Polymorphic type inference and assignment

Transcription

1 Proc. 18th Symp. Principls of Programming Languags, 1991, pags Polymorphic typ infrnc and assignmnt Abstract Xavir Lroy Ecol Normal Supériur W prsnt a nw approach to th polymorphic typing of data accpting in-plac modification in ML-lik languags. This approach is basd on rstrictions ovr typ gnralization, and a rfind typing of functions. Th typ systm givn hr lads to a bttr intgration of imprativ programming styl with th purly applicativ krnl of ML. In particular, gnric functions that allocat mutabl data can safly b givn fully polymorphic typs. W show th soundnss of this typ systm, and giv a typ rconstruction algorithm. 1 Introduction Polymorphic typ disciplins originat in th study of λ- calculus and its connctions to constructiv logic [7, 14], so it is no surpris it fits vry nicly within purly applicativ languags, without sid ffcts. Howvr, polymorphism bcoms problmatic whn w mov toward convntional imprativ languags (Algol, Pascal), and allow physical modification of data structurs. Th problm appard at an arly stag of th dsign of ML [8, p. 52], whn assignmnt oprators wr providd for th primitiv data typs of rfrncs and vctors. Considr th following xampl, in ML 1 : lt r = rf [ ] in r := [1]; if had(!r) thn... ls... If w naivly giv typ α. α list rf to th rfrnc r, w can first us it with typ int list rf, and stor in it th list with 1 as singl lmnt an int list, indd. Givn its typ, w can also considr r as having typ bool list rf, hnc had(!r) has typ bool, and th if statmnt is wll-typd. Howvr, had(!r) valuats to 1, which is not a valid boolan. This xampl shows Authors addrss: B.P.105, L Chsnay, Franc. xlroy@margaux.inria.fr, wis@margaux.inria.fr. 1 Survival kit for th radr unfamiliar with ML: [ ] is th (polymorphic) mpty list, [a 1 ;... ; a n ] th list with lmnts a 1... a n. rf x allocats a nw rfrnc (indirction cll), initializd to x. r := x updats th contnts of rfrnc r by x.!r rturns th currnt contnts of rfrnc r. Pirr Wis INRIA Rocquncourt that physical modification of data compromiss typ safty, sinc it can invalidat static typing assumptions. As dmonstratd hr, th us of polymorphic mutabl data (that is, data structurs that can b modifid in plac) must b rstrictd. An obvious way to tackl this problm, usd in arly implmntations of ML [3], is to rquir all such data to hav monomorphic, statically-known typs. This rstriction trivially solvs th problm, but it also maks it impossibl to writ polymorphic functions that crat mutabl valus. This fact has unfortunat consquncs. A first drawback is that it is not possibl to provid gnric, fficint implmntations of most data structurs (vctors, hash tabls, graphs, B-trs,... ), as thy rquir physical modification. Evn a trivial function such as taking a vctor of an arbitrary typ and rturning a copy of it is not wll-typd with th policy abov, sinc it crats a vctor with a statically unknown typ. Anothr drawback is that polymorphic mutabl valus ar prohibitd vn if thy ar not rturnd, but usd for intrnal computation only. As a consqunc, most gnric functions cannot b writtn in an imprativ styl, with rfrncs holding intrmdiat rsults. Considr th familiar map functional: lt rc applicativ map f l = if null l thn [ ] ls f (had l) :: applicativ map f (tail l) Hr is an altrnat implmntation of map in imprativ styl: lt imprativ map f l = lt argumnt = rf l and rsult = rf [ ] in whil not (null!argumnt) do rsult := f (had!argumnt) ::!rsult; argumnt := tail!argumnt don; rvrs!rsult Som ML typ systms rjct imprativ map as illtypd. Othrs giv it a lss gnral typ than its purly applicativ vrsion. In any cas, th imprativ vrsion cannot b substitutd for th applicativ on, Pag 1

2 vn though thy hav xactly th sam smantics. As dmonstratd hr, th programming styl (imprativ vs. applicativ) intrfrs with th typ spcifications. This clarly gos against modular programming. Som nhancmnts to th ML typ systm hav bn proposd [4, 16, 17, 1], that wakn th rstrictions ovr polymorphic mutabl data. Standard ML [11] incorporats som of ths idas. Ths nhancd typ systms mak it possibl to dfin many usful gnric functions ovr mutabl data structurs, such as th function that copis a vctor. Howvr, ths systms ar still not powrful nough: thy fail to infr th most gnral typ for th imprativ map xampl abov; and thy do not work wll in conjunction with highr-ordr functions. Bcaus of ths shortcomings, Standard ML dos not provid adquat support for th imprativ programming styl. This is a major waknss for a gnral-purpos programming languag. In this papr, w prsnt a nw way to typchck mutabl data within a polymorphic typ disciplin. Our typ systm is a simpl xtnsion of th on of ML. It prmits typ rconstruction and posssss th principal typ proprty. It rquirs minimal modifications to th ML typ algbra. Yt it dfinitly improvs th support for imprativ programming in ML. It allows gnric functions ovr mutabl data structurs to hav fully polymorphic typs. It is powrful nough to assign to a function writtn in imprativ styl th sam typ as its purly applicativ countrpart. For xampl, in this systm, th imprativ implmntation of map cannot b distinguishd from th usual, applicativ on, as far as typs ar concrnd. Th rmaindr of th papr is organizd as follows. In sction 2, w introduc informally our typ systm, and show th nd for a mor prcis typchcking of functions. Sction 3 formalizs th idas abov. W stat th typchcking ruls, show thir soundnss, and giv a typ rconstruction algorithm. Sction 4 brifly compars our approach with prvious ons. W giv a fw concluding rmarks in sction 5. 