Region analysis and the polymorphic lambda calculus

Transcription

1 Region analyi and the polymorphic lambda calculu Anindya Banerjee Steven Intitute of Technology Nevin Heintze Bell Laboratorie Jon G. Riecke Bell Laboratorie Abtract We how how to tranlate the region calculu of Tofte and Talpin, a typed lambda calculu that can tatically delimit the lifetime of object, into an extenion of the polymorphic lambda calculu called F #. We give a denotational emantic of F #, and ue it to give a imple and abtract proof of the correctne of memory deallocation. 1 Introduction Implementation of modern programming language divide dynamically allocated memory into two part. The tack i ued for data that ha a imple lat-in, firt-out lifetime determined by block tructure; the other part (often called the heap) i ued for data whoe lifetime extend beyond the cope of program block. The heap i periodically garbage collected to reclaim memory that i no longer needed. Tofte and Talpin region calculu [23] attempt to unify thee two tyle of memory management. The region calculu divide memory into region, and provide a local coping mechanim for thoe region. Every value created by the program i placed in one of thee region. When program execution enter the cope of a region, memory for that region i allocated, and when program execution leave the cope, memory for that region i deallocated. Hence memory i managed uing a imple, block-tructure allocation and deallocation mechanim. Thi paper decribe an abtract characterization of a verion of the region calculu and give a imple proof of it correctne. The proof relie on a tranlation into a verion of the polymorphic lambda calculu called F #. Our proof i more abtract than the verion preented by Tofte and Talpin [23]. Tofte and Talpin give a high-level operational emantic of a ubet of ML [11], a tranlation of that language into their region calculu, and an operational emantic of the region calculu that explicitly manipulate Member of Church Project, and upported in part by NSF grant EIA memory. Their main theorem i a proof of adequacy: evaluation in either operational emantic yield the ame reult. The proof relie on greatet fixed point and coinduction, partly becaue of the coupling of the region calculu and the tranlation, and partly becaue deallocation can yield dangling reference. In contrat, we work at a higher level: we prove that once the program leave the cope of a region, value held in that region do not affect the final reult of the program. Thi fact implie that once the program leave a block aociated with a region, the memory for that region may be afely reclaimed. The connection between the region calculu and parametric polymorphim can be brought out by a few example. In the region calculu, each type i annotated with a region, and the type ytem track all read and write to region. For example, the typing judgement x : e : mean that there i a free variable of type called x which live in region. The body e alo produce an in region, and during the computation, value may be both read from and written into. In general, a computation that accee a region ha an aociated effect ; converely, any computation that tore into region ha an aociated effect. In the above judgement, e may arbitrarily read from and write to. Contrat thi with the typing judgement x : e : In thi cae, e can acce the value of x, but it may not tore any value in region. We can view thi type a a retriction on the behavior of the function computed by the expreion. In thi cae, if there i a value produced by the computation, it mut be the value of x; the computation may diverge or converge to x depending on the value of x. The typing judgement x : e : /0 i even more retrictive: it pecifie that there are no accee to the region. The only function of thi type are the identity function and the function that diverge on all 1

