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Formal Methods Laboratory Romanian Academy
FML Technical Report
TR no. FML-10-02 Printed:
August 2013
Continuation Semantics for
Asynchronous Concurrency
G.Ciobanu and E.N.Todoran
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Formal Methods Laboratory.
Series of
Technical Reports (Institute of Computer Science Ia¸si)
ISSN 1842 - 1490
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Series of Technical Reports (Institute of Computer Science Ia¸si)
ISSN 1842 - 1490
Continuation Semantics for
Asynchronous Concurrency
by
Gabriel Ciobanu 1,2
Institute of Computer Science,
Romanian Academy, Ia¸si
gabriel@iit.tuiasi.ro
Eneia Nicolae Todoran
Technical University of Cluj-Napoca
Department of Computer Science
400027 Cluj-Napoca, Romania
Eneia.Todoran@cs.utcluj.ro
ABSTRACT
The paper presents a method of reasoning about the behavior of asynchronous programs in
denotational models designed with metric spaces and continuation semantics for concurrency.
1Head of Formal Methods Laboratory (FML)
2Contact person
Contents
1 Introduction 2
2 Preliminaries 4
2.0.1 Examples. ................................. 4
3 Syntax and continuation structure for L6
4 Continuation semantics for L8
5 Concurrency laws in continuation semantics 11
5.1 Continuations and configurations . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Continuations and syntactic contexts . . . . . . . . . . . . . . . . . . . . . . 18
5.3 Concurrencylaws................................. 23
6 Concluding remarks and future work 35
Continuation Semantics for Asynchronous Concurrency 2
1 Introduction
Metric spaces provide a convenient framework for designing denotational models of con-
currency [4]. The main mathematical tool used in this approach to semantics is Banach’s
theorem, which states that a contracting function on a complete metric space has a unique
fixed point. Contractions play a central role in designing and relating concurrency semantics
[21] and there is a general method of solving reflexive domain equations in a category of
complete metric spaces [1]. By using the methodology of metric semantics in [31] we intro-
duced a ”continuation semantics for concurrency” (CSC). The CSC technique can be used
to model both sequential and parallel composition in interleaving semantics while providing
the general advantages of the classic technique of continuations [12].
In continuation semantics a program is conceptually divided into a current statement
and the rest of the program. A continuation is a representation of the behavior of the rest
of the program, an evaluation context for the denotation of the current statement [16].
The distinctive characteristic of the CSC technique is the modeling of continuations as
application-specific structures of computations (partially evaluated denotations) rather than
the functions to some answer type that are used in the classic technique of continuations
[30]. Intuitively it is a semantic formalization of a (process) scheduler, a denotational sched-
uler. Computations are grouped in a continuation. There is one active computation (the
denotation of the current statement). Each computation remains active only until it per-
forms and atomic action. Next, another computation taken from the continuation is planned
for execution. In this way it can be obtained the desired interleaving behavior for parallel
composition.
We do not know whether the domain of CSC is fully abstract.1However, as continuations
are semantic evaluation contexts [16], it is easy to prove that a denotational semantics
designed with CSC is correct with respect to a corresponding operational model (see section
5 of [31]). Also, the domain of CSC is very general. Only the structure of continuations needs
to be adapted to the (concurrent) language under study [34]. In this sense the CSC technique
provides flexibility in the denotational design of concurrent languages. For example, in the
case of a sequential language a continuation can be a stack of computations. It is also
natural to use the concept of a multiset to represent parallel computations. For a general
combination of sequential and parallel composition a continuation is a tree of computations.
In this paper we use CSC to investigate the semantics of a simple concurrent language L
embodying a mechanism of asynchronous communication. We work within the mathemat-
ical framework of 1-bounded complete metric spaces, by following the approach advocated
in [3]. In theory syncronization is used because it is simple to express. Based on the re-
sults expressed in process algebra, asynchronous interaction is primitive, and syncronous
interaction could be expressed in terms of asyncronous interaction; see, e.g. [19]. Moreover,
asynchronous interaction is easier to implement. The relation between synchronous and
1In fact, we are not aware of any full abstractness result for a concurrent language designed with continuations,
although various papers employ continuations in the denotational description of concurrent languages [3, 13,
2, 28].
Continuation Semantics for Asynchronous Concurrency 3
asynchronous communication is a topic of recent and current research [25, 35]. Although
synchronous interaction cannot always be expressed in terms of asynchronous primitives,
asynchronous interaction is a basic mechanism in many distributed computing systems, in-
cluding Internet and Web aplications.
The language Lthat we study in this paper embodies the paradigm of asynchronous
communication introduced in [10]. The paradigm consists of a language based on a set of
atomic actions and operators for sequential composition, nondeterministic choice and paral-
lel composition. The semantics of atomic actions is defined with respect to an abstract set
of states; the model includes special states indicating suspension and deadlock. The atomic
actions are interpreted as state transformations. As explained in [10], various asynchronous
computing models can be obtained as instances of this paradigm, including concurrent con-
straint programming [29], and also in other languages like dataflow and asynchronous CSP.
Lextends the paradigmatic language studied in [10] with recursion.
For the language under consideration, we show that the semantic operators designed with
CSC satisfy some concurrency laws, such as the associativity and commutativity of parallel
composition. Continuation-based models rely heavily on manipulations of higher-order func-
tions. It may be difficult to reason directly in terms of higher-order functions. Therefore we
introduce a left merge operator and we obtain a finite axiomatization of the parallel com-
position (or merge) operator. Any nonrecursive asynchronous concurrent program is thus
provably equivalent to a corresponding nondeterministic sequential program. Obviously, the
approach is inspired by classic process algebra theories [23, 8] this approach being adapted
by us to a continuation-based framework.
The main contribution is given by the proofs that continuation semantics satisfies some
basic laws such as the associativity of the parallel and sequential composition operators. Each
semantic property, also called a law here, can be proved by identifying a corresponding invari-
ant of the computation. Such an invariant is expressed as a relation between continuation
structures. The identification of semantic properties from the invariants of the computa-
tion is common in classic bisimulation semantics [22]. The idea is adapted to a denotational
framework based on continuations, by using arguments of the kind ’ 1
2·ε≤ε⇒ε= 0’, which
are standard in metric semantics [3]. In our case εis the distance between two behavioraly
equivalent continuations, before and after a computation step, respectively. The effect of
each computation step is given by the 1
2contracting factor. Therefore ε= 0 and the desired
property follows.
The rest of the paper is organized as follows. Section 2 presents some theoretical pre-
liminaries. Section 3 defines formally the language Land presents a mathematical structure
that we use to define the domain of continuations. In section 4 we present a denotational
semantics for Ldesigned with CSC. In section 5 we prove the various laws that are satis-
fied by this semantics; we obtain a finite axiomatization of parallel composition. Section 6
provides some concluding remarks and objectives of future research.
Continuation Semantics for Asynchronous Concurrency 4
2 Preliminaries
The notation (x∈)Xintroduces the set Xwith typical element xranging over X. Let
f∈X→Ybe a function. The function [ f|x7→ y] : X→Y, is defined (for x, x′∈X, y∈Y)
by: [ f|x7→ y](x′) = if x′=xthen yelse f(x′). We use the notation [ f|x17→y1|x27→ y2]
as an abbreviation for [ [ f|x17→ y1]|x27→ y2]. If f:X→Xand f(x) = xwe call xafixed
point of f. When this fixed point is unique (see Theorem 2.1) we write x=f ix(f).
The denotational semantics given in this paper is built within the mathematical frame-
work of 1-bounded complete metric spaces, following the approach advocated in [3]. We work
with the following notions which we assume known: metric and ultrametric space, isometry
(distance preserving bijection between metric spaces, denoted by ’∼
=’), complete metric space,
and compact set.
2.0.1 Examples.
1. Let Xbe an arbitrary set. The discrete metric on X(d:X×X→[0,1]) is defined
(for x, y ∈X) as follows:
d(x, y) = 0 if x=y
= 1 if x6=y
(X, d) is a complete ultrametric space.
2. Let (a∈)Abe a nonempty set, and let (x, y ∈)A∞=A∗∪Aω, where A∗(Aω) is the set
of all finite (infinite) sequences over A. One can define a metric over A∞as follows:
d(x, y) = 2−sup{n|x(n)=y(n)}
where x(n) denotes the prefix of xof length n, in case length(x)≥n, and xotherwise
(by convention, 2−∞ = 0). dis a Baire-like metric. (A∞, d) is a complete ultrametric
space.
We recall that if (X, dX), (Y, dY) are metric spaces, a function f:X→Yis a contraction
if ∃c∈R, 0 ≤c < 1, ∀x1, x2∈X:dY(f(x1), f (x2))≤c·dX(x1, x2). In metric semantics it
is customary to attach a 1
2-contracting factor to each computation step. When c= 1 the
function fis called nonexpansive. In what follows we denote the set of all nonexpansive
functions from Xto Yby X1
→Y. Banach’s fixed point theorem [7] is at the core of metric
semantics.
Theorem 2.1 (Banach) Let (X, dX)be a complete metric space. Each contraction f:
X→Xhas a unique fixed point.
Continuation Semantics for Asynchronous Concurrency 5
Definition 2.2 Let (X, dX),(Y, dY)be (ultra) metric spaces. On (x∈)X,(f∈)X→Y(the
function space), (x, y)∈X×Y(the Cartesian product), u, v∈X+Y(the disjoint union of X
and Y, which can be defined by: X+Y= ({1} × X)∪({2} × Y)), and U, V ∈P(X)(the
power set of X) one can define the following metrics:
(a) d1
2·X:X×X→[0,1] d1
2·X(x1, x2) = 1
2·dX(x1, x2)
(b) dX→Y: (X→Y)×(X→Y)→[0,1]
dX→Y(f1, f2) = supx∈XdY(f1(x), f2(x))
(c) dX×Y: (X×Y)×(X×Y)→[0,1]
dX×Y((x1, y1),(x2, y2)) =
max{dX(x1, x2), dY(y1, y2)}
(d) dX+Y: (X+Y)×(X+Y)→[0,1]
dX+Y(u, v) = if (u, v ∈X)then dX(u, v)
else if (u, v∈Y)then dY(u, v)else 1
(e) dH:P(X)× P(X)→[0,1]:
dH(U, V ) = max{supu∈Ud(u, V ),supv∈Vd(v, U)}
where d(u,W )= infw∈Wd(u, w)and by convention sup ∅=0,inf ∅=1 (dHis the Haus-
dorff metric).
We use the abbreviation Pnco(·) to denote the power set of non-empty and compact subsets
of ’·’. Also, we often suppress the metrics part in domain definitions, and write, e.g., 1
2·X
instead of (X, d 1
2·X).
Remark 2.3 Let (X, dX),(Y, dY), d 1
2·X,dX→Y, dX×Y,dX+Yand dHbe as in Definition
2.2. In case dX, dYare ultrametrics, so are d1
2·X, dX→Y, dX×Y, dX+Yand dH. Moreover,
if (X, dX),(Y, dY)are complete then 1
2·X,X→Y,X1
→Y,X×Y,X+Y, and Pnco(X)
(with the metrics defined above) are also complete metric spaces [3].
We also use the abbreviation Pfinite (·) to denote the power sets of finite subsets of ’·’. In
general, the construct Pf inite(·) does not give rise to a complete space. In our study, we use
it to create a structure that we endow with the discrete metric. Any set endowed with the
discrete metric is a complete ultrametric space.
Continuation Semantics for Asynchronous Concurrency 6
3 Syntax and continuation structure for L
The syntax of Lis given in BNF in Definition 3.1. The basic components are a set (a∈)Act of
atomic actions and a set (x∈)RV ar of recursion variables. There is a special symbol δ∈Act,
whose behavior is explained below. ;, + and kare operators for sequential, nondeterministic
and parallel composition, respectively. kis also called a merge operator, and ⌊⌊ is the left
merge operator.
