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Join-Calculus
Sylvain Conchon, François Pottier
To cite this version:
Sylvain Conchon, François Pottier. JOIN(X): Constraint-Based Type Inference for the Join-Calculus.
Proceedings of the 10th European Symposium on Programming (ESOP’01), Apr 2001, Genova,
pp.221–236. �inria-00000029�
SylvainConhonandFrançois Pottier
INRIARoquenourt,{Sylvain.Conhon,Franois.Pottie r}i nria. fr
Abstrat. We present a generi onstraint-basedtype system for the
join-alulus.Thekeyissueistypegeneralization,whih,inthepresene
ofonurreny,mustberestrited.Werstdenealiberalgeneralization
riterion,andproveitorret.Then,wendthatithinderstypeinfer-
ene,andproposearuderone,reminisentofML'svaluerestrition.
Weestablishtypesafetyusingasemi-syntati tehnique,whihwebe-
lieveis ofindependentinterest.Itonsistsininterpretingtypingjudge-
mentsas (sets of) judgementsin anunderlying system, whihitself is
givenasyntatisoundnessproof.
1 Introdution
The join-alulus [3℄ is a name-passing proess alulus related to the asyn-
hronous -alulus.Theoriginal motivation forits introdutionwasto dene
aproessalulusamenabletoadistributedimplementation.Inpartiular,the
join-alulusmergesreeption,restritionandrepliationintoasinglesyntati
form,thedefonstrut,avoidingtheneedfordistributedonsensus.Thisdesign
deisionturns outtoalsohaveanimportantimpatontyping.Indeed,beause
thebehaviorofahannelisfullyknownatdenitiontime,itstypeanbesafely
generalized.Thus,defonstrutsbeomeanalogoustoML'sletdenitions.For
instane,thefollowingdenition:
def apply(f,x) = f(x)
denesahannelapplywhihexpetstwoargumentsfandxand,uponreeipt,
sendsthe messagef(x).InFournetet al.'stypesystem[4℄,applyreeivesthe
parametritypesheme8:hhi;i, wherehiisthehanneltypeonstrutor.
1.1 Motivation
Whydevelopanewtypesystemforthejoin-alulus?Theuniation-basedsys-
tem proposed by Fournet et al. [4℄shares many attrativefeatures with ML's
typesystem:itissimple,expressive,andeasytoimplement,asshownbytheJo-
Camlexperiment[1℄.LikeML,itispresriptive,i.e.intendedtoinferreasonably
simpletypesandto enforeaprogrammingdisipline.
Typesystemsareoftenusedasanieformalbasisforvariousprogramanal-
on. These systems,however,tend to beessentiallydesriptive, i.e. intended to
inferaurate typesand to rejetasfew programsas possible. Toahievethis
goal,itisommonto desribethebehaviorofprogramsusingarih onstraint
language,possibly involvingsubtyping, set onstraints,onditionalonstraints,
et.Wewishto denesuh adesriptivetypesystemforthejoin-alulus,asa
vehileforfuturetype-based analyses.
FollowingOderskyetal.[6℄,weparameterizeourtypesystemwithanarbi-
traryonstraintlogiX,makingitmoregeneriandmoreeasilyre-useable.Our
workmaybeviewedasan attempt toadapt their onstraint-basedframework
tothejoin-alulus,muhasFournetetal. adaptedML'stypedisipline.
1.2 TypeGeneralization Criteria
The def onstrut improves on let expressions by allowing synhronization
between hannels. Thus, we an dene a variant of apply that reeives the
hannel fandtheargumentxfromdierenthannels.
def apply(f) | args(x) = f(x)
Thissimultaneouslydenesthenamesapplyandargs.Themessagef(x)will
beemittedwheneveramessageisreeivedonbothofthesehannels.
Ina subtyping-onstraint-basedtype system, onewould expet apply and
args to be given types hi and hi, respetively, orrelated by the onstraint
hi.Theonstraintrequiresthehannelstobeusedinaonsistent way:the
typeofxmustmaththeexpetationsoff.Now,ifweweretogeneralizethese
typesseparately,wewouldobtainapply:8[hi℄:hiandargs:8[
hi℄:hi, whih are logiallyequivalentto apply:8:hhii and args:8:hi.
Thesetypesnolongerreettheonsistenyrequirement!
To addressthis problem, Fournet et al. statethat any typevariable whih
isshared betweentwojointlydened names(here, applyandargs),i.e.whih
ours free in their types, must not be generalized. However, this riterion is
based onthe syntax of types, and makes littlesense in the presene of an ar-
bitraryonstraint logi X. In the example above, applyand args havetypes
hi and hi, sothey share notype variables. Theorrelation isonly apparent
in the onstraint hi. Whentheonstraintlogi X is known, orrelations
anbedeteted byexamining the(syntaxof the) onstraint,looking for paths
onnetingand .However,wewantourtypesystemtobeparametriinX,
sothesyntax(andthemeaning)ofonstraintsis,ingeneral,notavailable.This
leads us to dene auniform, logial generalization riterion(Set. 5.2), whih
weprovesound.
