Weak updates and separation logic
Gang Tan1 , Zhong Shao2 , Xinyu Feng3 , and Hongxu Cai4
1 Lehigh University, 2 Yale University
3 Toyota Technological Institute at Chicago, 4 Google Inc.

Abstract. Separation Logic (SL) provides a simple but powerful technique forreasoning about imperative programs that use shared data structures. Unfortunately, SL supports only “strong updates”, in which mutation to a heap location
is safe only if a unique reference is owned. This limits the applicability of SL
when reasoning about the interaction between many high-level languages (e.g.,
ML, Java, C#) and low-level ones since these high-level languages do not support
strong updates. Instead, they adopt the discipline of “weak updates”, in which
there is a global “heap type” to enforce the invariant of type-preserving heap updates. We present SLw , a logic that extends SL with reference types and elegantly
reasons about the interaction between strong and weak updates. We also describe
a semantic framework for reference types; this framework is used to prove thesoundness of SLw .

1 Introduction
Reasoning about mutable, aliased heap data structures is essential for proving properties
or checking safety of imperative programs. Two distinct approaches perform such kindof reasoning: Separation Logic, and a type-based approach employed by many highlevel programming languages.
Extending Hoare Logic, the seminal work of Separation Logic (SL [10, 13]) is a
powerful framework for proving properties of low-level imperative programs. Throughits separating conjunction operator and frame rule, SL supports local reasoning aboutheap updates, storage allocation, and explicit storage deallocation.
SL supports “strong updates”: as long as a unique reference to a heap cell is owned,
the heap-update rule of SL allows the cell to be updated with any value:{(e 7→ −) ∗ p}[e] := e′ {(e 7→ e′ ) ∗ p}

(1)

In the above heap-update rule, there is no restriction on the new value e′ . Hereafter, we
refer to heaps with strong updates as strong heaps. Heap cells in strong heaps can holdvalues of different types at different times of program execution.
Most high-level programming languages (e.g., Java, C#, and ML), however, support
only “weak updates”. In this paradigm, programs can perform only type-preserving
heap updates. There is a global “heap type” that tells the type of every allocated heap
location. The contents at a location have to obey the prescribed type of the location in
the heap type, at any time. Managing heaps with weak updates is a simple and typesafe mechanism for programmers to access memory. As an example, suppose an ML

variable has type “τ ref” (i.e., it is a reference to a value of type τ). Then any update
through this reference with a new value of type τ is type safe and does not affect othertypes, even in the presence of aliases and complicated points-to relations. Hereafter, we
refer to heaps with weak updates as weak heaps.
This paper is concerned with the interaction between strong and weak updates.
Strong-update techniques are more precise and powerful, allowing destructive memory updates and explicit deallocation. But aliases and uniqueness have to be explicitly tracked. Weak-update techniques allow type-safe management of memory without
tracking aliases, but types of memory cells can never change. A framework that mixes
strong and weak updates enables a trade-off between precision and scalability.
Such a framework is also useful for reasoning about multilingual programs. Most
real-world programs are developed in multiple programming languages. Almost all
high-level languages provide foreign function interfaces for interfacing with low-level
C code (for example, the OCaml/C FFI, and the Java Native Interface). Real-world
programs consist of a mixture of code in both high-level and low-level languages. A
runtime state for such a program conceptually contains a union of a weak heap and astrong heap. The weak heap is managed by a high-level language (e.g., Java), accepts
type-preserving heap updates, and is garbage-collected. The strong heap is managed
by a low-level language, accepts strong updates, and its heap cells are manually recollected. To check the safety and correctness of multilingual programs, it is of practicalvalue to have one framework that accommodates both strong and weak updates.
Since Separation Logic (SL) supports strong heaps, one natural thought to mix
strong and weak updates is to extend SL with types so that assertions can also describe weak heaps. That is, in addition to regular SL assertions, we add {e 7→ τ}, which
specifies a heap with a single cell and the cell holds a value of type τ. This scheme,
however, would encounter two challenges.
First, allowing general reference types in {e 7→ τ} would make SL unsound. An
example demonstrating this point is as follows:{{x 7→ 4} ∗ {y 7→ even ref}} [x] := 3 {{x 7→ 3} ∗ {y 7→ even ref}}(2)

The example is an instantiation of the heap-update rule in (1) and uses the additional
assertion {e 7→ τ}. The precondition states that y points to a heap cell whose contents
are of type “even ref” (i.e., a reference to an even integer). Therefore, the precondition
is met on a heap where y points to x. However, the postcondition will not hold on the
new heap after the update because x will point to an odd number. Therefore, the above
rule is sound only if y does not point to x.
The second challenge of adding types to SL is how to prove its soundness with
mixed SL assertions and types. Type systems are usually proved sound following a
syntactic approach [19], where types are treated as syntax. Following the tradition of
Hoare Logic, SL’s soundness is proved through a denotational model, and SL assertions
are interpreted semantically. There is a need to resolve the conflict between syntactic
and semantic soundness proofs.
In this paper, we propose a hybrid logic, SLw , which mixes SL and a type system.
Although the logic is described in a minimal language and type system, it makes a solid
step toward a framework that reasons about the interaction between high-level and lowlevel languages. The most significant technical aspects of the logic are as follows:2