Definition 3.1 (Syntax of L)
(a) (Statements) s(∈Stat) ::= a|x|s;s|s+s|s⌊⌊ s|sks
(b) (Guarded statements) g(∈GStat) ::= a|g;s|g+g|g⌊⌊ s|gkg
(d) (Declarations) (D∈)Decl =RV ar →GStat
(e) (Programs) (ρ∈)L=Decl ×Stat
The meaning of atomic actions is defined by an interpretation function I:Act →Σ→
({↑} ∪ Σ), where (σ∈)Σ is a set of states. If I(a)(σ) =↑the action acannot proceed in state
σ; its execution is suspended. When all processes are suspended deadlock occurs. Notice that
I(δ)(σ) =↑,∀σ∈Σ, i.e. the action δsuspends in all states. Lincorporates the mechanism of
asynchronous communication studied in [10]. As explained in [10], this form of asynchronous
communication can be encountered in concurrent constraint programming, and also in other
languages like dataflow or asynchronous CSP.
We employ an approach to recursion based on declarations and guarded statements [3]. In
a guarded statement each recursive call is preceded by at least one elementary action, which
guarantees the fact that the semantic operators are contracting functions in the present
metric setting. For the sake of brevity (and without loss of generality) in what follows we
assume a fixed declaration D∈Decl, and all considerations in any given argument refer to
this fixed D.
For inductive proofs we introduce a complexity measure ςthat decreases upon recursive
calls. ςis well-defined due to our restriction to guarded recursion.
Definition 3.2 (Complexity measure) The function ς:S tat →Nis given by
ς(a) = 1
ς(x) = 1 + ς(D(x))
ς(s1op s2) = 1 + ς(s1)op ∈ {;,⌊⌊ }
ς(s1op s2) = 1+max{ς(s1), ς(s2)}op ∈ {+,k}
In the CSC approach a continuation is a structured configuration of computations. For
example, in the case of a sequential language a continuation is a stack of computations. It is
also natural to use the concept of a multiset to represent parallel computations. For a general
Continuation Semantics for Asynchronous Concurrency 7
combination of parallel and sequential composition a continuation is a tree of computations
with active computations at the leaves. For example, when the denotation of a program
fragment (s1ks2); s3is computed, the denotations of s1and s2become leaves in such a tree
and the denotation of s3becomes an inner node. This behavior is inspired by the concept
of a cactus stack [9], a stack with multiple tops that can be active concurrently. In order to
define such domains of trees of computations we employ a (partially ordered) set of identifiers
Id. (α∈)Id is the set of all finite, possibly empty,sequences over {1,2}, and α≤α′iff αis
a prefix of α′.
In this paper we use the symbol ’·’ as a concatenation operator over sequences, hence we
can represent any nonempty identifier α∈Id by a finite sequence α=i1·...·in, where
i1,...,in∈ {1,2}. We use the symbol λto represent the empty sequence over {1,2}(λ∈Id).
Definition 3.3
(a) Let (α∈)Id ={1,2}∗be a set of identifiers, equipped with the following partial ordering:
α≤α’ iff α′=α·i1·...·infor i1,··· , in∈ {1,2}, n ≥0. We define α < α′iff α≤α′
and α6=α′. If A∈ P(I d), we denote by ≤Athe restriction of ≤to A.
(b) We define a function max :P(Id)→P(Id)by:
max(A) = {α|αis a maximal element of (A, ≤A)}
Remark 3.4 λ≤α, for any α∈Id, which means that λis the least element of Id. Also,
when A∈ P(Id),αis a maximal element of (A, ≤A)if α∈Aand ∀α′∈A:¬(α < α′).
The concept of a tree that we use in this paper is taken from set theory, where a tree is a
partially ordered set in which the predecessors of each element are well-ordered. A set is
well-ordered if it is linearly ordered and every nonempty subset has a least element. A set is
linearly ordered if any two elements are comparable. There are several books on set theory
that provide formal definitions of these concepts; see, e.g., [20]. Here we only explain the
concept of a tree by means of an example.
(Id, ≤)is a partially ordered set, i.e. ≤is a binary relation over Id which is reflexive,
transitive and antisymmetric. In this paper we only work with finite trees. If A⊆I d is a
finite subset of Id then (A, ≤A)is a finite tree. For example, let A={α·1, α ·2, α·1·1, α ·1·2,
α·2·1, α ·2·2, α·1·1·1}, for some α∈Id.(A, ≤A)is a finite tree. The maximal elements of
(A, ≤A)are exactly the leaves of the tree: max(A) = {α·1·1·1, α ·1·2,α·2·1, α ·2·2}.
The predecessors of each element in Aare well-ordered. For example, α·1·1·1> α ·1·1
and α·1·1> α ·1. The set of predecessors of α·1·1·1is {α·1·1, α ·1}, which is linearly
ordered, i.e. any two elements in {α·1·1, α ·1}are comparable. In general, α1and α2are
comparable iff α1≤α2or α2≤α1. Obviously, every nonempty subset of {α·1·1, α ·1}
has a least element. In fact, every finite linearly ordered set is well-ordered.
Let (x∈)Xbe a metric domain, i.e. a complete metric space. We use the following
notation:
Continuation Semantics for Asynchronous Concurrency 8
{|X|} not.
=Pfinite(Id)×(Id →X)
Let α∈Id, (π, θ)∈ {|X|} with π∈ Pf inite (Id), θ∈Id →X. We define id :{|X|} → Pfinite(Id),
id(π, θ) = π. We also use the following abbreviations:
(π, θ)(α)not.
=θ(α) (∈X)
(π, θ)\αnot.
= (π\ {α}, θ) (∈ {|X|})
[ (π, θ)|α7→ x]not.
= (π∪ {α},[θ|α7→ x]) (∈ {|X|})
The basic idea is that we treat (π, θ) as a ’function’ with finite graph {(α, θ(α)) |α∈π},
thus ignoring the behaviour of θfor any α /∈π(πis the ’domain’ of (π, θ)). We use this
mathematical structure to represent finite partially ordered bags (or multisets)2of compu-
tations. The set Id is used to distinguish between multiple occurrences of a computation
in such a bag. We endow both sets Id and Pfinite (Id) with discrete metrics; every set with
a discrete metric is a complete ultrametric space. By using the composite metrics given in
Definition 2.2 {|X|} becomes also a metric domain. The operators behave as follows. id(π, θ)
returns the collection of identifiers for the valid computations contained in the bag (π, θ),
(π, θ)(α) returns the computation with identifier α, (π, θ)\αremoves the computation with
identifier α, and [ (π, θ)|α7→ x] replaces the computation with identifier α.
By a slight abuse, we use the same notations (including the operator id and the abbre-
viations (·)(α),(·)\α, [· | α7→ x]) when (x∈)Xis an ordinary set: {|X|} =Pf inite(Id)×
(Id →X); in this case we do not endow {|X|} with a metric.
4 Continuation semantics for L
We design a continuation-based denotational semantics for L. As a semantic universe for
the final yield of our denotational model we employ a standard linear-time domain (p∈)P=
Pnco(Σ∗∪Σ∗· {δ} ∪ Σω). Here Σ∗(Σω) denotes the collection of all finite (infinite) sequences
over Σ. An element of Σ∗· {δ}is a finite sequence terminated with the symbol δ, which
denotes deadlock. Also, we use the symbol λto represent the empty sequence. This is a slight
abuse of notation since we also use the symbol λto represent the empty sequence over {1,2};
however it is always clear from the context which is the type of λ(either λ∈Id (= {1,2}∗),
or λ∈Σ∗). We view (q∈)Σ∗∪Σ∗· {δ} ∪ Σωas a complete ultrametric space by endowing
it with the Baire metric (see section 2). We use the notation σ·p={σ·q|q∈p}, for any
σ∈Σ and p∈P. The type of the denotational semantics [[·]] for Lis Stat →D, where:
D∼
=Cont 1
→Σ→P
(γ∈)Cont =Id ×Kont
(κ∈)Kont ={| 1
2·D|}
2We avoid using the notion of a partially ordered multiset which is a more refined structure – see [6], or chapter
16 of [3].
Continuation Semantics for Asynchronous Concurrency 9
In the equations given above the sets Σ, Id (and Pfinite (Id)) are endowed with the discrete
(ultra)metric. The composed metric spaces are built up using the metrics of Definition 2.2.
To conclude that such a system of equations has a solution, which is unique up to isometry,
we rely on the general method developed in [1]. The solution for Dis obtained as a complete
ultrametric space. In [1], the family of complete (ultra)metric spaces is made into a category
C. It is proved that a generalized form of Banach’s fixed point theorem holds, stating that a
functor F:C → C that is contracting (in a sense) has a unique fixed point (up to isometry).
Intuitively, in the equation above the relevant functor is contracting as a consequence of the
fact that the recursive occurence of Dis preceded by a 1
2factor.
The construction {| 1
2·D|} =Pfinite(Id)×(Id →1
2·D) was introduced in section 3. In the
sequel ϑranges over Id →1
2·D. We call an element of Kont aclosed continuation and an
element of Cont an open continuation. A (closed or open) continuation is a representation
of what remains to be computed from the program [30]. A closed continuation κ∈Kont
is a self-contained structure of computations. An open continuation (α, κ)∈Cont behaves
like an evaluation context [16] for the denotational mapping [[·]]. In an expression [[s]](α, κ),
[[s]] is the active computation which is evaluated with respect to (α, κ). Intuitively, an open
continuation (α, κ) is a structured configuration of computations which contains a hole,
indicating the conceptual position of the active computation. The position of the ’hole’
is given in this representation by the identifier α, which is not an element of id(κ) ([[·]] is
designed to preserve this invariant property: α /∈id(κ) and α∈max({α} ∪ id(κ))).
The denotational function [[·]] is defined in 4.1 with the aid of a mapping kc, which
is called a scheduler. The denotational function maps an open continuation to a program
behavior. After producing an elementary step the denotational function transmits the control
to the scheduler. The scheduler receives as parameter a closed continuation κwhich it
maps to a corresponding program behavior. If the continuation κis empty (id(κ) = ∅) the
scheduler terminates the computation. Otherwise the scheduler activates a computatation
(denotation) selected in a nondeterministic manner from the continuation; it decomposes a
closed continuation into a computation and a corresponding open continuation and then it
executes the former with the latter as continuation.
The semantics of nondeterministic choice in Lis given by the operator + : (P×P)→P.
This definition reflects that p1+p2blocks only if both p1and p2block. It is easy to check that
+ is well-defined, nonexpansive, associative, commutative and idempotent. Also, p+{δ}=p,
for any p∈P.
p1+p2={q|q∈p1∪p2, q 6=δ} ∪ {δ|δ∈p1∩p2}
Definition 4.1 (Denotational semantics for L)
(a) Let kc :Kont →Σ→Pbe given by:
kc(κ)(σ) = if (id(κ) = ∅)then {λ}else +α∈max(id(κ)) κ(α)(α, κ \α)(σ)
Continuation Semantics for Asynchronous Concurrency 10
(b) We define Φ : (Stat →D)→(Stat →D)(for φ∈(Stat →D)) by:
Φ(φ)(a)(α, κ)(σ) = if (I(a)(σ) =↑)then {δ}
else I(a)(σ)·kc(κ)(I(a)(σ))
Φ(φ)(x)(α, κ)(σ) = Φ(φ)(D(x))(α, κ)(σ)
Φ(φ)(s1;s2)(α, κ)(σ) = Φ(φ)(s1)(α·1,[κ|α7→ φ(s2) ])(σ)
Φ(φ)(s1+s2)(α, κ)(σ) = Φ(φ)(s1)(α, κ)(σ) + Φ(φ)(s2)(α, κ)(σ)
Φ(φ)(s1⌊⌊ s2)(α, κ)(σ) = Φ(φ)(s1)(α·1,[κ|α·27→ φ(s2) ])(σ)
Φ(φ)(s1ks2)(α, κ)(σ) = Φ(φ)(s1)(α·1,[κ|α·27→ φ(s2) ])(σ) +
Φ(φ)(s2)(α·1,[κ|α·27→ φ(s1) ])(σ)
(c) We put [[·]] = fix(Φ). Let α0=λ, and κ0= (∅, ϑ0), where ϑ0(α) = [[δ]],∀α∈I d.
