Table Of ContentELECTRONIC WORKSHOPS IN COMPUTING
Series edited by Professor C.J. van Rijsbergen
David N. Turner (Ed)
Functional Programming,
Glasgow 1995
Proceedings of the 1995 Glasgow Workshop on Functional
Programming,
Ullapool, Scotland, 10-12 July 1995
Paper:
Uniqueness Typing in Natural
Deduction Style
Erik Barendsen and Sjaak Smetse
Published in collaboration with the
British Computer Society
BCS
(cid:211) Copyright in this paper belongs to the author(s)
Uniqueness Typing in Natural Deduction Style
ErikBarendsen SjaakSmetsers
UniversityofNijmegen,ComputingScienceInstitute
Toernooiveld1,6525ED Nijmegen,TheNetherlands
[email protected],[email protected]
Abstract
Wepresenttwotypesystemsforgraphrewriting: conventionaltypingand(polymorphic)uniquenesstyping. The
latterisintroducedasanaturalextensionofsimplealgebraicandhigher-orderuniquenesstyping. Thesystemsare
giveninnaturaldeductionstyleusinganinductivesyntaxofgraphdenotationswithfamiliarconstructssuchaslet
andcase.
TheconventionalsystemresemblestraditionalCurry-styletypingsystemsinfunctionalprogramminglanguages.
Uniquenesstypingextendsthiswithreferencecountinformation. Inbothtypesystems,typingispreservedduring
evaluation,andtypescanbedeterminedeffectively.
Duetothepresentformalization,thesystemcaneasilybecomparedwithotherproposalsbasedonlinearandaffine
logic.
1 Introduction
Inrecentyears,variousproposalshavebeenbroughtupassolutionstothe—atfirstsightparadoxical—desiretoallow
destructiveoperationsinafunctionalcontext.
Indeed,byadmittingtheseoperations(suchasfilemanipulations)onelosesreferentialtransparency. Theessence
ofcommonsolutions(e.g., Wadler(1990), Guzma´nandHudak(1990))istherestrictionofdestructiveoperationsto
argumentswithreferencecount .
Theuniquenesstypesystemf1orgraphrewritesystems(presentedinBarendsenandSmetsers(1993a)and (1993b))
offersthepossibilitytoindicatesuchreferencecountrequirementsoffunctionsinthetypesofthecorrespondingargu-
ments. Thesespecialso-calleduniquenesstypesareannotatedversionsoftraditionalCurry-liketypes. E.g.theoper-
ation whichwritesacharactertoafileistypedwith . Here,
(cid:2) (cid:15) (cid:15)
standWforr‘iuteniCquhea’rand‘non-unique’respectively. WriteChar:(Char ;File ) File (cid:15);(cid:2)
Uniquenesstypingcanberegardedasacombinationoflineartyping(dealingwithuniqueobjects)andtraditional
typing(fornon-uniqueobjects,withoutrestrictionsontheirreferencecounts),connectedbyasubtypingmechanism.
Infact,theparthandlinguniquenessallowsdiscardingofobjects, soitcorrespondsmorecloselytoaffinelogic, see
Blass(1992). Alogical/categoricalproposalforarelatedcombinationappearsinBenton(1994).
Thepresentpaperdescribesasimplifiedversionofthesysteminnaturaldeductionstyle,usinganinductivesyntax
forgraphexpressions. Theemphasisongraphdenotationscontraststheoriginalpresentation,whichreferreddirectly
tothenode/referencestructureof(non-inductive)graphobjects. Thegraphsyntaxissimilartotheobjectlanguagein
theequationalapproachtowardsTermGraphRewritingofAriolaandKlop(1995).
WestartwithaspecificationoftheformallanguageanddefineaCurry-like(conventional)typesystem. Aftera
verybriefintroductiontouniquenesstyping,adescriptionoftheuniquenesstypeassignmentsystemisgiven. Forboth
systemsweprovepreservationoftypingduringreductionandtheexistenceofprincipaltypes.