2 parameter. Thi lat example reemble the argument that the only function one may write of type α α α are the identity and the everywhere divergent function. It i perhap not urpriing that there hould be ome connection between parametric polymorphim and region; Reynold and other have made the ame point about the connection between block tructure in Idealized Algol and polymorphim [13, 1]. Other, too, have noted connection between garbage collection and polymorphim. Fradet, for intance, note that if a ubterm of a term can be typed with an abtract type α and α doe not appear in the type of the term the ubterm mut be garbage []. That i, no matter what ubterm i replaced for the original ubterm, it will not affect the final reult computed by the program. Hooya and Yonezawa note a imilar phenomenon [9]. The region calculu doe, however, raie new emantic iue. One might, for intance, ugget that type uch a be repreented by type variable. The type i not, however, completely abtract: even if the program cannot acce value of thi type, thee value mut till be integer. Alo, value of different type may be tored in the ame region. For example, type and can co-exit in the ame program. To model the behavior of the region calculu, our verion of the polymorphic lambda calculu contain a new type operator #. One of the novel feature of our calculu i that it combine a mechanim to hide the tructure of a type (i.e., make the type abtract) with another mechanim to reveal the tructure of uch a hidden type. Intuitively, when a value i put into a region, it tructure i hidden; when it i removed from the region, thi tructure i recovered. For example, conider the region type. Thi i modelled in F # by the type #, where now denote a type variable intead of a region variable. The preence of in the type # make the type of abtract. The language F # alo include a pecial type which i the unit of the # operation; a type of the form # can then be converted back to by binding to. We formalize the connection via a type-directed tranlation from the region calculu to F #. Uing the emantic of F #, we how that the tranlation i adequate, i.e., the anwer produced by ource and target program are the ame. Our main reult i a proof of the afety of region deallocation: we how that if the effect aociated with an expreion do not mention a region, then evaluation of the expreion i independent of the value placed in. Our proof i baed on a relational-tyle emantic [12, 18]. While the domain theory underlying our proof i complex, the adaptation of thi tandard machinery for proving correctne of region deallocation i mall and imple. The benefit of our tranlation approach, a oppoed to the operational technique of Tofte and Talpin, are twofold. Firt, the tranlation bring out the eential detail of the region calculu. In our verion of the calculu, for intance, we implify the rule for lambda abtraction, dropping a ide condition from the original rule of Tofte and Talpin [23]. An additional benefit of our approach i that it can be more eaily modified to account for extenion of the underlying language. Second, the proof of correctne i more modular. The main part of the proof i the type correctne of the tranlation. We believe, for intance, that other region-like calculi could be proven correct uing an appropriate tranlation. We might alo ue the approach to prove computational propertie of term, imilar to parametricity. 2 Region calculu The region calculu of Tofte and Talpin i the target of a type-directed tranlation from an ML-like core language [23]. Intead of preenting the original tranlation, we omit the tranlation and decribe the region calculu on it own. Thi implification make the typing rule eaier to read, and alo allow u to modify the type ytem o that the rule are uitable for type checking rather than type recontruction. The region calculu aociate a region with every type. Let range over region variable. The type and primitive effect are defined by the grammar t ε :: :: t where range over et of primitive effect ε. Effect are either read from a region, denoted, or write to a region, denoted. The et above the function pace denote the et of latent effect of the function, and i a familiar feature of effect ytem [10, 21]. Here, the latent effect of a function are the poible read to or write from region after an argument ha been upplied to the function. The value and expreion of the region calculu are defined by the grammar v :: n λx : e e :: v x e e e e!" e # e $ % e '! f x : ( e where x range over variable and n range over natural number. The annotation on integer and function value denote the region into which the value i placed; the annotation on and pecifie the region into which the reult will be placed. Mot of the expreion are familiar from PCF [15, 19], with the exception of the contruct. Thi annotation demarcate the cope of a region : within the region, value may be read from or written into. Upon exit from the, all value in the region become garbage and the memory may be reclaimed. e 2