(Command) c ::= · · · | x := [e] | [x] := e | x := alloc(e) | free(e)(Expression) e ::= x | v | op(e1 , . . . , en )(Value) v ::= n | ℓFig. 1. Language syntax– SLw extends SL with a simple type system. It employs SL for reasoning aboutstrong updates, and employs the type system for weak updates. Most interestingly,
SLw mixes SL assertions and types, and accommodates cross-boundary pointers(from weak to strong heaps and vice versa). This is achieved by statically maintaining the distinction between pointers to weak heaps and pointers to strong heaps.
SLw is presented in Section 2.
– To resolve the conflict between syntactic types and semantic assertions, we propose a semantic model of types. Our model of reference types follows a fixed-point
approach and allows us to define a denotational model of SLw and prove its soundness. The model of SLw is presented in Section 3.

2 SLw : Separation logic with weak updates
We next describe SLw , an extension of SL that incorporates reasoning over weak heaps.
In Section 2.1, we describe a minimal language that enables us to develop SLw . Rules
of SLw are presented in Section 2.2 and examples of using the logic in Section 2.3.
We first describe some common notations. For a map f , we write f [x ; y] for a new
map that agrees with f except it maps x to y. For two finite maps f1 and f2 , f1 ⊎ f2 is theunion of f1 and f2 when their domains are disjoint, and undefined otherwise. We write
f \ X for a new map after removing elements in X from the domain of f . We write ~x fora sequence of xs and ε for an empty sequence.
2.1 Language syntax and semantics
Figure 1 presents the syntax of the programming language in which we will develop
SLw . The language is the imperative language used by Hoare [6], augmented with aset of commands for manipulating heap data structures. It is similar to the one used
in Reynolds’ presentation of SL [13]. Informally, the command “x := [e]” loads thecontents at location e into variable x; “[x] := e” updates the location at x with the valuee; “x := alloc(e)” allocates a new location, initializes it with e, and assigns the newlocation to x; “free(e)” deallocates the location e.
In the syntax, we use n for integers, x for variables, ℓ for heap locations, and op forarithmetic operators. We assume there is an infinite number of variables and locations.
Figure 2 presents a formal operational semantics of the language. A state consistsof a map r from variables to values, a heap h, and a sequence of commands. Commands
bring one state to another state and their semantics is formally defined by a step relation
7−→ . We write 7−→∗ for the reflexive and transitive closure of 7−→ .
A state may have no next state, i.e., “getting stuck”. For example, a state whose next
instruction to execute is [x] := e gets stuck when x does not represent a location or the3

(State) s ::= (r, h,~c)(Locals) r ::= Var → Value(Heap) h ::= {ℓ1 7→ v1 , . . . , ℓn 7→ vn }if c =···
x := [e][x] := ex := alloc(e)free(e)(r, h, c · c~1 ) 7−→ (r2 , h2 , c~2 )then (r2 , h2 , c~2 ) =···(r[x ; h(ℓ)], h, c~1 )when r(e) = ℓ and ℓ ∈ dom(h)(r, h[ℓ ; r(e)], c~1 )when r(x) = ℓ and ℓ ∈ dom(h)(r[x ; ℓ], h ⊎ {ℓ 7→ r(e)}, c~1 ) when ℓ 6∈ dom(h)(r, h \ {ℓ}, c~1 )when r(e) = ℓ and ℓ ∈ dom(h) r(x)
where r(e) = v
op(r(e1 ), . . . , r(en ))when e = xwhen e = v
when e = op(e1 , . . . , en )Fig. 2. Operational semantics

location is not in the domain of the state’s heap. A state is a terminal state when thesequence of commands is empty.
Definition 1. (Stuck and terminal states)
stuck(s), ¬(∃s′ . s 7−→ s′ )terminal(r, h,~c) , ~c = εBelow we define the usual notions of safety and termination:
Definition 2. (Safety and termination)safe(s), ∀s′ . (s 7−→∗ s′ ) ∧ ¬terminal(s′ ) ⇒ ∃s′′ . s′ 7−→ s′′

terminate(s) , ∀s′ . s 7−→∗ s′ ⇒ ∃s′′ . s′ 7−→∗ s′′ ∧ terminal(s′′ )2.2 The logic SLw
Figure 3 presents assertions and types used in SLw . Assertions in SLw include all
formulas in predicate calculus (not shown in the figure), and all SL formulas. The only
additional assertion form in SLw is {e : τ}, which denotes that e has type τ.
SLw is equipped with a simple type system that classifies integers and locations.
Although the type system does not include many types in high-level languages, by
including reference types it is already sufficient to show interesting interactions between
strong and weak heaps. Reference types are the most common types when high-level
languages interoperate with low-level languages because in this setting most data arepassed by references.
4