Theoriginaluniquenesstypesystemisrathercomplex. Toavoidthatthereadergetsentangledintechnicaldetails,
thereferenceanalysisiskeptassimpleaspossible: itdoesnottaketheevaluationorderintoaccount. Fordetails,see
BarendsenandSmetsers(1993a).
Independently, Turneretal. (1995) havedevelopedastronglyrelated systembased on -calculus. Theresearch
reportedtherehasbeenguidedbydifferentobjectives. (cid:21)
FunctionalProgramming,Glasgow1995 1
UniquenessTypinginNaturalDeductionStyle
Uniqueness typing is now part of the functional programming languageClean, see Plasmeijer and van Eekelen
(1995).
2 Syntax
Wepresentasyntaxofaformallanguagewhichincorporatessomeessentialaspectsofgraphrewriting:sharing,cycles
andpatternmatching. Theobjectsareexpressionsgeneratedbythefollowingsyntax.
let in letrec in case of
0 0
E ::= xjS(E1;:::;Ek)j x=E E j ~x=E~ E j E P~jE~;
P ::= C(x1;:::;xk):
Here rangeover(sequencesof)termvariables,and oversomesetofsymbolsoffixedarity(wewillsometimes
suggexst;i~xvelyuse forfunctionsand fordataconstrucStors). Patterns(indicatedby intheabovesyntax)aresup-
posedtobelinearF: novariableisoccuCrringmorethanonceinthesamepattern. P
Theexpressionsareinterpretedasgraphs. Sharingmanifestsitselfbymultipleoccurrencesofthesamevariables.
Sharingofcompoundstructuresisexpressedbyaletbinding,whereas introducescyclicdependencies. E.g.,the
expression letrec
let inletrec in
x=0 z =F(Cons(x;G(x;z))) z
denotesthegraph
Fjj
(cid:15)(cid:15)
Cons
~~~~~~~ BBBBB
0‘‘ G
Operationsonobjectsaredefinedusingfunctiondefinitions. Thegeneralformoftheseis
F(x1;:::;xk)=E:
Forexample,
case of
Append(‘1;‘2)= ‘1 Cons(h;t) j Cons(h;Append(t;‘2))
Nil j ‘2:
The operational semantics of our language is given by the concept of Term Graph Rewriting, see Barendregt et al.
(1987). Each expression is translated to agraph denoted by . Function definitionsaretranslated into rewrite
rules, inducing a (multisteEp) rewrite relation on graphs (cf.[[BEa]]rendsen and Smetsers (1993a)). In contrast with
AriolaandKlop(1995),werefrainfromdefin!i!ngareductionrelationontheexpressionsthemselves.
3 Conventional Typing
Typesarebuiltupfromtypevariablesandtypeconstructors.
(cid:27) ::= (cid:11)jT~(cid:27) j(cid:27)1!(cid:27)2:
Here, rangesoveralgebraictypeconstructorswhichareassumedtobeintroducedbyanalgebraictypesystem
containTingspecificationslike A
List((cid:11))=Cons((cid:11);List((cid:11)))jNil:
Thefunctionspacetypeconstructor isusedwhendealingwithhigher-orderfunctions. Wewilltreattheseonlyin
theuniquenesscase. !
FunctionalProgramming,Glasgow1995 2
UniquenessTypinginNaturalDeductionStyle
Thealgebraictypesystem givesthetypesofthealgebraicconstructors. Let
A
T(cid:11)~ =C1(cid:27)~1j(cid:1)(cid:1)(cid:1)
beadeclarationin . Then
A
A‘Ci :(cid:27)~i T(cid:11)~
Forexample,forlistsonehas
A‘Nil:List((cid:11)); A‘Cons:((cid:11);List((cid:11))) List((cid:11)):
Thefunctionsymbolsaresupplied with atype by afunction typeenvironment , containingdeclarations of the
form F
F:((cid:27)1;:::;(cid:27)k) (cid:28);
where isthearityof . Inthiscasewewrite
k F
F ‘F:~(cid:27) (cid:28):
Thesymboltypesobtainedsofararereferredtoasthestandardtypes(in )ofthesymbols. Theseareregarded
astypeschemes: othertypesareobtainedbyinstantiation,usingthefollowinFg;rAule( denotessubstitution).