3 4 Table 1. Type ytem for the region calculu. Var Γ x : Γ x /0 Cont Γ n : Cont Γ e : c or Γ! c " e : "# $% ' "# If Γ e : Γ e i : t i Γ () e *+, e 1 -, e 2 : t $ 1 $ 2 $%, Lam Region Γ x : 1 e : 2 Γ!. λx : 1 / e : Γ e : 3 FRV Γ $ FRV Γ 5-,6 8 e : 9: ; < App Rec Γ e 1 : Γ e 2 : 1 2 Γ e 1 e 2 : 2 $ 1 $ 2 $%, Γ f : x : 1 e : 2 = Γ!.<( f x : 1 e > : The typing rule for the region calculu appear in Table 1. Type context Γ are finite map from variable to type, and judgement have the form Γ e :. We ue FRV Γ and FRV to denote the free region of a type context or et of effect. The type rule differ from the original region calculu of Tofte and Talpin [23] in three way. Firt, the original calculu ha no contant or operation on the integer. We include them here to give ome more computational content to the calculu. Second, we include no polymorphim on region variable or type variable, and the calculu i tuned for type checking rather than type inference. In particular, the rule for recurion i not a complicated, and there are no effect variable for ue in unifying two latent effect. We will dicu region polymorphim later in Section 6. Third, the original typing rule for abtraction include a ide condition requiring that the body of the abtraction have no free region variable not occurring in the type context [23]. The original proof of correctne of the region calculu eem to require thi ide condition, but we do not. The region calculu i a call-by-value language, whoe operational emantic can be defined in the evaluationcontext tyle of Felleien [6]. Evaluation context are defined by the grammar #? E :: E e v E E E!" E # e $ e E and the local operational rule are λx : e v e x : v n n!" n # e $ e e n 0!" n # e $ e e n A 0! f x : 1 e '! λx : 1 e f : f x : 1 e v when e rewrite to v and v cannot be rewrit- We write e B ten. The original correctne theorem of the region calculu talk about an abtract implementation with memory, and how that deallocation at the end of i afe : no data in the deallocated region can affect the final reult. Our main reult tate the property more abtractly: if the effect aociated with an expreion do not mention the region, then evaluation of the expreion i independent of the value placed in. Thi jutifie the rule, and therefore, region reue. Specifically, we hall prove the following property: Suppoe x : e : 0 where A 0 and CA. If /0 v i : for i 1 2, then λx : e λx : e 2 2 v 1 :B v 2 :B n 0 n 0 The proof will rely upon the tranlation of the region calculu into F #. We potpone the proof until Section 5. 3 Syntax and emantic of F # 3.1 Overview The target of the tranlation i F #, an extenion of the polymorphic lambda calculu [8, 16] with bae type, product, an aociative, commutative, idempotent operation # for making a type abtract, and a pecial contant which i the unit of #. For example, # #. Algebraically, # behave like an interection or union type operator. Unlike the cae for union and interection type operator, however, our calculu ha no introduction or elimination rule for # and alo no ubtyping rule for #, and that ha a ignificant impact. For example, with interection and union, we can contruct cloed term of type E t and F t. In F #, however, we cannot contruct cloed term for # t or # t. In fact, it i preciely thi feature of F # that allow u to model iff 3

4 4 Table 2. Typing rule for F #. Var Γ x : Γ x Unit Γ : 6 8 Int Γ n : Succ Γ Γ e : 6 e : Pred Γ e : Γ!' e : If Γ e : Γ e i : t Γ! 8() e *+8 e 1 -, e 2 : t Pair Γ e 1 : 1 Γ e 2 : 2 Γ e 1 e 2 : 1 2 Lam Γ x : 1 e : 2 Γ λx : 1 / e : Lift Γ e : Γ 5-8( e : Proj ForallI Γ e : 1 2 Γ!' i e : i Γ e : α 3 FTV Γ Γ! Λα/ e : α/ App Rec Γ e : Γ e" : 1 Γ e e" : 2 Seq Γ f : e : Γ µ f : / e : EqType Γ e : Γ x : e" : t Γ C, x e e" : t Γ e :! t Γ e : t ForallE Γ e : α/ Γ e t : α : t region. A value of type t in a region will have the F # type t # under the tranlation. The action of toring a value of type t in region (denoted by the effect ) correpond to a function of type t t #. Converely, getting a value of type t out of a region (denoted by the effect ) correpond to a function of type t # t. So, in order to put a value of type t into region, one mut poe the capability function t t #. To get a value of type t from a region, one mut poe the capability function t # t. The interpretation of # in the model of F # alo depart from the tandard interpretation of union and interection. The operation # i bet viewed a a form of abtraction operation. For example, the type # (correponding to region type ) i a hidden or opaque verion of. It retain the tructure of, but that tructure i not viible. To recover thi tructure, we intantiate with, ince #. The control of how and where may be intantiated i critical, and i intimately tied to the tranlation of the contruct. 