Unfortunately,andsomewhatsurprisingly,thisriterionturnsouttohinder
type inferene. As aresult, we will propose aruder one, reminisent of ML's
so-alledvaluerestrition [10℄.
1.3 Overview
Werstreallthesyntaxandsemantisofthejoin-alulus,andintroduesome
P jQQjP D1;D2D2;D1
P j0P D;D
P j(QjR )(PjQ)jR D1;(D2;D3)(D1;D2);D3
(defDinP)jQdefDin(P jQ) ifdn(D)\fn (Q)=?
defD
1
indefD
2
inPdefD
1
;D
2
inP iffn(D
1
)\dn(D
2 )=?
defD;J.P inQj'J!defD;J.PinQj'P ifdom(')=ln(J)
Fig.1.Operationalsemantis
alled B(T),andestablishitsorretnessin asyntatiway(Set. 4).Building
on this foundation, Set.5 introdues JOIN(X) and provesit orret withre-
spetto B(T). Set.6studies typereonstrution,suggestingthat arestrited
generalizationriterionmustbeadoptedinordertoobtainaompletealgorithm.
Bylakofspae,weomitallproofs,exeptthatofthemain typesoundness
theorem(Theorem5.9).Theinterestedreaderisreferredto[2℄.
2 The Join-Calulus
Weassumegivenaountable set ofnamesN,ranged overbyx;y;u;v;::: We
write~uforatuple(u
1
;::: ;u
n
)anduforaset fu
1
;:::;u
n
g,wheren0.The
syntaxofthejoin-alulusisasfollows.
P ::=0j(P jP)juh~vijdefDinP
D::=jJ.P jD;D
J ::=uh~yij(J jJ)
Thedenednames dn(J)(resp.dn(D))ofajoin-patternJ (resp.ofadenition
D)arethe hannelsdened byit. InaproessdefD inP,thedened names
ofD areboundwithinD andP.Moredetailsaregivenin[2℄.
Redution!isdenedasthesmallestrelationthatsatisesthelawsinFig.1.
(-onversionand ongruenerulesomittedforbrevity.) 'rangesoverrenam-
ings,i.e.one-to-onemapsfromN intoN.standsfor!\ .Itisustomary
todistinguishstruturalequivaleneandredution,butthisisunneessaryhere.
3 Notation
Denition3.1. GivenasetT,aT-environment,usuallydenoted ,isapartial
mappingfromN intoT.IfN N, j
N
denotestherestritionof toN. + 0
istheenvironmentwhihmapseveryu2N to 0
(u),ifitisdened,andto (u)
otherwise.When and 0
agreeondom( )\dom(
0
), + 0
iswritten 0
.
IfT isequippedwithapartialorder,itisextendedpoint-wisetoT-environments
Denition3.2. Given aset T,rangedover by t,
~
t denotes atuple (t
1
;:::t
n ),
of lengthn0; weletT
?
denote the set ofsuh tuples.If T isequipped witha
partialorder, itisextendedpoint-wise totuples of idential length.
Denition3.3. Given a set I,(x
i : t
i )
i2I
denotes the partial mapping x
i 7!
t
i
of domain x = fx
i
;i 2 Ig. (P
i )
i2I
denotes the parallel omposition of the
proesses P
i .(D
i )
i2I
denotesthe onjuntion of the denitionsD
i .
Denition3.4. The Cartesian produt of a labelled tuple of sets A = (x
i :
s
i )
i2I
,writtenA, isthe setof tuplesf(x
i :t
i )
i2I
;8i2I t
i 2s
i g.
Denition3.5. Givenapartially ordered setT andasubsetV ofT,the one
generated byV within T,denotedby "V,isft2T;9v 2V vtg. V issaid
tobe upward-losedif andonlyif V ="V.
4 The System B(T)
This setion denes an intermediate type system for the join-alulus, alled
B(T). It is aground type system: it does not have anotion of type variable.
Instead, it has monotypes, taken to be elements of some set T, and polytypes,
merelydenedas ertainsubsetsofT.
Assumptions. Weassumegiven aset T,whose elements, usuallydenoted by
t,arealledmonotypes.T mustbeequippedwithapartialorder.Weassume
givenatotalfuntion,denoted hi,from T
?
into T,suh thath
~
tih
~
t 0
iholdsif
andonlyif
~
t 0
~
t.
Denition4.1. A polytype, usually denoted by s, is a non-empty, upward-
losedsubset of T.Let S be the set of all polytypes. We order S by , i.e. we
writess 0
ifandonly ifss 0
.
Note that and hi operateon T. Furthermore,S is dened on top of T;
thereisnowaytoinjetS bakintoT.Inotherwords,thispresentationallows
rank-1 polymorphismonly;imprediativepolymorphismisruledout.Thisisin
keepingwiththeHindley-Milnerfamilyoftypesystems[5,6℄.
Denition4.2. Amonotypeenvironment, denotedby B,isaT-environment.
A polytypeenvironment,denotedby orA, isan S-environment.
