A Theory of Substructural Types and Control

Jesse A. Tov Riccardo Pucella Northeastern University, Boston, Massachusetts, USA {tov,riccardo}@ccs.neu.edu

Abstract cause the method raises an exception but does not return the Exceptions are invaluable for structured error handling in successfully allocated reference, there is no way for recovery high-level languages, but they are at odds with linear types. code that catches the exception to free the reference. More generally, control effects may delete or duplicate por- In short, exceptions and linear types refuse to get along, tions of the stack, which, if we are not careful, can invalidate because linear types make promises that exceptions do not all substructural usage guarantees for values on the stack. let them keep. We have developed a type-and-effect system that tracks con- With affine rather than linear types, however, divRef is trol effects and ensures that values on the stack are never not a problem, because such a type system does not require wrongly duplicated or dropped. We present the system first that references be freed explicitly. In a language with affine with abstract control effects and prove its soundness. We types, implicitly dropping a value is just fine—presumably then give examples of three instantiations with particular there is a garbage collector—and only duplication is forbid- control effects, including exceptions and delimited contin- den. Consider, however, adding delimited continuation oper- uations, and show that they meet the soundness criteria for ators such as shift and reset to a language with affine types. specific control effects. Assuming a method unref that dereferences and deallocates a reference, we might attempt to define a method squareRef Categories and Subject Descriptors D.3.3 [Programming that takes a reference to an integer, frees it, and returns its Languages]: Language Constructs and Features contents, squared: General Terms Languages def twiceTo(x: Int) = shift { (k: Int ⇒ Int) ⇒ k(k(x)) } 1. Substructural Types and Control def squareRef (r: Ref[Int]) = Consider, for example, a language like Scala (Odersky and reset { twiceTo(1) × r.unref () } Zenger 2005) with mutable references and arithmetic. Here Method twiceTo uses shift to capture its continuation up to is a method that takes two integers and divides each by the the nearest enclosing reset, and it then applies the captured other, returning a pair of references to their quotients: continuation k twice to the parameter x. Method squareRef def divRef (z 1: Int, z 2: Int) = provides the context for twiceTo to capture, which is to free (new Ref(z 1 / z 2), new Ref(z 2 / z 1)) r and multiply by its contents: Suppose that references in this language are linear, mean- [] × r.unref () . ing that they cannot be duplicated, and must be explicitly deallocated rather than implicitly dropped. In such a lan- Since twiceTo uses its continuation twice, the second use of guage, divRef has a memory leak. Most uses of divRef the continuation will access a dangling pointer that the first are harmless, but consider the expression divRef (0, 5). The use freed. method will raise a division-by-zero exception, but (assum- Typically, an affine type system works by imposing two ing one reasonable evaluation order) only after it has allo- syntactic requirements: a variable of affine type, such as cated a reference to hold the result of the first division. Be- r, cannot appear twice in its scope (up to branching), and a function that closes over an affine variable must itself have an affine type. The squareRef example violates neither dictum. In the presence of delimited continuations, we need Permission to make digital or hard copies of all or part of this work for personal or to add a third rule: that a captured continuation that contains classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation an affine value must not duplicated. A simple approximation on the first page. To copy otherwise, to republish, to post on servers or to redistribute of this rule is to give all captured continuations an affine to lists, requires prior specific permission and/or a fee. OOPSLA’11, October 22–27, 2011, Portland, Oregon, USA. (or in a linear system, linear) type. Such a rule would per- Copyright c 2011 ACM 978-1-4503-0940-0/11/10. . . $10.00 mit some limited uses of delimited continuations, such as coroutines, but we will show that this simple rule is overly about how continuations will be used in a non-substructural restrictive. source language, whereas we want to reason about continuations in order to safely use substructural types. Thielecke Our solution. The memory leak and dangling pointer in has linear types only in the object language of his translation, the above examples can be fixed by small changes to the whereas we are interested in linear (and other substructural) code. For divRef , it suffices to ensure that both divisions types in the source language. happen before both allocations: Other recent work relates substructural logics and control. def divRef (z 1: Int, z 2: Int) = { Kiselyov and Shan(2007) use a substructural logic to al- val z 12 = z 1 / z 2 low the “dynamic” control operator shift0 to modify answer val z 21 = z 2 / z 1 types in a typed setting. Unlike this work, their terms are (new Ref(z 12), new Ref(z 21)) structures in substructural logic, not their types. Mazurak } and Zdancewic’s Lolliproc (2010) relates double negation elimination in classical linear logic to delimited control. For squareRef , we need the dereferencing to happen once, We draw significantly on other work on control operators, outside the reset delimiter: effect systems, and substructural types as well. def squareRef (r: Ref[Int]) = { val z = r.unref () Control operators. The literature contains a large vocabu- reset { twiceTo(1) × z } lary of control operators, extending back to ISWIM’s J opera- } tor (Landin 1965), Reynolds’s escape (1972), and Scheme’s call/cc (Clinger 1985). However, for integration in a lan- Unfortunately, the conservative approximation suggested guage with substructural types, control operators with delim- above, that all continuations be treated linearly, would still ited extent, originating with Felleisen’s F (1988), are most disallow these repaired examples. We have designed a type- appropriate, because without some way to mask out control and-effect system (Lucassen and Gifford 1988) that permits effects, any use of control pollutes the entire program and these two repaired versions of the methods while forbidding severely limits the utility of substructural types. the original, erroneous versions. The key idea is to assign to As examples of control features to add to our calculus, we each expression a control effect that reflects whether it may consider the delimited continuation operators shift and reset duplicate or drop its continuation, and to prohibit using an (Danvy and Filinski 1989) and structured exception handling expression in a context that cannot be treated as the control (Goodenough 1975). Both shift/reset and structured excep- effect allows. In this paper, we tions have been combined with type-and-effect systems to • exhibit a generic type system for substructural types and make them more amenable to static reasoning. control defined in terms of an unspecified, abstract control effect (§4); Type-and-effect systems for control. Java (Gosling et al. 1996) has checked exceptions, an effect system for tracking • give soundness criteria for the abstract control effect and the exceptions that a method may raise. Our version of prove type safety for the generic system, relying on the exception effects is similar to Java’s, except that we offer ef- soundness of the abstract control effect (§5); and fect polymorphism, which makes higher-order programming • demonstrate three concrete instantiations of control ef- with checked exceptions more convenient. Our type system fects and prove that they meet the soundness criteria (§6). for exceptions appears in §6.3. Because Danvy and Filinski’s shift (1989) captures a de- The generic type-and-effect system in §4 is defined as limited continuation up to the nearest reset delimiter, typing an extension to λURAL (Ahmed et al. 2005), a substructural shift and reset requires some nonlocal means of communi- λ calculus, which we review in §3, after discussing related cating types between delimiters and control operators. They work in §2. realize this communication with a type-and-effect system, which allows shift to capture and compose continuations of 2. Related Work and Comparison varying types. Asai and Kameyama(2007) extend Danvy This work is not the first to relate substructural types to con- and Filinski’s (monomorphic) type system with polymor- trol operators and control effects. Thielecke(2003) shows phism, which includes polymorphism of answer types. We how to use a type-and-effect system to reason about how give two substructural type systems with shift and reset. expressions treat their continuations. In particular, he gives Section 6.1 presents a simpler version that severely limits the a continuation-passing style transform where continuations answer types of continuations that may be captured. Then, that will be used linearly are given a linear type. Thielecke in §6.2, we combine the simpler system with a polymor- notes that many useful applications of continuations treat phic version of Danvy and Filinski’s, similar to Asai and them linearly. However, his goals are different than ours. Kameyama’s, to allow answer-type modification and poly- He uses substructural types in his object language to reason morphism in a substructural setting. Substructural type systems. Researchers have proposed a v ::= x | λx.e | Λ.e | hi| inl v | inr v (values) plethora of substructural type systems. These range from e ::= v | e1 e2 | e | let hi = e1 in e2 (expressions) minimalistic models (Wadler 1992; Bierman 1993; Bar- | case e of inl x1 → e1; inr x2 → e2 ber 1996; Morrisett et al. 2005) based on Girard’s linear logic (1987), to real programming languages, which are q ∈{ U, R, A, L} (qualifier constants) often oriented toward specific problems such as safety in ξ ::= α | q (qualifiers) low-level languages (Grossman et al. 2002), typestate and τ ::= α | τ1 ( τ2 | ∀α:κ.τ | 1 | τ1 ⊕ τ2 (pretypes) protocol checking (Aldrich et al. 2009), or security (Swamy τ ::= α | ξτ (types) et al. 2010). We translate our substructural type-and-effect system into ι ::= ξ | τ | τ (type-level terms) URAL Ahmed et al.’s λ (2005), which is a polymorphic λ κ ::= QUAL | ? | ? (kinds) calculus that supports a variety of substructural typing dis- ciplines. We provide a primer on λURAL in §3. Figure 1. λURAL syntax Motivation. The software engineering case for structured exception handling is widely acknowledged and understood, terparts in the calculus with control effects (§4), which will but shift and reset (Danvy and Filinski 1989), the other appear in red.1 control operators discussed in this paper, are more obscure. The essential idea is simple: whereas raising an exception The expression level. Values include abstractions, type ab- discards the context up to some delimiter—the exception stractions, the unit value, and injections into a sum. (This handler—shift captures and reifies the context up to its delim- differs from Ahmed et al.’s presentation of λURAL by in- iter, reset, which allows reinstating the context later. These cluding sums—additive disjunctions, to be precise—rather control operators may be used to implement exceptions, by than multiplicative conjunctions. Our theorems handle both, capturing continuations but never reinvoking them, but they but we omit products in this paper for brevity. Sums are may also express other control structures, such as coroutines more useful for our purposes here.) Expressions include val- and cooperative multithreading, and they may be used to ues, application, type application, unit elimination, and sum abstract non-determinism and search in an elegant way. elimination. Following Ahmed et al., we elide the formal Our goal is to safely integrate control operators with parameter in type abstractions and the actual parameter in substructural types. A substructural type system regulates type applications. the order and number of uses of data by statically ensuring The type level. Expressions in λURAL are classified by that some values be used at most once, at least once, or ex- types (τ), but the language at the type level is much richer. actly once (Walker 2005). Like shift and reset, substructural Four constant qualifiers (q) distinguish four substructural types are a general facility that can express a variety of spe- properties that may be enforced for values: cific language features, mostly for the purpose of managing stateful resources, such as typestate, region-based memory L as in linear, for values that may be management, and session types. neither duplicated nor implicitly The direct impetus for this work is the design of the dropped; programming language Alms (Tov and Pucella 2011), which A as in affine, for values that may be L provides both exceptions and affine types, a variety of sub- dropped (weakening) but not dupli- @ structural type that can prohibit reusing particular values. cated; @ As demonstrated in §1, the combination of affine types and AR R as in relevant, for values that may exceptions is not a problem. However, as we observe in that @ be duplicated (contraction) but not @ previous work, “we anticipate that safely combining linear- U dropped; and ity with exceptions requires a type-and-effect system to track when raising an exception would implicitly discard linear U as in unlimited, for ordinary values values.” Our desire to add linear types to Alms motivates that allow both dropping and dupli- this development of a general theory of substructural types cation. and control effects. The four constant qualifiers form a lattice, whereby it is always safe to treat a value as if it has a higher qualifier than 3. Syntax and Semantics of λURAL its own. In this paper, we add control effects to Ahmed et al.’s λURAL Qualifiers (ξ) include both qualifier constants and type (2005), a substructural λ calculus. Our presentation of variables, allowing for qualifier polymorphism. Pretypes (τ ) URAL λ is heavily based on theirs, with a few small changes. 1 URAL This is the color version of this paper; a black-and-white version, which is The syntax of λ appears in Figure1. Those non- more suitable for printing, is available online at www.ccs.neu.edu/∼tov/ terminals that appear in blue are different from their coun- pubs/substructural-control. E ::=[] | E e2 | v1 E | E (evaluation contexts) ∆ ` ξ1 ξ2 (qualifier subsumption) | let hi = E in e2 | case E of inl x → e ; inr x → e QSUB-BOT QSUB-TOP QSUB-REFL 1 1 2 2 ∆ ` ξ : QUAL ∆ ` ξ : QUAL ∆ ` ξ : QUAL e 7−→ e0 (reduction) ∆ ` U ξ ∆ ` ξ L ∆ ` ξ ξ