3.2 Syntax The type of the target language are :: α # α where α range over type variable. To make the connection with the region calculu more imple, we will alo ue to range over type variable. Although F # permit type # t where and t are arbitrary type, our tranlation from the region calculu doe not employ thee type in their full generality e.g., we do not ue type uch a #. The contruct and # have a nontrivial equality theory given by the following axiom cheme: # # # t # t # u t # # t # u In other word, # and atify the ACUI equation. We write t when the equality can be derived from the equation above and the uual rule of reflexivity, ymmetry, tranitivity, and congruence. The term of F # are thoe of the polymorphic lambda calculu over the extended type. More preciely, the term are defined by the grammar e :: x µ f : e!" n e e e $ e 1 e $! e x e e 2 e e # e $ e λx : e Λα e e e e The typing rule for F # appear in Table 2. Neither the type nor type of the form # t have any pecial introduction or elimination operation aociated with them. Of coure, type variable alo have no introduction or elimination operation. Thi fact give ome evidence that value of type # t may not be created except by pecial operation. 3.3 Untyped denotational emantic The emantic of F # i given in two tep. Firt, we pecify a meaning function on term in an untyped model of the lambda calculu with contant. Second, we carve up the untyped model into ubet, and how how to contruct relation between the ubet from certain primitive relation. The relation play a key role in the definition of the polymorphic type. 4

5 Table 3. Some claue in the untyped denotational emantic of F #. Γ λx : / e : η inj 4 f where f d ; Γ x : 1 e : 2 η x 1 d Γ e e" : 2 η Γ e : η Γ e" : 1 η Γ! - 8( e : η inj 5 Γ e : η 0 Γ, x e e" : t η Γ! Λα/ e : α/ η Γ e : η Γ e t : α : t η Γ e : α/ η 1 Γ x : e" : t : η x d if Γ e : : η inj 5 d 0 otherwie The untyped emantic i traightforward and familiar. Let O be the Sierpinki pace. Let U be the initial olution, in the category of dcpo and embedding-projection pair, of the following domain equation [20]: U U U U The U tand for univere. Unlike certain partial equivalence relation model, we do not require any pecial propertie of the olution for U (cf. [2]). Let inj 1 : O U, inj 2 : N U, inj 3 : U U U, inj 4 : U U U, and inj 5 : U U be the injection correponding to the domain equation above. It i probably poible to contruct a domain U out of term rather than via domain theory, but then reaoning about recurion would become more difficult. Typing judgement may be interpreted directly in thi domain. Let η be a map from variable to element of U. To implify the notation, define the application operation on element of U by d d ( f d if d inj 4 otherwie Alo, element of U are either d 0 for d U or. The meaning of term i determined by induction on the typing derivation. A few of the claue appear in Table 3. Theorem 3.1 Suppoe Γ e : i derivable. Then Γ e : η U. 3.4 Typed denotational emantic To give the typed emantic, we need only pecify the meaning of type; the meaning of term i the ame a in the untyped model. The meaning of a type ha two component: a et of element and a mean of relating two interpretation of a type. efine a U-domain to be a ubet of U that contain and i directed-complete. More pecifically, X i a U-domain if, whenevery i a directed ubet of X, the f leat upper bound a calculated in U i an element of X. Given a type environment ι i.e., a map from type variable to U-domain we can then calculate the meaning of a type a a U-domain. The relational meaning of a type i more involved. efine a U-relation between U-domain A B to be a ubet R A B uch that the following two propertie hold: Pointedne: A. irected-completene: Suppoe X x i i I A and Y y i i I B are directed, and x i y i R for all i I. Then X Y R. We write the relation R : A B when R i uch a relation. A relation environment χ between type environment ι 1 ι 2, written χ : ι 1 ι 2, i a map from type variable to U-relation uch that for all α, χ α : ι 1 α ι 2 α. The domain and relational meaning of type are defined by imultaneou induction on the tructure of type. For type variable and the bae type, α ι ι α R α χ χ α ι inj 1 x x O R χ inj 1 x inj 1 x x O ι inj 2 x x N R χ inj 2 x inj 2 x x N ι U R χ For product type, 1 2 ι R 1 2 χ For function type, t ι R t χ inj 3 x 1 x 2 x i i ι inj 3 x 1 x 2 inj 3 y 1 y 2 x i y i R i χ inj 4 f for all x ι, f x inj 4 f inj 4 g x y R χ, f x g y t ι R t χ 5