Denition4.3. ThetypesystemB (T)isgiveninFig.2.Byabuseofnotation,
intherstpremiseofruleb-Join,amonotypebinding(u:t)isimpliitlyviewed
asthe polytypebinding (u:"ftg).
Every typing judgement arriesa polytype environment on itsleft-hand
side, representing a set of assumptions under whih its right-hand side may
be used. Right-hand sides ome in four varieties. u : t states that the name
u has type t. D :: B (resp. D :: A) statesthat the denition D gives rise to
Names
b-Inst
(u)=s t2s
`u:t
b-Sub-Name
`u:t 0
t 0
t
`u:t
Denitions
b-Empty
`::
~
0
b-Join
+(~u
i :
~
t
i )
i2I
`P
`(xih~uii) i2I
.P::(xi:h
~
tii) i2I
b-Or
`D
1 ::B
1
`D
2 ::B
2
`D1;D2 ::B1B2
b-Sub-Def
`D::B BB 0
`D::B 0
b-Gen
8B2A `D::B
`D::A
Proesses
b-Null
`0
b-Par
`P `Q
`P jQ
b-Msg
`u:h
~
ti `~v:
~
t
`uh~vi
b-Def
+A`D::A +A`P
`defDinP
Fig.2.ThesystemB (T)
onstrution,dn(D).Lastly,aright-handsideof theformP simplystatesthat
theproessP iswell-typed.
Themost salient aspet of these rules is theirtreatment of polymorphism.
Ruleb-Instperformsinstantiation byallowingapolytypestobespeializedto
anymonotypet2s.Conversely,ruleb-Genperformsgeneralization byallowing
the judgement ` D :: (x
i : s
i )
i2I
to be formed if `D ::(x
i : t
i )
i2I
holds
whenever (x
i : t
i )
i2I
2 (x
i : s
i )
i2I
, i.e. whenever 8i2 I t
i 2 s
i
holds. In
otherwords,thissystemoersanextensional viewofpolymorphism:apolytype
sisdenitionallyequaltothesetofitsmonotypeinstanes.
Rulesotherthanb-Gen,b-Instandb-Defarefairlystraightforward;they
involvemonotypesonly,andaresimilartothosefoundinommontypedproess
aluli. Theonly non-syntax-diretedrules arethe subtyping rules, namelyb-
Sub-Name andb-Sub-Def.Ruleb-Gen must(andanonly)beapplied one
aboveeveryuseofb-Def,soitisnotasoureofnon-determinism.
The following lemmas will be used in the proof of Theorem 5.9. The rst
oneallowsweakeningtypejudgementsbystrengtheningtheirenvironment.The
seondoneistehnial.
Lemma4.4. If `P and 0
,then 0
`P.
Lemma4.5. Assume `(D;J.P)::B 0
andB 0
j B.Then `J.P ::B.
We establish type soundness for B(T) following the syntati approah of
WrightandFelleisen[11℄,i.e.byprovingthatB(T)enjoyssubjetredution and
progress properties.
Theorem4.6 (Subjet redution). `P andP !P 0
imply `P 0
.
Denition4.7. Aproess of the formdefD;J.P inQjuh~viis faulty ifJ
denes amessageuh~yi where~v and~y have dierent arities.
Theorem 4.8 (Progress).Nowell-typedproessisfaulty.
5 The System JOIN(X)
5.1 Presentation
LikeB(T),JOIN(X)isparameterizedbyasetofgroundtypesT,equippedwith
a typeonstrutor hi and asubtyping relation . It is further parameterized
by a rst-order logi X, interpreted in T, whose variables and formulas are
respetively alled type variables and onstraints. The logi allows desribing
subsets of T as onstraints. Provided onstraintsatisabilityis deidable, this
givesriseto atypesystemwheretypehekingisdeidable.
Our treatment is inspired by the framework HM(X) [6, 9,8℄. Our presen-
tationdiers,however,byexpliitlyviewing onstraintsasformulasinterpreted
in T, ratherthan as elements of anabstrat ylindri onstraint system. This
presentation is moreonise, and givesus the ability to expliitly manipulate
solutions of onstraints, an essential requirement in our formulation of type
soundness(Theorem5.9).Eventhoughwelose somegeneralitywithrespet to
theylindri-systemapproah,welaimtheframeworkremainsgeneralenough.
Assumptions. Weassumegiven(T;;hi) asin Set. 4.Furthermore,we as-
sumegivenaonstraintlogiXwhosesyntaxinludesthefollowingprodutions:
C::=truej=h
~
ij jC^Cj9 :C j:::
(;;::: rangeoveradenumerableset oftypevariablesV.)Thesyntaxofon-
straintsisonlypartiallyspeied;thisallowsustomonstraintforms,notknown
in thispaper,tobelaterintrodued.