2 Informal prsntation In this sction, w informally introduc our typing disciplin for mutabl data, focusing on rfrncs to b mor spcific. Unlik th Standard ML approach, w do not attmpt to dtct th cration of polymorphic rfrncs. What w prohibit is th us of a rfrnc with two diffrnt typs. Th only way to us a valu with svral typs is to gnraliz its typ first, that is, to univrsally quantify ovr som of its typ variabls. (In ML, th only construct that gnralizs typs is th lt binding.) Hnc, what w rstrict is typ gnralization: w nvr gnraliz a typ variabl that may b fr in th typ of a rfrnc. Such a typ variabl is said to b dangrous. Not gnralizing dangrous typ variabls nsurs that a mutabl valu always posssss a monotyp. It rmains to dtct dangrous typ variabls at gnralization-tim. Though this suggsts a complx and xpnsiv static analysis, this turns out not to b th cas: dangrous variabls can b dtrmind by mr xamination of th typ bing gnralizd, as w shall now illustrat. 2.1 Datatyps Considr th xampl givn in introduction: lt r = rf [ ] in r := [1]; if had(!r) thn... ls... Th xprssion rf [ ] has typ A = α list rf. This typ is a rf typ, and α is fr in it, hnc α is dangrous in A. Th lt construct dos not gnraliz α, hnc α gts instantiatd to int whn typing r := [1], and typing if had(!r)... lads to a typ clash. Rfrncs may b mbddd into mor complx data structurs (pairs, lists, concrt datatyps). Fortunatly, th typ of a data structur contains nough information to rtriv th typs of th componnts of th structur. For instanc, a pair with typ A B contains on valu of typ A and on valu of typ B. Thrfor, any variabl which is dangrous in A or in B is also dangrous in A B. For instanc, in lt r = ([ ], rf [ ]) in th xprssion ([ ], rf [ ]) has typ α list β list rf, whr β is dangrous but not α. Hnc in, variabl r has typ α. α list β list rf. Th tratmnt of usr-dfind datatyps is similar. Paramtrlss datatyps cannot contain polymorphic data, so thr is no dangrous variabl in thm. Paramtrizd datatyps com in two flavors: thos which introduc nw rf typs (such as typ α foo = A B of α rf ), and thos which don t. Th formr ar tratd lik rf typs: all variabls fr in thir paramtr(s) ar considrd dangrous. Th lattr ar tratd lik product typs: thir dangrous variabls ar th dangrous variabls of thir paramtr(s). 2.2 Functions Function typs ar tratd spcially. Th rason is that a valu with typ A B dos not contain a valu of Pag 2

3 typ A, nor a valu of typ B, in contrast with rgular datatyps such as A B or A list. Hnc it sms thr ar no dangrous variabls in typ A B, vn if A or B contain dangrous variabls thmslvs. For instanc, th function lt mak rf = function x rf x has typ α α rf, and α is not dangrous in it, so it is fully polymorphic. It is actually harmlss by itslf. What s harmful is to apply mak rf to a polymorphic argumnt such as [ ], bind th rsult with a lt construct, and us it with two diffrnt typs. But this is not possibl in th proposd typ systm, sinc mak rf [ ] has typ β list rf, and this typ will not b gnralizd, as β is dangrous in it. In our approach, gnric functions that crat and rturn mutabl objcts ar givn vry gnral typs. Typ safty is nsurd by controlling what can b don with th rsult of thir application, as dscribd abov. Th analysis abov is basd solly on th typ of th function rsult. Hnc, th usag of a polymorphic function is unrstrictd if th function dos not rturn any rfrncs, as witnssd by th absnc of rf typs in its codomain typ. This holds vn if th function allocats rfrncs with statically unknown typs for intrnal purposs, but dos not rturn thm. For instanc, this is th cas for th imprativ map functional givn in th introduction: it is givn typ α, β. (α β) α list β list, th vry sam typ as its purly applicativ countrpart applicativ map. Th imprativ map functional can b substitutd for applicativ map in any contxt. In particular, it can b applid to highly polymorphic argumnts: th xprssion imprativ map (function x x) [ ] is wll-typd, and rturns th fully polymorphic mpty list, vn though two rfrncs wr cratd with typ α list rf. Th typ systm guarants that ths rfrncs ar usd consistntly insid th function, and ar not xportd outsid. Anothr strngth of this typ-basd analysis is its good handling of highr-ordr functions and partial applications. Considr th partial application of applicativ map to mak rf : our typ systm corrctly rcogniz it as harmlss, and givs it th fully polymorphic typ α. α list α rf list. This is not th cas for ML typ systms that attmpt to control th cration of rfrncs (s sction 4). 