(λx.e1) v2 7−→ {v2/x}e2 ∆ ` τ ξ (qualiﬁer bound for types) (Λ.e) 7−→ e

let hi = hi in e 7−→ e B-VAR B-TYPE ∆ ` α : ? ∆ ` τ : ? ∆ ` ξ0 ξ case inl v of inl x1 → e1; inr x2 → e2 7−→ {v/x1}e1 0 ∆ ` α L ∆ ` ξ τ ξ case inr v of inl x1 → e1; inr x2 → e2 7−→ {v/x2}e2 e 7−→ e0 E[e] 7−→ E[e0] ∆ ` Γ ξ (qualiﬁer bound for type contexts)

B-NIL B-CONS Figure 2. λURAL operational semantics ∆ ` ξ : QUAL ∆ ` Γ ξ ∆ ` τ ξ ∆ ` • ξ ∆ ` Γ, x:τ ξ

∆ ` ι : κ (kinding type-level terms) Figure 4. λURAL statics (ii): qualiﬁers

K-VAR α:κ ∈ ∆ K-QUAL

∆ ` α : κ ∆ ` q : QUAL ∆ ` Γ Γ1 Γ2 (type context splitting)

K-ARR K-ALL S-CONSL S-NIL ∆ ` τ1 : ? ∆ ` τ2 : ? ∆, α:κ ` τ : ? ∆ ` Γ Γ1 Γ2 ∆ ` τ : ? ∆ ` τ1 ( τ2 : ? ∆ ` ∀α:κ.τ : ? ∆ ` • • • ∆ ` Γ, x:τ (Γ1, x:τ) Γ2

K-SUM S-CONSR K-UNIT ∆ ` τ1 : ? ∆ ` τ2 : ? ∆ ` Γ Γ1 Γ2 ∆ ` τ : ? ∆ ` 1 : ? ∆ ` τ1 ⊕ τ2 : ? ∆ ` Γ, x:τ Γ1 (Γ2, x:τ)

K-TYPE S-CONTRACT ∆ ` τ : ? ∆ ` ξ : QUAL ∆ ` Γ Γ1 Γ2 ∆ ` τ R ξ ∆ ` τ : ? ∆ ` Γ, x:τ (Γ1, x:τ) (Γ2, x:τ)

Figure 3. λURAL statics (i): kinding Figure 5. λURAL statics (iii): context splitting specify the representation of a value, and its introduction and 3.1 Operational Semantics elimination rules. Pretypes include type variables, function The operational semantics of λURAL is completely standard types, universal quantification, the unit type, and additive and appears in Figure2. Reduction is call-by-value and eval- disjunction. Types (τ) classify expressions. A type is either uates operators before operands, which is important when ξ a pretype decorated with its qualifier ( τ ) or a type variable. we consider the sequencing of effects in §4. We use non-terminal ι to refer to the three kinds of type-level terms as a group. 3.2 Static Semantics Type judgments for λURAL use two kinds of contexts: The kind level. Types in λURAL are classified by three ∆ ::= •| ∆, α:κ (kind contexts) kinds (κ): QUAL for qualifiers, ? for pretypes, and ? for Γ ::= •| Γ, x:τ (type contexts) types. Type variables may have any of these three kinds, which is why universal quantification (∀α:κ.τ ) specifies the Figure3 contains the kinding judgment ( ∆ ` ι : κ), kind of α. which assigns kinds to type-level terms. This judgment ∆;Γ ` e : τ (typing expressions)

T-WEAK T-VAR T-ABS ∆ ` Γ Γ1 Γ2 ∆ ` Γ2 A ∆;Γ 1 ` e : τ ∆ ` τ : ? ∆ ` Γ ξ ∆;Γ , x:τ1 ` e : τ2 ξ ∆;Γ ` e : τ ∆; •, x:τ ` x : τ ∆;Γ ` λx.e : (τ1 ( τ2)

T-INL T-INR T-TABS T-UNIT ∆ ` τ1 ξ ∆ ` τ2 : ? ∆ ` τ2 ξ ∆ ` τ1 : ? ∆ ` Γ ξ ∆, α:κ;Γ ` e : τ ∆ ` ξ : QUAL ∆;Γ ` v1 : τ1 ∆;Γ ` v2 : τ2 ξ ξ ξ ξ ∆;Γ ` Λ.e : ∀α:κ.τ ∆; • ` hi : 1 ∆;Γ ` inl v1 : (τ1 ⊕ τ2) ∆;Γ ` inr v2 : (τ1 ⊕ τ2)