ThelogiX mustbeequipped withaninterpretation in T,i.e. atwo-plae
prediate`whoserstargumentisanassignment,i.e.atotalmappingfromV
into T,andwhose seondargumentis aonstraintC.Theinterpretationmust
bestandard,i.e.satisfythefollowinglaws:
`true
`
0
=h~
1
i i (
0
)=h(~
1 )i
`
0
1
i (
0 )(
1 )
`C
0
^C
1
i `C
0
^`C
1
0 0 0
(ndenotestherestritionofto Vn .) Theinterpretationofanyunknown
onstraintformsisleftunspeied.WewriteCC 0
ifandonlyifCentailsC 0
,
i.e.ifandonlyifeverysolutionofC satisesC 0
aswell.
JOIN(X)hasonstrainedtypeshemes,whereanumberoftypevariables
areuniversallyquantied,subjettoaonstraintC.
Denition5.1. Atypeshemeisatripleofasetofquantiers , aonstraint
C, and a type variable ; we write = 8[C ℄:. The type variables in are
bound in ; type shemesare onsidered equal modulo -onversion. By abuse
of notation,atypevariable may beviewedasatype sheme 8?[true℄:. The
setof typeshemesiswritten S.
Denition5.2. A polymorphityping environment, denoted by or A, is a
S-environment. A monomorphi typing environment, denoted by B, is a V-
environment.
Denition5.3. JOIN(X) is dened by Fig. 3. Every judgement C ; ` J is
impliitly aompaniedby theside onditionthat C mustbesatisable.
JOIN(X)diersfromB(T)byreplaingmonotypeswithtypevariables,poly-
typeswithtypeshemes,andparameterizingeveryjudgementwithaonstraint
C,whihrepresentsanassumptionaboutitsfreetypevariables.RuleWeaken
allowsstrengtheningthisassumption,while9Introallowshidingauxiliarytype
variableswhihappearnowherebutintheassumptionitself.Theserules,whih
areommontonames,denitions,andproesses,allowonstraintsimpliation.
Beausewedonothavesyntaxfortypes,rulesJoinandMsguseonstraints
oftheform =h~i toenodetypestrutureintoonstraints.
Ourtreatmentofonstrainedpolymorphismisstandard.WhereasB(T)takes
an extensional view of polymorphism, JOIN(X) oers the usual, intensional
view.TypeshemesareintroduedbyruleDef,andeliminatedbyInst.Beause
impliit -onversionis allowed, every instane of Inst is able to renamethe
boundvariablesat will.
Forthesakeofreadability,wehavesimpliedruleDef,omittingtwofeatures
presentinHM(X)'s8 Introrule[6℄.First,wedonotforetheintrodutionof
existentialquantiersinthejudgement'sonlusion.InthepreseneofWeaken
and 9 Intro, doingsowould not aet theset of valid typing judgements, so
wepreferasimplerrule.Seond,wemovethewholeonstraintCinto thetype
shemes 8[C℄B, whereas it would be suient to opy only the part of C
whereatuallyours.Thisoptimizationanbeeasilyaddedbakinifdesired.
5.2 A Look atthe Generalization Condition
The most subtle (and, it turns out, questionable; see Set. 6.1) aspet of this
systemisthegeneralizationondition,i.e.thethirdpremiseofruleDef,whih
determineswhihtypevariablesmaybesafelygeneralized.Wewillnowdesribe
Denition5.4. IfB=(x
i :
i )
i2I
,then8[C℄B isthepolymorphienviron-
ment(x
i
:8[C℄:
i )
i2I
.This mustnot be onfused with the notation 8[C ℄:B,
wherethe universalquantierlies outsideof the environmentfragmentB.
Names
Inst
(u)=8[C℄:
C ; `u:
Sub-Name
C ; `u: 0
C
0
C ; `u:
Denitions
Empty
C ; `::
~
0
Join
C ; +(u~
i :~
i )
i2I
`P 8i2I C
i
=h~
i i
C ; `(xih~uii) i2I
.P::(xi:i) i2I
Or
C ; `D1 :B1 C ; `D2:B2
C ; `D1;D2::B1B2
Sub-Def
C ; `D::B 0
CB 0
B
C ; `D::B
Proesses
Null
C ; `0
Par
C ; `P C ; `Q
C ; `PjQ
Msg
C ; `u: C ; `~v:~ C=h~i
C ; `uh~vi
Def
C ; +B`(Ji.Pi) i2I
::B \fv( )=?
8i2I C8[C ℄:Bj
dn(J
i )
8[C℄Bj
dn(J
i )
C 0
; +8[C℄B`P C
0
C
C 0
; `def(Ji.Pi) i2I
inP
Common
Weaken
C 0
; `J CC 0
C ; `J
9Intro
C ; `J \fv( ;J)=?
9:C ; `J
Fig.3.ThesystemJOIN(X)(withatentativeDefrule)
Theexisteneofthese twonotations,andthequestionofwhether itislegal to
onfusethetwo,ispreiselyattheheartofthegeneralizationissue.Letushave
alookatruleDef.Itsrstpremiseassoiatesamonomorphienvironmentfrag-
mentBto thedenitionD=(J
i .P
i )
i2I
.Ifthetypevariablesdonotappear
freein ,then itissurelyorretto generalizethefragmentasawhole,i.e. to
assertthatDhastype8[C℄:B.However,thisisnolongeravalidenvironment
fragment, beause the quantier appears in front of the whole vetor; so, we
quantier down into eah binding, yielding 8[C℄B , whih is awell-formed
environmentfragment,andanbeusedto augment .