2.3 Functions with fr variabls Th carful radr may hav noticd a flaw in th discussion abov: w hav nglctd th fact that a function may possss fr variabls. Suppos that a function f has a fr variabl r which is bound to a rfrnc outsid th function. Function f can accss and updat th rfrnc, yt th typ of r dos not ncssarily appar in th typ of f. A classic xampl is th functional prsntation of rfrncs as a pair of functions, on for accss, th othr for updat: lt functional rf x = lt r = rf x in (function ( )!r), (function z r := z) Th xprssion functional rf [ ] has typ (unit α list) (α list unit), whr no typ variabl is dangrous, so it is fully polymorphic, yt it braks typ safty just as rf [ ] dos. Th problm is that th typs of th fr variabls of a function do not appar in th typ of th function. In this contxt, it is usful to think of functions as closurs. Closurs, in contrast with any othr data structur, ar not adquatly dscribd by thir (functional) typ: w do not know anything about th typs of th valus containd in th nvironmnt part of a closur. Our solution is to kp track of what is insid a closur. W associat with any function typ a st of typs, th typs of all variabls fr in th function. W could put this st of typs as a third argumnt to th arrow typ constructor. For tchnical rasons, w find it mor convnint to add an xtra lvl of indirction in rcording this st of typs. Thrfor, ach function typ is adornd with a labl. Labls ar writtn L, M, and labld function typs ar writtn A L B. Sparatly, w rcord constraints on thos labls. Constraints hav th format A L, maning that objcts of typ A ar allowd to occur in nvironmnts of typ L. An nvironmnt of typ L is not rquird to contain a valu of typ A. It is not allowd to contain a valu of a typ B unlss B L is on of th rcordd constraints. This way, function typ labls allow typings of th nvironmnt parts of closurs. For instanc, functional rf now has typ: α L (unit M α) (α N unit) with α rf M, α rf N hnc functional rf [ ] has typ: (unit M β list) (β list N unit) with β list rf M, β list rf N thus rvaling th prsnc of a polymorphic rfrnc in ach function of th pair, and prvnting th gnralization of β. Th rul is that in a functional typ Pag 3

4 A M B, a variabl α is dangrous iff thr xists a constraint C M with α dangrous in C. This typing of closurs givs a prcis account of th intrlaving of intrnal computation and paramtr passing in (currid) functions, and of th possibl data sharing btwn invocations. For instanc, it succds in distinguishing th following two functions: function ( ) rf [ ] : unit L α list rf lt r = rf [ ] in function ( ) r : unit L α list rf with α list rf L Th formr is harmlss, sinc it rturns a frsh rfrnc ach tim, so it can b gnralizd. Th lattr always rturn th sam rfrnc, which is thrfor shard btwn all calls to th function. It must rmain monomorphic, which is indd th cas, sinc α is a dangrous variabl in its typ. Bfor prsnting th typ systm formally, w now giv th main intuitions bhind th typing of closurs. Constraints ar synthsizd during th typing of abstractions, as follows: whn giving typ A M B to th abstraction = function x..., for ach variabl y fr in, w look up th typ C y of y in th typing nvironmnt and rcord th constraint C y M. It is always saf to add nw constraints, sinc th constraints ovr a labl L ar intndd to giv an uppr bound for what can go insid closurs of typ L. (This may lad to lss gnral typs, howvr, sinc mor variabls will b dclard dangrous.) Unifying two labld function typs A L B and C M D is asy: it suffics to idntify both labls L and M, rsulting in a singl labl which bars th prvious constraints on M as wll as thos on L. For instanc, assuming f : int L int, th xprssion if... thn f ls lt z = 1 in function x x + z has typ int L int as wll, with th additional constraint that int L. Finally, to giv th most gnral typ to functionals, w must b abl to gnraliz ovr labls, in th sam way as w gnraliz ovr rgular typ variabls. Considr th functional: function f 2 + (f 1) It must b possibl to apply it to any function mapping intgrs to intgrs, whatvr its closur may contain. Yt if w giv it th typ (int L int) M int without gnralizing ovr L, w could only apply it to functions without fr variabls, assuming thr is no constraint ovr L in th currnt nvironmnt. Instad, th right typing is L. (int L int) M int. In mor complx situations, th currnt nvironmnt contains constraints ovr th labl to b gnralizd. Ths constraints ar dischargd in th typ schma, and rintroducd at spcialization tim. Typ schmas thrfor hav th format V 1... V n. A with Γ, whr th V i ar ithr labls or typ variabls, and Γ is a squnc of constraints. 3 Formalization In this sction, w formaliz a calculus basd on th idas abov. 3.1 Syntax Th languag w considr is th cor ML languag, λ- calculus plus a distinguishd lt construct. W shall assum a built-in int typ, with intgr constants. Th stor is prsntd through th typ A rf of rfrncs to a trm of typ A, and th oprations rf(a), to allocat a nw rfrnc to trm a,!a to gt th contnts of th rfrnc a, and a := b to updat th contnt of a by b. W assum givn a countabl st Var of trm variabls, with typical lmnts x, y. In th following, i rangs ovr intgrs. Th syntax of trms, with typical lmnts a, b, is as follows: a ::= x i λx. a (b a) lt x = a in b rf(a)!a a := b 3.2 Typchcking Typ xprssions, with typical lmnts A, B, hav th following syntax: A ::= X int A L B A rf In th dfinition abov, X stands for a typ variabl, ranging ovr a givn countabl st TVar. Function typs A L B ar annotatd by a labl L, takn from a countabl st Lbl. Labls ar distinct from trm variabls and typ variabls. Typ schmas, with typical lmnt Σ, ar composd of typ xprssions with som typ variabls and som labls univrsally quantifid. Thy also contain a squnc of constraints Γ. Constraints hav th format Σ L. Σ ::= A V 1... V n. A with Γ Γ ::= ɛ Σ L, Γ V ::= X L Pag 4