(α0, κ0)∈Cont is the empty continuation. Notice that kc(κ0)(σ) = {λ}, for any
σ∈Σ. We also define D[[·]] : Stat →Σ→Pby:
D[[s]] = [[s]](α0, κ0)
The semantics of atomic actions is defined with the aid of the interpretation function I
introduced in the paragraph that follows after Definition 3.1.
A continuation is a tree of computations with active elements at the leaves (the maximal
elements with respect to ’≤’). In the case of a sequential composition (s1;s2) the computa-
tions [[s1]] and [[s2]] are given the identifiers α·1 and α, respectively (α·1> α). The scheduler
function kc gives priority to the computations at the leaves of the tree that represents the
continuation. Therefore [[s2]] will only be evaluated after the completion of the evaluation
of [[s1]]. In the case of a parallel composition (s1ks2) the computations [[s1]] and [[s2]] are
given identifiers (α·1 and α·2) that are incomparable (with respect to ≤) therefore the
computations [[s1]] and [[s2]] are evaluated in an interleaved manner.
The denotational semantics [[·]] is defined as the (unique) fixed point of the higher-order
mapping Φ. It may not be obvious why on the right-hand sides of the equations given in
Definition 4.1(b), in some places we use Φ(φ) while in others we use φ. The definition of Φ(φ)
is organized by induction on ς(s) (see Definition 3.2). The computations φ(s) only occur in
the continuation and are always executed after an elementary step performed by the active
computation. This step ensures the contractiveness of Φ and is reflected by the 1
2·factor in
the definition of the domain of computations. Definition 4.1 is justified by Lemmas 4.2 and
4.3, whose proofs are omitted. Similar Lemmas are given in [31, 32]. See, e.g., the proofs of
Lemmas 3.13 and 3.14 in [31].3
3In Lemma 4.3(b), Φ(φ)(s) is (only) nonexpansive (rather than contractive) in the continuation. Still, this
implies that Φ is 1
2·-contractive in φ. Intuitively, the distance between denotations halves while they are
stored into a continuation. This is a consequence of the 1
2·-contracting factor in our domain equation. This
also explains the occurence of the multiplication factor ’ 2·’ (rather than ’ 1
2·’) in Lemma 4.2(b).
Continuation Semantics for Asynchronous Concurrency 11
Lemma 4.2
(a) The mapping kc (see Definition 4.1) is well-defined.
(b) ∀κ1, κ2∈Kont :d(kc(κ1), kc(κ2)) ≤2·d(κ1, κ2)
Lemma 4.3 For all φ∈(Stat →D), s ∈S tat, α ∈I d, κ ∈Kont, σ ∈Σ:
(a) Φ(φ)(s)(α, κ)(σ)∈P(it is well defined),
(b) Φ(φ)(s)is nonexpansive (in (α, κ)), and
(c) Φis 1
2- contractive (in φ).
5 Concurrency laws in continuation semantics
We present a method of describing the behavior of concurrent systems in a denotational
model designed with CSC, using a representation of continuations as structured configura-
tions of computations. For the language Lwe show that the semantic operators satisfy laws
that are usually included in concurrency theories, such as the associativity and commuta-
tivity of parallel composition.
Various properties can be proved for all continuations by simple manipulations of the
equations that define the denotational mapping [[·]].
Lemma 5.1 For all s, s1, s2, s3∈Stat :
(a) [[s1+s2]] = [[s2+s1]] (commutativity of +)
(b) [[(s1+s2) + s3]] = [[s1+ (s2+s3)]] (associativity of +)
(c) [[s+s]] = [[s]] (idempotency of +)
(d) [[(s1+s2); s3]] = [[s1;s3+s2;s3]] (right distributivity of ; over +)
(e) [[s+δ]] = [[s]]
(f) [[δ;s]] = [[δ]]
(g) [[s1ks2]] = [[s1⌊⌊ s2+s2⌊⌊ s1]]
(h) [[(s1+s2)⌊⌊ s3]] = [[s1⌊⌊ s3+s2⌊⌊ s3]] (right distributivity of ⌊⌊ over +)
(i) [[s1ks2]] = [[s2ks1]] (commutativity of k)
Proof: In order to prove that [[s]] = [[s]], for s, s ∈Stat, it is enough to show that
[[s]](α, κ)(σ) = [[s]](α, κ)(σ), for arbitrary (α, κ)∈Cont,σ∈Σ.
(a) [[s1+s2]](α, κ)(σ)
= [[s1]](α, κ)(σ) + [[s2]](α, κ)(σ) [+ is commutative]
= [[s2]](α, κ)(σ) + [[s1]](α, κ)(σ)
Continuation Semantics for Asynchronous Concurrency 12
= [[s2+s1]](α, κ)(σ)
(b) [[(s1+s2) + s3]](α, κ)(σ)
= [[s1+s2]](α, κ)(σ) + [[s3]](α, κ)(σ)
= ([[s1]](α, κ)(σ) + [[s2]](α, κ)(σ)) + [[s3]](α, κ)(σ) [+ is associative]
= [[s1]](α, κ)(σ) + ([[s2]](α, κ)(σ) + [[s3]](α, κ)(σ))
= [[s1]](α, κ)(σ) + [[s2+s3]](α, κ)(σ)
= [[s1+ (s2+s3)]](α, κ)(σ)
(a) [[s+s]](α, κ)(σ)
= [[s]](α, κ)(σ) + [[s]](α, κ)(σ) [+ is idempotent]
= [[s]](α, κ)(σ)
(d) [[(s1+s2); s3]](α, κ)(σ)
= [[s1+s2]](α·1,[κ|α7→ [[s3]] ])(σ)
= [[s1]](α·1,[κ|α7→ [[s3]] ])(σ) + [[s2]](α·1,[κ|α7→ [[s3]] ])(σ)
= [[s1;s3]](α, κ)(σ) + [[s2;s3]](α, κ)(σ)
= [[(s1;s3) + (s2;s3)]](α, κ)(σ)
(e) [[s+δ]](α, κ)(σ)
= [[s]](α, κ)(σ) + [[δ]](α, κ)(σ)
= [[s]](α, κ)(σ) + {δ}
= [[s]](α, κ)(σ)
(f) [[δ;s]](α, κ)(σ)
= [[δ]](α·1,[κ|α7→ [[s]] ])(σ)
={δ}
= [[δ]](α, κ)(σ)
(g) [[s1ks2]](α, κ)(σ)
= [[s1]](α·1,[κ|α·27→ [[s2]] ])(σ) + [[s2]](α·1,[κ|α·27→ [[s1]] ])(σ)
Continuation Semantics for Asynchronous Concurrency 13
= [[s1⌊⌊ s2]](α, κ)(σ) + [[s2⌊⌊ s1]](α, κ)(σ)
= [[s1⌊⌊ s2+s2⌊⌊ s1]](α, κ)(σ)
(h) [[(s1+s2)⌊⌊ s3]](α, κ)(σ)
= [[s1+s2]](α·1,[κ|α·27→ [[s3]] ])(σ)
= [[s1]](α·1,[κ|α·27→ [[s3]] ])(σ) + [[s2]](α·1,[κ|α·27→ [[s3]] ])(σ)
= [[s1⌊⌊ s3]](α, κ)(σ) + [[s2⌊⌊ s3]](α, κ)(σ)
= [[(s1⌊⌊ s3) + (s2⌊⌊ s3)]](α, κ)(σ)
(i) [[s1ks2]](α, κ)(σ)
= [[s1]](α·1,[κ|α·27→ [[s2]] ])(σ) + [[s2]](α·1,[κ|α·27→ [[s1]] ])(σ)
[+ is commutative]
= [[s2]](α·1,[κ|α·27→ [[s1]] ])(σ) + [[s1]](α·1,[κ|α·27→ [[s2]] ])(σ)
= [[s2ks1]](α, κ)(σ)
5.1 Continuations and configurations
All the above proofs are straightforward. However, the flexibility provided by continuations
comes at a price. Some properties may require more complex arguments and can be obtained
for continuations that contain only denotations of program statements. We introduce the
auxiliary notion of a configuration and a notion of isomorphism over configurations. A
configuration is a structure of Lstatements. A continuation may contain arbitrary values
of the type D. We prove the desired properties for continuations that can be obtained as
semantified versions of configurations, i.e. for continuations that contain only denotations
of statements. This represents an invariant of the denotational semantics, and ensures its
consistency just because the initial continuation is empty and the denotational semantics
adds to the continuation only denotations of statements. The function Kdefined in 5.2(b)
maps a configuration to a corresponding continuation.
Definition 5.2
(a) We define the set of closed configurations (k∈)Konf ={|Stat|}.4A closed config-
uration is a finite (partially ordered) bag (multiset) of statements (∈Stat). Also, we
define the set Conf of open configurations by:
4In this case the construct {| · |} is used to define an ordinary set; see the explanation given in the final part
of section 3.
Continuation Semantics for Asynchronous Concurrency 14
Conf ={(α, k)|(α, k)∈(Id ×K onf ), α /∈id(k), α ∈max({α} ∪ id(k))}
(b) We define K:Konf →Kont as follows: K(k) = (id(k), ϑ), where ϑ(α) = [[k(α)]],∀α∈
Id.
Definition 5.3
(a) We say that two closed configurations k1, k2∈Konf are isomorphic, and we write
k1∼
=k2, iff there exists a bijection µ:id(k1)→id(k2)such that:
(i) µ(α′)≤µ(α′′)⇔α′≤α′′,∀α′, α′′ ∈id(k1)
(ii) k2(µ(α)) = k1(α),∀α∈id(k1)
(b) We say that two open configurations (α1,k1),(α2,k2)∈Conf are isomorphic, and write
(α1, k1)∼
=(α2, k2)iff there exists a bijection µ: ({α1} ∪ id(k1)) →({α2} ∪ id(k2)) such
that:
(i) µ(α1) = α2
(ii) µ(α′)≤µ(α′′)⇔α′≤α′′ ,∀α′
, α′′ ∈ {α1} ∪ id(k1)
(iii) k2(µ(α)) = k1(α),∀α∈id(k1)
Obviously, ∀(α, k)∈Conf :(α, k)∼
=(α, k) and if (α1, k1),(α2, k2)∈C onf then (α1
, k1)∼
=(α2
, k2)
⇒k1∼
=k2. Also, the following Lemma is given without proof (which is simple enough and
can be used by the reader as proof exercise).
Lemma 5.4 For all k, k1, k2∈Konf
(a) For any α∈I d, s ∈Stat :K[k|α7→ s] = [ K(k)|α7→ [[s]] ]
(b) id(k) = id(K(k))
(c) If k1∼
=k2then id(k1) = ∅ ⇔ id(k2) = ∅
(d) K(k)\α=K(k\α), for any α∈Id.
(e) If α /∈id(k)then ([ k|α7→ s]\α)∼
=k.
(f) For any s∈Stat and α, α′∈I d, α′6=α:[k|α7→ s]\α′= [ k\α′|α7→ s].
In Corollary 5.6 we show that any two continuations that correspond to isomorphic
configurations behave the same. This result is obtained by combining Lemma 5.5 with
an argument ’ε≤1
2·ε⇒ε= 0’. Lemma 5.5 identifies the property - in this case the
isomorphism between configurations - that is preserved by each computation step. The
effect of each computation step is given in the present metric setting by the 1
2-contracting
factor.