[(cid:11):=(cid:26)]
F;A‘S:~(cid:27) (cid:28)
F;A‘S:~(cid:27)[(cid:11):=(cid:26)] (cid:28)[(cid:11):=(cid:26)]
Oursystemdealswithtypingstatementsoftheform
B ‘E :(cid:27);
where issomefinitesetofvariabledeclarationsoftheform calledabasis. Suchastatementisvalidifitcanbe
producBedusingthefollowingderivationrules. x:(cid:28)
B;x:(cid:27)‘x:(cid:27) (variable)
F;A‘S:~(cid:27) (cid:28) B‘E~ :~(cid:27)
(application)
B‘SE~ :(cid:28)
0
B‘E:(cid:27) B;x:(cid:27)‘E :(cid:28)
let in 0 (sharing)
B‘ x=E E :(cid:28)
0
B;~x:~(cid:27)‘Ei :(cid:27)i B;~x:~(cid:27)‘E :(cid:28)
letrec in 0 (cycle)
B‘ ~x=E~ E :(cid:28)
0
B‘E :(cid:28) A‘Ci :(cid:27)~i (cid:28) B;x~i:(cid:27)~i ‘Ei:(cid:28) (patternmatching)
case of 0 (if )
B ‘ E P~jE~ :(cid:28) Pi=Cix~i
Thisconcludesthetreatmentofexpressions. Astofunctiondefinitions,theenvironment shouldbeconsistent
withthese. Wesaythatthefunction F
F~x=E;
saywithstandardtype ,istypecorrectif
F:~(cid:27) (cid:28)
~x:~(cid:27) ‘E :(cid:28):
Typingispreservedduringreduction(theso-calledsubjectreductionproperty).
FunctionalProgramming,Glasgow1995 3
UniquenessTypinginNaturalDeductionStyle
SUBJECTREDUCTIONTHEOREM. Supposethefunctiondefinitionsaretypecorrect. Then
0 0 0
B ‘E :(cid:27); [[E]]!!g ) 9E [[E ]]=g & B ‘E :(cid:27):
Thesystemhastheprincipaltypeproperty: eachtypableexpression hasatypefromwhichallothertypesfor
canbeobtainedbyinstantiation. E E
PRINCIPALTYPE THEOREM. Let betypable. Thenthereexist (computablefrom )suchthat
andforany and E B0;(cid:27)0 E B0 ‘E :(cid:27)0
B (cid:27) forsomesubstitution .
(cid:3) (cid:3)
B ‘E :(cid:27) ) B (cid:19)B0;(cid:27) =(cid:27)0 (cid:3)
4 Simple Uniqueness Typing
Uniquenesstypingcombinesconventionaltypingandlineartyping,throughareferencecountadministration. Anen-
vironmenttype meansthat ’sargumentshould beuniquefor , i.e., shouldhavereferencecount1.
InthesamewayF,un:i(cid:27)qu(cid:15)e nes(cid:1)s(cid:1)o(cid:1)fresultsissFpecified: if ,thenaweFll-typedexpression remains
type-correct,evenif issubjecttocomputation. SGom:e(cid:1)ti(cid:1)m(cid:1) es,(cid:27)u(cid:15)niquenessisnotrequired. If F(G(E))thenstill
istypecorGre(cEt.)Thisisexpressedinasubtyperelation,suchthatroughly . OFff:e(cid:27)ri(cid:2)ng a(cid:1)n(cid:1)o(cid:1)n-unique
(cid:15) (cid:2)
Far(gGum(Een)t)ifafunctionrequiresauniqueonefails: . Thesubtyperelationis(cid:27)de(cid:20)fin(cid:27)edintermsoftheordering
(cid:2) (cid:15)
onattributes. (cid:27) (cid:2) (cid:27)
(cid:15)
(cid:20)T(cid:2)henon-unique(‘conventional’)andunique(‘linear’)typesarealsoconnectedbyacorrectionmechanism:aunique
resultmaybeusedmorethanonce,aslongasonlynon-uniquesupertypesarerequired.