T-APP ∆ ` Γ Γ1 Γ2 T-TAPP ξ ξ ∆;Γ 1 ` e1 : (τ1 ( τ2)∆;Γ 2 ` e2 : τ1 ∆;Γ ` e : ∀α:κ.τ ∆ ` ι : κ ∆;Γ ` e1 e2 : τ2 ∆;Γ ` e : {ι/α}τ

T-LETUNIT T-CASE ξ ∆ ` Γ Γ1 Γ2 ∆ ` Γ Γ1 Γ2 ∆;Γ 1 ` e : (τ1 ⊕ τ2) ξ ∆;Γ 1 ` e1 : 1 ∆;Γ 2 ` e2 : τ ∆;Γ 2, x1:τ1 ` e1 : τ ∆;Γ 2, x2:τ2 ` e2 : τ

∆;Γ ` let hi = e1 in e2 : τ ∆;Γ ` case e of inl x1 → e1; inr x2 → e2 : τ

Figure 6. λURAL statics (iv): typing enforces the type/pretype structure, whereby type construc- that both branches of a case expression share the same tors such as ⊕ form a pretype from types (ruleK-S UM), context. and decorating a pretype with a qualifier forms a type • Rule T-ABS selects a qualifier ξ for a function type based (ruleK-T YPE). on bounding the context, Γ. This means that the qualifier In Figure4, three judgments relate qualifiers to each of a function type must upper bound the qualifiers of the other, to types, and to type contexts. Qualifier subsumption types of the function’s free variables. As we will see in (∆ ` ξ1 ξ2) defines the qualifier order, with top L and §5, this property is key to our soundness theorem. bottom U. The next judgment bounds a type by a qualifier; judgment ∆ ` τ ξ means that values of type τ may safely 4. Generic Control Effects be used according to the structural rules implied by ξ. Finally, Rather than add a specific control effect, such as exceptions bounding a type context by a qualifier (∆ ` Γ ξ) means URAL that every type in context Γ is bounded by qualifier ξ. or delimited continuations, to λ , we aim to design a sub- Figure5 gives rules for splitting a type context into structural type system with a general notion of control effect. Thus, in this section, we define a new calculus, λURAL(C), two (∆ ` Γ Γ1 Γ2), which is necessary for dis- tributing typing assumptions to multiple subterms of a term. parameterized by an unspecified control effect. Any variable may be distributed to one side or the other. 4.1 The Control Effect Parameter RuleS-C ONTRACT implements the contraction structural In this subsection, we give the form of the parameter that rule, whereby variables whose type is unlimited or relevant stands for a particular control effect. Our definition of may be duplicated to both contexts. λURAL(C) relies only on this abstract specification of the Finally, Figure6 gives the judgment for assigning types formal parameter. In §5, we specify several properties of the to expressions (∆;Γ ` e : τ). Several points are worthy of parameter that are sufficient for a generic soundness theorem note: to hold, and in §6 we give three examples of actual control • The weakening rule, T-WEAK, allows discarding por- effect parameters. tions of the context that are upper-bounded by A, which Definition 4.1 (Control effect). means that all the values dropped are either affine or A control effect instance is a triple (C, ⊥C, ) where C is unlimited. 5 a set of control effects (c), ⊥C ∈ C is a distinguished pure • The rules for application and for unit and sum elimina- effect that denotes no actual control, and : C × C * C tion, T-APP, T-LETUNIT, and T-CASE, split the con- is an associative, partial, binary operation5 denoting effect text to distribute assumptions to subterms. Note, however, sequencing. D `C i : k (kinding type-level terms) D `C c ξ (qualifier bound for control effects)

C-K-BOT C-B-PURE C-B-UNL

D `C ξ : QUAL D `C c : CTL D `C ⊥C : CTL D `C ⊥C ξ D `C c U C-K-ARR

D `C t1 : ? D `C t2 : ? D `C c : CTL c D `C c1 c2 (control effect subsumption) D `C t1 (− t2 : ?

CSUB-REFL CSUB-TRANS C-K-ALL 0 0 D `C c : CTL D `C c1 c D `C c c2 D, α:k `C t : ? D `C c : CTL c D `C c c D `C c1 c2 D `C ∀ α:k.t : ?

Figure 7. λURAL(C) statics (i): updated kinding rules Figure 8. λURAL(C) statics (ii): control effect judgments

For example, in §6.3 we add exception handling to and splitting type contexts (D `C G G1 G2) are URAL λ (C). An exception effect is the set of exceptions that isomorphic to the λURAL versions of those judgments from may be raised by an expression, the distinguished pure effect Figures4 and5. They are merely updated with new non- ⊥C is the empty set, and sequencing is set union. A non- terminals as appropriate (i.e., κ to k, τ to t, and τ to t). empty effect indicates that an expression may discard part of its continuation, whereas the empty effect guarantees that Kinding. We identify control effects as the type-level an expression treats its continuation linearly. terms (i) that are assigned kind CTL by the kinding judg- In simple cases, as with exceptions, effects form a join ment. Figure7 shows one new kinding rule,C-K-B OT, semilattice where sequencing is the join, but this is not which assigns kind CTL to the pure effect ⊥C. We update necessarily true in general (§6.2). rulesC-K-A RR andC-K-A LL to account for latent effects in function and universal pretypes. The remaining kinding 4.2 Updated Syntax rules are the same as for λURAL, with non-terminals mutatis In λURAL(C), control effects constitute a fourth kind of type- mutandis. Specific control effect instances (§6) must define level term, in addition to qualifiers, pretypes, and types. We additional kinding rules for their particular effects. add a new kind, CTL, and include abstract control effects Control effect judgments. The first new judgment for con- (c ∈ C) among the type-level terms: trol effects (D `C c ξ, Figure8) relates control effects to k ::= QUAL | ? | ? | CTL (kinds) qualifiers. This gives the meaning of a control effect in terms i ::= ξ | t | t | c (type-level terms) of a lower bound for how an expression with that effect may e Function and universal pretypes now have latent effects, treat its own continuation. For example, if an expression has some effect c such that D ` c A, this indicates that which record the effect that will happen when an abstraction C e is applied. We update the definition of pretypes to include may drop but not duplicate its continuation. We give two these latent effects: rules here: c c • RuleC-B-P URE says that the pure effect is bounded by t ::= ···| t1 (− t2 | ∀ α:k.t (pretypes) t ::= α | ξt (types) any qualifier, which means that a pure expression satisfies any requirement for how it treats its continuation. The other pretype (t) productions remain unchanged. • RuleC-B-U NL says that all control effects are bounded For non-terminal symbols that differ between λURAL by U, which means that we may assume, conservatively, and λURAL(C), we use red Roman letters (t, k, G,...) that any expression might freely duplicate or drop its for λURAL(C) to distinguish them from λURAL, where they continuation. appeared in blue Greek (τ, κ, Γ, . . . ). Specific instances of the control effect parameter will extend URAL 4.3 Static Semantics of λ (C) this judgment to take into account the properties of a partic- All type system judgments from λURAL are updated for ular control effect. URAL URAL λ (C), and λ (C) adds two new judgments as well. The second judgment for control effects (D `C c1 The kinding and expression typing judgments are the only c2) defines a subsumption order for control effects. This two to change significantly. The judgments for bounding means that an expression whose effect is c1 may be safely types (D `C t ξ), bounding type contexts D `C G ξ), considered to have effect c2. Only two rules for the judgment D;G `C e : t ; c (typing expressions)

C-T-SUBSUME C-T-WEAK 0 0 D;G `C e : t ; c D `C c c D `C G G1 G2 D;G 1 `C e : t ; c D `C G2 A

D;G `C e : t ; c D;G `C e : t ; c

C-T-ABS C-T-TABS C-T-VAR D `C G ξ D `C G ξ C-T-UNIT

D `C t : ? D;G , x:t1 `C e : t2 ; c D, α:k;G `C e : t ; c D `C ξ : QUAL ξ c ξ c ξ D; •, x:t `C x : t ; ⊥C D;G `C λx.e : (t1 (− t2); ⊥C D;G `C Λ.e : ∀ α:k.t ; ⊥C D; • `C hi : 1 ; ⊥C

C-T-INL C-T-INR

D `C t1 ξ D `C t2 : ? D `C t1 : ? D `C t2 ξ

D;G `C v1 : t1 ; ⊥C D;G `C v2 : t2 ; ⊥C ξ ξ D;G `C inl v1 : (t1 ⊕ t2); ⊥C D;G `C inr v2 : (t1 ⊕ t2); ⊥C

