LL LL LL LL

FranoisPottier

INRIARoquenourt,BP105,78153LeChesnayCedex,Frane.

Franois.Pottierinria.fr

Abstrat. Extendingasubtyping-onstraint-basedtypeinfereneframe-

workwithonditionalonstraints androws yieldsapowerfultypeinfer-

ene engine. We illustrate this laim by proposing solutions to three

deliatetypeinferene problems:\aurate"patternmathings, reord

onatenation,and\dynami"messages.Untilnow,knownsolutionsre-

quiredsigniantlydierent tehniques;our theoretialontributionis

inusingonly asingle (and simple)set of tools. Onthe pratial side,

thisallowsallthreeproblemstobenetfromaommonsetofonstraint

simpliationtehniques,leadingtoeÆientsolutions.

1 Introdution

Typeinfereneisthetaskofexaminingaprogramwhihlakssome(orevenall)

typeannotations,andreoveringenoughtypeinformationtomakeitaeptable

by a type heker. Its original, and most obvious, appliation is to free the

programmer from the burden of manually providing these annotations, thus

making statityping alessdreary disipline. However,type inferenehas also

seenheavyuseasasimple,modularwayofformulatingprogramanalyses.

Thispaperpresentsaommonsolutiontoseveralseeminglyunrelatedtype

infereneproblems, byunifying inasingletypeinferene systemseveralprevi-

ouslyproposedtehniques,namely:asimpleframeworkforsubtyping-onstraint-

based type inferene [15℄, onditional onstraints inspired by Aiken, Wimmers

andLakshman[2℄,androws alaRemy[18℄.

Constraint-Based Type Inferene

Subtypingisapartialorderontypes,denedsothatanobjetofasubtypemay

safelybesuppliedwhereveranobjetofasupertypeisexpeted.Typeinferene

in the presene of subtyping reets this basi priniple. Every time a piee

of data is passed from a produer to a onsumer, the former's output type is

requiredtobeasubtypeofthelatter'sinputtype.Thisrequirementisexpliitly

reordedbyreatingasymbolisubtypingonstraintbetweenthesetypes.Thus,

eah potentialdata owdisoveredin the programyields one onstraint. This

fatallowsviewingaonstraintsetasadiretedapproximationoftheprogram's

dataowgraph{regardlessofourpartiulardenitionofsubtyping.

Various type inferene systems based on subtyping onstraints exist. One

and Smith [22℄, as well as an abstrat framework by Odersky, Sulzmann and

Wehr [12℄.Related systemsinlude set-basedanalysis [8,6℄and typeinferene

systemsbasedonfeatureonstraints[9,10℄.

Conditional Constraints

Inmostonstraint-basedsystems,theexpressionif e

0

then e

1

else e

2 may,

atbest,bedesribedby

1

^

2

where

i

stands for e

i

's type, and stands for the whole expression's type.

This amounts to stating that \e

1

's (resp. e

2

's) value may beome the whole

expression'svalue",regardlessofthetest'soutome.Amorepreisedesription

{ \if e

0

may evaluate to true (resp. false),then e

1

's (resp e

2

's) value may

beomethewholeexpression's value"{maybegivenusingnaturalonditional

onstraints:

true

0

?

1

^ false

0

?

2

Introduingtests intoonstraintsallowskeepingtrakofthe program'sontrol

ow{that is, mirroringthe way evaluation is aeted byatest's outome, at

theleveloftypes.

Conditional set expressions were introdued by Reynolds [21℄ as a means

of solving set onstraints involving strit type onstrutors and destrutors.

Heintze [8℄ usesthem to formulate an analysis whih ignores \dead ode".He

also introduesase onstraints,whih allowignoringtheeet of abranh,in

aaseonstrut,unless it isatually liableto be taken.Aiken, Wimmersand

Lakshman [2℄ use onditional types, together with intersetion types, for this

purpose.

In the present paper, we suggest a single notion of onditional onstraint,

whihisomparableinexpressivepowertotheaboveonstruts,andlendsitself

to asimpleand eÆientimplementation. (A similarhoie wasmade indepen-

dentlybyFahndrih[5℄.)Weemphasizeitsuseasawaynotonlyofintroduing

ontrolintotypes,butalsoofdelayingtypeomputations,thusintroduingsome

\laziness"intotypeinferene.

Rows

Designingatype systemforaprogramminglanguagewithreords, orobjets,

requires some way of expressing labelled produts of types, where labels are

eld ormethodnames.Dually,ifaprogramminglanguageallowsmanipulating

strutureddata, then itstype systemshall likely requirelabelled sums, where

labelsarenamesofdataonstrutors.