(Assertion) p ::= · · · | emp | {e1 7→ e2 } | p1 ∗ p2 | p1 −∗p2 | {e : τ}(Type) τ ::= int | ref | wref τ(HeapType) Ψ ::= {ℓ1 : τ1 , . . . , ℓn : τn }(LocalVarType) Γ ::= {x1 : τ1 , . . . , xn : τn }Fig. 3. Assertions and types
Ψ, Γ ⊢ e : τx ∈ dom(Γ)Ψ, Γ ⊢ x : Γ(x)Ψ, Γ ⊢ n : int

Ψ, Γ ⊢ ℓ : ref∀i ∈ [1..n]. Ψ, Γ ⊢ ei : int
Ψ, Γ ⊢ op(e1 , . . . , en ) : int

Ψ(ℓ) = τΨ, Γ ⊢ ℓ : wref τ

Fig. 4. Typing rules for expressions

Type int is for all integers and ref for all locations. Type “wref τ” is for locations ina weak heap, but not in a strong heap. A heap type Ψ tells the type of every location ina weak heap; mathematically, it is a finite map from locations to types. Given heap type
Ψ, location ℓ has type “wref τ” if Ψ(ℓ) equals τ. A local variable type, Γ, tells the typeof local variables.
Figure 4 presents typing rules for expressions, which are unsurprising. Notice thatthe typing rule for “wref τ” requires that the location ℓ is in the domain of the heap type
Ψ and Ψ(ℓ) has to be the same as τ. This rule and the later weak-update rule enforce
type-preserving updates on weak heaps.
The following schematic diagram helps to understand the relationship between weakheaps, strong heaps, local variables, assertions and various kinds of types in SLw :As shown in the diagram, SLw conceptually divides a heap into a weak heap hw
and a strong heap hs . The weak heap is specified by a heap type Ψ, and the strong heap
by SL formula p. Pointers to weak-heap cells (in solid lines) have type “wref τ” or ref.
Pointers to strong heap cells (in dotted lines) can have only type ref.
Figure 5 presents rules for checking commands. These rules use the judgmentΨ ⊢ {Γ, p} ~c {Γ′ , p′ }. In this judgment, Ψ, Γ and p are preconditions and specify
conditions on the weak heap, local variables, and the strong heap respectively. Postcon5Ψ ⊢ {Γ, p} ~c {Γ′ , p′ }(Well-formed statements)Ψ, Γ ⊢ e : ref Ψ, Γ ⊢ y : τ( S - LOAD )Ψ ⊢ {Γ, {e 7→ y}} x := [e] {Γ[x ; τ], x = y ∧ {e 7→ x}}
where x 6∈ FV(e)Ψ, Γ ⊢ x : ref( S - UPDATE )Ψ ⊢ {Γ, {x 7→ −}} [x] := e {Γ, {x 7→ e}}Ψ ⊢ {Γ, emp} x := alloc(e) {Γ[x ; ref], {x 7→ e}}( S - ALLOC )where x 6∈ FV(e)Ψ, Γ ⊢ e : ref( S - FREE )Ψ ⊢ {Γ, {e 7→ −}} free(e) {Γ, emp}Ψ ⊢ {Γ, p} ~c {Γ′ , p′ }( FRAME )Ψ ⊢ {Γ, p ∗ p1 } ~c {Γ′ , p′ ∗ p1 }where no variable occurring free in p1 is modified by ~c↑ T HE WORLD OF STRONG HEAPS−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−↓ T HE WORLD OF WEAK HEAPSΨ, Γ ⊢ e : wref τ( W- LOAD )Ψ ⊢ {Γ, emp} x := [e] {Γ[x ; τ], emp}
Ψ, Γ ⊢ x : wref τ Ψ, Γ ⊢ e : τ( W- UPDATE )Ψ ⊢ {Γ, emp} [x] := e {Γ, emp}
Ψ, Γ ⊢ e : τ( W- ALLOC )Ψ ⊢ {Γ, emp} x := alloc(e) {Γ[x ; wref τ], emp}
Fig. 5. Rules for commands (Rules for assignments, conditional statements, loops, and sequencing are the same as the ones in Hoare Logic and are omitted.)

ditions are Γ′ and p′ ; they specify conditions on local variables and the strong heap of
the state after executing ~c. Readers may wonder why there is no postcondition specification of the weak heap. As common in mutable-reference type systems, the implicit
semantics of the judgment is that there exists an extended heap type Ψ′ ⊇ Ψ and theweak heap of the poststate should satisfy Ψ′ . In terms of type checking, the particular Ψ′
does not matter. The formal semantics of the judgment will be presented in Section 3.
Rules in Figure 5 are divided into two groups. One group is for the world of strong
heaps, and another for the world of weak heaps. The rules for strong heaps are almost
the same as the corresponding ones in standard SL, except that they also update Γ whennecessary.
The rules for weak heaps are the ones that one would usually find in a type system for mutable-reference types. The weak-update rule W- UPDATE requires that the6(r, h, s2w(x) ·~c) 7−→ (r, h,~c)