C-T-APP ξ1 c D;G 1 `C e1 : (t1 −( t2); c1 D;G 2 `C e2 : t1 ; c2 C-T-TAPP ξ c0 D `C G2 ξ2 D `C c1 ξ2 D `C c2 ξ1 D;G `C e : ∀ α:k.t ; c 0 D `C G G1 G2 D `C c1 c2 c : CTL D `C i : k D `C c c : CTL 5 5 5 0 D;G `C e1 e2 : t2 ; c1 c2 c D;G `C e : {i/α}t ; c c 5 5 5 C-T-CASE ξ1 C-T-LETUNIT D;G 1 `C e : (t1 ⊕ t2); c ξ1 0 0 D;G 1 `C e1 : 1 ; c1 D;G 2 `C e2 : t ; c2 D;G 2, x1:t1 `C e1 : t ; c D;G 2, x2:t2 `C e2 : t ; c 0 D `C G2 ξ2 D `C c1 ξ2 D `C G2 ξ2 D `C c ξ2 D `C c ξ1 0 D `C G G1 G2 D `C c1 c2 : CTL D `C G G1 G2 D `C c c : CTL 5 5 0 D;G `C let hi = e1 in e2 : t ; c1 c2 D;G `C case e of inl x1 → e1; inr x2 → e2 : t ; c c 5 5

Figure 9. λURAL(C) statics (iii): typing appear in Figure8, which together ensure that control effect (1) The first premise, as in λURAL, splits the type context G subsumption is a preorder. As with control effect bounding, into G1 for typing e1 and G2 for typing e2. specific control effect instances will extend this judgment. (2–3) As in λURAL, these premises assign types to expressions e and e , but they assign control effects c and c Expression typing. The expression typing judgment for 1 2 1 2 as well. λURAL(C) (Figure9) assigns not only a type t but an effect (4) This premise relates the type of e1 to the effect of e2 c to expressions: D;G `C e : t ; c. Having seven premises, the rule for applications (C-T-APP) is unwieldy, but it likely to ensure that e2’s effect does not violate e1’s invariants. gives the most insight into how λURAL(C) works: Because we fix a left-to-right evaluation order, by the time e2 gets to run, e1 has reduced to a value of type ξ1 c (t1 (− t2), which thus may be treated according to qual- (1) D `C G G1 G2 ifier ξ1. Because that value is part of e2’s continuation, ξ1 c (2) D;G 1 `C e1 : (t1 (− t2); c1 we require that e2’s effect, c2, be lower-bounded by ξ1.

(3) D;G 2 `C e2 : t1 ; c2 In other words, e2 will treat its continuation no more liberally than ξ1 allows. (4) D `C c2 ξ1 (5–6) These premises relate the free variables of e2 to the (5) D `C G2 ξ2 effect of e1. Due to the evaluation order, e2 appears (6) D `C c1 ξ2 unevaluated in e1’s continuation, which means that if