6 For lifted type, ( ι inj 5 x x ι R χ inj 5 x 0 inj 5 y 0 x y R χ For polymorphic type, define the relation environment ι : ι ι by ι α x x x ι α and define α ι R α χ x A ι α A R : A 1 A 2, x x R ι α R x y R:A 1 A 2 R χ α R Finally, for the # operation, if χ : ι 1 ι 2, define # t ι ι t ι R # t χ x y R χ R t χ x ι 1 t ι 1 and y ι 2 Propoition 3.2 Suppoe i a type. t ι 2 If ι i a type environment, then ι i a U-domain. If χ : ι 1 ι 2 i a relation environment, then R χ : ι 1 ι 2 i a U-relation. Proof: The only real difficulty lie in howing that # work properly on relation, i.e., R # t χ i pointed and directed-complete. Note firt that R # t χ. Next, uppoe X i a directed ubet of R # t χ. Let X A X R χ and X B X R t χ (thee et might overlap). There are three cae. In the firt cae, for any tuple x x X A, there i a tuple y y X B uch that x y and x y. We claim that X B i directed. To ee why, uppoe y 1 y1 y 2 y2 X B. Then there mut be a tuple z z X uch that y i z and yi z. Now, if that tuple z z i in X B, we are done. If the tuple i in X A, on the other hand, we know that we can pick ome y y X B uch that z y and z y. Thi tuple y y i alo an upper bound for the y i yi. Let Y y y y X B and Y y y y X B. Both of thee et are directed becaue X B i. Thu, z z X B Y Y exit. Note that z ι 1 t ι 1 and imilarly for z. Alo, becaue of the directed completene of R t χ, z z R χ R t χ. Finally, z z i the leat upper bound of X B and, by the hypothei, i an upper bound for all of the X A ; thu, it i the leat upper bound of X, and o we are done. In the econd cae, for any tuple y y X B, there i a tuple x x X A uch that y x and y x. Thi cae i imilar to the previou cae. Finally, uppoe neither of the two cae above hold. That i, there i a tuple x 1 x 2 X A uch that no y y X B i above it, and there i a tuple y 1 y 2 X B uch that no x x X A i above it. Since X i directed, there mut be a tuple z 1 z 2 uch that x i y i z i. Now, either z 1 z 2 i in X A or X B. But both cae are ruled out by the hypothei. Thu, thi cae hold vacuouly. Thi complete the cae analyi and hence the proof. One may alo prove that the equality theory on type i atified by the interpretation: Propoition 3.3 Suppoe t u are type. Then the equation above hold, e.g., # ι ι R # χ R χ Finally, we can how that the meaning of any typing judgement i in the appropriate type. Suppoe η are environment, Γ i a type environment, and ι i a map from type variable to U-domain. We ay η i compatible with Γ ι if for all x Γ, η x Γ x ι. Then Theorem 3.4 Suppoe Γ e : i derivable and η i compatible with Γ ι. Then Γ e : η ι. The proof require additional induction hypothee, one of which amount to Reynold Abtraction Theorem [18]. 4 Tranlation of region calculu into F # We are now ready to connect the region calculu with F #. We ue a type-directed tranlation: type in the region calculu are repreented by type in F #, and typing derivation in the region calculu are tranlated into typing derivation in F #. We then how how to ue the emantic model to prove the correctne property tated in Section 2 for the region calculu. We begin with the tranlation of type, which involve two function. efine the tranlation? from type with region to F # -type, and?, from type without region to F # -type, a follow: t # Thee type are related to the interpretation of call-by-value in call-by-name (ee [14]). The tranlation alo involve the tranlation of et of effect to F # -type. efine β β # ( β β β β # t 6