Operational semantics:Rule:Ψ, Γ ⊢ x : ref Ψ, Γ ⊢ e : τS2WΨ ⊢ {Γ, {x 7→ e}} s2w(x) {Γ[x ; wref τ], emp}

Fig. 6. Rule for converting a location from the strong heap to the weak heap
⊢ {Γ′1 , p′1 } ⇒ {Γ1 , p1 } Ψ ⊢ {Γ1 , p1 } ~c {Γ2 , p2 } ⊢ {Γ2 , p2 } ⇒ {Γ′2 , p′2 }Ψ ⊢ {Γ′1 , p′1 } ~c {Γ′2 , p′2 }WEAKENING⊢ {Γ, p} ⇒ {Γ′ , p′ }⊢ {Γ, p} ⇒ {Γ, p ∧ {x : Γ(x)}}W1⊢ p ⇒ p′W2
⊢ {Γ, p} ⇒ {Γ, p′ }Fig. 7. Weakening rules

pointer be of type “wref τ”, and that the new value be of type τ. This rule enforces
type-preserving updates. Once these conditions hold, Γ remains unchanged after the
update. Notice in this rule there is no need to understand separation and aliases as the
S - UPDATE rule does. The W- ALLOC rule does not need to extend the heap type Ψ because Ψ is only a precondition. When proving the soundness of the rule, we need to finda new Ψ′ that extends Ψ and is also satisfied by the new weak heap after the allocation.
Finally, there is no rule for free(e) in the world of weak heaps. Weak heaps should be
garbage-collected.1
Figures 6 and 7 present some rules that show the interaction between weak andstrong heaps. Figure 6 adds a new instruction “s2w(x)” for converting a location from astrong heap to a weak heap. Operationally, this instruction is a no-op (so it is an annotation, rather than a “real” instruction). Its typing rule, however, involves transforming
the ownership in the strong heap to a pointer of weak-reference types. Notice that there
is no rule for converting a location from the weak heap to the strong heap; this is similar
to deallocation in weak heaps and requires the help of garbage collectors.
Figure 7 presents weakening rules. Rule W 1 converts type information in Γ to information in assertion p. This is useful since information in Γ might be overwritten due
to assignments to variables. One of examples in later sections will show the use of thisrule. Rule W 2 uses the premise ⊢ p ⇒ p′ ; any valid SL formula p ⇒ p′ is acceptable.
2.3 Examples
We now show a few examples that demonstrate the use of SLw . In these examples,
we assume an additional type even for even integers. For clarity, we will also annotate
1 We do not formally consider the interaction between garbage collectors and weak heaps. Whenconsidering a garbage collector, SLw has to build in an extra level of indirection for crossboundary references from strong heaps to weak heaps as objects in weak heaps may get moved(this is how the JNI implements Java references in native code). We leave this as future work.

7

the allocation instruction to indicate whether the allocation happens in the strong heap
or in the weak heap. We write x := allocs (e) for a strong-heap allocation. We write
x := allocw,τ (e) for a weak-heap allocation, and the intended type for e is τ. These
annotations help in guiding the type checking of SLw .
The first example shows how the counterexample in the introduction (formula (2)on page 2) plays out in SLw . The following program first initializes the heap to a form
such that y points to a location of type “wref even” and x points to 4, and then performsa heap update through x. The whole program is checkable in SLw with respect to any
heap type (remember the heap type specifies the initial weak heap). Below we also
include conditions of the form “Γ, p” between instructions.
{}, empz := allocw,even (2){z : wref even}, emp
y := allocs (z){y : ref, z : wref even}, {y 7→ z}
by rule (w1){y : ref, z : wref even}, {y 7→ z} ∧ {z : wref even}by rule (w2){y : ref, z : wref even}, ∃v. {y 7→ v} ∧ {v : wref even}z := 0{y : ref, z : int}, ∃v. {y 7→ v} ∧ {v : wref even}
x := allocs (4){x : ref, y : ref, z : int}, ∃v. ({y 7→ v} ∧ {v : wref even}) ∗ {x 7→ 4}[x] := 3{x : ref, y : ref, z : int}, ∃v. ({y 7→ v} ∧ {v : wref even}) ∗ {x 7→ 3}
Different from the counterexample, the condition before “[x] := 3” limits where y canpoint to. In particular, y cannot point to x because (1) by the type of v, variable y must
point to a weak-heap location; (2) x represents a location in the strong heap. Therefore,
the update through x does not invalidate the type of v. We could easily construct anexample where y indeed points to x. But in that case the type of v would be ref, which
would also not be affected by updates through x.
One of the motivations of SLw is to reason about programs where code in high-level
languages interacts with low-level code. Prior research [4, 14] has shown that it is error
prone when high-level code interoperates with low-level code. All kinds of errors may
occur. One common kind of errors occurs when low-level code makes type misuses ofreferences that point to objects in the weak heap. For instance, in the JNI, types of allreferences to Java objects are conflated into one type in native code—jobject. Consequently, there is no static checking of whether native code uses these Java references
in a type-safe way. Type misuses of these Java references can result in silent memory
corruption or unexpected behavior.
The first example already demonstrates how SLw enables passing pointers fromhigh-level to low-level code. In the example, the first allocation is on the weak heap andcan be thought of as an operation by high-level code. Then, the location is passed to the
low level by being stored in the strong heap. Unlike foreign function interfaces wheretypes of cross-boundary references are conflated into a single type in low-level code,
SLw can track the accurate types of those references and enable type safety.
8