5 (int) (varspc) (fun) (app) (ltgn) (rf) (assign) E i : int with Γ E(x) = X 1... X n, L 1... L m. A with Γ E x : A{X i B i, L j M j } with Γ{X i B i, L j M j } Γ E[x A] b : B with Γ for all y fr in λx. b, (E(y) L) Γ E λx. b : A L B with Γ E b : A L B with Γ E a : A with Γ E (b a) : B with Γ E a : A with Γ (Σ, Γ ) = Gn(A, E, Γ) E[x Σ] b : B with Γ Γ E lt x = a in b : B with Γ Γ E a : A with Γ E a : A rf with Γ (drf) E rf(a) : A rf with Γ E!a : A with Γ E a : A rf with Γ E b : A with Γ E a := b : A with Γ Figur 1: Th typing ruls. Givn a typ A in th contxt of a constraint squnc Γ, w dfin its fr variabls (labls as wll as typ variabls) F V (A with Γ) and its dangrous fr variabls DV (A with Γ) as follows. Th intuition bhind th dfinition of F V is that th componnts of a functional typ A L B ar not only A, B and L, but also any typ xprssion C such that C L is on of th rcordd constraints. F V (X with Γ) = {X} F V (int with Γ) = F V (A rf with Γ) = F V (A with Γ) F V (A L B with Γ) = {L} F V (A with Γ) F V (B with Γ) F V (Σ with Γ) DV (X with Γ) = DV (int with Γ) = (Σ L) Γ DV (A rf with Γ) = F V (A with Γ) DV (A L B with Γ) = DV (Σ with Γ) (Σ L) Γ To complt th dfinition abov, w xtnd F V and DV to typ schmas and to typing nvironmnts in th obvious way: F V ( ( V 1... V n. A with Γ ) with Γ ) = F V (A with Γ Γ ) \ {V 1... V n } DV ( ( V 1... V n. A with Γ ) with Γ ) = DV (A with Γ Γ ) \ {V 1... V n } F V (E with Γ) = F V (E(x) with Γ) x Dom(E) Th typing ruls ar givn in figur 1. Thy ar vry similar to th ruls for ML, xcpt for th additional handling of constraints, rminiscnt of th tratmnt of subtyping hypothss in typ infrnc systms with subtyps [12, 6]. Th ruls dfin th proposition trm a has typ A undr assumptions E and constraints Γ, writtn E a : A with Γ. Th typing nvironmnt E is a partial mapping from trm variabls to typ schmas. W writ E[x A] for th nvironmnt idntical to E, xcpt that x is mappd to A. W assum th usual st oprations ar dfind ovr constraints Γ in th obvious way. As in Standard ML [11, p. 21], th Gn oprator usd in th (ltgn) rul is rsponsibl for gnralizing as many typ variabls as possibl in a typ. Hr, w also gnraliz ovr labls whnvr possibl. In addition, w prohibit gnralization ovr dangrous typ variabls. A tntativ dfinition would thrfor b Gn(A, E, Γ) = V 1... V n. A whr th st {V 1... V n } is F V (A with Γ) \ DV (A with Γ) \ F V (E with Γ). Howvr, it would b incorrct to gnraliz ovr a variabl which rmains fr in th constraint squnc usd Pag 5

6 i[s] = int(i)[s] x[s] b[s] = (x)[s] (λx. a)[s] = clos(x, c, 1 )[s 1 ] a[s 1 ] = v 2 [s 2 ] c[s 2 ] 1[x v 2 ] = v 3 [s 3 ] (b a)[s] = v 3 [s 3 ] = clos(x, a, F V (a) )[s] a[s] a[s]!a[s] = v 1 [s 1 ] b[s 1 ] [x v1] = v 2 [s 2 ] (lt x = a in b)[s] = loc(l)[s ] = s (l)[s ] = v 2 [s 2 ] a[s] rf(a)[s] a[s] = v[s ] l / Dom(s ) = loc(l)[s [l v]] = loc(l)[s 1 ] b[s 1 ] = v[s 2 ] (a := b)[s] = v[s 2 [l v]] Figur 2: Th valuation ruls latr. Thrfor, Gn also dischargs in th schma all gnric constraints, i.. th constraints in Γ whr on of th V i is fr, and rturns th rmaining constraints, to b usd furthr in th typing drivation. Dfinition 1 (Gnralization) Lt A b a typ xprssion, E b a typing nvironmnt, Γ b a constraint squnc. Dfin: {V 1... V n } = F V (A with Γ) \ DV (A with Γ) \ F V (E with Γ). Lt Γ b th squnc of thos constraints Σ L in Γ such that L is on of th V i. Thn: Gn(A, E, Γ) = ( V 1... V n. A with Γ ), (Γ \ Γ ) 3.3 Evaluation W giv hr an valuation mchanism for trms of our calculus, using structural oprational smantics. Th valuation rlation a[s] = v[s ] maps a trm a, in th contxt of an valuation nvironmnt and a stor s, to som valu v, and a modifid stor s. Valus hav th following syntax: v ::= int(i) clos(x, a, ) loc(l) Hr, l rangs ovr a countabl st Loc of locations. Stors ar partial mappings from locations to valus. Sinc w assum call-by-valu, valuation nvironmnts ar partial mappings from trm variabls to valus. Th ruls givn in figur 2 dfin prcisly th valuation rlation, assuming standard lft-to-right valuation ordr. To account for run-tim typ rrors, w introduc th spcial rsult wrong, and stat that if non of th valuation ruls match, thn a[s] = wrong. This way, w can distinguish btwn typ rrors and nontrmination. 3.4 Soundnss of typing Th typ systm prsntd hr is snsibl with rspct to th valuation mchanism abov: no wll-typd trm can valuat to wrong. Proposition 1 Lt a b a trm, A b a typ, Γ b a squnc of constraints such that w can driv a : A with Γ. Thn, for all stors s 0, a[s 0 ] dos not valuat to wrong; that is, w cannot driv a[s 0 ] = wrong. Th proof can b found in appndix. It closly follows Toft s [16, chaptr 5]. Th crucial point is that closur labls and constraints giv a bttr control ovr th typs of stor locations than in ML. As a consqunc, a variabl cannot b fr in th typ of a location rachabl from a valu without bing dangrous in th typ of that valu. This guarants th soundnss of typ gnralization. 3.5 Typ rconstruction In this sction, w considr an adaptation to our languag of th wll-known Damas-Milnr typ rconstruction algorithm for ML (algorithm W of [5]). In th following, w writ mgu(a, B) for th principal unifir of typs A and B, if A and B ar unifiabl. (Othrwis, th typ infrnc algorithm fails.) Typ xprssions ar trms of a two-sortd fr algbra, hnc this nsurs th xistnc of a principal unifir, that can b obtaind by th classical unification algorithm btwn trms of a fr algbra. Lt a b a trm, E b a typing nvironmnt, and Γ b an initial squnc of constraints. W dfin infr(e, a, Γ) as th tripl (A, σ, ), whr A is a typ xprssion, σ a substitution and a constraint squnc, as follows: Pag 6