Continuation Semantics for Asynchronous Concurrency 15
Lemma 5.5
(a) For all k1, k2∈Konf with k1∼
=k2and σ∈Σ, there exists s∈Stat,(α1, k1),(α2, k2)∈
Conf with (α1, k1)∼
=(α2, k2)such that:
d(kc(K(k1))(σ), kc(K(k2))(σ)) ≤
d([[s]](α1, K(k1))(σ),[[s]](α2, K (k2))(σ))
(b) For all s∈Stat, (α1, k1),(α2, k2)∈Conf with (α1, k1)∼
=(α2, k2)and σ∈Σ, there
exists s∈Stat, (α1, k1),(α2, k2)∈Conf with (α1, k1)∼
=(α2, k2)and σ∈Σsuch that:
d([[s]](α1, K(k1))(σ),[[s]](α2, K (k2))(σ)) ≤
1
2·d([[s]](α1, K(k1))(σ),[[s]](α2, K (k2))(σ))
Proof: For 5.5(a) we distinguish two subcases. If (by Lemma 5.4(b) and (c)) id(K(k1)) =
∅=id(K(k1)) then
d(kc(K(k1))(σ), kc(K(k2))(σ)) = d({λ},{λ}) = 0.
Otherwise,
d(kc(K(k1))(σ), kc(K(k2))(σ))
=d(+α∈max(id(K(k1)))K(k1)(α)(α, K (k1)\α)(σ),
+α∈max(id(K(k2)))K(k2)(α)(α, K (k2)\α)(σ)) [Lemma 5.4(b)]
=d(+α∈max(id(k1))K(k1)(α)(α, K (k1)\α)(σ),
+α∈max(id(k2))K(k2)(µ(α))(µ(α), K(k2)\µ(α))(σ))
[+ is nonexpansive]
=max{d(K(k1)(α)(α, K(k1)\α)(σ), K(k2)(µ(α))(µ(α), K(k2)\µ(α))(σ))
|α∈max(id(k1))}[Lemma 5.4(d)]
=max{d([[s]](α, K(k1\α))(σ),[[s]](µ(α), K(k2\µ(α)))(σ))
|α∈max(id(k1)), s =k1(α) = k2(µ(α))}
where µ:id(k1)→id(k2) is a bijection that satisfies the properties given in Definition 5.3(a).
As k1∼
=k2, (α, k1\α)∼
=(µ(α), k2\µ(α)) ∈Conf, for any α∈max(id(k1)). Therefore
∃s∈Stat, α ∈Id,5such that s=k1(α) = k2(µ(α)), (α, k1\α)∼
=(µ(α), k2\µ(α)) and
d(kc(K(k1))(σ), kc(K(k2))(σ))
5α∈max(id(k1)).
Continuation Semantics for Asynchronous Concurrency 16
≤d([[s]](α, K(k1\α))(σ),[[s]](µ(α), K(k2\µ(α)))(σ))
which concludes the proof of 5.5(a).
Next, we treat 5.5(b) by induction on ς(s). Four subcases.
Case [s=a] when I(a)(σ) =↑.
d([[a]](α1, K(k1))(σ),[[a]](α2, K (k2))(σ))
=d({δ},{δ}) = 0
Case [s=a] when I(a)(σ) = σ∈Σ.
d([[a]](α1, K(k1))(σ),[[a]](α2, K (k2))(σ))
=d(σ·kc(K(k1))(σ), σ ·kc(K(k2))(σ))
=1
2·d(kc(K(k1))(σ), kc(K(k2))(σ)) (5.5.b.1)
By Lemma 5.5(a), ∃s∈Stat, (α1, k1)∼
=(α2, k2)∈Conf such that:
(5.5.b.1) ≤1
2·d([[s]](α1, K(k1))(σ),[[s]](α2, K (k2))(σ))
Case [s=x]
d([[x]](α1, K(k1))(σ),[[x]](α2, K (k2))(σ))
=d([[D(x)]](α1, K (k1))(σ),[[D(x)]](α2, K(k2))(σ)) (5.5.b.2)
One can use the induction hypothesis (ς(D(x)) < ς(x)) and infer that ∃s∈Stat,
(α1, k1)∼
=(α2, k2)∈Conf, such that:
(5.5.b.2) ≤1
2·d([[s]](α1, K(k1))(σ),[[s]](α2, K (k2))(σ))
Case [s=s1⌊⌊ s2]
d([[s1⌊⌊ s2]](α1, K(k1))(σ),[[s1⌊⌊ s2]](α2, K(k2))(σ))
=d([[s1]](α1·1,[K(k1)|α1·27→ [[s2]] ])(σ),
[[s1]](α2·1,[K(k2)|α2·27→ [[s2]] ])(σ)) [Lemma 5.4(a)]
=d([[s1]](α1·1, K[k1|α1·27→ s2])(σ),
[[s1]](α2·1, K[k2|α2·27→ s2])(σ)) (5.5.b.3)
It is easy to check that (α1, k1)∼
=(α2, k2) implies
(α1·1,[k1|α1·27→ s2]) ∼
=(α2·1,[k2|α2·27→ s2])
For example, if µ: ({α1} ∪ id(k1)) →({α2} ∪ id(k2)) is a bijection that satisfies
the properties given in Definition 5.3(b), we can prove the above isomorhism by
defining a bijection µ′as follows:6
6Notice that:
id([ k1|α1·27→ s2]) = {α1·2} ∪ id(k1)
id([ k2|α2·27→ s2]) = {α2·2} ∪ id(k2)
Continuation Semantics for Asynchronous Concurrency 17
µ′: ({α1·1, α1·2} ∪ id(k1)) →({α2·1, α2·2} ∪ id(k2))
µ′(α1·1) = α2·1
µ′(α1·2) = α2·2
µ′(α) = µ(α), for any α∈id(k1)
Therefore, as ς(s1)< ς(s1⌊⌊ s2), we can use the induction hypothesis and we infer
that ∃s∈Stat, σ ∈Σ,(α1, k1)∼
=(α2, k2)∈Conf, such that
(5.5.b.3) ≤1
2·d([[s]](α1, K(k1))(σ),[[s]](α2, K (k2))(σ))
Case [s=s1ks2]
d([[s1ks2]](α1, K(k1))(σ),[[s1ks2]](α2, K (k2))(σ))
=d([[s1]](α1·1,[K(k1)|α1·27→ [[s2]] ])(σ) +
[[s2]](α1·1,[K(k1)|α1·27→ [[s1]] ])(σ),
[[s1]](α2·1,[K(k2)|α2·27→ [[s2]] ])(σ) +
[[s2]](α2·1,[K(k2)|α2·27→ [[s1]] ])(σ))
[Lemma 5.4(a), + is nonexpansive]
≤max{d([[s1]](α1·1, K[k1|α1·27→ s2])(σ),
[[s1]](α2·1, K[k2|α2·27→ s2])(σ)) (5.5.b.4) ,
d([[s2]](α1·1, K[k1|α1·27→ s1])(σ),
[[s2]](α2·1, K[k2|α2·27→ s1])(σ)) (5.5.b.5) }
It is easy to check that (α1, k1)∼
=(α2, k2) implies
(α1·1,[k1|α1·27→ s2]) ∼
=(α2·1,[k2|α2·27→ s2])
(α1·1,[k1|α1·27→ s1]) ∼
=(α2·1,[k2|α2·27→ s1])
For the first isomorphism we can define a bijection exactly in the same way as we
did for the subcase [s=s1⌊⌊ s2]. As ς(s1)< ς(s1ks2), we can use the induction
hypothesis for (5.5.b.4) and we infer that ∃s∈Stat, (α1, k1)∼
=(α2, k2)∈Conf
and σ∈Σ such that:
(5.5.b.4) ≤1
2·d([[s]](α1, K(k1))(σ),[[s]](α2, K (k2))(σ))
(5.5.b.5) can be handled in a similar manner and the desired result follows imme-
diately.
Corollary 5.6
(a) For all s∈Stat, (α1, k1)∼
=(α2, k2) (∈Conf):[[s]](α1, K(k1))=[[s]](α2, K(k2)).
(b) For all k1∼
=k2(∈Konf):kc(K(k1)) = kc(K(k2)).
Continuation Semantics for Asynchronous Concurrency 18
Proof: Let
(w∈)W={(s, (α1, k1),(α2, k2), σ)
|s∈Stat, (α1, k1),(α2, k2)∈Conf : (α1, k1)∼
=(α2, k2), σ ∈Σ}
For (s, (α1, k1),(α2, k2), σ)∈Wwe use the notation:
εI(s, (α1, k1),(α2, k2), σ)not.
=d([[s]](α1, K(k1))(σ),[[s]](α2, K (k2))(σ))
Let (s, (α1, k1),(α2, k2), σ)∈W. By 5.5(b) there exists (s, (α1, k1),(α2, k2), σ)∈Wsuch
that:
εI(s, (α1, k1),(α2, k2), σ)≤1
2·εI(s, (α1, k1),(α2, k2), σ)
and thus supw∈WεI(w)≤1
2·supw∈WεI(w), where w= (s, (α1, k1),(α2, k2), σ) and w=
(s, (α1, k1),(α2, k2), σ). This means that we have supw∈WεI(w) = 0, i.e.
d([[s]](α1, K(k1))(σ),[[s]](α2, K(k2))(σ)) = 0 and thus [[s]](α1, K(k1))(σ) = [[s]](α2, K(k2))(σ)
for any σ∈Σ, which implies Corollary 5.6(a). Corollary 5.6(b) follows immediately from
Lemma 5.5(a) and Corollary 5.6(a).
5.2 Continuations and syntactic contexts
We show that in the CSC approach continuations can be used to reason in a compositional
manner upon the behavior of concurrent programs. For this purpose we introduce a notion
of syntactic context for the class of Lstatements.
Definition 5.7 (Contexts for L)
C::= (·)|a|x|C;C|C+C|C⌊⌊ C|CkC
We denote by C(s)the result of substituting sfor all occurences of (·)in C. Formally, this
substitution can be defined inductively: (·)(s) = s, a(s) = a, x(s) = xand (C1op C2)(s) =
C1(s)op C2(s), where op ∈ {;,+,⌊⌊ ,k}.
Lemma 5.8 shows that program properties are preserved in any syntactic context by
all CSC continuations containing only denotations of statements. The proof relies on an
auxiliarry Lemma 5.9 which involves again the identification of an appropriate computing
invariant and the use of contraction ε≤1
2·ε⇒ε= 0.
Lemma 5.8 If s1, s2∈Stat are such that for all (α, k)∈C onf :
[[s1]](α, K(k)) = [[s2]](α, K(k))
then for all (α, k)∈Conf and for all contexts C:
[[C(s1)]](α, K (k)) = [[C(s2)]](α, K(k))
Continuation Semantics for Asynchronous Concurrency 19
Proof: By structural induction on C. Cases [C=a] and [C=x] are trivial. The
case [C= (·)] follows by the assumption. A non-trivial case is the context for parallel
composition [C=C1kC2]. Let (α, k)∈Conf and σ∈Σ. We have to prove that
[[(C1kC2)(s1)]](α, K (k))(σ) = [[(C1kC2)(s2)]](α, K(k))(σ). We compute as follows:
[[(C1kC2)(s1)]](α, K (k))(σ) = [[C1(s1)kC2(s1)]](α, K(k))(σ)
= [[C1(s1)]](α·1,[K(k)|α·27→ [[C2(s1)]] ])(σ)(5.8.1) +
[[C2(s1)]](α·1,[K(k)|α·27→ [[C1(s1)]] ])(σ)(5.8.2)
We handle (5.8.1) first.