Remy[18℄elegantlydealswithbothproblemsatonebyintroduingnotation

toexpress denumerable,indexed familiesoftypes,alledrows:

(Here, rangesovertypes, and a;b;::: range over indies.) An unknown row

mayberepresentedbyarow variable, exatlyasin theaseof types.(Bylak

ofsymbols,weshallnotsyntatiallydistinguishregulartypevariablesandrow

variables.) The term a: ; representsa row whose element at index ais ,

and whoseother elementsaregivenby.Theterm standsforarowwhose

elementat anyindexis.Theseinformalexplanationsaremadepreise viaan

equationaltheory:

a:

a

; (b:

b

; )=b:

b

; (a:

a

; )

=a:;

Formoredetails,wereferthereaderto[18℄.

Rowsoerapartiularlystraightforwardwayofdesribingoperationswhih

treatalllabels(exeptpossiblyanitenumberthereof)uniformly.Beauseevery

failityavailableattheleveloftypes(e.g.onstrutors,onstraints)analsobe

made available at thelevelof rows, adesriptionofwhat happens atthe level

ofasinglelabel{writtenusingtypes{analsobereadasadesriptionofthe

whole operation {written using rows.This interestingpoint will bedeveloped

further inthepaper.

Putting ItAll Together

Ourpointistoshowthattheombinationofthethreeoneptsdisussedabove

yields a very expressive system, whih allows type inferene for a number of

advaned language features. Among these, \aurate" pattern mathing on-

struts,reordonatenation,and\dynami"messageswillbedisussedinthis

paper.Oursystemallowsperformingtype infereneforall ofthese features at

one. Furthermore,eÆieny issuesonerning onstraint-basedtypeinferene

systemshavealreadybeenstudied [5, 15℄.Thisexisting knowledgebenetsour

system,whih maythus beusedto eÆiently perform typeinferenefor allof

theabovefeatures.

Inthispaper,wefousonappliations ofourtypesystem,i.e.weshowhow

it allowssolvingeah ofthe problemsmentioned above.Theoretialaspets of

onstraint solving are disussed in [15, 17℄. Furthermore, a robust prototype

implementationispublilyavailable [14℄.Wedonotprovethatthetypesgiven

to the three problemati operations disussed in this paperare sound, but we

believethis isastraightforwardtask.

Thepaperisorganizedasfollows.Setion 2givesabrieftehnialoverview

of the type system, fousing on the notion of onstrained type sheme, whih

should be enoughto gain an understanding ofthe paper.Setions 3, 4,and 5

disuss typeinferene for\aurate" pattern mathings, reordonatenation,

and \dynami" messages, respetively, within our system. Setion 6 sums up

ourontribution,thenbrieydisussesfutureresearhtopis.AppendixAgives

somemoretehnialdetails,inludingthesystem'stypeinferenerules.Lastly,

AppendixBgivesseveralexamples,whihshowwhatinferredtypeslooklikein

2 System's Overview

Theprogramminglanguageonsidered throughout thepaperis aall-by-value

-aluluswithlet-polymorphism,i.e.essentiallyoreML.

e::=x;y;:::jx:ej(ee)jX;Y;:::jlet X=e in e

The typealgebra needed to dealwith suh aore languageis simple. The set

ofgroundtermsontainsallregulartreesbuiltover?,>(witharity0) and!

(witharity2). Itisequippedwithastraightforwardsubtyping relationship[15℄,

denoted ,whihmakesit alattie. Itisthe logialmodelin whih subtyping

onstraintsareinterpreted.

Symbols,type variables,types andonstraintsaredened asfollows:

s::=?j!j> v::=;;:::

::=vj?j ! j> ::=

j sv?

A groundsubstitution isamapfrom typevariablesto groundterms.A on-

straintoftheform

1

2

,whihreads\

1

mustbeasubtypeof

2

",issatised

by if and only if (

1

) (

2

). A onstraint of the form s ?

1

2 ,

whihreads\ifexeedss,then

1

mustbeasubtypeof

2

",issatisedbyif

and onlyifs

S

head(()) implies(

1 )(

2

),where headmapsaground

termto itsheadonstrutor,and

S

istheexpeted orderingoversymbols.A

onstraintset Cis satisedbyifandonlyifallofitselementsare.

Atypesheme isoftheform

::=8C :

where isatypeandCisaonstraintset,whihrestritsthesetof'sground

instanes.Indeed,thelatter,whihweall'sdenotation,isdenedas

f 0

;9 satisesC^() 0

g

Beauseallofatypesheme'svariablesareuniversallyquantied,wewillusually

omitthe8quantierandsimplywrite\ whereC".