7 infr(e, i, Γ) = (int, Id, Γ) infr(e, x, Γ) = lt ( X 1... X n, L 1... L m. A with ) = E(x) lt Y 1,..., Y n b frsh typ variabls (not fr in E nor in A) and M 1,..., M m b frsh labls lt σ = {X i Y i, L j M j } in (σa, Id, Γ σ ) infr(e, λx. b, Γ) = lt X, L b frsh variabls lt Θ = {E(y) L y fr in λx. b} lt B, σ, = infr(e[x X], b, Γ Θ) in (σx L B, σ, ) infr(e, (b a), Γ) = lt B, ρ, Θ = infr(e, b, Γ) lt A, σ, = infr(ρe, a, Θ) lt X, L b frsh variabls lt µ = mgu(σb, A L X) in (µx, µσρ, µ ) infr(e, lt x = a in b, Γ) = lt A, σ, = infr(e, a, Γ) lt Σ, = Gn(A, σe, ) lt B, ρ, Θ = infr((σe)[x Σ], b, ) in (B, ρσ, Θ) infr(e, rf(a), Γ) = lt A, σ, = infr(e, a, Γ) in (A rf, σ, ) infr(e,!a, Γ) = lt A, σ, = infr(e, a, Γ) lt X b a frsh variabl lt µ = mgu(a, X rf ) in (µx, µσ, µ ) infr(e, a := b, Γ) = lt A, σ, = infr(e, a, Γ) lt B, ρ, Θ = infr(σe, b, ) lt µ = mgu(ρa, B rf ) in (µb, µρσ, µθ) This algorithm njoys th good proprtis of th Damas-Milnr algorithm: it is corrct and complt with rspct to th typing ruls, and th infrrd typ is th most gnral on. Th proof is vry similar to Damas proof [4]. 3.6 Rlation to ML W hav introducd closur typing as a way to kp track of mutabl valus mbddd in functions. As a consqunc, two xprssions having th sam functional typ in ML may now b distinguishd by thir closur typ, and w may far that this lads to a typ systm mor rstrictiv than th on of ML. Idally, w would lik th purly applicativ fragmnt of our calculus (that is, without th rf typ constructor, and th rf, := and! trm constructors) to b a consrvativ xtnsion of ML: any pur, closd trm that is wll-typd in ML should also b wll-typd in our calculus. Unfortunatly, this is not th cas but for mor subtl rasons than th on outlind abov. Actually, two typ xprssions in our systm cannot b distinguishd by thir closur labls only. Th rason is that unification cannot fail bcaus of th labls: givn th syntax of typ xprssions, a labl can only b matchd against anothr labl, and labls ar tratd as variabls as far as unification is concrnd. Thrfor, in our systm, unification btwn typs is consrvativ with rspct to unification btwn th corrsponding ML typs. To b mor prcis, w introduc th strip oprator, that maps typ xprssions in our calculus to ML typ xprssions, by rasing th labls from function typs: int = int X = X (A rf ) = A rf (A M B) = A B Thn, two typ xprssions A and B ar unifiabl if and only if A and B ar, and in this cas, taking σ = mgu(a, B) and ρ = mgu(a, B ), w hav (σc) = ρ(c ) for all typs C. This lmma, along with th clos rsmblanc btwn our algorithm infr and Damas-Milnr s W algorithm, lad us to bliv that, givn th sam pur trm, both algorithms infr th sam typ, modulo closur labls. Howvr, this dos not hold bcaus of th gnralization stp in th cas of a lt construct. Considr th typing of lt x = a in b. Assum that, starting from typing nvironmnt E, algorithm infr infrs typ A with Γ for a, whil algorithm W, starting from th corrsponding ML typing nvironmnt E, infrs th corrsponding ML typ A. W must chck that both algorithms gnraliz xactly th sam typ variabls, so that thy will typ b in compatibl nvironmnts. This could b not th cas for two rasons. Th first rason is that algorithm infr dos not gnraliz dangrous typ variabls, whil W dos. But this is no problm hr, sinc w considr only th pur fragmnt of our calculus, without th rf typ constructor, hnc th st of dangrous variabls of any typ is always mpty. Pag 7

8 Th scond, mor srious rason is that closur typing introducs additional fr variabls in a givn typ. In gnral, F T V (A with Γ), th st of fr typ variabls in A with Γ, is a suprst of F V (A ). (Tak for instanc A = int L int and Γ = X L.) So it is not obvious that F T V (A with Γ) \ F T V (E with Γ), th st of typ variabls gnralizd by infr, is th sam as F V (A ) \ F V (E ), th st of typ variabls gnralizd by W. Indd, thr ar cass whr infr dos not gnraliz som typ variabl X, whil W dos, bcaus X is fr in E with Γ, but not in E. Considr: λz. lt id = λx. ( if... thn z ls (λy. x; y) ) ; x in id id Assuming x : X and y : Y, th trm (λy. x; y) is givn typ Y M Y with X M, and th if construct forcs z to hav th sam typ. Thrfor, whn w attmpt to gnraliz X L X (th typ of λx.... ; x), w hav E(z) = Y M Y undr constraints Γ = X M, Y M Y L, and w cannot gnraliz ovr X, sinc it is fr in th typ of z. Hnc, id rmains monomorphic, and th application id id is ill-typd. In ML, w would hav z : Y Y, so w could frly gnraliz ovr X, gtting id : X. X X, and th whol trm would b wll-typd. Th xampl abov is quit convolutd, and it is th simplst on w know that xhibits this variabl captur through labls phnomnon. W nd mor xprinc with th proposd typ systm to find whthr this phnomnon happns in mor practical situations. Howvr, in cas this turns out to b a srious flaw of our closur typing systm, w ar invstigating two possibl improvmnts that sm to avoid variabl capturs. On dirction is to rcord lss constraints whn typing a λ-abstraction. In th xampl abov, on could argu that th constraint X M should not b rcordd, on th grounds that x is not actually