The next example demonstrates how low-level code can initialize a data structure in
the strong heap, and then transfer that structure to the weak heap so that the structure isusable by high-level code.
{}, empx := allocs (4){x : ref}, {x 7→ 4}
y := allocs (x){x : ref, y : ref}, {x 7→ 4} ∗ {y 7→ x}
s2w(x){x : wref even, y : ref}, {y 7→ x}s2w(y){x : wref even, y : wref (wref even)}, emp

3 Soundness of SLw
Soundness of SLw is proved by a semantic approach. We first describe a semantic
model for weak-reference types. Based on this model, semantics of various concepts
in SLw are defined. Every rule in SLw is then proved as a lemma according to thesemantics.
3.1 Modeling weak-reference types
Intuitively, a type is a set of values. This suggests that a semantic type should be a
predicate of the metatype “Value → Prop”. However, this idea would not support weakreference types. To see why, let us examine a naı̈ve model where “wref τ” in a heap
h would denote a set of locations ℓ such that h(ℓ) is of type τ. This simple model isunfortunately unsound, which is illustrated by the following example:
1. Create a reference of type “wref even”, and let the reference be x.
2. Copy x to y. By the naı̈ve model, a reference of type “wref even” also has type“wref int” (because an even number is also an integer). Let “wref int” be the type ofy.
3. Update the reference through y with an odd integer, say 3. As y has the type“wref int”, updating it with an odd integer is legal.
4. Dereference x. Alas, the dereference returns 3, although the type of x implies a
result of an even number!
The problem with the naı̈ve model is that, with aliases, it allows inconsistent viewsof memory. In the foregoing example, x and y have inconsistent views on the same
memory cell. To address this problem, SLw uses a heap type Ψ to type check a location.
This follows the approach of Tofte [16] and Harper [5]. An example Ψ is as follows:Ψ = {ℓ0 : even, ℓ1 : int, ℓ2 : wref even, ℓ3 : wref int}

(3)

A heap type Ψ helps to define two related concepts, informally stated below (theirformal semantic definitions will be presented in a moment):
9

(i) A location ℓ is of type “wref τ” if and only if Ψ(ℓ) equals τ.
(ii) A heap h is consistent with Ψ if for every ℓ, the value h(ℓ) has type Ψ(ℓ). For
the example Ψ, it means that h(ℓ0 ) should be an even number, h(ℓ1 ) should be an
integer, h(ℓ2 ) should be of type “wref even”, ...
The heap type Ψ prevents aliases from having inconsistent views of the heap. Aliases
have to agree on their types because the types have to agree with the type in Ψ. In particular, the example showing the unsoundness of the naı̈ve model would not work in theabove model because, in step 3 of the example, y cannot be cast from type “wref even”
to “wref int”: type “wref even” implies that Ψ(y) = even, which is a different type fromint.
A subtlety of the above model is the denotation of “wref τ” depends on the heap
type Ψ, but is independent of the heap h. A weak-reference type is connected to theheap h only indirectly, through the consistency relation between h and Ψ.
Example 3. Let h = {ℓ0 7→ 4, ℓ1 7→ 3, ℓ2 7→ ℓ0 , ℓ3 7→ ℓ1 }. It is consistent with theexample Ψ in (3). To see this, 4 at location ℓ0 is an even number and 3 at location ℓ1
is an integer. At location ℓ2 , ℓ0 is of type “wref even” because, by (i), this is equivalentto Ψ(ℓ0 ) = even—a true statement. Similarly, the value ℓ1 at location ℓ3 is of type“wref int”.
2Formalizing a set of semantic predicates following (i) and (ii) directly, however,would encounter difficulties because of a circularity in the model: by (ii), Ψ is a map
from locations to types; by (i), the model of types takes Ψ as an argument—Ψ is necessary to decide if a location belongs to “wref τ”. If defined naı̈vely, the model would
result in inconsistent cardinality, as described by Ahmed [1].
We next propose a fixed-point approach. We rewrite the heap type Ψ as a recursive
equation. After adding Ψ as an argument to types, the example in (3) becomes:Ψ = {ℓ0 : even(Ψ), ℓ1 : int(Ψ), ℓ2 : (wref even)(Ψ), ℓ3 : (wref int)(Ψ)}(4)