Pattern matching(expressed by the caseconstruction) is an essentialaspect of term graph rewriting, causing a
functiontohaveaccesstoargumentsviadatapathsinsteadofasinglereference. Thisgivesriseto‘hiddensharing’of
objectsbyaccessviaintermediatedatanodes. Forexample,ifafunction hasaccesstoalistwithnon-uniquespine,
thelistelementsshouldalsobeconsideredasnon-uniquefor : otherfunFctionsmayaccessthemviathespine. This
effectistakenintoaccountbyarestrictionontheuniquenessFtypesofdataconstructors: theresultofaconstructoris
uniquewheneveroneofitsargumentsis. Thisuniquenesspropagationcanbeexpressedusingthe relation. Inthe
caseoflists,forexample, (cid:20)
u v u v u
Cons:((cid:11) ;List ((cid:11) )) List ((cid:11) )
iswell-attributedif (thisindeedexcludesaconstructorfor ). Notethattheattributeofanon-variable
(cid:2) (cid:15)
typeisattachedtoitvs(cid:20)topumosttypeconstructor. List (Int )
InBarendsenandSmetsers(1993a),thetypingsystemisshowntobesoundwithrespecttotermgraphrewriting,
withtheabove-mentionedinterpretationofuniqueness.
AlgebraicUniqueness Types
Wewillfirstdescribethesystemwithoutthetypeconstructor . Below, rangeoveruniquenesstypes. The
outermostattributeof isdenotedby . ! S;T;:::
ThesubtyperelatioSnisverysimplep:Stqhevalidityof dependssubtypewiseonthevalidityof with
0 0
attributesin . Onehas,forexample, S (cid:20) S u (cid:20) u
0 0
u;u S;S
u v w u0 v0 w0 iff 0 0 0
List (List (Int ))(cid:20)List (List (Int )) u(cid:20)u;v (cid:20)v ;w(cid:20)w :
Inordertoaccountformultiplereferencestothesameobjectweintroduceauniquenesscorrection: iftheobjectin
questionhastype ,thenonlynon-uniqueversionsof maybeused. Given ,weconstructthesmallestnon-unique
supertypeof : S S S
S
u (cid:2)
[(cid:11) ] = (cid:11) ;
u (cid:2)
[T S~] = T S~:
FunctionalProgramming,Glasgow1995 4
UniquenessTypinginNaturalDeductionStyle
Thelastclausepossiblyintroducestypeslike . ContrastingTurneretal.(1995),weallowthesetypesinour
(cid:2) (cid:15)
system. Thisisharmlesssincethese‘inconsisLtiesntt’(tIynpte)shavenoinhabitants(forexample,thereisno yielding
type ). Cons
(cid:2) (cid:15)
CLyicsltic(oInbtje)cts(withtheirinherentsharing)aretreatedbycorrectingbothinternalandexternalreferencestotheir
roots.
Thenotionofstandardtypeisadaptedinthefollowingway. Asbefore,standardtypesoffunctionsymbols(
)arecollectedinanenvironment . F :
S~ ATscanbeseenfromthe exampleF,thereareseveralstandardtypesforeachdataconstructor. Saythealgebraic
environment contains List
A
T(cid:11)~ =C1(cid:27)~1j(cid:1)(cid:1)(cid:1)
Asetofstandardtypesfor consistsofattributedversionsoftheconventionaltype ,suchthat
(1) multipleoccurrenceCsoifthesamevariableandoftheconstructor havethesam(cid:27)~eiu niqTu(cid:11)~enessattributethrough-
outeachversion; T
(2) eachversionisuniquenesspropagating;
(3) thesetcontainsatmostoneversionforeachattributedvariantof .
ThisleavessomefreedomastothechoiceofattributesonpositionsnotcorTre(cid:11)~spondingto . BarendsenandSmetsers
(1993a)offerageneralmethodforconstructingareasonablesetofstandardtypesforeaTc;h(cid:11)~constructor. Inmostcases
(like , see above), however, the choice of attributes of fixes those for the . From now on we assume that
standaLridsttypeshavebeendetermined. ForthesestandardtypTe(cid:11)~s weset (cid:27)~i asbefore.