However,8[C℄B maybestritly moregeneralthan8[C℄:B, beauseit
binds separately in eah entry,rather thanone in ommon.We must avoid
this situation, whih would allow inonsistent uses of the dened names, by
properlyrestriting. (When isempty,thetwonotionsoinide.)
Toensurethat8[C℄B and8[C℄:B oinide,previousworks[4,7℄propose
syntati riteria, whih forbid generalization of a type variable if it appears
free in twodistint bindingsin B. Inan arbitraryonstraintlogi, however,a
syntatiourreneof atypevariable doesnotneessarilyonstrainitsvalue.
So,itseemspreferabletodenealogial,ratherthansyntati,riterion.Todo
so,werstgivelogialmeaningto thenotations8[C ℄B and8[C℄:B.
Denition5.5. The denotation of a type sheme = 8[C ℄: under an as-
signment,writtenJK
,isdenedas"f 0
();( 0
n=n ) ^ 0
`Cgif this
setis non-empty;itisundenedotherwise.
This denition interprets a typesheme asthe set of its instanes in T,
or, more preisely, as the upper one whih they generate. (Taking the one
aountsforthesubtypingrelationshipambientinT.)Itisparameterizedbyan
assignment,whih givesmeaningtothefreetypevariablesof.
Denition5.6. The denotation of an environmentfragment A= (u
i :
i )
i2I
under anassignment , written LAM
, isdened asJAK
=(u
i : J
i K
)
i2I
.
The denotation of 8[C℄:B under an assignment , written L8[C℄:BM
,is de-
nedas"f 0
(B);( 0
n=n ) ^ 0
`Cg.
This denition interprets environment fragments as a whole, rather than
point-wise. Thatis,LM
maps environmentfragmentstosetsoftuplesofmono-
types. A polymorphi environment fragment A maps eah name u
i
to a type
sheme
i
. Thefat thatthese typeshemesare independent ofoneanotheris
reetedinourinterpretationofAastheCartesianprodutoftheirinterpreta-
tions. Onthe other hand,8[C ℄:B is just a type sheme whose body happens
to beatuple, soweinterpretitas(theupperone generatedby)theset ofits
instanes,asinDenition 5.5.
Interpreting thenotations 8[C ℄B and 8[C℄:B within the samemathe-
matialspaeallowsustogivealogialriterionunderwhihtheyoinide.
Denition5.7. Bydenition, C 8[C℄:B 8[C℄B holds if andonly if,
under everyassignment suhthat`C,L8[C ℄:BM
L8[C℄B M
holds.
ThestrengthofthisriterionistobeindependentoftheonstraintlogiX.
ThisallowsustoproveJOIN(X)orretinapleasantgeneriway(seeSet.5.3).
Asanal remark,letuspointoutthat, independentlyofhowto dene the
generalizationriterion,there isalsoaquestionof howto apply it.It would be
orret for ruleDef to requireC 8[C℄:B 8[C℄B, asin [4℄. However,
redued,soitissuienttoseparatelyensurethatthemessageswhihappearin
eahlausehaveonsistenttypes.Asaresult,wesuessivelyapplytheriterion
to eah lauseJ
i .P
i
, byrestritingB to thesetof itsdened names,yielding
Bj
dn(J
i )
.Inthisrespet,weloselyfollowtheJoCamlimplementation[1℄aswell
asOderskyetal.[7℄.
5.3 TypeSoundness, Semi-Syntatially
ThissetiongivesatypesoundnessproofforJOIN(X)byshowingthatitissafe
withrespettoB(T).Thatis,weshowthateveryjudgementC ; `J desribes
the set of all B(T) judgementsof the form ( ` J), where ` C. Thus, we
give logial(rather than syntati) meaning to JOIN(X) judgements, yielding
a onise and natural proof. As a whole, the approah is still semi-syntati,
beauseB(T)itself hasbeenprovenorretinasyntatiway.
Denition5.8. When dened (f. Denition 5.5), JK
is a polytype, i.e. an
elementofS.ThedenotationfuntionJK
isextendedpoint-wisetotypingenvi-
ronments.Asaresult,if isanS-environment,thenJ K
isanS-environment.
Theorem5.9 (Soundness).Let(u:),(D::B), (P)stand for u:(),
D::(B), P,respetively.Then, `C andC ; `J imply J K
`(J).
Proof. Bystruturalindutiononthederivationoftheinputjudgement.Weuse
exatlythe notationsof Fig.3.In eah ase,weassumegiven somesolution
oftheonstraintwhihappearsin thejudgement'sonlusion.
CaseInst.WehaveJ K
(u)=J8[C ℄:K
3()beause`C.Theresult
followsbyb-Inst.