Notice that Ψ appears on both the left and the right side of the equation. Once Ψ iswritten as a recursive equation, it follows that any fixed point of the following functional
is a solution to the equation (4):λΨ.{ℓ0 : even(Ψ), ℓ1 : int(Ψ), ℓ2 : (wref even)(Ψ), ℓ3 : (wref int)(Ψ)}(5)

To get a fixed point of (5), we follow the indexed model of recursive types by Appeland McAllester [2]. We first introduce some domains:(SemHeapType) F ∈ Loc ⇀ SemIType(SemIType) t ∈ SemHeapEnv → Nat → Value → Prop
(SemHeapEnv) φ ∈ Loc ⇀ Nat → Value → Prop
We use F for a semantic heap type (it is the metatype of the denotation of heap types,
as we will see). It maps locations to indexed types. An important point is that from F
we can define λφ, ℓ. F(ℓ) φ, which has the metatype SemHeapEnv → SemHeapEnv.
10

Therefore, a semantic heap type is effectively a functional similar to the one in (5), anda fixed point of F is of the metatype SemHeapEnv.
A semantic type t is a predicate over the following arguments: φ is a semantic
heap environment; k is a natural-number index; v is a value. The heap environmentφ ∈ SemHeapEnv is used in our model of WRef(t) to constrain reference types. The
index k comes from the indexed model and is a technical device that enables us todefine the fixed point of a semantic heap type F.
Following the indexed model, we introduce a notion of contractiveness.
Definition 4. (Contractiveness)
contractive(F) , ∀ℓ ∈ dom(F). contractive(F(ℓ))
contractive(t) , ∀φ, k, j ≤ k, v. (t φ j v) ↔ (t (approx(k, φ)) j v)
approx(k, φ) , λℓ, j, v. j < k ∧ φ l j v.

We define (℘F) = λφ, ℓ. F(ℓ) φ. That is, it turns F into a functional of type
SemHeapEnv → SemHeapEnv.
Theorem 5. If contractive(F), then the following µF is the least fixed point2 of the functional (℘F):µF , λℓ, k, v. (℘F)k+1 (⊥) ℓ k v,
where ⊥ = λℓ, k, v. false, and (℘F)k+1 applies the functional k + 1 times.
The theorem is proved by following the indexed model of recursive types [2]. We
present the proof in our technical report [15].
The following lemma is an immediate corollary of Theorem 5.

Lemma 6. For any contractive F, any ℓ, k, v, we have F(ℓ) (µF) k v ↔ (µF)(ℓ) k v
Most of the semantic types ignore the φ argument. For example,
Even , λφ, k, v. ∃u. v = 2 × u.
We use capitalized Even to emphasize that it is a predicate, instead of the syntactic typeeven. The model of weak-reference types uses the argument φ.
Definition 7. WRef(t) , λφ, k, ℓ. ∀ j < k, v. φ ℓ j v ↔ t φ j v
In words, a location ℓ is of type WRef(t) under heap environment φ, if φ(ℓ) equals t
approximately, with index less than k.
Example 8. Let F0 = {ℓ0 : Even, ℓ1 : WRef(Even)}. Then “WRef(Even) (µF0 ) k ℓ0 ”
holds for any k. To see this, for any j < k and v, we have(µF0 ) ℓ0 j v ↔ F0 (ℓ0 )(µF0 ) j v ↔ Even (µF0 ) j v
The first step is by lemma 6, and the second is by the definition of F0 at location ℓ0 .
We can similarly show “WRef(WRef(Even)) (µF0 ) k ℓ1 ” holds.
2wNote that the definition of WRef(t) is more general than the “wref τ” type in SL ,as τ is syntactically defined, while t can be any (contractive) semantic predicate.

2 Since F is contractive in the sense that “F(ℓ) φ k w” performs only calls to φ on arguments

smaller than k, it is easy to show by induction that any two fixed points of F are identical;
therefore, the least fixed point of F is also its greatest fixed point.