(5.8.1) = [[C1(s1)]](α·1,[K(k)|α·27→ [[C2(s1)]] ])(σ) [Lemma 5.4(a)]
= [[C1(s1)]](α·1, K [k|α·27→ (C2(s1)) ])(σ) [Ind. hyp.]
= [[C1(s2)]](α·1, K [k|α·27→ (C2(s1)) ])(σ) [Lemma 5.4(a)]
= [[C1(s2)]](α·1,[K(k)|α·27→ [[C2(s1)]] ])(σ)
(α, k)∈Conf implies (α·1, k)∈Conf . By the induction hypothesis:
[[C2(s1)]](α·1, K (k))(σ) = [[C2(s2)]](α·1, K(k))(σ)
Hence, by Lemma 5.9(b) we have:
[[C1(s2)]](α·1,[K(k)|α·27→ [[C2(s1)]] ])(σ)
= [[C1(s2)]](α·1,[K(k)|α·27→ [[C2(s2)]] ])(σ)
Similarly:
(5.8.2) = [[C2(s1)]](α·1,[K(k)|α·27→ [[C1(s1)]] ])(σ)
= [[C2(s2)]](α·1,[K(k)|α·27→ [[C1(s2)]] ])(σ)
Therefore:
[[(C1kC2)(s1)]](α, K (k))(σ) = (5.8.1) +(5.8.2)
= [[C1(s2)]](α·1,[K(k)|α·27→ [[C2(s2)]] ])(σ)+
[[C2(s2)]](α·1,[K(k)|α·27→ [[C1(s2)]] ])(σ)
= [[C1(s2)kC2(s2)]](α, K (k))(σ) = [[(C1kC2)(s2)]](α, K(k))(σ)
Lemma 5.9 If s1, s2∈Stat are such that [[s1]](α, K(k))(σ) = [[s2]](α, K(k))(σ)for all
(α, k)∈C onf, σ ∈Σthen
Continuation Semantics for Asynchronous Concurrency 20
(a) for all k∈K onf, α∈Id, σ ∈Σ
kc[K(k)|α7→ [[s1]] ](σ) = kc[K(k)|α7→ [[s2]] ](σ)
(b) for all s∈S tat, (α, k)∈Conf , σ ∈Σ,α∈I d, ¬(α≥α):
[[s]](α, [K(k)|α7→ [[s1]] ])(σ) = [[s]](α, [K(k)|α7→ [[s2]] ])(σ)
Proof: For (a) we show that ∀k∈K onf, σ ∈Σ,α∈Id,∃s′∈Stat, (α′, k′)∈Conf such
that ¬(α≥α′) and:
d(kc[K(k)|α7→ [[s1]] ](σ), kc[K(k)|α7→ [[s2]] ](σ))
≤d([[s′]](α′,[K(k′)|α7→ [[s1]] ])(σ),[[s′]](α′,[K(k′)|α7→ [[s2]] ])(σ))
not.
=εc(s′,(α′, k′),α, s1, s2, σ)(5.9.1)
For (b) we show that ∀s∈Stat, (α, k)∈Conf , α∈Id, σ ∈Σ with ¬(α≥α),∃s′∈
Stat, (α′, k′)∈Conf, σ′∈Σ such that ¬(α≥α′) and:7
εc(s, (α, k),α, s1, s2, σ)
=d([[s]](α, [K(k)|α7→ [[s1]] ])(σ),[[s]](α, [K(k)|α7→ [[s2]] ])(σ))
≤1
2·d([[s′]](α′,[K(k′)|α7→ [[s1]] ])(σ′),[[s′]](α′,[K(k′)|α7→ [[s2]] ])(σ′))
=1
2·εc(s′,(α′, k′),α, s1, s2, σ′)(5.9.2)
The invariant property that is preserved by the computation steps is given here by the
condition that (α, k)∈C onf and ¬(α≥α).
Let
(w∈)W={(s, (α, k),α, s1, s2, σ)
|s, s1, s2∈Stat, (α, k)∈Conf, α∈Id, ¬(α≥α), σ ∈Σ}
We infer that
supw∈Wεc(w)≤1
2·supw′∈Wεc(w′)
where w= (s, (α, k),α, s1, s2, σ), w′= (s′,(α′, k′), α, s1, s2, σ′). But
supw∈Wεc(w) = supw′∈Wεc(w′)
Therefore we infer supw∈Wεc(w) = 0, which implies Lemma 5.9(b). Next, by using this
result and (5.9.1) we obtain immediately Lemma 5.9(a). In the sequel we prove (5.9.1) and
(5.9.2) .
First, we handle (5.9.1) . Let κ1, κ2∈Kont,k1, k2∈Kont
7¬(α≥α) is the invariant property that is preserved by the computation steps.
Continuation Semantics for Asynchronous Concurrency 21
κ1= [ K(k)|α7→ [[s1]] ]
κ2= [ K(k)|α7→ [[s2]] ]
k1= [ k|α7→ s1]
k2= [ k|α7→ s2]
By Lemma 5.4(a), κ1=K(k1), κ2=K(k2). Obviously, id(k1) = id(k2) = id(k)∪ {α}. Also,
by Lemma 5.4(b), id(κ1) = id(k1) = id(k2) = id(κ2) = id(k)∪ {α}. We distinguish two
subcases. If α /∈max(id(k)∪ {α}) we compute as follows:
d(kc(κ1)(σ), kc(κ2)(σ))
=d(+α∈max(id(k))κ1(α)(α, κ1\α)(σ),+α∈max(id(k)) κ2(α)(α, κ2\α)(σ))
[+ is nonexpansive]
=max{d(κ1(α)(α, κ1\α)(σ), κ2(α)(α, κ2\α)(σ))
|α∈max(id(k))}(5.9.3)
By Lemma 5.4(a) and 5.4(d)
κ1\α=K(k1)\α=K(k1\α) = K[k\α|α7→ s1] = [ K(k\α)|α7→ [[s1]] ]
κ2\α=K(k2)\α=K(k2\α) = K[k\α|α7→ s2] = [ K(k\α)|α7→ [[s2]] ]
Also, notice that for any α∈max(id(k)): κ1(α) = κ2(α) = [[k(α)]]. Therefore we have:
(5.9.3) =max{d([[k(α)]](α, [K(k\α)|α7→ [[s1]] ])(σ),
[[k(α)]](α, [K(k\α)|α7→ [[s2]] ])(σ))
|α∈max(id(k))}
Clearly, this means that ∃s′∈Stat, (α′, k′)∈Conf with ¬(α≥α′) such that:8
d(kc(κ1)(σ), kc(κ2)(σ))
≤d([[s′]](α′,[K(k′)|α7→ [[s1]] ])(σ),[[s′]](α′,[K(k′)|α7→ [[s2]] ])(σ))
Next, we treat the subcase α∈(max(id(k)) ∪ {α}). In this subcase:
d(kc(κ1)(σ), kc(κ2)(σ))
=d((+α∈max(id(k)),¬(α≥α)κ1(α)(α, κ1\α)(σ)) + [[s1]](α, κ1\α)(σ),
8More precisely, for some α∈max(id(k)), s′=k(α), α′=α, k′=k\α.
Continuation Semantics for Asynchronous Concurrency 22
(+α∈max(id(k)),¬(α≥α)κ2(α)(α, κ2\α)(σ)) + [[s2]](α, κ2\α)(σ))
[+ is nonexpansive, κ1=K(k1), κ2=K(k2), k1\α∼
=k∼
=k2\α,
Corollary 5.6(a)]
max{d([[s1]](α, K(k))(σ),[[s2]](α, K(k))(σ)),
max{d(κ1(α)(α, κ1\α)(σ), κ2(α)(α, κ2\α)(σ))
|α∈max(id(k)),¬(α≥α)}} (5.9.4)
By assumption [[s1]](α, K(k))(σ) = [[s2]](α, K(k))(σ). Also, for any α∈max(id(k)), ¬(α≥
α) : κ1(α) = κ2(α) = k(α), and κ1\α= [ K(k\α)|α7→ [[s1]] ], κ2\α= [ K(k\α)|α7→ [[s2]] ].
Therefore:
(5.9.4) =max{d([[k(α)]](α, [K(k\α)|α7→ [[s1]] ])(σ),
[[k(α)]](α, [K(k\α)|α7→ [[s2]] ])(σ))
|α∈max(id(k)),¬(α≥α)}
Clearly, this means that ∃s′∈Stat, (α′, k′)∈Conf with ¬(α≥α′) such that:
d(kc(κ1)(σ), kc(κ2)(σ))
≤d([[s′]](α′,[K(k′)|α7→ [[s1]] ])(σ),[[s′]](α′,[K(k′)|α7→ [[s2]] ])(σ))
This concludes the proof of 5.9(1).
In the sequel we prove (5.9.2) . More precisely, we show that ∀s∈Stat, (α, k)∈Conf, α∈
Id with ¬(α≥α) and σ∈Σ, ∃s′∈Stat, (α′, k′)∈Conf with ¬(α≥α′) and σ′∈Σ such
that:
εc(s, (α, k),α, s1, s2, σ)≤1
2·εc(s′,(α′, k′), α, s1, s2, σ′)
We proceed by induction on ς(s). Two subcases.
Case [s=a] when I(a)(σ) = σ′∈Σ.
εc(a, (α, k),α, s1, s2, σ)
=d([[a]](α, [K(k)|α7→ [[s1]] ])(σ),[[a]](α, [K(k)|α7→ [[s2]] ])(σ))
=d(σ′·kc([ K(k)|α7→ [[s1]] ])(σ′), σ′·kc([ K(k)|α7→ [[s2]] ])(σ′)) (5.9.5)
By (5.9.1) ∃s′∈Stat, (α′, k′)∈Conf with ¬(α≥α′) such that:
(5.9.5) ≤1
2·d([[s′]](α′,[K(k′)|α7→ [[s1]] ])(σ′),
[[s′]](α′,[K(k′)|α7→ [[s2]] ])(σ′))
Continuation Semantics for Asynchronous Concurrency 23
=1
2·εc(s′,(α′, k′),α, s1, s2, σ′)
Case [s=s1ks2]
εc(s1ks2,(α, k),α, s1, s2, σ)
=d([[s1ks2]](α, [K(k)|α7→ [[s1]] ])(σ),
[[s1ks2]](α, [K(k)|α7→ [[s2]] ])(σ))
=d([[s1]](α·1,[K(k)|α7→ [[s1]] |α·27→ [[s2]] ])(σ) +
[[s2]](α·1,[K(k)|α7→ [[s1]] |α·27→ [[s1]] ])(σ),
[[s1]](α·1,[K(k)|α7→ [[s2]] |α·27→ [[s2]] ])(σ) +
[[s2]](α·1,[K(k)|α7→ [[s2]] |α·27→ [[s1]] ])(σ))
[’+’ is nonexpansive; ¬(α≥α)⇒ ¬(α≥α·1),¬(α≥α·2);
Lemma 5.4(a)]
≤max{d([[s1]](α·1,[K[k|α·27→ s2]|α7→ [[s1]] ])(σ),
[[s1]](α·1,[K[k|α·27→ s2]|α7→ [[s2]] ])(σ)) (5.9.6)
d([[s2]](α·1,[K[k|α·27→ s1]|α7→ [[s1]] ])(σ),
[[s2]](α·1,[K[k|α·27→ s1]|α7→ [[s2]] ])(σ)) (5.9.7) }
As ¬(α≥α·1) and ¬(α≥α·2), we can apply the induction hypothesis and
we infer that ∃s′
1∈Stat, (α′
1, k′
1)∈Conf with ¬(α≥α′
1) and σ′
1∈Σ, and
∃s′
2∈Stat, (α′
2, k′
2)∈Conf with ¬(α≥α′
2) and σ′
2∈Σ , such that:
(5.9.6) ≤1
2·d([[s′
1]](α′
1,[K(k′
1)|α7→ [[s1]] ])(σ′
1),
[[s′
1]](α′
1,[K(k′
1)|α7→ [[s2]] ])(σ′
1)) (5.9.6′)
(5.9.7) ≤1
2·d([[s′
2]](α′
2,[K(k′
2)|α7→ [[s1]] ])(σ′
2),
[[s′
2]](α′
2,[K(k′
2)|α7→ [[s2]] ])(σ′
2)) (5.9.7′)
Finally, we obtain:
εc(s1ks2,(α, k),α, s1, s2, σ)
≤max{(5.9.6′),(5.9.7′)}
=max{1
2·εc(s′
1,(α′
1, k′
1), α, s1, s2, σ′
1),1
2·εc(s′
2,(α′
2, k′
2), α, s1, s2, σ′
2)}
This implies immediately the desired result.