7 C Table 4. Tranlation of the region calculu. ) ) ) ) : 9 Var Γ x : Γ x /0 Γ λp : /0 x : /0 Γ x Cont Γ n : Γ λp : p n : Γ e : x b b- freh Γ e : c "!$## or %(' Γ λx : Cont 1 )*+, -. 1!/(0 b " e π 1 x b-(1 π Γ c - e : x b 1 π 1-2 x c b-2 : 1 -, Γ e : Γ e Γ e : Γ e i : t i : i t x b b- freh i Γ λx : If )*+ 4 Γ 3 e (5 3!/(0 b " e π 3 x b- " π e 1!/ e 2 : t 3 + x b 3 4 (5 b- e 1 π 3 1 x 6!/ e 2 π 3 2 x : 3 t Lam Γ x : 1 e : Γ x : 1 e : 2 p freh Γ λx : 1 e : Γ λp : p 1 2 λx : 1 e : App Γ e 1 Γ e 1 : Γ e 2 : Γ e 2 : 2 1 x f g v freh 2 Γ 8 λx : )*+ 3!/(0 f1 e 1 π 3 1 x Γ 3 e 1 e 2 : 2 g " π 3 + x 1 2 f 3!/(0 v e 2 π 3 2 x g v π 3 x : 3 2 Region Γ e : FRV Γ ) FRV Γ e : x freh 1 ;<+ =, Γ 8 %(+ 6> Γ λx : e : 1 Λ e add 1 x : 1 1 Rec Γ f : x : 1 e : 2 " 1 2 Γ 8 f x : 1? e : Γ 8 λp : Γ f : x : 1 e 2 p freh / µ f : A p 1 2 λx : 1 e B : To be precie about the tranlation of et of effect, aume ome canonical ordering of effect. Suppoe that the et ε 1 ε n i ordered by thi ordering. efine ε 1 ε n ε 1 ε n if n 0 otherwie Note that the ordering enure that two different mean of writing the ame effect et yield the ame type. The tranlation of typing derivation appear in Table 4. To implify the tranlation, we ue let notation (which can be deugared into abtraction and application), I for Λβ λx : β x, and the notation π 1 2 : 1 2, where 2 1, for the canonical projection of the firt et onto the econd et. We alo ue the notation add 2 1 : 2 1 for a function that take a tuple in 2 and put in I for the miing component to produce a tuple in 1. Propoition 4.1 Suppoe Γ e : i derivable, and the derivation tranlate to Γ e :. Then the tranlation of the judgement i a legal typing derivation in F #. Theorem 4.2 (Adequacy) Suppoe /0 e : derivable, and the derivation tranlate to /0 e : # where C are the free region variable. Then e B ΛC e I! I n. n i iff To ee how the tranlation work, we conider a mall example. Let 1 1, 2 2, 3 1, and Alo, /0 let Conider the following cloed expreion given with their type and effect. e 0 e 1 e 2 e 3 % e 2 e 3 : 2 1 e 1 λx : 2 x 2 2 : 2 : : 1 2 Then the tranlation of e 3 i λp : 2! p 2 which ha type 2 t with t # 2. To tranlate e 2, note # 2 ( # 2 Then # 1, and the tranlation for e 2 i! λp : 1 p λx : t λp :! x which ha type 1. The term e 1 get tranlated a λz : f e2 π 1 z g π 3 z f v e3 π 2 z g v π z with type t. Finally, e 0 judgement i tranlated into λu : 2 Λ 1 e1 I u I

8 C C with type 2 t. Thi example alo erve to illutrate the adequacy theorem. The judgement for e0 can be implified to! /0 λu : 2 u 2 : 2 t Note that e 0 B 2 2 iff! Λ 2 e0 I 2. 5 Application The # operation of F # make a type abtract (i.e. hide it content) in a manner that i imilar to quantification over type variable in the polymorphic lambda calculu. In direct analogy to the property that function of type α α α are either the identity function or the everywhere- function, we have the following property for function of type α # α # α : Propoition 5.1 If f For any x α ι, f x # α ; or # α ι, then For any x ι, f x x. Proof: Conider any x y ι. efine the U-domain A 1 x and A 2 y, and et R : A 1 A 2 by R x y. Then we know that f x f y R # α ι α R by the definition of. Thi mean that for all x y ι, either f x( or f y( y. In other word, For any x For any y ι, f x ι, f y( y., or For another example of how the model can be ued to etablih propertie, recall the judgement x : e : from Section 1, which i tranlated into F # a: x : # e : β β # β( # The following propoition how that any function of thi type mut either return it argument or diverge. Propoition 5.2 Suppoe f α # α β β # α β( # α ι For any U-domain A, any x # α ι α A, and any g β β # α β ι α A, then either f x g x or f x g. The next propoition and theorem ue the model to etablih the correctne of region deallocation. Propoition 5.3 If f α # α ι, A i a U- domain, and x y # α ι α A, then f x f y. Proof: Conider the relation R : A A where R A A. We know that f f R # α ι α R Pick any element x y # α ι α A. Then x y R # α ι α R, o But then f x f x f y f y R ι α R% a deired. Theorem 5.4 Suppoe x : 0 and %A. If /0 v i 1 2, then λx : e λx : e 2 v 1 B 2 v 2 B e : 0 : n 0 n 0 where A for i Proof: We ue the model of F # in a crucial way to carry out the proof. Let C be the region variable other than. t 1 e λx : e 2 e i ΛC Λ λz : 1 f e π 1 2 z g π 1 2 z t f v vi π 1 z g v π 1 z I I u C : F Λ λv : u f e C : C I f v I I Note firt that u u # for ome u. Alo, a imple inpection of the tranlation of value how that C : C λp :! β β β # vi vi for ome term vi. Let Note that F ha type v i λp : β β β # vi u # iff. Now λx : e 2 v 1 :B n 0! iff e 1 ι n ι iff F! v 1 : I ι n ι iff F! v 2 : I ι n ι! iff e 2 ι n ι iff λx : e 2 v 2 :B n 0 8