11

Heap allocation. We need an additional idea to cope with heap allocation in the weak
heap. Our indexed types take the fixed point of a semantic heap type F as an argument.
But F changes after heap allocation. For example, fromF = {ℓ0 : Even, ℓ1 : WRef(Even)} to F′ = {ℓ0 : Even, ℓ1 : WRef(Even), ℓ2 : Even},
after ℓ2 is allocated and initialized with an even number.
After a new heap location is allocated, any value that has type t before allocation should still have the same type after allocation. This is the monotonicity condition
maintained by type systems. To model it semantically, our idea is to quantify explicitly
outside of the model of types over all future semantic heap types and assert that the typein question is true over the fixed point of any future semantic heap type.
First is a semantic notion of type-preserving heap extension from F to F′ :Definition 9. F′ ≥ F ,contractive(F′ ) ∧ contractive(F) ∧ ∀ℓ ∈ dom(F), φ, k, v. F′ (ℓ) φ k v ↔ F(ℓ) φ k v
Lemma 10. The relation F′ ≥ F is reflexive, anti-symmetric, and transitive (thus a partial order).
Next, we define the consistency relation between h and F, and also a relation thatstates a value v is of type t under F. Both relations quantify over all future semantic
heap types, and require that the type in question be true over the fixed point of any
future semantic heap type.
Definition 11. |= h : F , dom(h) ⊆ dom(F) ∧ ∀ℓ ∈ dom(h).F |= h(ℓ) : F(ℓ)
F |= v : t , ∀F′ ≥ F.∀k. t (µF′ ) k v
With our model, the following theorem for memory operations can be proved (pleasesee our technical report [15] for proofs).
Theorem 12.
(i) (Read) If |= h : F, and ℓ ∈ dom(h), and F |= ℓ : WRef(t), then F |= h(ℓ) : t.
(ii) (Write) If |= h : F, and ℓ ∈ dom(h), and F |= ℓ : WRef(t), and F |= v : t, then|= h[ℓ ; v] : F.
(iii) (Allocation) If |= h : F, and F |= v : t, and contractive(t), and ℓ ∈/ dom(F), then|= h ⊎ {ℓ 7→ v} : F ⊎ {ℓ 7→ t}.
3.2 Semantic model of SLw
To show the soundness of SLw , we define semantics for judgments in SLw and thenprove each rule as a lemma according to the semantics. Figure 8 presents definitions
that are used in the semantics.
The semantics of types is unsurprising. In particular, the semantics of [[wref τ]] isdefined in terms of the predicate WRef(t) in Definition 7. All these types are contractive.
The semantics of Ψ and Γ is just the point-wise extension of the semantics of types.
The predicate “F, r, h |= p” interprets the truth of assertion p. When p is a standard
SL formula, the interpretation is the same as the one in SL. When p is {e : τ}, the12[[τ]] ∈ SemIType[[int]] , λφ, k, v. ∃n.v = n.

[[ref]] , λφ, k, v. ∃ℓ.v = ℓ.

[[wref τ]] , WRef([[τ]])[[Ψ]] ∈ Loc ⇀ SemIType[[{ℓ1 : τ1 , . . . , ℓn : τn }]] , {ℓ1 : [[τ1 ]], . . . , ℓn : [[τn ]]}[[Γ]] ∈ Var ⇀ SemIType[[{x1 : τ1 , . . . , xn : τn }]] , {x1 : [[τ1 ]], . . . , xn : [[τn ]]}F, r, h |= pF, r, h |= {e : τ} , F |= r(e) : [[τ]]F, r, h |= emp , dom(h) = 0/
F, r, h |= {e1 7→ e2 } , dom(h) = r(e1 ) ∧ h(r(e1 )) = r(e2 )F, r, h |= p1 ∗ p2 , ∃h1 , h2 . (h = h1 ⊎ h2 ) ∧ (F, r, h1 |= p1 ) ∧ (F, r, h2 |= p2 )/ ∧ (F, r, h1 |= p1 )) ⇒ (F, r, h1 ⊎ h |= p2 )F, r, h |= p1 −∗p2 , ∀h1 . ((dom(h1 ) ∩ dom(h) = 0)
F |= r : Γ, ∀x ∈ dom(Γ). F |= r(x) : [[Γ(x)]]r, h |= F ∗ p , ∃h1 , h2 . (h = h1 ⊎ h2 ) ∧ (dom(h1 ) = dom(F)) ∧ (|= h1 : F) ∧ (F, r, h2 |= p)Fig. 8. Semantic definitions

interpretation depends on F. Notice that the interpretation of {e : τ} is independent of
the heap; it is a pure assertion (that is, it does not depend on the strong heap).
The definition of F |= r : Γ is the point-wise extension of F |= v : t to local variabletypes. The definition of “r, h |= F ∗ p” splits the heap into two parts. One for the weak
heap, which should satisfy F, and the other for the strong heap, which is specified by p.
With the above definitions, we are ready to define the semantics of the judgments in
SLw . The following definitions interpret “Ψ, Γ ⊢ e : τ”, “⊢ p ⇒ p′ ”, and“⊢ {Γ, p} ⇒ {Γ′ , p′ }”.