Of ourse, the type algebra given above is very muh simplied. In gen-

eral,thesystemallowsdeningmoretypeonstrutors,separatingsymbols(and

terms)intokinds,andmakinguseofrows.(Afulldenition{withoutrows{ap-

pearsin[17℄.)However,forpresentation'ssake,wewillintroduethesefeatures

onlystepbystep.

Theoreprogramminglanguagedesribedaboveisalsolimited.Toextendit,

wewilldenenewprimitiveoperations,equippedwithanoperationalsemantis

and an appropriate type sheme. However, no extension to the type system {

e.g. in the form of new typing rules { will bemade. This explainswhy we do

notfurther desribethesystemitself. (Somedetailsare givenin Appendix A.)

3 Aurate Analysis of Pattern Mathings

When faed with a pattern mathing onstrut, most existing type inferene

systemsadoptasimple,onservativeapproah:assumingthateahbranhmay

betaken,theyletitontributetothewhole expression'stype.Amoreaurate

system should use types to prove that ertain branhes annot be taken, and

preventthemfromontributing.

Inthissetion, wedesribesuh asystem.Theessentialidea{introduing

a onditional onstrut at the level of types { is due to [8, 2℄. Some novelty

residesin ourtwo-steppresentation,whihwebelievehelpsisolateindependent

onepts. First, we onsider the ase where only one data onstrutor exists.

Then, we easily move to the general ase, by enrihing the type algebra with

rows.

3.1 The Basi Case

Weassumethelanguageallowsbuildingandaessingtaggedvalues.

e::=:::jPrejPre 1

Asingledataonstrutor,Pre,allowsbuildingtaggedvalues,whilethedestru-

tor Pre 1

allowsaessing theirontents.This relationshipis expressedbythe

followingredutionrule:

Pre 1

v

1 (Prev

2

) redues to (v

1 v

2 )

TherulestatesthatPre 1

rsttakesthetagothevaluev

2

,thenpassesit to

thefuntionv

1 .

At the levelof types,we introdue a (unary) varianttype onstrutor [℄.

Also,weestablishadistintionbetweenso-alled\regulartypes,"written,and

\eldtypes,"written.

::=;;;:::j?j>j ! j[℄

::='; ;:::jAbsjPre jAny

A subtypeorderingovereldtypesisdened straightforwardly:Absisitsleast

element,Anyisitsgreatest,andPreisaovarianttypeonstrutor.

ThedataonstrutorPreisgiventhefollowingtypesheme:

Pre:![Pre℄

Notie that there is no way of building a valueof type [Abs℄. Thus, if anex-

pressionhasthis type,thenitmustdiverge.Thisexplainsourhoieofnames.

Ifanexpression hastype [Abs℄, thenitsvaluemustbe\absent";ifithastype

ThedatadestrutorPre 1

isdesribedasfollows:

Pre 1

:(!)!['℄!

where 'Pre

Pre'?

The onditional onstraintallows (Pre 1

e

1 e

2

)to reeive type ? when e

2 has

type[Abs℄, reetingthefat that Pre 1

isn't invokeduntile

2

produes some

value. Indeed,aslongas'equals Abs, theonstraintis vauouslysatised, so

is unonstrainedand assumesitsmost preisevalue, namely?. However,as

soon as Pre ' holds, must be satised as well. Then, Pre 1

's type

beomesequivalentto(!)![Pre℄!,whihisitsusualMLtype.

3.2 The General Case

Wenowmovetoalanguagewithadenumerablesetofdataonstrutors.

e::=:::jKjK 1

jlose

(We letK,L; ::: stand for data onstrutors.)An expression maybe tagged,

as before, by applying a data onstrutor to it. Aessing tagged values be-

omesslightlymoreomplex,beausemultiple tagsexist.Thesemantisofthe

elementarydatadestrutor,K 1

,isgivenbythefollowingredutionrules:

K 1

v

1 v

2 (Kv

3

) reduesto (v

1 v

3 )

K 1

v

1 v

2 (L v

3

) reduesto (v

2 (Lv

3

)) whenK6=L

Aordingtotheserules,ifthevaluev

3

arriestheexpetedtag,thenitispassed

tothefuntionv

1

.Otherwise,thevalue{stillarryingitstag{ispassedtothe

funtion v

2

. Lastly, a speial value, lose, is added to the language, but no

additionalredutionruleisdened forit.