usd in th body of th function, as witnssd by th fact that th typ variabl X is not fr in th typ of th function rsult, nor in th typ of th paramtr. Th othr dirction is to chck that two function typs ar compatibl without actually idntifying thir closur typs. This way, w could avoid th propagation of th constraint X M to th typ of z. 3.7 Pragmatics of constraint handling Practically, th main concrn with th typ infrnc algorithm abov is th additional ovrhad introducd by th handling of constraints. First, it is possibl to simplify squncs of constraints. Hr ar som possibl simplification ruls: Σ M, Σ M Σ M Σ M ɛ if Σ is closd A B M A M, B M Ths thr simplifications ar obviously sound with rspct to th computation of fr and dangrous variabls. Th third rul actually gnrat mor constraints, but this may opn th way to furthr simplifications,.g. if A = B, or if A is closd. In addition, constraint handling bcoms quit chap if w distribut constraints insid typs, instad of handling a singl list of constraints. W suggst grouping togthr all constraints ovr th sam labl L, arrangd as a list of typ schms. This list rprsnts th labl L itslf. To idntify two labls, w just hav to concatnat th two lists, and this can b don in constant tim, using.g. diffrnc lists. Hnc, whn unifying two typ xprssions A and B, th additional work of idntifying labls taks tim at most proportional to th siz of A, B. Thrfor, unification can b prformd in linar tim, as in ML. 4 Comparison with othr typ systms In this sction, w compar our typ systm with prvious proposals of polymorphic typ systms for languags faturing physical modification. 4.1 Standard ML W considr first th systms proposd for ML. All ths systms rly on dtcting th cration of mutabl valus (.g. by spcial typchcking ruls for th rf construct), and nsuring that th rsulting mutabl valus hav monomorphic typs. Th first systm w considr is th on proposd by Toft [16, 17], and adoptd in Standard ML [11]. It maks us of wak typ variabls (writtn with a suprscript) to prohibit polymorphic rfrncs. Wak typ variabls cannot b gnralizd by a lt binding, unlss th corrsponding xprssion is guarantd to b non-xpansiv, i.. that it dos not crat any rfrncs. Damas [4] proposd a rlatd, but slightly diffrnt schm, that givs similar rsults in most cass. Th othr systm is an xtnsion of Toft s, usd in th Standard ML of Nw Jrsy implmntation [1]. In this schm, typ variabls ar no longr partitiond into wak and non-wak variabls; instad, ach typ Pag 8

9 Standard ML SML-NJ Our systm mak rf α α rf α 1 α 1 rf α L α rf mak rf [ ] rjctd rjctd rjctd imprativ map (α β ) α list β list (α 2 β 2 ) α 2 list β 2 list imprativ map id [ ] rjctd rjctd α list (α L β) M α list N β list with α L β N applicativ map mak rf rjctd rjctd α list L α rf list id mak rf rjctd rjctd α L α rf (rais Exit : α rf ) α rf α rf rjctd Figur 3: Comparison with othr typ systms variabl has an associatd intgr, its dgr of waknss, or strngth. This dgr masurs th numbr of abstractions that hav to b applid bfor th corrsponding rfrnc is actually cratd. Rgular, unconstraind typ variabls hav strngth infinity. Variabls with strngth zro cannot b gnralizd. Variabls with strngth n > 0 can b gnralizd, but ach function application dcrmnts th strngth of variabls. Th comparativ rsults ar givn in figur 3. For ach tst program, w giv its most gnral typ in ach systm. W assum ths ar top-lvl phrass. As in most ML implmntations, w rjct top-lvl phrass whos typs cannot b closd (bcaus som fr variabls cannot b gnralizd). Th first tst is th mak rf function, dfind as function x rf x. It xrciss th possibility of writing gnric functions that crat and rturn updatabl data structurs. Most functions ovr vctors, matrixs, doubly linkd lists,... ar typd similarly. All typ systms considrd hr captur th fact that mak rf should b applicabl to monomorphic valus only. Th scond tst is th imprativ map functional givn in th introduction. It illustrats th us of polymorphic rfrncs as auxiliaris insid a gnric function. Our typ systm is th only on which givs a fully polymorphic typ to imprativ map. Th othrs rstrict it to b usd with monomorphic typ. Th third tst is th partial application of applicativ map, dfind in th introduction, to th mak rf function. It xrciss th compatibility btwn functions that crat mutabl data and highrordr functions. SML and SML-NJ rfus to gnraliz its typ, hnc rjct it. Thy ar unabl to dtct that applicativ map dos not apply mak rf immdiatly, hnc that applicativ map mak rf dos not crat any polymorphic rfrnc. A simplifid vrsion of this tst, shown blow, is to apply th idntity function id to th mak rf function. Our typ systm givs th sam typ to id mak rf and to mak rf. Th othrs don t, and w tak this as strong vidnc that thy do not handl full functionality corrctly. Th last xampl illustrats a waknss of our typbasd approach. By using xcptions, for instanc, on may mimic th cration of a rfrnc as far as typs ar concrnd, without actually crating on. In th xprssion (rais Exit : α rf ), our typ systm considrs that α is dangrous and dos not gnraliz it. Othr typ systms rcogniz that no rfrncs ar cratd, hnc thy corrctly gnraliz it. This flaw of our mthod has littl impact on actual programming, howvr. 4.2 Qust In our systm, w mad no attmpt at rstricting th cration of mutabl valus, and concntratd on typ gnralization instad. W wr inspird by Cardlli s Qust languag [2], which dparts significantly from ML, but faturs mutabl data structurs and polymorphic typing. Qust maks almost no typing rstrictions for mutabl valus. Soundnss is nsurd by diffrnt smantics for typ spcialization. Namly, an xprssion with polymorphic typ is valuatd ach tim its typ is spcializd, in contrast with ML, whr it would b valuatd only onc. This is consistnt with th fact that polymorphism is xplicit in Qust programs: polymorphic objcts ar actually prsntd as functions that tak a typ as argumnt and rturn a spcializd vrsion of th objct. But ths smantics ar incompatibl with ML, whr gnralization and spcialization ar kpt implicit in th sourc program. Pag 9