Definition 13.
Ψ, Γ |= e : τ , ∀F ≥ [[Ψ]] . ∀r. F |= r : Γ ⇒ F |= r(e) : [[τ]].
|= p ⇒ p′ , ∀F, r, h. (F, r, h |= p) ⇒ (F, r, h |= p′ )|= {Γ, p} ⇒ {Γ′ , p′ } ,∀F, r, h. (F |= r : Γ ∧ r, h |= F ∗ p) ⇒ (F |= r : Γ′ ∧ r, h |= F ∗ p′ )Now we are ready to interpret Ψ ⊢ {Γ, p} ~c {Γ′ , p′ }. Following Hoare Logic, we
define both partial and total correctness:13

Definition 14. (Partial and total correctness)
Ψ |=p {Γ, p} ~c {Γ′ , p′ } ,∀F ≥ [[Ψ]], r, h. (F |= r : Γ) ∧ (r, h |= F ∗ p) ⇒safe(r, h,~c) ∧∀r′ , h′ . (r, h,~c) 7−→∗ (r′ , h′ , ε) ⇒ ∃F′ ≥ F. (F′ |= r′ : Γ′ ) ∧ (r′ , h′ |= F′ ∗ p′ )
Ψ |=t {Γ, p} ~c {Γ′ , p′ } ,(Ψ |=p {Γ, p} ~c {Γ′ , p′ }) ∧∀F ≥ [[Ψ]], r, h. (F |= r : Γ) ∧ (r, h |= F ∗ p) ⇒ terminate(r, h,~c)
In the partial-correctness interpretation, it assumes a state that satisfies the condition{Γ, p} and requires that the state be safe (see Definition 2 on page 4 for safety). In
addition, it requires that, for any terminal state after the execution of ~c, we must beable to find a new semantic heap type F′ so that F′ ≥ F and the new state satisfies{Γ′ , p′ }. Note that F′ may be larger than F due to allocations in ~c. The total-correctness
interpretation requires termination in addition to the requirements of partial correctness.
Theorem 15. All rules in Figures 5, 6 and 7 are sound for both partial and total correctness.
The proof uses Theorem 12. It is largely straightforward and omitted. We referinterested readers to our technical report [15] for the proof.

4 Related work
We discuss related work in three categories: (1) work related to language interoperation;(2) work related to integrating SL with type systems; and (3) work related to semantic
models of types.
Most work in language interoperation focuses on the design and implementationof foreign function interfaces. Examples are plenty. Given a multilingual program, one
natural question is how to reason about the program as a whole. This kind of reasoning
requires models, program analyzers, and program logics that can work across language
boundaries. Previous work has addressed the question of how to model the interoperation between dynamically typed languages and statically typed languages [9], and theinteroperation between two safe languages when they have different systems of computational effects [18]. By integrating SL and type systems, SLw can elegantly reason
about properties of heaps that are shared by high-level and low-level code.
Previous systems of integrating SL with type systems [11, 8] assume that programs
are well-typed according to a syntactic type system, and SL is then used as an addon to reason about more properties of programs. Honda et al’s program logic [7, 20]
for higher-order languages supports reference types but also requires a separate typesystem (in addition to the Hoare assertions); Reus et al [12] presented an extension of
separation logic for supporting higher-order store (i.e., references to higher-order functions), but their logic does not support weak heaps which we believe embodies the key
feature of reference types (i.e., the ability to perform safe updates without knowing the14exact aliasing relation). Compared to previous systems, SLw targets the interoperation
between high-level and low-level code. It allows cross-boundary references and mixes
SL formulas and types.
The soundness of SLw is justified by defining a semantic model, notably for types.
Ahmed [1] and Appel et al [3] presented a powerful index-based semantic model for arich type system with ML-style references. They rely on constructing a “dependently
typed” global heap type to break the circularity discussed in Section 3. Our current
work, in contrast, simply takes a fixed point of the recursively defined heap type predicate and avoids building any dependently typed data structures. Our work also differsfrom theirs in that we are reasoning about reference types in a program logic. Appelet al. [3] can also support impredicative polymorphism which is not addressed in ourcurrent work.

5 Discussion and future work
This work aims toward a framework for reasoning about language interoperation, but
a lot remains to be done. A realistic high-level language contains many more language
features and types. We do not foresee much difficulty in incorporating language features
and types at the logic level as their modeling is largely independent from the interaction
between weak and strong heaps. One technical concern is how to extend our semantic
model to cover a complicated type system, including function types and OO classes.
SLw does not formally consider the effect of a garbage collector. A garbage collector would break the crucial monotonicity condition of the weak heap that our semantic
model relies on. We believe a possible way to overcome this problem is to use a regionbased type system [17]. A garbage collector would also imply that there cannot bedirect references from strong heaps to weak heaps; an extra level of indirection has tobe added.

6 Conclusion
In his survey paper of Separation Logic [13], Reynolds asked “whether the dividing line
between types and assertions can be erased”. This paper adds evidence that the typebased approach has its unique place when ensuring safety in weak heaps and when reasoning about the interaction between weak and strong heaps. The combination of types
and SL provides a powerful framework for checking safety and verifying properties ofmultilingual programs.

Acknowledgments
We thank anonymous referees for suggestions and comments on an earlier version ofthis paper. Gang Tan is supported in part by NSF grant CCF-0915157. Zhong Shao is
supported in part by a gift from Microsoft and NSF grants CCF-0524545 and CCF0811665. Xinyu Feng is supported in part by NSF grant CCF-0524545 and National
Natural Science Foundation of China (grant No. 90818019)
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