5.3 Concurrency laws
This section concludes with Theorem 5.12, which presents the main results of the paper.
Theorem 5.12 allows us to reason in a compositional manner upon the behavior of Lasyn-
chronous programs. The denotational semantics [[·]] preserves the following invariant prop-
erty: continuations contain only computations denotable by program statements. The initial
Continuation Semantics for Asynchronous Concurrency 24
continuation (α0, κ0) (see Definition 4.1(c)) is empty (contains no computations) and each
equation in the definition of [[·]] adds only denotations of statements to the continuation.
The properties given in Theorem 5.12 hold for continuations containing only computations
denotable by program statements, which is sufficient in practice. The proof of Theorem 5.12
uses some auxiliary results given as Lemma 5.10 and Lemma 5.11. Essentially, Lemma 5.10
and Lemma 5.11 identify (non-isomorphic) continuation structures - specific of sequential
composition and parallel composition, respectively - that behave the same.
Lemma 5.10 For all ˜s, s1, s2∈Stat, σ ∈Σ,˜α, α ∈Id, k ∈Konf such that ( ˜α, k)∈
Conf, (α, k)∈Conf,( ˜α6=α),¬(˜α≤α)and ¬(α≤˜α)we have:
(a) kc[K(k)|α7→ [[s1ks2]] ](σ) = kc[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ](σ)
(b) [[˜s]]( ˜α, [K(k)|α7→ [[s1ks2]] ])(σ) =
[[˜s]]( ˜α, [K(k)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ)
Proof: We use the notation Q(˜α, α), for ˜α, α ∈I d, to express the fact that ˜αand αare
different9and incomparable with respect to ’≤’.
Q(˜α, α)not.
= (¬(˜α≤α)) ∧(¬(α≤˜α))
Also, we use the notation
Pk(˜α, α, k)not.
=Q(˜α, α)∧((˜α, k)∈Conf)∧((α, k)∈Conf )
Pkis the invariant property which is preserved by the computation steps.
We proceed as follows. For 5.10(a) we show that ∀s1, s2∈Stat, σ ∈Σ, α ∈I d, k ∈Konf
with (α, k)∈Conf ,∃s′∈Stat, α′∈Id, k′∈Konf with Pk(α′, α, k′) such that:
d(kc[K(k)|α7→ [[s1ks2]] ](σ), kc[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ](σ))
≤d([[s′]](α′,[K(k′)|α7→ [[s1ks2]] ])(σ),
[[s′]](α′,[K(k′)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ)) (5.10.1)
not.
=εk(s′, α′, k′, α, s1, s2, σ)
Also, for 5.10(b) we show that ∀˜s, s1, s2∈Stat, σ ∈Σ,˜α, α ∈I d, k ∈Konf such that Pk(˜α, α, k),
∃s′∈Stat, σ′∈Σ, α′∈Id, k′∈Konf with Pk(α′, α, k′) such that:
εk(˜s, ˜α, k, α, s1, s2, σ)
=d([[˜s]]( ˜α, [K(k)|α7→ [[s1ks2]] ])(σ),
[[˜s]]( ˜α, [K(k)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ))
9Notice that Q(˜α, α)⇒˜α6=α.
Continuation Semantics for Asynchronous Concurrency 25
≤1
2·d([[s′]](α′,[K(k′)|α7→ [[s1ks2]] ])(σ′),
[[s′]](α′,[K(k′)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ′)) (5.10.2)
not.
=εk(s′, α′, k′, α, s1, s2, σ′)
If we put (w∈)W=Stat ×Id ×Conf ×Id ×Stat ×Stat ×Σ we infer that:
sup w∈W:Pk( ˜α,α,k)εk(w)≤1
2·sup w′∈W:Pk(α′,α,k′)εk(w′)
where w= (˜s, ˜α, k, α, s1, s2, σ) and w′= (s′, α′, k′, α, s1, s2, σ′).
But obviously, sup w∈W:Pk( ˜α,α,k)εk(w) = sup w′∈W:Pk(α′,α,k′)εk(w′), so
sup w∈W:Pk( ˜α,α,k)εk(w) = 0
which implies immediately 5.10(b). Next, by using this result and (5.10.1) we obtain imme-
diately 5.10(a). In the sequel we prove (5.10.1) and (5.10.2) .
First, we prove (5.10.1) . Let s1, s2∈Stat, σ ∈Σ, α ∈I d, k ∈Konf with (α, k)∈Conf .
Let also κ1= [ K(k)|α7→ [[s1ks2]] ], κ2= [ K(k)|α·17→ [[s1]] |α·27→ [[s2]] ],
k1= [ k|α7→ (s1ks2) ] and k2= [ k|α·17→ s1|α·27→ s2]. By Lemma 5.4(a)
κ1=K(k1), κ2=K(k2). We compute as follows:
=d(kc(κ1)(σ), kc(κ2)(σ)) [Lemma 5.4(b)]
=d([[s1ks2]](α, [K(k)|α7→ [[s1ks2]] ] \α)(σ) +
(+α′∈max(id(k1)),α′6=ακ1(α′)(α′, κ1\α′)(σ)),
[[s1]](α·1,[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ] \α·1)(σ) +
[[s2]](α·2,[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ] \α·2)(σ) +
(+α′∈max(id(k1)),α′6=α·1,α′6=α·2κ2(α′)(α′, κ2\α′)(σ))) (5.10.3)
By Lemma 5.4(a) and 5.4(d)
[K(k)|α7→ [[s1ks2]] ] \α= (K[k|α7→ (s1ks2) ]) \α
=K([ k|α7→ (s1ks2) ] \α)
By Lemma 5.4(e), [ k|α7→ (s1ks2) ] \α∼
=k. Similarly,
[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ] \α·1
=K([ k|α·17→ s1|α·27→ s2]\α·1)
and [ k|α·17→ s1|α·27→ s2]\α·1∼
=[k|α·27→ s2]. Also
[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ] \α·2
Continuation Semantics for Asynchronous Concurrency 26
=K([ k|α·17→ s1|α·27→ s2]\α·2)
and [ k|α·17→ s1|α·27→ s2]\α·2∼
=[k|α·17→ s1]. In addition, it is easy to
check that (α·2,[k|α·17→ s1]) ∼
=(α·1,[k|α·27→ s1]). Therefore, by Corollary
5.6(a) and taking into account that the semantic operator + is nonexpansive we can
compute as follows:
(5.10.3) ≤max{d([[s1ks2]](α, K(k))(σ),
[[s1]](α·1,[K(k)|α·27→ [[s2]] ])(σ) +
[[s2]](α·1,[K(k)|α·27→ [[s1]] ])(σ)) (5.10.4) ,
d(+α′∈max(id(k1)),α′6=ακ1(α′)(α′, κ1\α′)(σ),
+α′∈max(id(k1)),α′6=α·1,α′6=α·2κ2(α′)(α′, κ2\α′)(σ)) (5.10.5) }
Obviously, (5.10.4) = 0 from the definition of the denotational mapping [[·]]. Next, we
treat (5.10.5) . As (α, k)∈Conf and α·1≥α, α ·2≥α, it is easy to see that:
max(id(k1)) = {α} ∪ {α′|α′∈max(id(k)), Q(α′, α)}
max(id(k2)) = {α·1, α ·2} ∪ {α′|α′∈max(id(k)), Q(α′, α)}
Therefore
(5.10.5) =d(+α′∈max(id(k)),Q(α′,α)κ1(α′)(α′, κ1\α′)(σ)
+α′∈max(id(k)),Q(α′,α)κ2(α′)(α′, κ2\α′)(σ)
[+ is nonexpansive ]
≤max{d(κ1(α′)(α′, κ1\α′)(σ), κ2(α′)(α′, κ2\α′)(σ))
|α′∈max(id(k)), Q(α′, α)}
This means that we have:
d(kc(κ1)(σ), kc(κ2)(σ))≤max{(5.10.4) ,(5.10.5) }=max{0,(5.10.5) }
≤max{d(κ1(α′)(α′, κ1\α′)(σ), κ2(α′)(α′, κ2\α′)(σ))
|α′∈max(id(k)), Q(α′, α)}(5.10.6)
We only treat the case when the above set (the argument of max in (5.10.6) ) is
nonempty. Recall that κ1= [ K(k)|α7→ [[s1ks2]] ] and κ2= [ K(k)|α·17→
[[s1]] |α·27→ [[s2]] ], so
(5.10.6) =max{d(K(k)(α′)(α′,[K(k)|α7→ [[s1ks2]] ] \α′)(σ)
Continuation Semantics for Asynchronous Concurrency 27
K(k)(α′)(α′,[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ] \α′)(σ))
|α′∈max(id(k)), Q(α′, α)}
[α′∈max(id(k)),(α, k)∈C onf, Q(α′, α)⇒
α′6=α, α′6=α·1, α′6=α·2, Lemma 5.4(a), 5.4(d) and 5.4(f)]
=max{d([[k(α′)]](α′,[K(k\α′)|α7→ [[s1ks2]] ])(σ),
[[k(α′)]](α′,[K(k\α′)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ))
|α′∈max(id(k)), Q(α′, α)}(5.10.7)
As (α, k)∈Conf , it is easy to check that ∀α′∈max(id(k)) with Q(α′, α) : Pk(α′, α, k \
α′). Therefore, by taking the maximal element of (5.10.7) we obtain immediately the
desired result (5.10.1) , i.e. we infer that ∃s′∈Stat, σ′∈Σ, α′∈Id, k′∈Konf(k′=
k\α′, s′=k(α′)) satisfying the invariant property Pk(α′, α, k′) and such that
d(kc[K(k)|α7→ [[s1ks2]] ](σ), kc[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ](σ))
≤d([[s′]](α′,[K(k′)|α7→ [[s1ks2]] ])(σ),
[[s′]](α′,[K(k′)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ))
which concludes the proof of (5.10.1) .
Next, we prove (5.10.2) . We proceed by induction on ς(˜s) using (5.10.1) . In the compu-
tations given below, by assumption Pk(˜α, α, k). Three subcases.
Case [˜s=a] when I(a)(σ) = σ′∈Σ.
d([[a]](˜α, [K(k)|α7→ [[s1ks2]] ])(σ),
[[a]](˜α, [K(k)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ))
=d(σ′·kc[K(k)|α7→ [[s1ks2]] ](σ′),
σ′·kc[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ](σ′))
=1
2·d(kc[K(k)|α7→ [[s1ks2]] ](σ′),
kc[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ](σ′)) (5.10.8)
As (α, k)∈Conf , by (5.10.1) ,∃s′∈S tat, α′∈Id, k′∈Konf such that Pk(α′, α, k′)
and:
(5.10.8) ≤1
2·d([[s′]](α′,[K(k′)|α7→ [[s1ks2]] ])(σ′),
[[s′]](α′,[K(k′)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ′))
Case [˜s=x].