10 4.3 FX Th FX ffct systm [9] is a polymorphic typ systm that prforms purity analysis as wll: th typ of an xprssion indicats what kind of sid-ffcts its valuation can prform. This provids a simpl way to dal with th problm of mutabl valus: th typ of an xprssion cannot b gnralizd unlss this xprssion is rfrntially transparnt. It is asy to s that such pur xprssions can safly b usd with diffrnt typs. Though this approach is attractiv for othr purposs (.g. automatic program paralllization), it dfinitly dos not addrss th main issu considrd hr: how to giv th sam typ to smantically quivalnt functions, whthr writtn in applicativ styl or in imprativ styl. Th rason is that th purity analysis of FX maks it apparnt in th typs whthr a function allocats mutabl data for local purposs. For instanc, imprativ map and applicativ map do not hav th sam typ in FX. 5 Conclusion W hav prsntd hrin an xtnsion of th ML typ systm that considrably nhancs th support for data accpting in-plac modification. This is a significant stp toward th intgration of polymorphic typ disciplin and imprativ programming styl. W hav introducd th notion of closur typing, which is ssntial to th soundnss of this approach. W hav givn on typ systm that prforms this closur typing, basd upon labls and constraints. Our systm is simpl, but falls short of bing a consrvativ xtnsion of ML. Furthr work includs invstigating altrnat, mor subtl ways to typchck closurs, that would not rjct any wll-typd ML program, whil corrctly kping track of mutabl data. Th scop of this work is not strictly limitd to th problm of mutabl data structurs. For instanc, it is wll-known that polymorphic xcptions rais th sam issus as rfrncs, and can b handld in th sam way. Mor gnrally, similar problms aris in th intgration of polymorphic typing within svral othr programming paradigms. For instanc, on approach to th intgration of functional and logic programming is to allow partially dfind valus containing logical variabls within a convntional functional languag [15]. Polymorphic logical variabls brak typ safty just as polymorphic rfrncs do, and ar amnabl to th sam tratmnt. In objct-orintd programming, status variabls of objcts can b sn as rfrncs systmatically ncapsulatd in functions (th mthods). Closur typing sms rlvant to th polymorphic typchcking of such objcts. Finally, som calculi of communicating systms fatur channls as first-class valus [10]. Polymorphic typing of ths channls must guarant that sndrs and rcivrs agr on th typs of transmittd valus, and this is similar to nsuring that writrs and radrs of a rfrnc us it consistntly [13]. Acknowldgmnts Didir Rémy suggstd th us of constraints in th typing ruls. W also bnfitd from his xprtis in typ infrnc and unification problms. Many thanks to Brad Chn for his ditorial hlp. A Proof of soundnss In this appndix, w sktch th proof of soundnss of th typ systm with rspct to th oprational smantics givn in sction 3.3. W formaliz th fact that a valu v smantically blongs to a typ xprssion A undr constraints Γ. W writ this S = v : A with Γ. Th hypothsis S is a stor typing, that is, a partial mapping from locations to typ xprssions. Th stor typing is ndd to tak into account th sharing of valus introducd by th stor. Dfinition 2 (Smantic typing judgmnts) Lt S b a stor typing, v a valu, A a typ, Γ a constraint squnc. Th prdicat S = v : A with Γ is dfind by induction on v: S = int(i) : int with Γ. S = loc(l) : (A rf) with Γ iff l blongs to th domain of S, and S(l) = A. S = clos(x, b, ) : (A M B) with Γ iff for all x Dom(), thr xists a typ schma Σ such that Σ M is in Γ, and S = (x) : Σ with Γ; and thr xists a typing nvironmnt E such that E[x A] b : B with Γ. Th prdicat abov is xtndd to typ schmas by taking S = v : ( V 1... V n. A with Γ ) with Γ iff for all substitutions µ ovr V 1... V n, w hav S = v : µa with Γ µγ Lt b an valuation nvironmnt, and E b a typing nvironmnt. W say that S = : E with Γ iff Dom() Dom(E) and for all x Dom(), w hav S = (x) : E(x) with Γ. Pag 10