Symboltypesareinstantiatedviatherule S~ T A‘C:S~ T
F;A‘S:S~ T p(cid:11)q=pRq (instantiation)
F;A‘S:S~[(cid:11):=R] T[(cid:11):=R]
Auniquenesstypingstatement(in )hastheform
F;A
B ‘E :S:
Inourlanguage,sharingappearsasmultipleoccurrencesofthesamevariable. Likeinlinearlogic,wehavetobeprecise
whendealingwithbasesusedfortypingsubterms: thedenotation standsforadisjointunionofbases.
Therulesfortypeassignmentarethefollowing. B1;B2
x:S‘x:S (variable)
F;A‘S:S~ T Bi ‘Ei :Si
(application)
B~ ‘SE~ :T
0 0
B‘E :S B ;x:S‘E :T
0 let in 0 (sharing)
B;B ‘ x=E E :T
0 0
Bi;~x:[S~]‘Ei :Si B ;~x:[S~]‘E :T
0 letrec in 0 (cycle)
B~;B ‘ ~x=E~ E :T
0 0
B‘E :T F;A‘Ci:S~ T B ;x~i :S~i ‘Ei:T (patternmatching)
0 case of 0 (if )
B;B ‘ E P~jE~ :T Pi =Cix~i
0
B‘E:S S(cid:20)S
0 (subsumption)
B‘E:S
FunctionalProgramming,Glasgow1995 5
UniquenessTypinginNaturalDeductionStyle
Additionally,wehavethefollowing‘structuralrules’. Weakeningexpressesthatonecandiscard(uniqueornon-
unique)input. Thecontractionruledealswithcorrectionoftypesofsharedobjects: multipleuseofthesameobjectis
allowedaslongasonlynon-uniquevariantsofthetypesareused.
B‘E :T
(weakening)
B;x:S‘E:T
B;y:[S];z:[S]‘E:T
(contraction)
B;x:S ‘E[y:=x;z:=x]:T
Likeintheconventionalcase, shouldbeconsistentwiththefunctiondefinitions. Afunction
F
F~x=E;
saywithstandardtype ,istypecorrectif
F:S~ T
~x:S~ ‘E :T:
Higher-OrderUniqueness Types
Wewillnowdescribetheincorporation ofhigher-orderfunctions. Higher-order functions giveriseto partial(often
calledCurried)symbolapplications. Infunctionalprogramminglanguages,theseapplicationsareusuallywrittenas
(with )),denotingthefunction
FE1(cid:1)(cid:1)(cid:1)Ek k <arity(F
(cid:21)~x:F(E1;:::;Ek;xk+1;:::;xarity(F)):
Ifthesepartialapplicationscontainuniquesubexpressionsonehastobecareful. Consider,forexample,afunction
FIfowniethatlylopwesFth:a(t(cid:27)th(cid:15);is(cid:28)a(cid:2)p)p lic(cid:27)a(cid:15)tioinntihseuaspepdlmicaotrieonthFanEo.nCcele,aornlye,cthanenroestuglutatyrapneteoeftthhiesaarpgpulmiceantitoni(swoifththteypfoerm)(cid:28)r(cid:2)em!ua(cid:27)in(cid:15)s.
(cid:15)
uniqueduringevaluation. E.g.if ispassedtoafunction ,theoccurrencEesof willr(cid:27)esultintwo
applicationsof sharingthesamFeEexpression . ApparentlGy,(tfh)e=(fe0x;pfre1s)sionisnecessarilyunfique: itsreference
countshouldneFverbecomegreaterthan1. TheEreareseveralwaysFtEopreventsuchanexpressionfrombeingcopied.
For instance, one might introduce a new uniqueness attribute, say , for any unique object that does not coerce to
4
a non-unique variants. This has been described in Barendsen and Smetsers (1993a). An alternative solution is the
administrationofso-calledregionsintroducedbyReynolds(1995).