(7) D `C c1 c2 c : CTL e1 drops or duplicates its continuation then e2 may be 5 5 D;G `C e1 e2 : t2 ; c1 c2 c evaluated never or more than once. Premise (5) says that 5 5 the type context for typing e2, and thus e2’s free variables, We consider the premises in order: are bounded above by some qualifier ξ2, and this qualifier thus indicates how many times it is safe to evaluate e2. The development of this section is constrained by several Premise (6) lower bounds e1’s effect, c1, by ξ2, ensuring dependencies, so we provide an outline: e e that 1’s effect treats 2 properly. The Translation Parameter (§5.1). A control effect instance (7) The net effect of the application expression is a sequence must supply a few definitions to fully specify its particu- of the effect of e1 (c1), then the effect of e2 (c2), and lar CCoS translation. e finally the latent effect of the function to which 1 must The Translation (§5.2). The definition of the CCoS transla- evaluate (c): c1 c2 c. This premise checks that those 5 5 tion relies on the definitions supplied by the control effect three effects may be sequenced in that order according to parameter. a particular control effect’s definition of sequencing and the kinding judgment. Parameter Properties (§5.3). A control effect instance must satisfy several properties on which the generic type safety Rules C-T-LETUNIT and C-T-CASE (unit and sum elimi- theorem relies. nation) are similar, since they need to safely sequence two Generic Type Safety (§5.4). The section culminates in a subexpressions. Both rules follow rule C-T-APP in relating generic proof of type safety for λURAL(C). the effect of the first subexpression to the type context of the second and effect of the second to the qualifier of the first. 5.1 The Translation Parameter Rule C-T-TAPP (type application), while dealing with only Definition 5.1 (Translation parameter). one effectful subexpression, needs to sequence the effect of evaluating the expression in a type application with the latent The definition of the generic CCoS translation relies on the effect of the resulting type abstraction value. following effect-specific definitions: The subsumption rule C-T-SUBSUME implements con- • a metafunction (·)∗ from effects to qualifiers, such that ∗ ∗ trol effect subsumption, whereby an expression of effect c ⊥C = L and α = α; 0 0 may also be considered to have effect c if c is less than c • a value doneC, to use as the initial continuation for a in the control effect subsumption order. C-T-WEAK, which CCoSed program; and handles weakening, is unchanged from λURAL. − + • a pair of answer-type metafunctions hh·, ·iic and hh·, ·iic , The remaining rules are for typing values, which always each of which maps a λURAL type and a λURAL(C) effect URAL have the pure effect ⊥C. Rules C-T-UNIT, C-T-INL, and to a λ type. C-T-INR, for unit and sum introduction, are unchanged from ∗ − λURAL, except that each of them assigns the pure effect. Intuitively, we can understand metafunctions (·) , hh·, ·iic , + URAL Rules C-T-ABS and C-T-TABS also assign the pure effect and hh·, ·iic as relating the effect of a λ (C) expression URAL to their values, but each records the effect of the abstraction to the type of its translation into λ . Typically, the CPS body as the latent effect in the resulting type. translation of an expression of some type τ yields a type like (τ → Answer) → Answer. 5. The Generic Theory Given a λURAL(C) expression whose translated type is τ and λURAL(C) To prove type safety for , we define a type-preserv- whose effect is c, our translation yields type ing translation to λURAL. Rather than provide a reduction semantics for λURAL(C), we define its operational seman- c∗ − + (τ ( hhτ0,c iiC ) ( hhτ0,c iiC tics in terms of the translation and the reduction semantics URAL ∗ of λ (§3.1). Thus, if we can show that all well-typed for some answer type τ0. That is, (·) gives the qualifier of λURAL(C) programs translate to well-typed λURAL programs, the continuation, and the other two metafunctions give the then λURAL’s type safety theorem applies to λURAL(C) as answer types, which may depend on the nature of the control well. effect. Because they give the answer types in negative and − The translation is into what Danvy and Filinski(1989) positive positions, respectively, we call hhτ,c iiC the negative + call continuation-composing style (henceforth “CCoS”). It is answer type and hhτ,c iiC the positive answer type. similar to continuation-passing style, but unlike continuation- passing style it still relies on the object language’s order of 5.2 The Translation evaluation. In this subsection, we specify the CCoS translation from In order to specify the translation and prove the proposi- λURAL(C) to λURAL. In several places, we rely on the defi- ∗ − + tions specified later in this section, we impose several more nitions of c , doneC, hhτ,c iiC , and hhτ,c iiC supplied by the requirements on the abstract control effect parameter. As the control effect parameter. semantics of λURAL(C) was parameterized by an abstract The translation for kinds and kind contexts appears in Fig- control effect, so is the theory of λURAL(C) parameterized by ure 10. The control effect kind CTL translates to QUAL, and several definitions and properties that a control effect must the other three kinds translate to themselves. The translation satisfy. of a kind context merely translates each kind in its range. QUAL∗ = QUAL (kinds) additional type abstraction added to the front, which matches ?∗ = ? the addition of the universal quantifier in the type translation, ?∗ = ? and both translate the body according to the expression translation e . The expression translation is standard except for CTL∗ = QUAL C two unusualJ K aspects of the translation of applications and •∗ = • (kind contexts) type applications: (D, α:k)∗ =D ∗, α:k∗ • The result of evaluating e1, bound to x1, is in each case instantiated by a type application, which compensates for Figure 10. CCoS translation (i): kinds and kind contexts the new type abstraction in the translation of abstractions. For the type application case, x1 is instantiated then α∗ = α (pretypes) again, corresponding to the instantiation from the source expression. 1∗ = 1 ∗ ∗ ∗ • Curiously, the continuation y is η-expanded to λx.y x. (t1 ⊕ t2) = t1 ⊕ t2 While η-expanding a variable may seem useless, it is (t −c t )∗ = 1 ( 2 actually necessary to obtain a type-preserving translation. L ∗ L c∗ ∗ − + ∀α: ?. (t1 ( ( (t2 ( hhα,c iiC ) ( hhα,c iiC )) η c ∗ In particular, the reason for this expansion is to handle (∀ β:k.t ) = effect subsumption. Effects in λURAL(C) are translated to ∗ L ∗ L c ∗ − + URAL URAL ∀α: ?. ∀β:k . ( (t ( hhα,c iiC ) ( hhα,c iiC ) qualifiers in λ , and while λ (C) supports effect sub- ∗ sumption directly, there is no analogous qualifier subsump- α = α (types) URAL ξ ∗ ξ ∗ tion in λ . However, qualifier subsumption for function ( t) = t types can be done explicitly using η expansion: ∗ • = • (type contexts) Lemma 5.2 (Dereliction). ∗ ∗ ∗ (G, x:t) =G , x:t ξ 0 If ∆;Γ ` v : (τ1 ( τ2) and ∆ ` ξ ξ then ∆;Γ ` ξ0 λx.v x : (τ1 ( τ2). Figure 11. CCoS translation (ii): type-level terms and contexts The proof of Lemma 5.2 relies on another lemma: Lemma 5.3 (Value strengthening). Figure 11 presents the translation for pretypes, types, and Any qualifier that upper bounds the type of a value also type contexts. Most of this translation is straightforward: bounds the portion of the type context necessary for typing type variables and the unit pretype translate to themselves, that value. That is, if ∆;Γ ` v : τ and ∆ ` τ ξ then there sum types translate both disjuncts, types composed of a qual- exist some Γ1 and Γ2 such that ifier and a pretype translate the pretype, and type contexts translate all the types in their range. The two interesting • ∆ ` Γ Γ1 Γ2, cases are for function and universal pretypes. These follow • ∆;Γ 1 ` v : τ, the usual CPS translation for function and universal types, • ∆ ` Γ1 ξ, and with several refinements: • ∆ ` Γ2 A. • Each adds an extra universal quantifier in front of its result, which is used to make (type) abstractions polymor- Proof. By induction on the typing derivation for v. phic in their answer types. • Because the effect of an expression limits how it may Proof of Lemma 5.2. Choose type contexts Γ and Γ ac- use its continuation, the translation c∗ of latent effect c 1 2 cording to Lemma 5.3. Then ∆;Γ , x:τ ` v x : τ by becomes the qualifier of the continuation. 1 1 2 rule T-APP. By induction on the length of Γ1 and transitivity 0 • All other qualifiers of the translated pretype are L. (This is of qualifier subsumption, we know that ∆ ` Γ1 ξ . Then ξ0 because the translation never needs to duplicate partially- by rule T-ABS, ∆;Γ 1 ` λx.v x : (τ1 ( τ2), and we applied continuations, so L is a sufficiently permissive change Γ1 to Γ by rule T-WEAK. qualifier for those continuations. Furthermore, because the type rules for abstractions always allow a qualifier of L, using L wherever possible simplifies the proof.) Operational semantics of λURAL(C). Having defined the Translation of values and expressions is defined by mu- translation, we run a program e by applying the CCoS trans- ∗ tual induction in Figure 12. Value translation (v ) is mostly lation and passing it the initial continuation doneC. We de- straightforward. Both value and type abstraction have an fine the operational semantics of λURAL(C) as a partial func- ∗ x = x (values) 4. If D `C c1 c2, then for every type τ there exists some ∗ 0 0 − − 0 + (λx.e) =Λ .λx. e type τ such that hhτ ,c 1iiC = hhτ,c 2iiC and hhτ ,c 1iiC = C + ∗ J K hhτ,c 2iiC . (Λ.e) =Λ .Λ. e C RATIONALE. For control effect subsumption to be (inl v)∗ = inl v∗J K valid, related control effects must generate related ∗ ∗ (inr v) = inr v answer types. hi∗ = hi Parameter Property 2 (Done). ∗ L v C = λy.y v (expressions) If ∆ ` τ A then ∆; • ` done : (τ hhτii ). J K C ( C e1 e2 C = λy. e1 C (λx1. e2 C (λx2.x1 x2 (λx.y x))) RATIONALE. The doneC value must be well typed for J K J K J K e1 C = λy. e1 C (λx1.x1 (λx.y x)) the translation of a whole program to be well typed. J K J K Parameter Property 3 (Effect sequencing). let hi = e1 in e2 C J K ∗ ∗ ∗ = λy. e1 (λx1. let hi = x1 in e2 y) If D `C c1 c2 : CTL then D ` (c1 c2) c1 and C C ∗ ∗ ∗ J K J K D ` (c1 5c2) c2 . 5 case e of inl x → e ; inr x → e 1 1 2 2 C RATIONALE5 . Sequencing lowers the translation of J K = λy. e C (λx. case x of inl x1 → e1 C y; control effects in the qualifier order. This makes sense, J K J K inr x2 → e2 C y) because if either of two sequenced expressions may J K duplicate or discard their continuations, then the com- Figure 12. CCoS translation (iii): values and expressions pound expression may do the same. Parameter Property 4 (Bottom and lifting). tion eval : Expressions * Values ∪ {WRONG}: 1. c1 c2 = ⊥C if and only if c1 = c2 = ⊥C.  ∗ 5 v if e done 7−→ v; RATIONALE. Sequencing impure expressions should  C C  J K ∗ 0 not result in a pure expression. eval(e) = WRONG if e C doneC 7−→ e 0 2. If D ` c c : CTL and c c 6= ⊥ , then there exist  JsuchK that e is not a value C 1 2 1 2 C  0 5 0 5 and ¬∃e00.e0 7−→ e00. some c1 6= ⊥C and c2 6= ⊥C such that 0 • D `C c1 c1, 5.3 Parameter Properties 0 • D `C c2 c2, Having defined the CCoS translation, we are now ready to 0 0 • c1 c2 = c1 c2, and state the additional properties that the abstract control effect 5 0 0 5 • D `C c c : CTL. parameter must satisfy for the generic type safety theorem 1 2 RATIONALE5 . This assumption is likely not necessary, (§5.4) to hold: but it significantly simplifies the proof by allowing the Parameter Property 1 (Answer types). effects in a sequence to be considered either all pure