Case Sub-Name. The indution hypothesis yields J K
` u : ( 0
). The
seondpremiseimplies( 0
)().Applyb-Sub-Nametoonlude.
CaseEmpty.Immediate.
CaseJoin. LetB =(x
i :
i )
i2I
.Applying theindution hypothesis to the
rstpremiseyieldsJ K
+(u~
i :J~
i K
)
i2I
`P.SineJK
is"f()g,thismaybe
written J K
+(u~
i :(~
i ))
i2I
`P.(Reallthe abuseofnotationintroduedin
Denition4.3.)Theseondpremiseimplies8i2I (
i
)=h(~
i
)i .Asaresult,
byb-Join,J K
`D:(B)holds.
Case Or.Then, D is D
1
^D
2
and B is B
1 B
2
. Applying the indution
hypothesistothepremisesyieldsJ K
`D
i :(B
i
).Applyb-Ortoonlude.
Case Sub-Def. The indution hypothesis yields J K
` D : (B 0
). The
seondpremiseimplies(B 0
)(B).Applyb-Sub-Def toonlude.
CasesNull,Par.Immediate.
CaseMsg.Applyingtheindutionhypothesistothersttwopremisesyields
J K
` u: () and J K
`~v : (~ ).The last premise entails ()= h(~ )i.
Applyb-Msgto onlude.
Case Def. By hypothesis, ` C 0
; aording to the last premise, ` C
also holds. Let A = 8[C℄B. Take B 2 LAM
. Take i 2 I and dene B
i
=
Bj
dn(Ji)
. Then, B
i
is a memberof L8[C ℄Bj
dn(Ji) M
, whih, aordingto the
thirdpremise,isasubsetofL8[C℄:Bj
dn(J
i )
M
.Thus, thereexistsanassignment
0
suh that ( 0
n = n) ^ 0
` C and 0
(Bj
dn(J
i )
) B
i
. The indution
hypothesis,appliedtotherstpremiseandto 0
,yieldsJ +BK
0
`D::
0
(B).
ByLemma4.5,thisimpliesJ +BK
0 `J
i .P
i ::B
i .
Now,beause\fv( )=?,J K
0 isJ K
.Furthermore,giventheproperties
of 0
, we have JBK
0 J8[C ℄BK
= JAK
. As a result, by Lemma 4.4, the
judgementaboveimpliesJ K
+JAK
`J
i .P
i ::B
i .
Beausethisholds foranyi2I,repeateduseofb-Oryieldsaderivationof
J K
+JAK
`D::B.Lastly,beausethisholdsforanyB2LAM
,b-Genyields
J K
+JAK
`D::JAK
.
ApplyingtheindutionhypothesistothefourthpremiseyieldsJ K
+JAK
`
P.Applyb-Deftoonlude.
Case Weaken. The seond premise gives ` C 0
. Thus, the indution hy-
pothesismaybeappliedtotherstpremise,yieldingthedesiredjudgement.
Case 9 Intro. We have ` 9 :C . Then, there exists an assignment 0
suh that ( 0
n = n) ^ 0
` C. Consideringthe seond premise, wehave
J K
0
=J K
and
0
(J)=(J).Thus,applyingtheindutionhypothesistothe
rstpremiseandto 0
yieldsthedesiredjudgement.
This proof is, in our opinion, fairly readable. Infat, all asesexept Def
arenexttotrivial.
IntheDefase,wemustshowthat thedenition D hastypeJAK
, where
A=8[C℄B .BeauseB(T)hasextensionalpolymorphism(i.e.ruleb-Gen),
itsues toshowthatit haseverytypeB2JAK
. Notiehowwemustut
B into piees B
i
,orrespondingto eah lauseJ
i
,in order to makeuse ofthe
per-lausegeneralizationriterion.Weusetheindution hypothesisatthelevel
ofeahlause,thenreombinetheresultingtypederivationsusingb-Or.Notie
how we use Lemma 4.4; proving an environment strengthening lemma at the
levelofJOIN(X)wouldbemuhmoreumbersome.
Theeight non-syntax-diretedrulesare easily provenorret. Indeed,their
onlusiondenotesfewer(Sub-Name,Sub-Def,Weaken)orexatlythesame
(9Intro)judgementsinB(T)astheirpremise.Inasyntatiproof,thepresene
oftheseruleswouldrequireseveralnormalizationlemmas.
Corollary 5.10. Nowell-typedproess getsfaulty throughredution.
Proof. Assume C ; ` P. Beause C mustbesatisable, it must haveat least
onesolution.ByTheorem5.9, J K
`P holdsin B(T).Theresultfollowsby
Theorems4.6and4.8.
6 Type Inferene
6.1 Troublewith Generalization
Twosevere problems quikly arise when attempting to dene aomplete type
infereneproedureforJOIN(X).Bothareausedbythefragility ofthelogial
Def
C ; +B`(Ji.Pi) i2I
::B \fv( )=?
(9i2I jdn(Ji)j>1))=?