Continuation Semantics for Asynchronous Concurrency 28
d([[x]](˜α, [K(k)|α7→ [[s1ks2]] ])(σ),
[[x]](˜α, [K(k)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ))
=d([[D(x)]]( ˜α, [K(k)|α7→ [[s1ks2]] ])(σ),
[[D(x)]]( ˜α, [K(k)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ)) (5.10.9)
By the induction hypothesis (ς(D(x)) < ς(x)) ∃s′∈Stat, σ′∈Σ, α′∈Id, k′∈
Konf such that Pk(α′, α, k′) and:
(5.10.9) ≤1
2·d([[s′]](α′,[K(k′)|α7→ [[s1ks2]] ])(σ′),
[[s′]](α′,[K(k′)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ′))
Case [˜s= ˜s1k˜s2].
d([[˜s1k˜s2]](˜α, [K(k)|α7→ [[s1ks2]] ])(σ),
[[˜s1k˜s2]](˜α, [K(k)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ))
=d([[˜s1]]( ˜α·1,[K(k)|α7→ [[s1ks2]] |˜α·27→ [[˜s2]] ])(σ) +
[[˜s2]]( ˜α·1,[K(k)|α7→ [[s1ks2]] |˜α·27→ [[˜s1]] ])(σ),
[[˜s1]]( ˜α·1,
[K(k)|α·17→ [[s1]] |α·27→ [[s2]] |˜α·27→ [[˜s2]] ])(σ) +
[[˜s2]]( ˜α·1,
[K(k)|α·17→ [[s1]] |α·27→ [[s2]] |˜α·27→ [[˜s1]] ])(σ))
[ + is nonexpansive ]
≤max{d([[˜s1]]( ˜α·1,[K(k)|α7→ [[s1ks2]] |˜α·27→ [[˜s2]] ])(σ),
[[˜s1]]( ˜α·1,
[K(k)|α·17→ [[s1]] |α·27→ [[s2]] |˜α·27→ [[˜s2]] ])(σ)),
d([[˜s2]]( ˜α·1,[K(k)|α7→ [[s1ks2]] |˜α·27→ [[˜s1]] ])(σ),
[[˜s2]]( ˜α·1,
[K(k)|α·17→ [[s1]] |α·27→ [[s2]] |˜α·27→ [[˜s1]] ])(σ))}
[Pk(˜α, α, k)⇒˜α·1,˜α·2/∈ {α, α ·1, α ·2}; Lemma 5.4(a)]
=max{d([[˜s1]]( ˜α·1,[K[k|˜α·27→ ˜s2]|α7→ [[s1ks2]] ])(σ),
[[˜s1]]( ˜α·1,
[K[k|˜α·27→ ˜s2]|α·17→ [[s1]] |α·27→ [[s2]] ])(σ)),(5.10.10)
d([[˜s2]]( ˜α·1,[K[k|˜α·27→ ˜s1]|α7→ [[s1ks2]] ])(σ),
[[˜s2]]( ˜α·1,
[K[k|˜α·27→ ˜s1]|α·17→ [[s1]] |α·27→ [[s2]] ])(σ)) (5.10.11) }
Continuation Semantics for Asynchronous Concurrency 29
Pk(˜α, α, k) implies Pk(˜α·1, α, [k|˜α·27→ ˜si]) for i= 1,2. Therefore, we can
use the induction hypothesis (because ς(˜si)< ς(˜s1k˜s2) for i= 1,2) and we infer
that (1) ∃s′
1∈Stat, σ′
1∈Σ, α′
1∈Id, k′
1∈Konf such that Pk(α′
1, α, k′
1) and (2)
∃s′
2∈Stat, σ′
2∈Σ, α′
2∈Id, k′
2∈Konf such that Pk(α′
2, α, k′
2) and:
(5.10.10) ≤1
2·d([[s′
1]](α′
1,[K(k′
1)|α7→ [[s1ks2]] ])(σ′
1),
[[s′
1]](α′
1,[K(k′
1)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ′
1)) (5.10.10′)
(5.10.11) ≤1
2·d([[s′
2]](α′
2,[K(k′
2)|α7→ [[s1ks2]] ])(σ′
2),
[[s′
2]](α′
2,[K(k′
2)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ′
2)) (5.10.11′)
Therefore, by taking max{(5.10.10′),(5.10.11′)}we obtain the desired result (5.10.2) .
Lemma 5.11 For all ˜s, s1, s2∈Stat, σ ∈Σ,˜α, α ∈Id, k ∈Konf such that ( ˜α, k)∈
Conf, α /∈id(k), α ·1/∈id(k)and (¬( ˜α≤α·1)) we have:
(a) kc[K(k)|α7→ [[s1;s2]] ](σ) = kc[K(k)|α·17→ [[s1]] |α7→ [[s2]] ](σ)
(b) [[˜s]]( ˜α, [K(k)|α7→ [[s1;s2]] ])(σ) =
[[˜s]]( ˜α, [K(k)|α·17→ [[s1]] |α7→ [[s2]] ])(σ)
Proof: The proof of this Lemma is similar to the proof of Lemma 5.10. In this case the
invariant property which is preserved by the computation steps is P;:
P;(˜α, α, k)not.
= ((˜α, k)∈Conf)∧(α /∈id(k)) ∧(α·1/∈id(k)) ∧(¬(˜α≤α·1))
We proceed as follows. For 5.11(a) we show that ∀s1, s2∈Stat, σ ∈Σ, α ∈I d, k ∈Konf
such that α /∈id(k) and α·1/∈id(k), ∃s′∈Stat, α′∈I d, k′∈Konf such that P;(α′, α, k′)
such that:
d(kc[K(k)|α7→ [[s1;s2]] ](σ), kc[K(k)|α·17→ [[s1]] |α7→ [[s2]] ](σ))
≤d([[s′]](α′,[K(k′)|α7→ [[s1;s2]] ])(σ),
[[s′]](α′,[K(k′)|α·17→ [[s1]] |α7→ [[s2]] ])(σ)) (5.11.1)
not.
=ε;(s′, α′, k′, α, s1, s2, σ)
Also, for 5.11(b) we show that ∀˜s, s1, s2∈Stat, σ ∈Σ,˜α, α ∈I d, k ∈Konf such that P;(˜α, α, k),
∃s′∈Stat, σ′∈Σ, α′∈Id, k′∈Konf such that P;(α′, α, k′) and:
ε;(˜s, ˜α, k, α, s1, s2, σ)
=d([[˜s]]( ˜α, [K(k)|α7→ [[s1;s2]] ])(σ),
[[˜s]]( ˜α, [K(k)|α·17→ [[s1]] |α7→ [[s2]] ])(σ))
Continuation Semantics for Asynchronous Concurrency 30
≤1
2·d([[s′]](α′,[K(k′)|α7→ [[s1;s2]] ])(σ′),
[[s′]](α′,[K(k′)|α·17→ [[s1]] |α7→ [[s2]] ])(σ′)) (5.11.2)
not.
=ε;(s′, α′, k′, α, s1, s2, σ′)
If we put (w∈)W=Stat ×Id ×Konf ×Id ×Stat ×Stat ×Σ we infer that:
sup w∈W:P;( ˜α,α,k)ε;(w)≤1
2·sup w′∈W:P;(α′,α,k′)ε;(w′)
where w= (˜s, ˜α, k, α, s1, s2, σ) and w′= (s′, α′, k′, α, s1, s2, σ′).
But obviously, sup w∈W:P;( ˜α,α,k)ε;(w) = sup w′∈W:P;(α′,α,k′)ε;(w′). So we infer that:
sup w∈W:P;( ˜α,α,k)ε;(w) = 0
which implies immediately 5.11(b). Next, by using this result and (5.11.1) we obtain imme-
diately 5.11(a).
The proofs of Lemma 5.11 and Lemma 5.10 are very similar. The main difference is
given by the computing invariants P;and Pk, which are specific of sequential and parallel
composition, respectively. Here we skip the proof of (5.11.1) (which is similar to the proof of
(5.10.1) ). We only give the proof of (5.11.2) . We proceed by induction on ς(˜s), using (5.11.1) .
In the computations given below it is assumed that P;( ˜α, α, k). We treat two subcases.
Case [˜s=a] when I(a)(σ) = σ′∈Σ.
d([[a]](˜α, [K(k)|α7→ [[s1;s2]] ])(σ),
[[a]](˜α, [K(k)|α·17→ [[s1]] |α7→ [[s2]] ])(σ))
=d(σ′·kc[K(k)|α7→ [[s1;s2]] ](σ′),
σ′·kc[K(k)|α·17→ [[s1]] |α7→ [[s2]] ](σ′))
=1
2·d(kc[K(k)|α7→ [[s1;s2]] ](σ′),
kc[K(k)|α·17→ [[s1]] |α7→ [[s2]] ](σ′)) (5.11.3)
By (5.11.1) ∃s′∈Stat, α′∈I d, k′∈Konf such that P;(α′, α, k′) and:
(5.11.3) ≤1
2·d([[s′]](α′,[K(k′)|α7→ [[s1;s2]] ])(σ′),
[[s′]](α′,[K(k′)|α·17→ [[s1]] |α7→ [[s2]] ])(σ′))
Case [˜s= ˜s1; ˜s2].
d([[˜s1; ˜s2]](˜α, [K(k)|α7→ [[s1;s2]] ])(σ),
[[˜s1; ˜s2]](˜α, [K(k)|α·17→ [[s1]] |α7→ [[s2]] ])(σ))
=d([[˜s1]]( ˜α·1,[K(k)|α7→ [[s1;s2]] |˜α7→ [[˜s2]] ])(σ),
[[˜s1]]( ˜α·1,[K(k)|α·17→ [[s1]] |α7→ [[s2]] |˜α7→ [[˜s2]] ])(σ))
Continuation Semantics for Asynchronous Concurrency 31
[P;(˜α, α, k)⇒˜α6=α, ˜α6=α·1; Lemma 5.4(a) ]
=d([[˜s1]]( ˜α·1,[K[k|˜α7→ ˜s2]|α7→ [[s1;s2]] ])(σ),
[[˜s1]]( ˜α·1,[K[k|˜α7→ ˜s2]|α·17→ [[s1]] |α7→ [[s2]] ])(σ)) (5.11.4)
P;(˜α, α, k) implies P;( ˜α·1, α, [k|˜α7→ ˜s2]). Therefore, we can use the induction
hypothesis (ς(˜s1)< ς (˜s1; ˜s2)) and we infer that ∃s′∈Stat, σ′∈Σ, α′∈Id, k′∈
Konf such that P;(α′, α, k′) and
(5.11.4) ≤1
2·d([[s′]](α′,[K(k′)|α7→ [[s1;s2]] ])(σ′),
[[s′]](α′,[K(k′)|α·17→ [[s1]] |α7→ [[s2]] ])(σ′))
In the sequel we use the notation
s≃s(s, s ∈Stat)
to express that [[C(s)]](α, K(k)) = [[C(s)]](α, K(k)) for all contexts Cand for all isomorphic
configurations (α, k)∼
=(α, k) (∈Conf ).
Theorem 5.12 For all s, s1, s2, s3∈Stat :
(a) s1+s2≃s2+s1(commutativity of +)
(b) (s1+s2) + s3≃s1+ (s2+s3) (associativity of +)
(c) s+s≃s(idempotency of +)
(d) (s1+s2); s3≃s1;s3+s2;s3(right distributivity of ; over +)
(e) s1; (s2;s3)≃(s1;s2); s3(associativity of ; )
(f) s+δ≃s
(g) δ;s≃δ
(h) s1ks2≃s1⌊⌊ s2+s2⌊⌊ s1
(i) a⌊⌊ s≃a;s
(j) (a;s1)⌊⌊ s2≃a; (s1ks2)
(k) (s1+s2)⌊⌊ s3≃s1⌊⌊ s3+s2⌊⌊ s3(right distributivity of ⌊⌊ over +)
(l) s1ks2≃s2ks1(commutativity of k)
(m) s1k(s2ks3)≃(s1ks2)ks3(associativity of k)
;binds stronger than k,⌊⌊ . Also, k,⌊⌊ bind stronger than +.