− + or all impure. 1. For all τ, hhτ, ⊥CiiC = hhτ, ⊥CiiC . RATIONALE. For pure expressions, the negative and The final property concerns four lemmas that we state positive answer types agree, because a pure expression and prove for the generic system in the next subsection. An finishes by calling its continuation. Henceforth, we actual control effect instance needs to extend these lemmas to cover any additional rules added to the relevant judgments: are justified defining the pure answer type hhτiiC , + hhτ, ⊥CiiC . ∗ ∗ − 2. If D ` τ : ? and D `C c : CTL then D ` hhτ,c iiC : ? Parameter Property 5 (New rules). and D∗ ` hhτ,c ii+ : ?. C 1. Lemma 5.4 (§5.4) must be extended, by induction on RATIONALE. For the translation to be well typed, well-kinded types and effects must become well- derivations, for any rules added to the kinding judgment kinded answer types. D `C i : k. 3. For all D, τ, c 6= ⊥ , and c 6= ⊥ such that D ` 2. Lemma 5.5 (§5.4) must be extended, by induction on 1 C 2 C C derivations, for any rules added to the control effect c1 c2 : CTL, 5 − − bounding judgment D `C c ξ. (a) hhτ,c 1 c2iiC = hhτ,c 2iiC , 5 + + 3. Lemma 5.6 (§5.4) must be extended, by induction on (b) hhτ,c 1 c2iiC = hhτ,c 1iiC , and 5− + derivations, for any rules added to the control effect sub- (c) hhτ,c 1iiC = hhτ,c 2iiC . sumption judgment D `C c1 c2. RATIONALE. Effect sequencing must maintain an- 4. Lemma 5.7 (§5.4) must be extended, by induction on swer types in order for the continuations of sequenced derivations, for any rules added to the expression typing expressions to compose. judgment D;G `C e : t ; c. ∗ ∗ In §6, we give several example control effects and show the lemma assumption, D ` ξ0 c , and by transitivity ∗ 0∗ that they satisfy the above properties. of qualifier subsumption, D ` ξ0 c . Thus, we can 0 apply the induction hypothesis at D;G `C e : t ; c , using 5.4 Generic Type Safety 0 the same ξ0 but with τ0 for τ0, yielding Assuming that the above properties hold of the control ∗ ∗ L ξ0 ∗ 0 0 − 0 0 + effect parameter, we can now prove a type safety theorem D ;G ` e C : ( (t ( hhτ0,c iiC ) ( hhτ0,c iiC ). J K for λURAL(C) that leaves the control effect abstract. We Then it suffices to substitute hhτ ,c ii− for hhτ 0 ,c 0ii− and sketch the proof here, but the full proof is available online at 0 C 0 C hhτ ,c ii+ for hhτ 0 ,c 0ii+, which we know to be equal by www.ccs.neu.edu/∼tov/pubs/substructural-control. 0 C 0 C Property1(part 4). We begin with a lemma that ensures that control effects translate to well-formed qualifiers: D `C G G1 G2 D `C G2 ξ2 ξ1 c D;G 1 `C e1 : (t1 (− t2); c1 D `C c1 ξ2 Lemma 5.4 (Translation of kinding). D;G 2 `C e2 : t1 ; c2 D `C c2 ξ1 For all D, i, and k, if D ` i : k then D∗ ` i∗ : k∗. C D `C c1 c2 c : CTL Case 5 5 . We continue with two lemmas concerning how the trans- D;G `C e1 e2 : t2 ; c1 c2 c 5 5 lation of control effects to qualifiers relates to qualifier sub- For rule C-T-APP, we want to show that e1 e2 C has sumption. The former ensures that the control effect bound type J K L ξ0 ∗ − + used by typing rules such as C-T-APP matches the qualifier ( (t2 ( hhτ0,c 1 c2 ciiC ) ( hhτ0,c 1 c2 ciiC ). assigned to the type of a continuation by the CCoS trans- 5 5 5 5 Consider the translation of e e , lation. The latter shows that a larger control effect, which 1 2 indicates more liberal treatment of a continuation, maps to a λy. e1 C (λx1. e2 C (λx2.x1 x2 (λx.y x))). smaller qualifier, which indicates more liberal treatment of J K J K any value. The type derivation is too large to show here in detail, but it hinges on giving the right qualifiers to the types of con- Lemma 5.5 (Translation of effect bounds). ∗ ∗ tinuations. We will consider the continuation passed to If D `C c ξ then D ` ξ c . the whole expression and the continuations constructed Lemma 5.6 (Translation of effect subsumption). for e1, e2, and the function application itself, in turn. ∗ ∗ ∗ If D `C c1 c2 then D ` c2 c1 . First we consider y, the continuation of the whole application expression. Given the type that we need to derive Proofs of Lemmas 5.4, 5.5, and 5.6. By induction on deriva- for the whole expression, the qualifier of y’s type must be tions. ξ0. Furthermore, from the assumptions of the lemma, we ∗ ∗ The most difficult lemma, and the heart of the proof, know that D ` ξ0 (c1 c2 c) . By Property3, each ∗ ∗ ∗ 5 5 ∗ λURAL(C) of c1 , c2 , and c is greater than (c1 c2 c) , so by is about typing translated expressions. Given a 5 5 expression whose control effect is c, the translation of the transitivity, ξ0 is less than each of these. ∗ control effect, c , is the qualifier of the continuation of the Expression e1 has effect c1, so by the induction hypothe- ∗ translated expression: sis, its continuation may have qualifier c1 . The continu-

ation passed to e1 C is Lemma 5.7 (Translation of term typing). J K

If D;G `C e : t ; c then λx1. e2 C (λx2.x1 x2 (λx.y x)), J K ∗ ∗ L c∗ ∗ ∗ − ∗ + D ;G ` e C : ( (t ( hht ,c iiC ) ( hht ,c iiC ). whose free variables are {y} ∪ fv(e2). Thus, the quali- J K fier of this function must upper bound both ξ0 and the Proof. By induction on the typing derivation, generalizing qualifiers of the types in G2 (the type context for e2). ∗ ∗ the induction hypothesis thus: We have D ` ξ0 c1 from the previous paragraph. ∗ Furthermore, looking at the premises of rule T-APP, we If D;G `C e : t;c, then for all τ0 such that D ` τ0 : ?, ξ D∗ ` ξ c∗ see that ξ2 upper bounds the types in G2 and is less than and for all 0 such that 0 , we have ∗ ∗ D∗;G ∗ ` e : L(ξ0 (t∗ hhτ ,c ii−) hhτ ,c ii+). c1 (by Property5(part 2)), so by transitivity, D ` C ( 0 C ( 0 C ∗ ∗ J K G2 c1 , as desired. We consider two cases here: Expression e2 has effect c2, so similarly, its continuation 0 0 ∗ D;G `C e : t ; c D `C c c should have qualifier c . The free variables of e ’s con- Case . 2 2 D;G `C e : t ; c tinuation are only y and x1, which is the value of e1. We ∗ ∗ 0∗ ξ1 c ∗ By Property5(part 3), D ` c c , and thus by handle y as before. The type of x1 is ((t1 (− t2) ), so 0 ∗ ∗ Property1(part 4), there exists some type τ0 such that it remains to show that D ` ξ1 c2 , by Property5 0 0 − − 0 0 + + hhτ0,c iiC = hhτ0,c iiC and hhτ0,c iiC = hhτ0,c iiC . By (part 2) applied to the premise D `C c2 ξ1. ξ1 c ∗ Finally, given that x1 has type ((t1 −( t2) ), it expects d ::= ⊥D (no effect) a continuation whose qualifier is c∗. The type of y has | α (an effect variable) ∗ qualifier ξ0, which is less than c . Then by Lemma 5.2 | ξ (treats continuation like ξ) (Dereliction), the type of the η expansion λx.y x may be | d t d (effect join) given qualifier c∗. 1 2 Let D be the set of delimited continuation effects (d) quo- Corollary 5.8 (Translation of program typing). tiented by the following equivalences: If D;G ` e : t ; ⊥ where D ` t A, then C C C L t ξ = ξ t L = ξ; ∗ ∗ ∗ D ;G ` e C doneC : hht iiC. d t ⊥D = ⊥D t d = d t d = d. J K Proof. By Lemma 5.4, Lemma 5.7, Property2, and rules (The quotient simplifies defining other functions and rela- QSUB-REFL and T-APP. tions on delimited continuation effects.) Then we define

delimited continuation effects as the triple (D, ⊥D, t). URAL Lemma 5.9 (λ safety). We extend the type system of λURAL(C) with the new ∗ If •; • ` e1 : τ and e1 7−→ e2, then either ∃v2.e2 ≡ v2 or rules in Figure 13. The new kinding rules say that qualifiers- ∃e3.e2 7−→ e3. as-effects (ξ) and joins (d1 t d2) are well-kinded if their components are. The new control effect bound rules say that Proof. See the proof in Ahmed et al.(2005). a control effect ξ0 is bounded by all qualifiers ξ that are less 0 Theorem 5.10 (λURAL(C) safety). than ξ and that any bound of both effects in a join bounds the join as well. The rules added for effect subsumption ef- If •; • ` e : t ; ⊥ , and • ` t A then eval(e) 6= WRONG. C C C fectively axiomatize the delimited continuation effect lattice. ∗ Finally, we add two rules for typing shift and reset. To type Proof. By Corollary 5.8, •; • ` e C doneC : hht iiC. Then by Lemma 5.9, either e doneJ Kreduces to a value v, in an expression reset e, subexpression e may have any effect C C U which case eval(e) = Jv,K or e done diverges, in which whatsoever, but must return type 1. (We lift this restriction C C U case eval(e) is undefined. J K in §6.2.) Then reset e is pure and also has type 1. To ξ ⊥D U type shift x in e, we give x type (t −−( 1) for checking 6. Example Control Effects e, where ξ is joined with the effect of e to get the effect of the whole shift expression. That is, because shift captures its In the previous section, we proved type safety for λURAL(C), continuation and gives the reified continuation qualifier ξ, its a substructural λ calculus parameterized by abstract control effect must be at least ξ, since that qualifier determines how effects. In this section, we give three instances of control it might treat its captured continuation. effects as described by Definition 4.1 and show that they satisfy the properties on which the generic type safety theorem Type safety. To prove type safety for λURAL(C) extended depends. with delimited continuation effects, we need to give the translation parameter as described by Definition 5.1. We 6.1 Shift and Reset define the translation parameter as follows: We define here a control effect instance for delimited con- hhτ,d ii− = hhτ,d ii+ = U1 tinuations. In this example, we restrict answer types to the D D U unit type 1 in order to keep the effects simple. In §6.2, we doneD = λx.hi show how to define a more general control effect instance L d = ⊥ that allows answer-type modification.  if D URAL  We add shift and reset to λ (C) as follows. First, we ∗ α if d = α extend the syntax: d = ξ if d = ξ  e ::= ···| shift x in e | reset e (expressions) U otherwise We give the dynamics of the new expressions by defining Then, we must show that this definition satisfies the prop- their CCoS translations, which are standard: erties of §5.3:

reset e D = λy.y ( e D (λx.x)) Theorem 6.1 (Delimited continuation properties). J K J K 0 0 Delimited continuation effects (D, ⊥D, t) satisfy Proper- shift x in e D = λy.(λx. e D (λx .x )) J K (Λ.λx.λyJ K 0.y0 (y x)) ties1–5. Proof. To type shift and reset, we define delimited continuation effects d as the dual lattice of the qualifier lattice ξ with a Property1(Answer types) . We must show several equal- − + new point ⊥D: ities on answer types, such as hhτ,d 1iiD = hhτ,d 2iiD , D `D i : k (kinding delimited control effects) D `D d ξ (qualifier bound for delimited control effects)