Insteadof introducinganewattribute, thepresentpaperconsidersthe constructorincombinationwiththe
(cid:15)
attributeasspecial: itisnotpermittedtodiscarditsuniqueness. Theleadst!oanadjustmentofthesubtypingrelations
aswellasofthetypecorrectionoperator .
Astothesubtypingrelation,theattribu[(cid:1)t]esofcorrespondingoccurrencesofthe constructors(intheleft-handand
theright-handsideofaninequality)shouldbeidentical. Thesameisrequired(toe!nsuresubstitutivityofthesubtyping
relation)forvariables.
Thesubtypingrelationbecomesinherentlymorecomplexthaninthealgebraiccasebecauseoftheso-calledcon-
travarianceof initsfirstargument:
!
u 0 u 0 0 0
S!S (cid:20)T !T , T (cid:20)S; S (cid:20)T :
Since may appear inthedefinitionsofalgebraictypeconstructors, theseconstructorsmayinherittheco-orcon-
travari!antsubtyping behaviour with respectto their arguments. Wecanclassify the‘sign’ of thearguments of each
typeconstructoras (positive,covariant), (negative,contravariant)or (bothpositiveandnegative). Ingeneral
thisisdonebyanaly(cid:8)zingthe(possiblymutua(cid:9)llyrecursive)algebraictypede>finitionsbyafixedpointconstruction,with
basis . Thesubtypingrelation isdefinedbyinduction.
sign(!)=((cid:9);(cid:8)) (cid:20)
FunctionalProgramming,Glasgow1995 6
UniquenessTypinginNaturalDeductionStyle
Notation.
(cid:8) (cid:9)
S (cid:20) T , S (cid:20)T; anSd(cid:20) T , T (cid:20)S;
> (cid:8) (cid:9)
S (cid:20) T , S (cid:20) T forSeac(cid:20)h T;
Nowset S~ (cid:20)~s T~ , Si (cid:20)si Ti i:
u v
(cid:11) (cid:20)(cid:11) , u=v;
and
TuS~ (cid:20)TvT~ , u(cid:20)v S~ (cid:20)sign(T) T~;
u 0 v 0 and (cid:9) and 0 (cid:8) 0
S!S (cid:20)T !T , u=v S (cid:20) T S (cid:20) T :
E.g.
u v w u0 v0 w0 iff 0 0 0
Int !(cid:12) (cid:20)Int !(cid:12) u (cid:20)u;v =v ;w =w :
Adjustingthetypecorrectionoperatoriseasy: correctionof (cid:15) typessimplyfails. Thustheoperator becomesa
partialfunction: ! [(cid:1)]
if ,
u (cid:2)
[(cid:11) ] = (cid:11) u=(cid:2)
u (cid:2)
[T S~] = T S~;
u (cid:2) if ,
[S!T] = S!T u=(cid:2)
(undefined) inallothercases.
[S] = "
Partialapplicationscanbeincorporatedinourformalsystemasfollows. Anapplication iswrittenas
(so ). Moreoverweaddtheapplicationoperator tooursynFtaEx1w(cid:1)it(cid:1)h(cid:1)Eefkfect(roughly)
Fk(E1;:::;Ek) Farity(F) =F Ap
Ap(Fk(E~);E0)=Fk+1(E~;E0):
Thetypeof isdefinedintermsthetypeof bythefollowingrule.
Fk Fk+1
0
F;A‘Fk+1 :(S~;T) T u(cid:20)(cid:5)pS~q
u 0 (Curry)
F;A‘Fk :S~ (T !T )
Here, stands for the so-called cumulativeuniqueness attributeof : itequals whenever some is , and
otherw(cid:5)is~ue. ~u (cid:15) ui (cid:15) (cid:2)
Thetypingrulefor isdefinedstraightforwardly.