Howdowemodifyourtypealgebratoaommodatemultipledataonstru-

tors? InSetion 3.1, weused eld typesto enode informationaboutatagged

value's presene or absene. Here, we need exatlythe same information, but

this time about every tag. So, we need to manipulate a family of eld types,

indexed by tags.Todo so,weadd onelayerto the typealgebra:rows of eld

types.

::=;;;:::j?j>j ! j[℄

::='; ;:::jK:; j

::='; ;:::jAbsjPre jAny

Weannowextendtheprevioussetion'sproposal, asfollows:

K:![K:Pre; Abs℄

K 1

:(!)!([K:Abs; ℄!)![K:'; ℄!

where 'Pre

Pre'?

K 1

's typesheme involvesthesameonstraintsasin thebasiase. Usinga

singlerowvariable,namely ,intwodistintpositionsallowsexpressingthefat

thatvaluesarryinganytagotherthanKshallbepassedunmodiedtoK 1

's

seondargument.

lose'sargumenttypeis[Abs℄,whihpreventsitfromeverbeinginvoked.

This aords with the fat that losedoes not have an assoiated redution

rule.Itplaystherole ofafuntiondened byzeroases.

Thissystemoers extensible patternmathings: k-ary ase onstruts may

be written using k nested destrutor appliations and lose, and reeive the

desired, aurate type. Thus, no spei languageonstrut or type inferene

ruleisneededtodealwiththem.

4 Reord Conatenation

Stati typing for reord operations is a widely studied problem [4, 13℄. Com-

monoperationsinludeseletion,extension,restrition,andonatenation.The

latteromesintwoavors:symmetriandasymmetri.Theformerrequiresits

argumentsto havedisjointsetsof elds,whereasthelattergivespreedeneto

theseondonewhen aonitours.

Of these operations, onatenation is probably the most diÆult to deal

with, beause itsbehaviorvaries aordingto thepresene orabsene of eah

eldin itstwoarguments.Thishasledmanyauthorstorestrittheirattention

totypeheking,andtonotaddresstheissueoftypeinferene[7℄.Aninferene

algorithm for asymmetri onatenation was suggestedby Wand [23℄. He uses

disjuntions of onstraints, however,whih giveshis system exponentialom-

plexity.Remy[19℄suggestsanenodingofonatenationinto-abstrationand

reordextension,wheneaninferenealgorithmmaybederived.Unfortunately,

itspowerissomewhatdereasedbysubtleinterationswithML'srestritedpoly-

morphism;furthermore,theenodingisexposedto theuser.Inlaterwork [20℄,

Remy suggests a diret, onstraint-based algorithm, whih involves a speial

formofonstraints.Ourapproahisinspiredfromthiswork,butre-formulated

in terms of onditional onstraints, thus showing that no ad ho onstrut is

neessary.

Again,ourpresentationis in twosteps.Thebasi ase,where reordsonly

haveoneeld,istakledusingsubtypingandonditionalonstraints.Then,rows

allowustoeasilytransferourresultstotheaseofmultipleelds.

4.1 The Basi Case

Weassumealanguageequipped withone-eldreords,whoseuniqueeld may

beeither\absent"or\present".Morepreisely,weassumeaonstantdataon-

strutorAbs,andaunarydataonstrutorPre;a\reord"isavaluebuiltwith

oneoftheseonstrutors.Adatadestrutor,Pre 1

,allowsaessingtheontents

onatenationprimitives,writtenand,respetively.

e::=:::jAbsjPrejPre 1

jj

The relationship between reord reation and reord aess is expressed by a

simpleredutionrule:

Pre 1

(Prev) reduesto v

Thesemantisofasymmetrireordonatenationisgivenasfollows:

v

1

Abs redues to v

1

v

1

(Prev

2

) redues to Prev

2

(Ineahoftheserules,thevaluev

1

isrequiredtobeareord.)Lastly,symmetri

onatenationisdenedby

Absv

2

reduesto v

2

v

1

Abs reduesto v

1

(In thesetworules,v

1 andv

2

arerequiredtobereords.)

TheonstrutionofourtypealgebraissimilartotheoneperformedinSe-

tion3.1.Weintroduea(unary)reordtypeonstrutor,aswellasadistintion

betweenregulartypesandeldtypes:

::=;;;:::j?j>j ! jfg

::='; ;:::jBotjAbsjPre jEither jAny

Letusexplain,stepbystep,ourdenitionofeldtypes.Ourrst,naturalstep

is to introdue type onstrutors Abs and Pre, whih allow desribing values

built with the data onstrutors Abs and Pre. The former is a onstant type

onstrutor,whilethelatterisunaryandovariant.