D-K-QUAL D-K-JOIN D-B-QUAL D-B-JOIN 0 D `D ξ : QUAL D `D d1 : CTL D `D d2 : CTL D `D ξ ξ D `D d1 ξ D `D d2 ξ 0 D `D ξ : CTL D `D d1 t d2 : CTL D `D ξ ξ D `D d1 t d2 ξ

D `D d1 d2 (delimited control effect subsumption)

DSUB-JOIN 0 0 DSUB-BOT DSUB-LIN DSUB-TOP D `D d1 d1 D `D d2 d2 0 0 D `D d : CTL D `D ξ : QUAL D `D d : CTL D `D d1 t d2 : CTL D `D d1 t d2 : CTL 0 0 D `D ⊥D d D `D L ξ D `D d U D `D d1 t d2 d1 t d2

D;G `D e : t ; d (delimited control expression typing)

D-T-RESET D-T-SHIFT U ξ ⊥D U U D;G `D e : 1 ; d D;G , x: (t −−( 1) `D e : 1 ; d U D;G `D reset e : 1 ; ⊥D D;G `D shift x in e : t ; d t ξ

Figure 13. Statics for delimited continuation effects

hold whenever d1td2 is well formed. All of the equalities These require a lemma about the translation of quali- − + U are trivial because hhτ,d iiD = hhτ,d iiD = 1. fier subsumption derivations. ∗ ∗ Property2(Done) . We need to show that ∆; • ` 2. We show that D `D d1 d2 implies that D ` d2 L ∗ doneD : (τ ( hhτiiD). Given the definition of doneD, d1 , again by induction on the derivation. The only L we can show ∆; • ` λx.hi : (τ ( hhτiiD) by a nontrivial case is when straightforward type derivation. D ` d d0 D ` d d0 Property3(Effect sequencing) . We need to show that D 1 1 D 2 2 D ` d t d : CTL D ` d0 t d0 : CTL D ` d t d : CTL implies that D∗ ` (d t d )∗ d ∗ D 1 2 D 1 2 D 1 2 1 2 1 0 0 . ∗ ∗ ∗ D ` d t d d t d and D ` (d1 t d2) d2 . By symmetry, it suffices to D 1 2 1 2 show the former: ∗ 0 0 ∗ ∗ We show that D ` (d1 t d2) (d1 t d2) by exhaustively enumerating the possibilities for d1, d2, (1) D `D d1 d1 byCS UB-REFL 0 0 d1, and d2 such that the premises hold. (2) D `D ⊥D d2 byDS UB-BOT 3. For translation of kinding, we show that D `C d : CTL (3) D ` d t ⊥ d t d by (1–2),DS UB-JOIN D 1 D 1 2 implies that D∗ ` d∗ : QUAL. We proceed, as usual, D ` d d t d d t ⊥ = d (4) D 1 1 2 by (3), 1 D 1 by a simple induction on the derivation, considering ∗ ∗ ∗ (5) D ` (d1 t d2) d1 by (4), Lemma 5.6. the two new kinding rules for delimited continuation effects. Property4(Bottom and lifting) . 4. For translation of typing, we use the generalized in- 1. To show that d1 td2 = ⊥D if and only if d1 = d2 = duction hypothesis as in the proof of Lemma 5.7. ⊥D, we consider the quotienting of D. There are two cases, for shift and reset, each of which 2. We must also show that if D `D d1 t d2 : CTL and requires a large type derivation. 0 d1 t d2 6= ⊥D, then there exist some d1 6= ⊥D 0 6.2 Shift and Reset with Answer-Type Modification and d2 6= ⊥D with particular properties. For each (i∈{1,2}) 0 di , if di = ⊥D then let di = L; otherwise, The type-and-effect system for shift and reset described in 0 0 let di = di . This ensures that 1–2) each D `D di di , §6.1 requires that all answer types—the type of all reset 0 0 0 0 U 3) d1 t d2 = d1 t d2, and 4) d1 t d2 is well formed. expressions—be 1. Our second example adds answer-type Property5(New rules) . modification (à la Danvy and Filinski 1989), which allows ∗ ∗ 1. We show that D `D d ξ implies that D ` ξ d , shift to capture and compose continuations of differing types by induction on the derivation. The only new cases and allows the answer delivered by reset to have any type. to consider are for rulesD-B-Q UAL andD-B-J OIN. Both the syntax and CCoS translation are as in §6.1, but D À i : k (kinding answer-type effects) D À a ξ (qualifier bound for answer-type effects)

A-K-EFFECT A-B-QUAL

D À ξ1 : QUAL ··· D À ξk : QUAL D À ξ ξ1 ··· D À ξ ξj

D À t1 : ? D À t2 : ? D À t1 : ? D À t2 : ? ξ ,...,ξ D ` 1 k (t t ): CTL ξ1,...,ξj A 1 2 D À (t1 t2) ξ

D `A a1 a2 (answer-type effect subsumption)

ASUB-BOT ASUB-L ASUB-TOP Ξ Ξ Ξ D À (t t): CTL D À (t1 t2): CTL D À (t1 t2): CTL Ξ L Ξ Ξ U D À ⊥A (t t) D À (t1 t2) (t1 t2) D À (t1 t2) (t1 t2)

ASUB-JOIN 0 Ξ1 Ξ1 D À (t1 t2) (t1 t2) 0 Ξ2 Ξ2 D À (t1 t2) (t1 t2) 0 0 Ξ1,Ξ2 Ξ1,Ξ2 D À (t1 t2) (t1 t2)

D;G `A e : t ; a (answer-type effect expression typing)

A-T-RESET A-T-SHIFT Ξ ξ ⊥A Ξ D;G À e : t0 ; (t0 t) D;G , x: (t1 −−( t2) À e : t0 ; (t0 t) Ξ,ξ D;G À reset e : t ; ⊥A D;G À shift x in e : t1 ; (t2 t)

Figure 14. Statics for answer-type effects we change the definition of control effects as follows. An For answer-type modification effects, we define the par- answer-type control effect a is either the pure effect ⊥A or tial sequencing operation as follows: a collection of qualifiers ξ1, . . . , ξj along with old and new answer types t1 and t2: ⊥A ◦ a = a

a ◦ ⊥A = a a ::= ⊥A (pure) 0 0 Ξ Ξ 0 Ξ 0 Ξ,Ξ | (t1 t2) (a control effect) (t t2) ◦ (t1 t ) = (t1 t2). where Ξ ::= ξ1, . . . , ξj Any other cases are undefined. Then we define answer-type ξ1,...,ξj A type derivation D;G À e : t ; (t1 t2) may be modification effects as the triple (A, ⊥A, ◦). understood as follows: The new type rules for answer-type effects appear in Figure 14. For the most part, these rules treat the collection • The collection of qualifiers ξ1, . . . , ξj keeps track of all of qualifiers ξ1, . . . , ξj similarly to the delimited continua- the ways that expression e may treat its context; expres- tion effect ξ1 t · · · t ξj from §6.1. However, there is some sion e may be considered to treat its context according subtlety to the definition of answer-type effect subsumption: to any qualifier ξ that lower bounds all of ξ1, . . . , ξj . We the only non-bottom effects related by subsumption are those need a collection of qualifiers because qualifiers do not, whose before and after answer types match, pairwise, but in the presence of qualifier variables, have greatest lower the pure effect ⊥A is less than any effect whose before and bounds. after answer types match each other (ruleAS UB-BOT). This • Evaluated in a context expecting type t whose original makes sense, as pure expressions do not change the answer answer type was t1, expression e changes the answer type. type to t2. This means that our type-and-effect judgment, The rules for typing shift and reset expressions are a disregarding substructural considerations, is equivalent hybrid of the rules from §6.1, which they follow for the to the type judgment that Danvy and Filinski write as qualifier portion, and the rules from Danvy and Filinski Γ,t 1 ` e : t,t 2. (1989), which they follow for maintaining answer types. URAL Type safety. To prove type safety for λ (C) extended D `X i : k (kinding exception effects) with answer-type modification, we define the translation parameter as follows: X-K-UNION X-K-SING D `X Ψ1 : CTL D `X Ψ2 : CTL − hhτ, ⊥AiiA = τ D `X {ψ} : CTL D `X Ψ1 ∪ Ψ2 : CTL Ξ − ∗ hhτ, (t1 t2)iiA = t1 + hhτ, ⊥AiiA = τ D `X Ψ ξ (qualifier bound for exception effects) Ξ + ∗ hhτ, (t1 t2)iiA = t2 X-B-RAISE

doneA = λx.x D `X Ψ: CTL

 D `X Ψ A L if a = ⊥  A ∗ ξ a = ξ if a = (t1 t2)  U otherwise D;G `X e : t ;Ψ (exception effect expression typing)