9 = Copy PolyFun PolyApp Γ e : t 0 )*+, -. Γ # - e : t - 0 Γ f : x : 1 e : Γ f x : 1 6 e -2 : -. Γ e 1 : 1 2 -, 1 Γ e 2 : ) 1 ) 2 ) + -2 Γ e 1 e 2 : 2 3 Table 5. Polymorphim and it tranlation. Γ e : t # Γ 8 λx : 0!/(0 f1 e π 0 x π 0 - x t π 0 + x freh x t f : 0 Γ f : x : 1 e 2 t " 1 2 Γ λp : -2 µ f : p t Λ λx : 1 e : Γ e 1 : Γ e 2 : 2 1 Γ 8 λx : 3!/(0 f e 1 π 3 1 x!/(0 v e 2 π 3 2 x w " π x f w v π 3 x : 3 2 t - p freh x freh where the firt and lat line follow from the Adequacy Theorem, the econd and fourth by an argument about the meaning of term, and the third by Propoition 5.3. Intuitively, thi reult ay that if a computation ha effect and produce a value in region 0 according to the region calculu, then that computation cannot be influenced by value in region unle contain. In contrat, Tofte and Talpin [23] give a low-level operational emantic of the region calculu with explicit memory allocation and deallocation, and preent region type ytem a a tranlation from a fragment of ML into the region calculu. Their main reult [23, Theorem 6.1] i that if e i tranlated into e, and e evaluate to v (in ML), then e (in the region calculu) evaluate to a value v uch that v i equivalent to v. Thi how that memory deallocation (in the region calculu) doe not go wrong. Clearly, thee two theorem are at oppoite end of the pectrum with repect to their level of abtraction. The Tofte-Talpin theorem i baed on an operational emantic that i cloe to an implementation model. However, the actual tatement of their theorem i complex, particularly the definition of conitency. In contrat, our theorem relie on a denotational model that i further removed from an implementation model, but it i more abtract and much impler to tate. The quetion of how thee reult relate, or which i more ueful in practice, are open. 6 Region Polymorphim In the full region calculu [23], it i poible to write region-polymorphic function in recurive function definition, where the polymorphim i over region and effect variable only. For intance, one may write f 1 2 x : 1 e 1 where 1 i the region where x i located, 2 i the region of the return reult, and i the region of the recurive function. Region polymorphim i ueful epecially e 2 becaue of recurive call: the actual parameter to a recurive call can be placed in region different from the region of the original parameter. For example, a recurive call f get the parameter from 3 and allocate pace for the reult in 4. We can extend our grammar for type, value, and expreion of the region calculu from Section 2 to allow region polymorphim over region variable. We redefine t v :: :: n! C t f x : e add the form e to expreion, and replace the expreion form e e with e C e. Thu, all function are now potentially recurive and polymorphic over region, and all application mut involve application of region variable. The copy operation copie a value from one region to another, and i ueful in allocating recurive call in different region. Both the new typing rule and the tranlation appear in Table 5. The tranlation of function type i modified o that C t C t. icuion We have hown how to prove the correctne of the region calculu via a tranlation into F #. The main noveltie of the paper are twofold: firt, we point out the cloe correpondence between the region calculu and the polymorphic lambda calculu; and econd, we how that there are certain difference that motivate the extenion preent in F #. Along the way, we how that the region calculu can be implified by eliminating a ide condition on the typing rule for functional abtraction. Our work on modelling type ytem that track dependency [1] originally led u to conider the region calculu. In the work on dependency, the target of the tranlation i a language called the dependency core calculu, or CC for hort. The emantic of CC ue logical relation, a we do here. However, our attempt to ue CC to model 9