Many type systems for reord languages dene Pre to be a subtype of

Abs. This allows a reord whose eld is present to pretend it is not, leading

to a lassi theory of reords whose elds may be \forgotten" via subtyping.

However,whenthelanguageoersreordonatenation,suhadenition isn't

appropriate.Why?Conatenation{asymmetriorsymmetri{involvesahoie

betweentworedutionrules,whihisperformedbymathingone,orboth,ofthe

argumentsagainstthe dataonstrutors AbsandPre. If,at theleveloftypes,

weallowanon-empty reordtomasquerade asan empty one,thenitbeomes

impossible, basedonthe arguments'types, tond outwhih ruleapplies, and

to determinethetypeoftheoperation'sresult.Insummary,in thepreseneof

reord onatenation,nosubtyping relationshipmust exist betweenPre and

Abs.(Thisproblemiswelldesribed{althoughnotsolved{in[4℄.)

ThisleadsustomakingAbsandPreinomparable.Onethishoiehasbeen

made,ompletingthedenitionofeldtypesisratherstraightforward.Beause

Bot, and agreatest element Any. Lastly, weintrodue aunary, ovarianttype

onstrutor,Either, whih wedeneastheleastupperboundofAbsandPre,

sothatAbst(Pre)equalsEither.Thisoptionalrenementallowsustokeep

trakofaeld's type,evenwhen itspreseneisnotasertained.Thelattieof

eld typesisshownin gure1.

Any

Either

OO

Abs

99 t

t t t t t t t t

Pre

ee LL

LL LL LL LL

Bot

ee JJ

JJ JJ JJ J

99 r

r r r r r r r r r

Fig.1.Thelattieofreordeldtypes

Letusnowassigntypestotheprimitiveoperationsoeredbythelanguage.

Reordreationandaess reeivetheirusualtypes:

Abs:fAbsg

Pre:!fPreg

Pre 1

:fPreg!

There remainstoomeupwithorret,preise typesforbothavorsofreord

onatenation. Thekeyidea issimple. As shown by its operationalsemantis,

(eitheravorof)reordonatenationisreallyafuntion dened byasesover

thedataonstrutorsAbsandPre{andSetion3hasshownhowtoaurately

desribesuh afuntion.Letusbegin,then,withasymmetri onatenation:

:f'

1 g!f'

2 g!f'

3 g

where '

2

Either

2

Abs'

2

?'

1 '

3

Pre'

2

?Pre

2 '

3

Clearly, eah onditional onstraint mirrors one of the redution rules. In the

seond onditional onstraint,we assume

2

isthe typeof theseond reord's

eld {if ithas one.The rstsubtyping onstraint representsthis assumption.

Notiethat weusePre

2

, ratherthan'

2

,astheseond branh's resulttype;

thisisstritly morepreise,beause' maybeoftheform Either .

Lastly,weturnto symmetrionatenation:

:f'

1 g!f'

2 g!f'

3 g

where Abs'

1

?'

2 '

3

Abs'

2

?'

1 '

3

Pre'

1

?'

2 Abs

Pre'

2

?'

1 Abs

Again, eah of the rst two onstraints mirrorsaredution rule. The last two

onstraintsdisallowtheasewherebothargumentsarenon-emptyreords.(The

areful reader will notie that any one of these two onstraints would in fat

suÆe;botharekeptforsymmetry.)

Inbothases,theoperation'sdesriptionin termsof onstraintsloselyre-

sembles its operationaldenition. Automatially derivingthe former from the

latterseemspossible;thisisanareaforfutureresearh.

4.2 The General Case

We now move to a languagewith adenumerable set of reord labels, written

l, m, et. The languageallows reatingthe empty reord, aswell asany one-

eldreord;italsooersseletionandonatenationoperations.Extensionand

restritionanbeeasily added,ifdesired;weshalldispensewiththem.

e::=?jfl=egje:ljj

Wedonotgivethelanguage'ssemantis,whihshouldhopefullybelearenough.

At thelevelof types,we againintroduerows ofeld types, denoted by.

Furthermore,we introdue rowsof regular types,denoted by%. Lastly, welift

theveeld typeonstrutorstothelevelofrows.