11 Similarly, lt s b a stor. W say that = s : S with Γ if Dom(s) Dom(S) and for all l Dom(s), w hav S = s(l) : S(l) with Γ. In th proof of soundnss, w will us th fact that th smantic typing judgmnt is stabl undr substitution. Lmma 1 (Smantic substitution) If S = v : A with Γ, thn µs = v : µa with µγ for all substitutions µ. Starting from a givn valu v, it is not possibl in gnral to accss any location in th stor. Th st R(v) of locations rachabl from v is dfind by induction on v, as follows. Dfinition 3 (Rachabl locations) R(int(i)) = R(loc(l)) = {l} R(clos(x, a, )) = y Dom() R((y)) In th dfinition of S = v : A with Γ, th typs of th locations that cannot b rachd from v ar irrlvant. W can actually assum any othr typ for thos locations: Lmma 2 (Garbag collction) Lt S, S b two stor typings such that S(l) = S (l) for all l R(v). If S = v : A with Γ, thn S = v : A with Γ Our typ xprssions ar informativ nough to allow us to connct th typ of a location l rachabl from a valu v with th typ A of v. Mor prcisly, w hav th following ky lmma which rlats th variabls fr in th typ of l with th dangrous variabls of A. Lmma 3 (Stor typing control) Assum S = v : A with Γ. Lt l b a location in R(v). Thn F V (S(l) with Γ) DV (A with Γ). Proof: By induction on v. Cas 1: A = int and v = int(i). Obvious, sinc thr ar no locations rachabl from v. Cas 2: A = B rf, v = loc(l ), and S(l ) = B. By dfinition of th rachabl locations, l = l. Hnc F V (S(l) with Γ) = F V (B with Γ) = DV (A with Γ). L Cas 3: A = A 1 A2, and v = clos(x, a, ). Lt y b a variabl such that l is rachabl from (y). Lt Σ b a typ schm such that S = (y) : Σ and (Σ L) Γ. W writ V 1... V n. B with Γ for Σ. For all substitutions µ ovr V 1... V n, sinc S = (y) : µb with Γ µγ, w gt by induction hypothsis: Taking wll-chosn substitutions µ, it follows that: F V (S(l) with Γ) DV (B with Γ Γ ) \ {V 1... V n } Hnc th dsird rsult: F V (S(l) with Γ) DV (Σ with Γ) DV (A with Γ) W ar now rady to show th soundnss of th typing ruls. Proposition 2 (Soundnss) Assum E a : A with Γ. Lt b an valuation nvironmnt, s a stor, S a stor typing such that: S = : E with Γ = s : S with Γ. Assum a[s] = w. Thn, w = v[s ], and thr xists a stor typing S xtnding S, such that: S = v : A with Γ = s : S with Γ Proof: By induction on th lngth of th valuation. All cass procd as in Toft s proof [16, chaptr 5], xcpt whn a is a lt binding. Thn, th last stp of th typing drivation is: E a : A with Γ (Σ, Γ ) = Gn(A, E, Γ) E[x Σ] b : B with Γ E lt x = a in b : B with Γ Applying th induction hypothsis to th first prmis, w gt v a, s a, S a xtnding S such that: a[s] = v a [s a ] S a = v a : A with Γ = s a : S a with Γ. To apply th induction hypothsis to th last prmis, w nd to show that: S a = [x v a ] : E[x Σ] with Γ whr w writ (Σ, Γ ) for Gn(A, E, Γ), and V 1... V n. A with Γ for Σ. Sinc S a xtnds S, and V 1... V n ar not fr in E, w alrady hav S a = : E with Γ as a corollary of lmma 2. It rmains to show that: S a = v a : ( V 1... V n. A with Γ ) with Γ. To do so, w must prov that, for any substitution µ ovr th V i, (1) S a = v a : µa with Γ µγ. Sinc S a = v a : A with Γ, w hav, by lmma 1, F V (S(l) with Γ µγ ) DV (µb with Γ µγ ) (2) µs a = v a : µa with µγ. Pag 11

12 By dfinition of Gn, non of th V i appars in any constraint of Γ, hnc (3) µ(γ) = µ(γ Γ ) = Γ µγ. By lmma 3, for any location l rachabl from v a, w hav F V (S a (l) with Γ) DV (A with Γ), and non of th V i is dangrous in A with Γ, hnc non of th V i is fr in S a (l). Thrfor, (4) (µs a )(l) = S a (l) for all l R(v a ) and th claim (1) abov follows from (2), (3), (4), and lmma 2. Thrfor, w can apply th induction hypothsis to th right branch of th typing drivation, gtting v b, s b, S b xtnding S a such that: b[s a ] [x v a] = v b [s b ] S b = v b : B with Γ = s b : S b with Γ which is th xpctd rsult. Rfrncs [1] A. W. Appl and D. B. MacQun. Standard ML rfrnc manual (prliminary). AT&T Bll Laboratoris, Includd in th Standard ML of Nw Jrsy distribution. [2] L. Cardlli. Typful programming. In E. J. Nuhold and M. Paul, ditors, Formal Dscription of Programming Concpts, pags Springr-Vrlag, [3] G. Cousinau and G. Hut. Th CAML primr. Tchnical rport 122, INRIA, [4] L. Damas. Typ assignmnt in programming languags. PhD thsis, Univrsity of Edinburgh, [8] M. J. Gordon, A. J. Milnr, and C. P. Wadsworth. Edinburgh LCF, volum 78 of Lctur Nots in Computr Scinc. Springr-Vrlag, [9] J. M. Lucassn and D. K. Gifford. Polymorphic ffct systms. In 15th symposium Principls of Programming Languags, pags ACM Prss, [10] R. Milnr, J. Parrow, and D. Walkr. A calculus of mobil procsss: part 1. Rsarch rport ECS- LFCS-89-85, Univrsity of Edinburgh, [11] R. Milnr, M. Toft, and R. Harpr. Th dfinition of Standard ML. Th MIT Prss, [12] J. C. Mitchll. Corcion and typ infrnc. In 11th symposium Principls of Programming Languags, pags ACM Prss, [13] J. H. Rppy. First-class synchronous oprations in Standard ML. Tchnical Rport TR , Cornll Univrsity, [14] J. C. Rynolds. Toward a thory of typ structur. In Programming Symposium, Paris, 1974, volum 19 of Lctur Nots in Computr Scinc, pags Springr-Vrlag, [15] G. Smolka. FRESH: a highr-ordr languag with unification and multipl rsults. In D. DGroot and G. Lindstrom, ditors, Logic Programming: Functions, Rlations, and Equations, pags Prntic-Hall, [16] M. Toft. Oprational smantics and polymorphic typ infrnc. PhD thsis CST-52-88, Univrsity of Edinburgh, [17] M. Toft. Typ infrnc for polymorphic rfrncs. Information and Computation, 89(1), [5] L. Damas and R. Milnr. Principal typ-schms for functional programs. In 9th symposium Principls of Programming Languags, pags ACM Prss, [6] Y.-C. Fuh and P. Mishra. Typ infrnc with subtyps. In ESOP 88, volum 300 of Lctur Nots in Computr Scinc, pags Springr Vrlag, [7] J.-Y. Girard. Intrprétation fonctionnll t élimination ds coupurs d l arithmétiqu d ordr supériur. Thès d État, Univrsité Paris VII, Pag 12