10 the region calculu failed. It would be intereting to ee if there i ome unification of the idea into one calculu; poibly F # already ha enough tructure to model notion of dependency. We believe that F # i a good candidate for modelling other region-baed calculi. For intance, Aiken, Fähndrich, and Levien [3] decribe a type ytem where the allocation and deallocation of region i plit from the coping declaration of. Other paper develop the more practical apect of region-baed memory management [4, 22]. A recent paper of Crary, Walker, and Morriett decribe a different type ytem [5] for region baed on capabilitie. We plan to contruct tranlation for thee calculi into F #. For example, the capabilitie in the calculu of Crary, Walker, and Morriett may be cloely connected with our ue of function to model the and effect; if the type doe not mention or, there i no capability for acceing the region. Such tranlation might uncover implification in thee other region calculi, a well a give greater confidence in their correctne. We conjecture that a tranlation from the original effect ytem for tate [21] may be poible for a verion of F # with recurive type. In thee effect ytem, only aignable reference cell are aociated with region, and term may be polymorphic over region. Reference [1] M. Abadi, A. Banerjee, N. Heintze, and J. G. Riecke. A core calculu of dependency. In Conference Record of the Twenty-Sixth Annual ACM Sympoium on Principle of Programming Language. ACM, [2] M. Abadi and G.. Plotkin. A PER model of polymorphim and recurive type. In Proceeding, Fifth Annual IEEE Sympoium on Logic in Computer Science, page , [3] A. Aiken, M. Fähndrich, and R. Levien. Better tatic memory management: Improving region-baed analyi of higher-order language. In Proceeding of the 1995 ACM SIGPLAN Conference on Programming Language eign and Implementation. ACM, [4] L. Birkedal, M. Tofte, and M. Vejltrup. From region inference to von Neumann machine via region repreentation inference. In Conference Record of the Twenty-Third Annual ACM Sympoium on Principle of Programming Language, page ACM, [5] K. Crary,. Walker, and G. Morriett. Typed memory management in a calculu of capabilitie. In Conference Record of the Twenty-Sixth Annual ACM Sympoium on Principle of Programming Language. ACM, [6] M. Felleien. The theory and practice of firt-cla prompt. In Conference Record of the Fifteenth Annual ACM Sympoium on Principle of Programming Language, page ACM, [] P. Fradet. Collecting more garbage. In Proceeding of the 1994 ACM Conference on Lip and Functional Programming, page ACM, [8] J.-Y. Girard. Une extenion de l interprétation de Gödel à l analye, et on application à l élimination de coupure dan l analye et la théorie de type. In J. E. Fentad, editor, Proceeding of the Second Scandinavian Logic Sympoium, volume 63 of Studie in Logic and the Foundation of Mathematic, page North-Holland, 191. [9] H. Hooya and A. Yonezawa. Garbage collection via dynamic type inference a formal treatment. In Proc. Second International Workhop on Type in Compilation (TIC 98), volume 143 of Lect. Note in Computer Sci. Springer- Verlag, [10] J. M. Lucaen and. K. Gifford. Polymorphic effect ytem. In Conference Record of the Fifteenth Annual ACM Sympoium on Principle of Programming Language, page 4 5. ACM, [11] R. Milner, M. Tofte, R. Harper, and. MacQueen. The efinition of Standard ML (Revied). MIT Pre, 199. [12] P. W. O Hearn and J. C. Reynold. From Algol to polymorphic linear lambda calculu. J. ACM, To appear. [13] P. W. O Hearn and R.. Tennent. Parametricity and local variable. J. ACM, 42:658 09, [14] C.-H. L. Ong. The Lazy Lambda Calculu: An Invetigation into the Foundation of Functional Programming. Ph thei, Imperial College, Univerity of London, [15] G.. Plotkin. LCF conidered a a programming language. Theoretical Computer Sci., 5:223 25, 19. [16] J. C. Reynold. Toward a theory of type tructure. In Proceeding Colloque ur la Programmation, volume 19 of Lecture Note in Computer Science, page , Berlin, 194. Springer-Verlag. [1] J. C. Reynold. The eence of Algol. In J. W. de Bakker and J. C. van Vliet, editor, Algorithmic Language, page North-Holland, Amterdam, [18] J. C. Reynold. Type, abtraction and parametric polymorphim. In R. E. A. Maon, editor, Information Proceing 83, page North Holland, Amterdam, [19]. Scott. A type theoretical alternative to CUCH, ISWIM, OWHY. Theoretical Computer Sci., 121: , Publihed verion of unpublihed manucript, Oxford Univerity, [20] M. B. Smyth and G.. Plotkin. The category-theoretic olution of recurive domain equation. SIAM J. Computing, 11:61 83, [21] J.-P. Talpin and P. Jouvelot. Polymorphic type, region and effect inference. Journal of Functional Programming, 2:245 21, [22] M. Tofte and L. Birkedal. A region inference algorithm. ACM Tran. Programming Language and Sytem, 20(4):34 6, July [23] M. Tofte and J.-P. Talpin. Region-baed memory management. Information and Computation, 132(2):109 16,