::=;;;:::j?j>j ! jfg

::='; ;:::jBotjAbsjPre jEither jAny

%::=;;;:::jl:; %j

::='; ;:::jl:; jjBotjAbsjPre%jEither%jAny

Thisallowswritingomplexonstraintsbetweenrows,suhas'Pre,where

'andarerowvariables.Aonstraintbetweenrowsstandsforaninnitefamily

ofonstraintsbetweentypes,obtainedomponent-wise.That is,

(l:' 0

; ' 00

)Pre(l: 0

; 00

) standsfor (' 0

Pre 0

)^(' 0 0

Pre 00

)

We may now give types to the primitive reord operations. Creation and

seletionareeasilydealtwith:

?:fAbsg

fl=g:!fl:Pre; Absg

Interestingly,thetypesofbothonatenationoperationsareunhangedfromthe

previoussetion { at least, syntatially. (For spaereasons, we donot repeat

them here.) Asubtle dierene liesin thefat that allvariables involvedmust

nowbereadasrowvariables,ratherthanastypevariables.Inshort,theprevious

setionexhibitedonstraintswhihdesribeonatenation,atthelevelofasingle

reordeld;here,therowmahineryallowsustorepliatetheseonstraintsover

aninnitesetoflabels.Thisinreaseinpoweromesalmostforfree:itdoesnot

addanyomplexityto ournotionofsubtyping.

5 Dynami Messages

So-alled\dynami"messageshavereentlyreeivednewattentioninthestati

typing ommunity. Bugliesi and Crafa [3℄ propose a higher-order type system

whihaountsforrst-lassmessages.Nishimura[11℄taklestheissueoftype

infereneandsuggestsaseond-ordersystemalaOhori[13℄.MullerandNishi-

mura[10℄proposeasimpliedapproah,basedonanextendedfeature logi.

Theproblemonsistsinperformingtypeinfereneforanobjet-orientedlan-

guagewheremessagesarerst-lassvalues,madeupofalabel andaparameter.

Here, we view objets asreords of funtions, and messagesas tagged values.

(Betterwaysofmodelingobjetsexist,butthatisanindependentissue.)Thus,

weonsideralanguagewithreordsanddataonstrutors,asdesribedinSe-

tions3.2and4.2.Furthermore,weletreordlabelsanddataonstrutorsrange

overasinglenamespae,thatofmessagelabels.(Tosavespae,wehoosetodeal

diretlywith thease ofmultiple messagelabels; however,ourusual, two-step

presentationwouldstillbepossible.)Lastly,wedeneaprimitivemessage-send

operation,written#,whosesemantisisasfollows:

#fm=v

1

;:::g(mv

2

) reduesto (v

1 v

2 )

Inplain words,# examinesitsseondargument,whih mustbesomemessage

m with parameter v

2

. It then looks up the method named m in the reeiver

objet,and appliesthemethod's ode,v

1

,to themessageparameter.

In a languagewith \stati" messages, a message-send operation may only

involveaonstantmessagelabel.So,insteadofasinglemessage-sendoperation,

afamily thereof,indexedby messagelabels, isprovided. Infat,in oursimple

model, these operationsare denablewithin the language.Theoperation#m,

whihallowssendingthemessagemtosomeobjetowithparameterp,maybe

dened aso:p:(o:mp). Then,typeinfereneyields

#m:fm:Pre(!); Anyg!!

Beausethemessagelabel,m,isstatiallyknown,itmaybeexpliitlymentioned

in the type sheme, making it easy to require the reeiver objetto arry an

appropriatemethod.Inalanguagewith\dynami"messages,ontheotherhand,

misnolongerknown.Theproblemthusappearsmoreomplex;ithas,infat,

sparked thedevelopment ofspeial-purpose onstraintlanguages[10℄. Yet,the

Considerthe partial appliation of the message send primitive # to some

reordr.Itisafuntionwhihaeptssometaggedvalue(mv),theninvokesan

appropriatepieeofode,seletedaordingtothelabelm.Thisshould ringa

bell{itismerelyaform ofpattern mathing,whihthispaperhasextensively

disussedalready.Therefore,wepropose

#:f'g![ ℄!

where Pre

Pre ?'Pre(!)

(Here,allvariablesexept arerowvariables.)Theoperation'srst(resp.se-

ond) argument is required to be an objet (resp. a message), whose ontents

(resp. possiblevalues) aredesribedby therow variable '(resp. ).The rst

onstraintmerelyletsstandforthemessageparameter'stype.Theonditional

onstraint,whihinvolvestworowterms,shouldagainbeunderstoodasafamily,

indexedbymessagelabels,ofonditionalonstraintsbetweenreordeldtypes.

Theonditionalonstraintassoiatedwithsomelabelmshallbetriggeredonly

if 'selementatindexmisoftheformPre ,i.e.onlyifthemessage'slabelmay

bem.Whenitistriggered,itsright-handsidebeomesative,withathree-fold

eet. First,''s element at index m mustbe ofthe form Pre( ! ),i.e. the

reeiverobjetmustarryamethodlabeledm.Seond,themethod'sargument

typemustbe(asupertypeof)'selementat labelm,i.e. themethodmust be

able to aeptthe message'sparameter.Third, themethod's resulttype must

be(asubtypeof),i.e.thewholeoperation'sresulttypemustbe(atleast)the

join ofallpotentiallyinvokedmethods'returntypes.