Theorem 6.2 (Answer-type effect properties). X-T-RAISE D `X t : ? Answer-type modiﬁcation effects (A, ⊥A, ◦) satisfy Proper- ties1–5. D; • `X raise ψ : t ; {ψ}

6.3 Exceptions X-T-HANDLE D ` G G G We add exceptions to λURAL(C) as follows. We assume a X 1 2 D;G ` e : t ; {ψ} ∪ Ψ set Exn of exception names ψ and extend the syntax of 1 X 1 D;G ` e : t ;ΨD ` G A expressions: 2 X 2 X 2 D;G `X e1 handle ψ → e2 : t ;Ψ ψ ∈ Exn (exceptions)

e ::= ···| raise ψ | e1 handle ψ → e2 (expressions) While these exceptions are simple tags, it would not be Figure 15. Statics for exception effects difficult to have exceptions carry values. As in the previous example, we define the dynamics by the CCoS translation. it. (Of course, the empty exception set ∅ is bounded by L by However, because the CCoS translation for exceptions is ruleC-B-P URE.) type directed, we show how the type system is extended first. To define the CCoS translation, we assume a run-time To type exceptions, we instantiate λURAL(C) as follows. representation of exceptions and exception sets as follows: Exception effects, Ψ, are sets of primitive exception names • There is an exception pretype exn such that ∆ ` exn : ?. ψ: • Each exception ψ is represented by a λURAL value ψ∗, Ψ ::= (pure) ∅ such that ∆; • ` ψ∗ : Uexn. | α (an effect variable) URAL • For each exception ψ and pair of λ values v1 and v2, |{ ψ} (a single exception) URAL there is a λ value [v1, v2]ψ such that | Ψ1 ∪ Ψ2 (exception set union) 0 Let X be the set of exception effect sets (Ψ). Then we ψ 6= ψ ∗ ∗ 0∗ 0∗ . define exception effects as the triple (X, ∅, ∪). We consider [v1, v2]ψ ψ 7−→ v1 ψ [v1, v2]ψ ψ 7−→ v2 ψ exception effects as true sets, not merely as the free algebra ξ1 U generated by the syntax. Thus, the subsumption order is set ∆;Γ ` v1 : ( exn ( τ)∆ ` ξ1 ξ ξ2 U containment: ∆;Γ ` v2 : ( exn ( τ)∆ ` ξ2 ξ ξ U D `X Ψ1 Ψ2 (exception effect subsumption) ∆;Γ ` [v1, v2]ψ : ( exn ( τ)

XSUB-SUBSET Intuitively, [v , v ] performs case analysis on exception Ψ ⊆ Ψ D ` Ψ : CTL D ` Ψ : CTL 1 2 ψ 1 2 X 1 X 2 values: when applied to exception ψ, it passes the exception D `X Ψ1 Ψ2 to v1, and when applied to any other exception, it passes the exception to v2. The other new type rules for exception effects appear in For exception effects, we use a typed CCoS translation Figure 15. Note that ruleX-B-R AISE says that all exception that takes an extra parameter: the exception effect of the ex- effects are bounded below by A; this is because exceptions pression to be translated. We assume that the generic CCoS allow an expression to discard its context but not duplicate has been updated to translate type derivations as well in order to propagate control effects correctly. Then we can give the Theorem 6.3 (Exception effect properties). CCoS translation for exceptions: Exception effects (X, ∅, ∪) satisfy Properties1–5. Ψ ∗ raise ψ X = λ . inl ψ The proofs of theorems in this section appear in the ex- J KΨ {ψ}∪Ψ tended version of this paper, available at www.ccs.neu.edu/ e1 handle ψ → e2 X = λy.[v, y]( e1 X (λx. inr x)) J K J K ∼tov/pubs/substructural-control. ( ∅ λ . e2 X y if Ψ= ∅; where v = J K Ψ 7. Conclusion [λ . e2 X y, λx. inl x]ψ if Ψ 6= ∅. J K We began this study with the desire to add linear types to Example. The first Scala example from §1 may be recast Alms, a general-purpose programming language with affine in λURAL (with references, pairs, and integer division) as types and exceptions. The treatment of exceptions in §6.3 follows: points the way toward that goal. One question that remains, however, concerns the pragmatics of checked exceptions in a λz1 z2.pair (ref (z1 / z2)) (ref (z2 / z1)) higher-order language such as Alms, where latent exception Let us assume the following (monomorphic, for brevity) effects are likely to appear on many function arrows. We types for the operations: believe that with appropriate defaults most function arrows will not require annotation, but more research is required in U U ∅ U U {DivBy0} U · / · : ( int −( ( int −−−−−( int)) that direction. U U ∅ L ref : ( int −( intref) Another potential direction for future research is to consider how other control effects fit into our general framework. pair : U(Lintref −∅ L(Lintref −∅ L(Lintref ⊗ Lintref))) ( ( We suspect that some control operators common to impera- tive languages, such as return, break, and goto, absent first- To type the application of term pair (ref (z1 / z2)) to term class labels, would be straightforward. More exotic forms ref (z2 / z1), according to premise (6) of rule C-T-APP, the effect of the operator must be bounded by the qualifier of control may be harder. Some control operators, such as of the type of the operand. The effect of the operator, shift0, are very difficult to type even in a simpler setting (Kiselyov and Shan 2007), which is why have not considered pair (ref (z1 / z2)), is {DivBy0}, based on the type of /; L them. Others, such as Felleisen’s prompt and control (1988) the type of the operand, ref (z2 / z1), is intref. But • `X {DivBy0} L is not derivable—a term that can raise an are probably tractable with a more expressive version of our exception does not necessarily treat it context linearly—so generic type system, because effects need to reflect not only the original code has a type error in λURAL(C). how an expression treats its continuation, but how a continu- We can repair the example, as we did in §1, by explicitly ation, if captured and reinvoked, treats its new continuation. URAL ordering the effects so that both divisions happen before any For the cases we consider, however, λ (C) provides references are allocated: a simple and generic framework for integrating substructural types and control effects. We have shown that our type URAL λz1 z2.(λx1 x2.pair (ref x1)(ref x2)) (z1 / z2)(z2 / z1) system for λ (C) is sound provided that the particular instantiation of control effects meets several criteria, and we Term λx1 x2.pair (ref x1)(ref x2) has an unlimited type: have exhibited three instances of control effects that meet these criteria. We contend that this provides a solid ground- U U ∅ U U ∅ L L L ( int −( ( int −( ( intref ⊗ intref))) ing for the extension of realistic substructural programming languages with control effects. Thus, it does not matter that its argument, z1 / z2, has nontrivial effect. Similarly, because the codomain of that type is unlimited, it is permissible that the second argument, z2 / z1, Acknowledgments has non-trivial effect as well. Thus, the repaired example is We wish to thank Vincent St-Amour, Sam Tobin-Hochstadt, URAL typeable in λ (C). Aaron Turon, and the anonymous referees for their helpful Type safety. To prove type safety for λURAL(C) extended comments, discussion, and corrections. This research was with exceptions, we define the translation parameter as fol- supported in part by AFOSR grant FA9550-09-1-0110. lows: References hhτ, Ψii− = hhτ, Ψii+ = L(Uexn ⊕ τ) X X A. Ahmed, M. Fluet, and G. Morrisett. A step-indexed

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