead of the
I So op enco ding of recursivetyp es using recursive constraints new GcdNum
then returns an ob ject of typ e GcdTyp e Without inheritance this typ e is
p erfectly adequate Wenow lo ok at the adequacy of this typ e with inheritance
We extend our example by dening ZGcdNum a sub class of GcdNum that also
includes a metho d that tests for zero
let ZGcdNum class s
sup er
u of GcdNum
inst
val uval
meth
gcd u gcd
zero sval zero is
In this case we did not override the gcd metho d instead we inherited it from
GcdNum denoted here by the sup erclass variable u in this enco ding we explic
itly state the sup erclass of each inherited metho d Using the inheritance is
subtyping paradigm the inherited instance variables and metho ds must have
the same typ es as in the sup erclass since these typ es are xed Thus the typ e
of ZGcdNum must b e
ZGcdNum ZGcdTyp eZGcdTyp e where
ZGcdTyp e
t val Nat ref gcd GcdTyp eGcdTyp e zero Bo ol
Note the gcd metho d still op erates on GcdTyp enotZGcdTyp eThus if gcd
were overridden in ZGcdNum with a function that used nums zero metho d
this typing would fail an undesirable fact Another problem with this typing
is illustrated in the following additional co de
let zgnum new ZGcdNum in zgnum gcd zgnum zero
The gcd metho d typ e is not parametric in the typ e of the ob ject given to it
Thus it will accept an ob ject of ZGcdTyp e as an argument since bysubtyping
ZGcdTyp e GcdTyp e but the result returned is only of GcdTyp eandthus is
not known to havea zero metho d The ab ove co de will thus not typ e check
Eifrig Smith and Trifonov
even though it executes without error
An alternativetyping is needed Since we inherit from GcdNum the ZGcd
Num ob jects that eventually are created will have more metho ds than just gcd
To capture this wemust takea parametric or openended view of the self typ e
in GcdNums typ e The parametricitywe desire in GcdNum is that t should
be any sub class with at least gcd and val and furthermore that gcd paramet
rically maps t to t To express the op en ended view as a typ e F b ounded
quantication is used as follows
GcdNum t GcdTyp eFt tGcdTyp eFt where
GcdTyp eFt val Nat ref gcd tt
ZGcdNum may then b e typ ed as
ZGcdNum t ZGcdTyp eFt tZGcdTyp eFt where
ZGcdTyp eFt val Nat ref gcd tt zero Bo ol
giving zgnum the typ e t ZGcdTyp eFtThus the ab ovecodetyp e checks In
addition it would have b een p ossible to override gcd in ZGcdNum imp ossible
in the simple recursive typ es view
The F b ounded typing has a drawback however ZGcdNum ob jects can
no longer b e lifted to b e GcdNum ob jects since their typ es are recursivetyp es
with t o ccurring negatively and thus the following co de will not typ e check
let gnum new GcdNum in
let zgnum new ZGcdNum in
gnum gcd zgnum
Note that the recursivetyping would allow this co de to typ e check
So b oth the F b ounded interpretation of inheritance and the recursive
typ es interpretation fail to typ echeck certain typable programs Our typ e
inference algorithm however infers typ es that will allow b oth of the ab ove
varieties of message send to b e typ ed in a single program
Types inferredinISoop
To simplify the presentation we will ignore the instance variable val in the
example We will also simplify the translation scheme to reect this by elim
inating the rst line from the macro expansion of class and replacing u by e
i
and dening new as Y
First consider the typ es inferred for the classes GcdNum and ZGcdNum
The simplied translations are
let GcdNum
s gcd num if then s
else if then s
else num gcd s
in let ZGcdNum
s let u GcdNum s in
Eifrig Smith and Trifonov
gcd u gcd
zero zero is
We rst sketchhow the inference system of rules infers GcdNums
inf
typ e These rules are deterministic mo dulo variants so pro of construction
is mechanical Starting from the leaves and using rules Record App and
Selinturnwe obtain
A jj s t t num t num gcd t n C
a b inf d
where C ft t t gcd t g
b c c d
Next using App
A jj s t t num t num gcd s t n C
a b inf e
where C ft t t gcd t t t t g
b c c d d e
Next expanding the conditional and using Abs and Matchtwice
A jj s t t num t t n C ft t t t g
a inf b e
FinallybyRecordandAbstwice
A GcdNum t t gcd t t n C ft t t t g
inf a b e
This is the typ e inferred by the inference rule system An actual im
plemented typ e inference algorithm would automatically p erform a number
of simplications on this typ e that do not change the meaning Here we
present these simplications informally by giving typings deduced in the gen
eral rules that are simplied forms of the inferred typ es For GcdNum t is
a
unconstrained so it may b e replaced by by subsumption t has only one
b
p ositive o ccurrence in the typ e so it may b e replaced with its upp er b ound