Our proposal shows that type inferene for \dynami" messages requires

nodediated theoretialmahinery.Italso showsthat\dynami"messagesare

naturallyompatible with alloperationsonreords, inludingonatenation{

aquestionwhih wasleftunansweredbyNishimura[11℄.

6 Conlusion

In this paper, we have advoated enrihing an existing onstraint-based type

inferene framework [15℄ with rows [18℄ and onditional onstraints [2℄. This

providesasingle (andsimple) solution to several diÆulttypeinferene prob-

lems, eah of whih seemed to require,until now,speial forms ofonstraints.

Fromapratialpointofview,itallowsthemtobenetfromknownonstraint

simpliationtehniques[17℄,leadingto aneÆientinferenealgorithm[14℄.

We believe our system subsumes Remy's proposal for reord onatena-

tion [20℄, as well asMuller and Nishimura's viewof \dynami" messages [10℄.

Aiken, Wimmers and Lakshman's \soft" type system [2℄ is more preise than

ours, beauseit interprets onstraintsin a riher logialmodel, but otherwise

oerssimilarfeatures.Infat,theideasdevelopedinthispaperouldhavebeen

presentedin the setting of Bane [5℄, or, more generally, of any systemwhih

Referenes

[1℄ AlexanderS. Aiken and Edward L. Wimmers. Type inlusion onstraints and

typeinferene. InFuntionalProgramming&Computer Arhiteture,pages31{

41. ACMPress, June1993. URL: http://http.s.berkeley.edu/~ aike n/ftp /

fpa93.ps.

[2℄ Alexander S. Aiken, Edward L. Wimmers, and T. K. Lakshman. Soft typing

withonditionaltypes. InPriniples ofProgrammingLanguages,pages163{173,

January1994. URL:http://http.s.berkeley.edu/ ~aik en/ft p/po pl94 .ps.

[3℄ Mihele Bugliesi and Silvia Crafa. Objet aluli for dynami messages. In

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edu/pub/kim/FOOL6/bugliesi.ps.

[4℄ LuaCardelliandJohnMithell.Operationsonreords.MathematialStrutures

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editors, Theoretial Aspets of Objet-Oriented Programming: Types, Semantis,

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[8℄ NevinHeintze. SetbasedanalysisofMLprograms. TehnialReportCMU-CS-

93-193,CarnegieMellonUniversity,ShoolofComputerSiene,July1993.URL:

ftp://reports.adm.s.mu.edu/u sr/an on/1 993/C MU-CS-9 3-19 3.ps .

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de/pub/papers/ProgrammingSysLa b/Fir stCl ass98 .ps. gz.

[11℄ SusumuNishimura. Statityping for dynami messages. InConferene Reord

of POPL '98: The 25th ACM SIGPLAN-SIGACT Symposium on Priniples

of Programming Languages, pages 266{278, San Diego, California, January

1998. URL: ftp://ftp.kurims.kyoto-u.a.j p/pub /pap er/m ember /nis imura /

dmesg-popl98.ps.gz.

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[20℄ DidierRemy.Aasestudyoftypehekingwithonstrainedtypes:Typingreord

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URL:ftp://ftp.s.neu.edu/pub/peop le/wa nd/p aper s/i-91.d vi.

A Rules

This appendix gives a short desription of the system's type inferene rules

(Figure2).Eventhoughonlytheorelanguageisexpliitlytreated,theserules

are suÆient to deal with a full-featured programminglanguage. Indeed, any

extralanguageonstrutmaybeviewedeither assyntatisugar,orasanew

primitiveoperation, whih anbe bound in an initial typing environment

0 .

Also,notethatthesetypeinferenerulesuseneitheronditionalonstraints,nor

rows;thesewillome onlyfrom

0 .

Forsimpliity, wedistinguish identiersbound by ,denoted x;y;::: from

those bound bylet, denoted X;Y;::: Furthermore,we expet -identiers to

fresh

`

I

x:8?:hx:i)

(Vari)

`

I

e:8C :A) 0

A(x)=

`Ix:e:8C :(Anx))! 0

(Abs

i )

`Ie1:8C1:A1)1 `Ie2:8C2:A2)2

fresh C=C

1 [C

2 [f

1

2

!g

`Ie1e2:8C :(A1uA2))

(App

i )

(X)= freshrenamingof

`

I

X :()

(LetVari)

`

I e

1 :

1

+[X7!