C 0
; +8[C℄B`P C 0
C
C 0
; `def(Ji.Pi) i2I
inP
Fig.4.DenitiveDefrule
Non-determinism.Tobeginwith,theriterionisnon-deterministi.Itstates
asuientonditionforagivenhoieoftobeorret.However,thereseems
tobe,ingeneral,nobest hoie.
Non-monotoniity.Moresubtly,strengthening theonstraintCmay,insome
ases,auseapparentorrelationstodisappear. Considertheenvironmentfrag-
ment B = (a: ;b : ) under the onstraint? = (assuming thelogi X
oerssuhaonstraint,tobereadifisnon-?,thenmustequal).Thereis
aorrelationbetweenaandb,beause,inertainases(thatis,when6=?),
and mustoinide.However,letus nowaddtheonstraint=?.Weobtain
?=^=?,whihis logiallyequivalentto =?.Itislearthat,under
thenewonstraint,aandbarenolongerorrelated.So,thesetofgeneralizable
typevariablesmayinreaseastheonstraintCismademorerestritive.
GivenadenitionD,anaturaltypeinferenealgorithmwillinfertheweakest
onstraintC under whih it is well-typed, then will useC to determinewhih
typevariables maybegeneralized.Beauseof non-monotoniity, thealgorithm
may nd apparent orrelations whih would disappear if the onstraint were
deliberatelystrengthened. However,there isno wayfor thealgorithm to guess
ifandhowitshoulddoso.
These remarks showthat it is diult to dene a omplete type inferene
algorithm,i.e.onewhih provablyyieldsasingle,mostgeneraltyping.
Previousworks[4,7℄useasimilartype-basedriterion,yetreportnodiulty
withtypeinferene.Thisleadsustoonjeturethattheseproblemsdonotarise
when subtyping is interpreted asequality and no ustom onstraint forms are
available. This may be truefor other onstraintlogis aswell. Thus, a partial
solution would be to dene a type inferene proedure only for those logis,
takingadvantageoftheirpartiularstruture toproveitsompleteness.
Inthegeneralase,i.e.underanarbitraryhoieofX,weknowofnosolution
otherthantoabandonthelogialriterion.Wesuggestreplaingitwithamuh
morenaïveone, basedon thestruture of the denition itself, ratherthan on
typeinformation.OnepossiblesuhriterionisgiveninFig.4.Itsimplyonsists
inrefusinggeneralizationentirelyifthedenitioninvolvesany synhronization,
i.e.ifanyjoin-patterndenesmorethanonename.(Itispossibletodoslightly
better,e.g.bygeneralizingallnamesnotinvolvedinasynhronizationbetween
twomessages of non-zeroarity.) It is learly safewith respet to theprevious
The new riterion is deterministi, and impervious to hanges in C, sine
it depends solely on the struture of the denition D. It is the analogue of
theso-alledvaluerestrition,suggestedbyWright[10℄,nowinuseinmostML
implementations.ExperienewithMLsuggeststhatsuharestritionistolerable
in pratie;aquikexperimentshowsthat allofthesampleodebundled with
JoCaml[1℄iswell-typedunder it.
Inthefollowing,weadopttherestritedDefruleofFig.4.
6.2 A Type Inferene Algorithm
Fig.5givesaset ofsyntax-diretedtypeinferenerules.Again,ineveryjudge-
mentC ; `
I
J,itisunderstoodthatCmustbesatisable.Therulesimpliitly
desribeanalgorithm,whoseinputsareanenvironment andasub-termu,D
or P, andwhoseoutput, inaseofsuess, isajudgement.Rulei-Orusesthe
followingnotation:
Denition6.1. The least upperbound of B
1 and B
2
, written B
1 tB
2 , is a
pair ofamonomorphi environmentandaonstraint.Itisdenedby:
B
1 tB
2
=(u:
u )
u2U
;
^
i2f1;2g;u2dom (Bi) B
i
(u)
u
whereU =dom(B
1
)[dom(B
2
)andthe typevariables (
u )
u2U
arefresh.
Following[9℄, wehavesaturatedeverytypeinferenejudgementbyexisten-
tialquantiation.Althoughslightlyverbose,thisstylenielyshowswhihtype
variablesareloalto asub-derivation,yieldingthefollowinginvariant:
Lemma6.2. IfC ; `
I
J holds,thenfv (C)fv( ;J)andfv (J)\fv( )=?.
We now prove the type inferene rules orret and omplete with respet
to JOIN(X). For the sake of simpliity, welimit the statement to the ase of
proesses(omittingthat ofnamesanddenitions).
Theorem 6.3. C ; `
I
P impliesC ; `P.Conversely,ifC ; `P holds,then
thereexistsaonstraintC 0
suhthat C 0
; `
I
P andCC 0
.
7 Disussion
JOIN(X) is losely related to HM(X) [6, 8℄, a similar type system aimed at
purelyfuntionallanguages.Italsodrawsinspirationfromprevioustypesystems
forthejoin-alulus [4,7℄, whihwerepurely uniation-based.JOIN(X)isan
attemptto bringtogetherthesetwoorthogonallinesofresearh.