t t and t may also each b e replaced The following typ e may then b e
c d e
deduced for GcdNum in the general rules
t t t gcd gcd t t t nft t g
Hereafter we present the simplied forms of typ es only An actual imple
mented typ e inference algorithm would automatically p erform these simpli
cations For ZGcdNum the simplied inferred typ e is
t t t gcd gcd t t t zero Bo ol nft t g
Contrast these typ es with the F b ounded typ e given GcdNum in the op en
self enco ding ab ove Observe that the parameter num is an ob ject with
a gcd metho d Since that is the only metho d of num that is used no more
elds are required in the inferred typ e Contrast that with the F b ounded
case where num has all metho ds of GcdNum the op en endedness here is more
precise each metho d that is passed the self requires that self to only have
the metho ds actually used Note also that this is not even an F b ounded
typ e the constraint t t is not recursive Recursive constraints maynot
arise in classes since the knot has not b een tied yet
Eifrig Smith and Trifonov
Consider now the ob ject typ es gnum and zgnum have the following sim
plied typ es inferred
gnum t t t nf gcd gcd t t t t t g
zgnum t t t nf gcd gcd t t t
zero Bo ol t t g
It is dicult to explain precisely what these typ es denote except to say
they are denitely not the recursivetyp es used in b oth enco dings for ob jects
ab ove
The message sends from the example have the following constrained typ es
zgnum gcd gnum t nf
gcd gcd t t t zero Bo ol t
a a
t gcd t t t gcd t t
a b b a
gcd gcd t t t t t t t t g
b b a b
zgnum gcd zgnum t nf
zero Bo ol t t t gcd gcd t
a a
t gcd t t t gcd t t
a b b a
gcd gcd t t t zero Bo ol t
b b
t t t t g
a b
can b e proved to never and t Note the function upp er b ounds of t t t
a b
b a
b e used a more complete set of simplication transformations would justify
their removal Eachuseofgnum and zgnum gives rise to fresh variables bythe
PVar rule if these ob jects were not let p olymorphic the two message sends
ab ovewould share typ e variables and generalitywould b e lost Observe there
are no contradictions in the constraint systems of either of these message sends
Also note the result typ e t is in eect the union of t and t since it is an
a b
upp er b ound of these twotyp es This corresp onds to the fact that the result of
gcd could b e either a gnum or a zgnum Consider sending a zero message to the
result of the second message send zgnum gcd zgnum zero The rules
force t zero Bo ol to b e added to the constraints but this is
still consistent On the other hand consider zgnum gcd gnum zero
This maygive a run time error so should not typ e check Indeed t
zero Bo ol by transitive closure also requires a record with
out zero to b e a subtyp e of a record with zero but this is by denition an
inconsistent constraint
Compared to other work on rigorously sound class based ob ject languages
neither Bruces TOOPLE or TOIL languages nor our Lo op language
allows the ab ove program to typ e check in fact weknowofnostatictyp e
Eifrig Smith and Trifonov
system for ob ject oriented programming that successfully typ e checks this ex
ample So not only do we obtain ob ject typ e inference wehavearicher typ e
language where it is not required to cho ose b etween inheritance is subtyping
and the op en ended view of self
Discussion
Wehavegiven a new p owerful metho d for typ e inference for ob ject oriented
languages that is in manyways more p owerful than previously existing meth
ods Wehave hop es that the core we present here will lead to developmentofa
full scale ob ject oriented programming language incorp orating typ e inference
What we presenthereonlyshows this metho d is feasible however Further
study is necessary to see if it can b e implementedeciently in practice There
also is the question of howwell other language features will combine with this
inference metho d Mo dules in particular will b e a challenge There also should
b e separate syntax and typ es added for OOP features such as class denition
and message send This will provide a uniform notion of what OOP is to all
programmers and limit incompatibility of co de Lastlyeven though this sys
tem is signicantly stronger than the existing HindleyMilner style inference
algorithms the typ es it pro duces are larger and less easily readable by pro
grammers Thus it is imp ortant to address b oth the problem of simplication
of these typ es and the problem of how a b etter descriptions of what led to a
typ e error can b e given to programmers
Acknow ledgemen ts
Wewould liketoacknowledge Jens Palsb erg for helpful discussions on related
work and AmyZwarico for contributions in the early phases of this pro ject
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