Ap
u 0 0
B‘E:S!T B ‘E :S (curriedapplication)
0 0
B;B ‘Ap(E;E ):T
5 Polymorphic Uniqueness Typing
Inordertodenoteuniquenessschemes,weextendtheattributesetwithattributevariables( ). Thisincreases
the expressivity of the type system. Moreover, attribute polymorphism is needed for thead;ebt;ear1m;i:n:a:tion of principal
types. Uniquenessconstraintsareindicatedby(finite)setsofattributeinequalitiescalledcoercionenvironments. For
example,thestandardtypeofthesymbol isnowexpressedby
Cons
a b a b a
Cons:((cid:11) ;List ((cid:11) )) List ((cid:11) ) j b(cid:20)a:
Allnotionsoftheprevioussection(typeenvironment,subtyping,typederivation)arere-definedrelativetocoercion
environments.
FunctionalProgramming,Glasgow1995 7
UniquenessTypinginNaturalDeductionStyle
Astotheattributerelation ,wesaythat isderivablefromthecoercionenvironment (notation )
if canbeproduced(cid:20)bytheaxiomsu(cid:20)v (cid:0) (cid:0)‘u(cid:20)v
(cid:0)‘u(cid:20)v
if
(cid:0)‘u(cid:20)v (u(cid:20)v)2(cid:0);
(cid:15)
(cid:0)‘u(cid:20)u; (cid:0)‘u(cid:20)(cid:2); (cid:0)‘ (cid:20)u
andrule
(cid:0)‘u(cid:20)v (cid:0)‘v (cid:20)w
:
(cid:0)‘u(cid:20)w
Thisdenotationisextendedtofinitesetsofinequalities: if foreach . Wesaythat is
0 0
consistentif . (cid:0)‘(cid:0) (cid:0)‘u(cid:20)v (u(cid:20)v)2(cid:0) (cid:0)
(cid:15)
Therelati(cid:0)on6‘(cid:2)is(cid:20)extendedtotypes. Onehas,forexample,
(cid:20)
u v w x u0 v0 w0 x0 iff 0 0 0 0
(cid:0)‘List (Int !(cid:11) )(cid:20)List (Int !(cid:11) ) (cid:0)‘u(cid:20)u;v (cid:20)v;w=w ;x=x:
Thecontextrulesbecome
0
F;A‘S:S~ T j(cid:0) (cid:0) ‘(cid:0)(attributeinstantiation)
0
F;A‘S:S~ T j(cid:0)
F;A‘S:S~ T j(cid:0) (cid:0)‘p(cid:11)q=pRq(instantiation)
F;A‘S:S~[(cid:11):=R] T[(cid:11):=R]j(cid:0)
0
F;A‘Fi+1 :(S~;T) T j(cid:0) (cid:0)‘u(cid:20)pS~q
u 0 (Curry)
F;A‘Fi :S~ (T !T )j(cid:0)
Thecorrectionoperationisalsorelativized: forexample, u (cid:2) if .
Therulesforproducingtypingstatements ar[eSo!btaTin]e(cid:0)d=froSm!thTepre(cid:0)vi‘ouuso=ne(cid:2)s,roughlybyreplacing
by . Theapplicationrulebecomes B ‘(cid:0) E :S
‘ ‘(cid:0)
F;A‘S:S~ T j(cid:0) Bi ‘(cid:0) Ei:Si
(application)
B~ ‘(cid:0) SE~ :T
andthesubsumptionrule
0
B ‘(cid:0) E:S (cid:0)‘S(cid:20)S
0 (subsumption)
B ‘(cid:0) E:S
The environments in the deduction system are global in the sense that they may contain auxiliary uniqueness
constraints(attributein(cid:0)equalitiesappearinginsomederivationstep,butnotoccurringinthefinalbasisandtype). In
ordertoeliminatethesesuperfluousconstraintsintheconclusionofadeduction,werefinethenotionofderivability.