Proof: First notice that [[s]] = [[s]] ⇒s≃s, for any s, s ∈Stat. Indeed, [[s]] = [[s]]
⇒[[C(s)]] = [[C(s)]] for any context C(by the compositionality of [[·]]) and [[C(s)]] = [[C(s)]] ⇒s≃s,
by Corolarry 5.6(a). Therefore, the properties stated by Theorem 5.12 (a)-(d), (f)-(h), (k)
and (l) follow immediately by Lemma 5.1.
On the other hand, the properties stated by Theorem 5.12(e), (i), (j) and (m) can be
proved for continuations containing only denotations of statements (not for arbitrary con-
tinuations) and require more involved arguments based on the identification of computing
invariants and the use of contraction. By Lemma 5.8 and Corolarry 5.6(a), in order to prove
Continuation Semantics for Asynchronous Concurrency 32
s≃sit is enough to show that [[s]](α, K(k)) = [[s]](α, K(k)) for any (α, k)∈Conf (or that
[[s]](α, K(k))(σ) = [[s]](α, K(k))(σ) for any (α, k)∈Conf and an arbitrary σ∈Σ).
Property 5.12(e) follows by using Lemma 5.11.
[[s1; (s2;s3)]](α, K (k))(σ)=[[s1]](α·1,[K(k)|α7→ [[s2;s3]] ])(σ)
[Lemma 5.4(a)]
= [[s1]](α·1, K[k|α7→ (s2;s3) ])(σ)(5.12.1)
It is easy to check that (α·1,[k|α7→ (s2;s3) ]) ∼
=(α·1·1,[k|α7→ (s2;s3) ]). Thus (5.12.1) =
[[s1]](α·1·1, K[k|α7→ (s2;s3) ])(σ), by Corollary 5.6(a), and
[[s1]](α·1·1, K[k|α7→ (s2;s3) ])(σ) [Lemma 5.4(a)]
= [[s1]](α·1·1,[K(k)|α7→ [[s2;s3]] ])(σ) [Lemma 5.11(b)]
= [[s1]](α·1·1,[K(k)|α·17→ [[s2]] |α7→ [[s3]] ])(σ)
= [[s1;s2]](α·1,[K(k)|α7→ [[s3]] ])(σ)
= [[(s1;s2); s3]](α, K(k))(σ)
Property 5.12(i) is an easy consequence of Corollary 5.6. We only treat the subcase when
I(a)(σ) = σ∈Σ.
[[a⌊⌊ s]](α, K(k))(σ)
= [[a]](α·1,[K(k)|α·27→ [[s]] ])(σ)
=σ·kc[K(k)|α·27→ [[s]] ](σ) [Lemma 5.4(a)]
=σ·kc(K[k|α·27→ s])(σ)
[ (α, k)∈Conf ⇒[k|α·27→ s]∼
=[k|α7→ s], Corollary 5.6(b)]
=σ·kc(K[k|α7→ s])(σ) [Lemma 5.4(a)]
=σ·kc[K(k)|α7→ [[s]] ](σ)
= [[a]](α·1,[K(k)|α7→ [[s]] ])(σ)
= [[a;s]](α, K(k))(σ)
In the proofs of 5.12(j) and (m) one can use Lemma 5.10. For 5.12(j) we only consider
the subcase when I(a)(σ) = σ∈Σ.
[[(a;s1)⌊⌊ s2]](α, K(k))(σ)
Continuation Semantics for Asynchronous Concurrency 33
= [[a;s1]](α·1,[K(k)|α·27→ [[s2]] ])(σ)
= [[a]](α·1·1,[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ)
=σ·kc[K(k)|α·17→ [[s1]] |α·27→ [[s2]] ])(σ) [Lemma 5.10(a)]
=σ·kc[K(k)|α7→ [[s1ks2]] ](σ)
= [[a]](α·1,[K(k)|α7→ [[s1ks2]] ])(σ)
= [[a; (s1ks2)]](α, K (k))(σ)
Next we prove the property stated by Theorem 5.12(m). First, we expand the expressions
involved as follows:
[[s1k(s2ks3)]](α, K (k))(σ)
= [[s1]](α·1,[K(k)|α·27→ [[s2ks3]] ])(σ) +
[[s2ks3]](α·1,[K(k)|α·27→ [[s1]] ])(σ) [+ is associative]
= [[s1]](α·1,[K(k)|α·27→ [[s2ks3]] ])(σ)(5.12.2) +
[[s2]](α·1·1,[K(k)|α·27→ [[s1]] |α·1·27→ [[s3]] ])(σ)(5.12.3) +
[[s3]](α·1·1,[K(k)|α·27→ [[s1]] |α·1·27→ [[s2]] ])(σ)(5.12.4)
[[(s1ks2)ks3]](α, K(k))(σ)
= [[s1ks2]](α·1,[K(k)|α·27→ [[s3]] ])(σ) +
[[s3]](α·1,[K(k)|α·27→ [[s1ks2]] ])(σ)
= [[s1]](α·1·1,[K(k)|α·27→ [[s3]] |α·1·27→ [[s2]] ])(σ)(5.12.5) +
[[s2]](α·1·1,[K(k)|α·27→ [[s3]] |α·1·27→ [[s1]] ])(σ)(5.12.6) +
[[s3]](α·1,[K(k)|α·27→ [[s1ks2]] ])(σ)(5.12.7)
We show that (5.12.2) =(5.12.5) ,(5.12.3) =(5.12.6) and (5.12.4) =(5.12.7) . First we prove
that (5.12.3) =(5.12.6) . Indeed:
[[s2]](α·1·1,[K(k)|α·27→ [[s1]] |α·1·27→ [[s3]] ])(σ) [Lemma 5.4(a)]
= [[s2]](α·1·1, K[k|α·27→ s1|α·1·27→ s3])(σ)
[(α, k)∈Conf ⇒
(α·1·1,[k|α·27→ s1|α·1·27→ s3]) ∼
=
Continuation Semantics for Asynchronous Concurrency 34
(α·1·1,[k|α·27→ s3|α·1·27→ s1]), Corollary 5.6(a)]
= [[s2]](α·1·1, K[k|α·27→ s3|α·1·27→ s1])(σ)
= [[s2]](α·1·1,[K(k)|α·27→ [[s3]] |α·1·27→ [[s1]] ])(σ)
Both (5.12.2) =(5.12.5) and (5.12.4) =(5.12.7) can be handled by using Lemma 5.10(b).
As the proofs are similar, we only treat here (5.12.2) =(5.12.5) .
[[s1]](α·1,[K(k)|α·27→ [[s2ks3]] ])(σ) [Lemma 5.10(b)]
= [[s1]](α·1,[K(k)|α·2·17→ [[s2]] |α·2·27→ [[s3]] ])(σ) [Lemma 5.4(a)]
= [[s1]](α·1, K[k|α·2·17→ s2|α·2·27→ s3])(σ)
[(α, k)∈Conf ⇒
(α·1,[k|α·2·17→ s2|α·2·27→ s3]) ∼
=
(α·1·1,[k|α·27→ s3|α·1·27→ s2]), Corollary 5.6(a)]
= [[s1]](α·1·1, K[k|α·27→ s3|α·1·27→ s2])(σ) [Lemma 5.4(a)]
= [[s1]](α·1·1,[K(k)|α·27→ [[s3]] |α·1·27→ [[s2]] ])(σ)
Remark 5.13
(a) As it is well-known, the properties stated by Theorem 5.12 provide a finite axiomati-
zation for the parallel composition operator k; see, e.g., [5].10 For any non-recursive
Lprogram (closed term) s∈Stat there is a non-recursive program s∈Stat that
contains only the operators for sequential composition (;) and nondeterministic choice
(+) and such that the above set of laws imply s≃s. The operators kand ⌊⌊ can
be eliminated from any non-recursive asynchronous Lprogram. Such an elimination
can always be performed without manipulating continuations explicitly. For example
a1ka2≃a1⌊⌊ a2+a2⌊⌊ a1≃a1;a2+a2;a1.
(b) The semantics of sequential composition (;) is defined using continuations, but it ulti-
mately relies on the prefixing operation σ·p. Notice that if δ /∈pthen σ·(p+{δ}) =
σ·p6=σ·p+{σδ}=σ·p+σ· {δ}. Therefore we cannot expect to get a model which
also satisfies the law s1; (s2+s3)≃s1;s2+s1;s3, i.e. the left distributivity of ;over +.
Recall that D[[s]]σ= [[s]](α0, κ0)(σ) (see Definition 4.1(c)). For any a∈Act,s, s′∈Stat and
for any context Cone can easily check the following:
•s≃s′implies D[[C(s)]] = D[[C(s′)]],
10Strictly speaking, properties 5.12(l) and 5.12(m) are not needed for this purpose.
Continuation Semantics for Asynchronous Concurrency 35
• D[[a;s]]σ=if (I(a)(σ) =↑)then {δ}else I(a)(σ)· D[[s]](I(a)(σ))11 and
• D[[s+s′]]σ=D[[s]]σ+D[[s′]]σ.
For example, if I(a1)(σ) = σ1, I (a2)(σ1) = σ2and I(a2)(σ) =↑then
D[[a1ka2]]σ=D[[a1⌊⌊ a2+a2⌊⌊ a1]]σ=D[[a1;a2+a2;a1]]σ
=D[[a1;a2]]σ+D[[a2;a1]]σ={σ1σ2}+{δ}={σ1σ2}
6 Concluding remarks and future work
We presented a method of reasoning about the behavior of programs in denotational models
designed with metric spaces and continuation semantics for concurrency (CSC) [31]. The
method was illustrated on the particular case of a simple asynchronous language [10]. We
proved that the semantic operators designed with continuations obey concurrency laws such
as the associativity and commutativity of parallel composition. The method is general; we
think it could be applied to every language designed by using CSC. The method relies on
the identification of computing invariants as relations between continuation structures in
combination with arguments of the kind ’ε≤1
2·ε⇒ε= 0’, which are standard in metric
semantics. The significance of the results is given mainly by the flexibility provided by the
continuations technique which can thus be used to describe concurrent behaviour.
In previous work, we developed CSC-based denotational models for a couple of advanced
concepts, including nondeterministic promotion in Andorra-like languages [33], and syn-
chronous communication on multiple channels [32] in the style of Join calculus [17]. In
ongoing work, we investigate the possibility to develop a denotational semantics designed
with CSC for membrane computing [26]. As far as we know such advanced control concepts
have not been modeled denotationally until now without CSC. It should be possible to use
the proof method presented in this paper to reason compositionally about the behavior of
such advanced concepts. Still, this has to be verified.
In this paper the domain of continuations (Kont ={| 1
2·D|}) was modelled with the aid
of a function space from a set of identifiers (endowed with a partial order) to the domain
of computations: Id →1
2·D(see Section 3 where the construction {| · |} was introduced).
According to Corollary 5.6, any two isomorphic continuations behave the same. Intuitively,
the domain of continuations could be defined in terms of isomorphism classes [I d →1
2·D]
of such structures. Since the existing models based on isomorphism classes of semantic
structures (in particular the metric pomset model [6]) do not involve domains defined by
reflexive equations (like D), such a construction also requires further work.
11Notice that κ0=K(k0), where k0= (∅, t0), with t0∈(I d →Stat), t0(α) = δ,∀α∈Id. The property follows
by using Corollary 5.6 and the fact that [ k0|α′7→ s]\α′∼
=k0,∀α′∈Id. In fact [ k|α′7→ s]\α′∼
=k,
∀k∈Konf if α′/∈id(k).
Continuation Semantics for Asynchronous Concurrency 36
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