1

℄`

I e

2 :

2

`Ilet X=e1 in e2:2

(Let

i )

Fig.2.Typeinferenerules

program.Lastly,in everyexpressionoftheformlet X =e

1 in e

2

,werequire

X to appearfreewithin e

2

.It would beeasytooveromethese restritions,at

theexpenseofheaviernotation.

Therules are fairly straightforward. Themain point of interest is the way

eahappliation node produesasubtypingonstraint. Theonlypeuliarityis

inthewaytypeenvironmentsaredealtwith.Theenvironment ,whihappears

ontheleftoftheturnstile,isalistofbindingsoftheformX :.Typeshemes

are slightlymoreomplex thaninitially shownin Setion 2.Theyare, in fat,

of the form ::= 8C :A ), where the ontext A is a set of bindings of the

formx:.Thepointofsuhaformulationistoobtainasystemwherenotype

sheme hasfreetype variables.This allowsasimplertheoretial desriptionof

onstraintsimpliation.

Asfar asnotation isonerned, hx:i representsaontextonsisting ofa

singleentry,whihbindsx to .Anx istheontext obtainedbyremovingx's

bindingfromA,ifitexists.Forthesakeofreadability,wehaveabusednotation

slightly. In rule (Abs

i

), A(x) stands for thetype assoiatedwith x in A, if A

ontains a binding for x; it stands for > otherwise. In rule (App

i ), A

1 uA

2

represents the point-wise intersetion of A

1 and A

2

. That is, whenever x has

a binding in A

1 or A

2

, its binding in A

1 uA

2 is A

1

(x)uA

2

(x). Beause we

donothaveintersetion types,this expressionshould in fat beunderstood as

afreshtypevariable,aompaniedbyanappropriateonjuntion ofsubtyping

onstraints.

Therulesimpliitlyrequireeveryonstraintsettoadmitatleastonesolution.

B Examples

Example 1. Wedeneafuntionwhihreadseldloutofareordr,returninga

defaultvaluedifrhasnosuheld,bysettingextrat=d:r:(fl=dgr):l.

Inoursystem,extrat'sinferredtypeis

extrat:!fl:'; g!

where 'Either Either

Abs'? Abs ?AbsAny

Pre'? Pre ?PreAny

Therstonstraintretrievesr:l'stypeand namesit, regardlessoftheeld's

presene.(Iftheeldturnsouttobeabsent, willbeunonstrained.)Theleft-

hand onditionalonstraintslearlyspeify thedependenybetweentheeld's

preseneandthefuntion'sresult.

Theright-handonditionalonstraintshavetautologousonlusions{there-

fore,theyaresuperuous.Theyremainonlybeauseoururrentonstraintsim-

pliation algorithms are \lazy" and ignore any onditional onstraints whose

onditionhasnotyetbeenfullled.Thisproblemouldbexedbyimplementing

slightlymoreaggressivesimpliationalgorithms.

Thetypeinferredforextrat0fl=1gandextrat0fm=1gisint.Thus,

inmanyases,oneneednotbeawareoftheomplexityhiddeninextrat'stype.

Example 2. Weassumegivenanobjeto,ofthefollowingtype:

o:f getText:Pre(unit!string); setText:Pre(string!unit);

selet:Pre(intint!unit); Abs g

o mayrepresent, forinstane, an editable text eld in agraphi userinterfae

system.Itsmethodsallowprogrammatiallygettingandsettingitsontents,as

wellas seletingaportionoftext.

Next,weassumealistdatastruture, equippedwithasimpleiterator:

iter:(!unit)! list!unit

Thefollowingexpressionreatesalistofmessages,andusesitertosendeahof

themin turntoo:

iter(#o)[setText\Hello!"; selet(0;5)℄

This expression is well-typed,beause o ontainsappropriate methods to deal

witheahofthesemessages,andbeausethesemethodsreturnunit,asexpeted

byiter.Theexpression's typeisofourseunit,iter'sreturntype.

Hereisasimilarexpression, whihinvolvesagetTextmessage:

iter(#o)[setText\Hello!"; getText()℄

This time, it is ill-typed. Indeed, sending a setText messageto o produesa

resultoftypeunit,whilesendingitagetTextmessageproduesaresultoftype

string.Thus,(#o)'sresulttypemustbe>,thejoinofthesetypes.Thismakes

(#o) an unaeptable argument for iter, sine the latter expets a funtion

whosereturntypeisunit.