By wedenotethat isderivableforsome‘conservative’extension of (withrespectto
attriBbu‘tesEo:ccSujrr(cid:0)ingin ,theenvBiron‘m(cid:0)e0nEt :Sprovesthesamestatementsastheoriginal ). A(cid:0)lt0erna(cid:0)tively,onecould
0
extendthedeductionsy(cid:0)stemwithrulesfor(cid:0)explicitsimplificationofenvironments. (cid:0)
Typecorrectnessoffunctiondefinitionscannowbeexpressedasfollows. Thefunction
F~x=E;
FunctionalProgramming,Glasgow1995 8
UniquenessTypinginNaturalDeductionStyle
saywith -standardtype ,istype-correctif
F S~ T j(cid:0)
~x:S~ ‘E :T j(cid:0):
Thesubjectreductionpropertyalsoholdsfortheuniquenesstypesystem,cf.BarendsenandSmetsers(1993a).
Weareinterestedinattributedversionsofconventionaltypederivations:given and andaderivationshowing
,wewishtoconstructauniquenessvariant(intheobvioussense:assigniBng;tEhesam(cid:27)eunderlyingconventional
Btyp‘esEto:th(cid:27)esubexpressionsof )ofthisderivation,yielding with and .
E B0;S0;(cid:0)0 jB0j=B;jS0j=(cid:27) B0 ‘E :S0j(cid:0)0
PRINCIPALATTRIBUTIONTHEOREM. Supposeagivenderivationof isattributable. Thenthereexistsan
attribution, (computablefromtheconventionalderivation, B ‘aEnd:(cid:27))suchthatforanyotherattribution,
,B0;S0;(cid:0)0 B;E (cid:27)
B1;S1;(cid:0)1
(cid:5) (cid:5) (cid:5)
B1 =B0; S1 =S0; (cid:0)1 ‘(cid:0)0
forsomeattributesubstitution .
(cid:5)
Thisdecidabilityresult(intheoriginalgraphframework)hasbeenaddressedinBarendsenandSmetsers(1995).
Example
The(higher-order)function isdefinedasusual:
Map
case of
Map(f;‘) = ‘ Cons(h;t) j Cons(Ap(f;h);Map(f;t))
Nil j Nil:
say
= E(f;‘); :
Then
a a d b c a e b
h:(cid:11) ;f:(cid:11) !(cid:12) ;t:List ((cid:11) )‘Cons(Ap(f;h);Map(f;t)):List ((cid:12) )j(cid:2)(cid:20)d: ((cid:3))
Moreover
a d b c a e b
f:(cid:11) !(cid:12) ;‘:List ((cid:11) )‘E(f;‘):List ((cid:12) )j(cid:2)(cid:20)d:
Thisalsovalidates((cid:11)a!(cid:2) (cid:12)b;Listc((cid:11)a)) Liste((cid:12)b)asstandardtypeforMapusedin((cid:3)).
6 Concluding Remarks
WehavepresentedClean’suniquenesstypingsysteminnaturaldeductionstyle. Theoriginalsystemshavebeenshown
to bedecidable in thesensethatprincipaltypes can bedetermined effectively, seeBarendsen and Smetsers(1993a)
and (1995). Thepresentframeworknotonlyprovidesmoredirectproofsoftheseresults,butalsofitscloselytomore
commonmethodsbasedonsubstructurallogics. TherelationwiththeapproachofBenton(1994)issubjecttofurther
research.
References
Ariola, Z.M. and J.W. Klop (1995). Equational term graph rewriting, Technical Report CS-R9552, Centrum voor
WiskundeenInformatica(CWI),ComputerScience/DepartmentofSoftwareTechnology.
Barendregt,H.P.,M.C.J.D.vanEekelen,J.R.W.Glauert,J.R.Kennaway,M.J.PlasmeijerandM.R.Sleep(1987).Term
graphreduction,in:J.W.deBakker,A.J.NijmanandP.C.Treleaven(eds.),ProceedingsoftheConferenceonPar-
allelArchitecturesandLanguagesEurope(PARLE)II,Eindhoven,TheNetherlands,LectureNotesinComputer
Science259,Springer-Verlag,Berlin,pp.141–158.
Barendsen,E.(1995). TypesandComputationsinLambdaCalculiandGraphRewriteSystems,Dissertation,Univer-
sityofNijmegen.
FunctionalProgramming,Glasgow1995 9