Ourresultsarepartlynegative:under anaturalgeneralizationriterion,the
existeneofprinipaltypingsisproblemati.Thisleadsus,in thegeneralase,
tosuggestamoredrastirestrition.Nevertheless,thelogialriterionmaystill
Names
i-Inst
(u)=8[C℄: fresh
9:(C ^); `
I u:
Denitions
i-Empty
true; `I::
~
0
i-Join
C ; +(u~
i :~
i )
i2I
`
I
P (~
i )
i2I
;(
i )
i2I
fresh
9(
i )
i2I
:(C^
^
i2I
i
=h~
i i); `
I (x
i h~u
i i)
i2I
.P ::(x
i :
i )
i2I
i-Or
C
1
; `
I D
1 :B
1 C
2
; `
I D
2 :B
2
B;C=B1tB2
=fv(B1;B2)
9
:(C
1
^C
2
^C); `
I D
1
;D
2 ::B
Proesses
i-Null
true ; `
I 0
i-Par
C
1
; `
I
P C
2
; `
I Q
C
1
^C
2
; `
I PjQ
i-Msg
C ; `
I u:
~
C; `
I
~v:~
9:(C ^
~
C^=h~i); `Iuh~vi
i-Def
B fresh
=fv(B)
C
1
; +B`
I (J
i .P
i )
i2I
::B 0
0
=fv(B 0
) C2=9
0
:(C1^B 0
B)
if9i2I jdn(J
i
)j>1then=?else =
C3; +8[C2℄B`IP
9
:(C
2
^C
3 ); `
I def(J
i .P
i )
i2I
inP
Fig.5.Typeinferene
still beahieved, orin situations where their existene isnot essential(e.g. in
programanalysis).
Toestablishtypesafety, weinterpret typing judgements as(sets of) judge-
mentsinanunderlyingsystem,whih isgivenasyntatisoundnessproof.The
formerstep,bygivingalogialviewofpolymorphismandonstraints,aptlyex-
pressesourintuitionsaboutthesenotions,yieldingaoniseproof.Thelatteris
amatterofroutine,beausethelow-leveltypesystemissimple.Thus,bothlogi
andsyntaxareputtobestuse.Wehavebaptizedthisapproahsemi-syntati;
wefeelitisperhapsnotpubliizedenough.
Aknowledgements
AlexandreFreysuggestedtheuseofextensionalpolymorphismintheintermedi-
ofonstraint-basedtypeinfereneinHM (X),whihwashelpfulinourownom-
pleteness proof.Wewould alsoliketoaknowledgeMartin OderskyandDidier
Rém yforstimulatingdisussions.
Referenes
[1℄ Sylvain Conhon and FabrieLeFessant. Joaml: Mobile agentsfor Objetive-
Caml. InFirstInternationalSymposiumonAgentSystemsandAppliationsand
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PalmSprings,California,Otober1999.URL:http://para.inria.fr/~onhon /
publis/asa99.ps.gz.
[2℄ Sylvain Conhon and François Pottier. JOIN(X): Constraint-based type infer-
ene for the join-alulus. Long version. URL: http://pauilla.inria.fr/
~fpottier/publis/onhon-fpotti er-e sop01 -lo ng.ps .gz, April2001.
[3℄ CédriFournet andGeorgesGonthier. Thereexive hemialabstratmahine
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ofProgrammingLanguages,pages372385,1996.URL:http://pauilla.inria.
fr/~fournet/papers/popl-96.ps.g z.
[4℄ CédriFournet,LuMaranget,CosimoLaneve,andDidierRémy.Impliittyping
àla ML for the join-alulus. In8th InternationalConferene onConurreny
Theory(CONCUR'97),volume1243ofLetureNotesinComputerSiene,pages
196212, Warsaw, Poland, 1997. Springer. URL: ftp://ftp.inria.fr/INRIA/
Projets/ristal/Didier.Remy/typ ing-join .ps.g z.
[5℄ RobinMilner. Atheoryoftypepolymorphisminprogramming.JournalofCom-
puterandSystemSienes,17(3):348375,Deember1978.
[6℄ MartinOdersky,MartinSulzmann,andMartinWehr. Typeinferene withon-
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http://www.s.mu.oz.au/~sulzmann /pub liat ions/ tapo s.ps.
[7℄ MartinOdersky, ChristophZenger, Matthias Zenger, and Gang Chen. Afun-
tionalviewofjoin.TehnialReportACRC-99-016,UniversityofSouthAustralia,
1999. URL: http://lampwww.epfl.h/~zenge r/pap ers/ tr-a r-99-0 16.ps .
gz.
[8℄ Martin Sulzmann. A general framework for Hindley/Milner type systems with
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2000. URL:http://www.s.mu.oz.au/~sulzmann /publ iat ions/ diss. ps.g z.
[9℄ MartinSulzmann, Martin Müller, and Christoph Zenger. Hindley/Milner style
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[11℄ AndrewK.WrightandMatthiasFelleisen. Asyntatiapproahtotypesound-
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