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Chapter 4: Actors

Programming Distributed Computing Systems: A Foundational Approach

Carlos Varela

Rensselaer Polytechnic Institute

Open Distributed Systems

• Addition of new components

• Replacement of existing components

• Changes in interconnections

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Actor Configurations

• Model open system components:

– Set of indivitually named actors

– Messages “en-route”

– Interface to environment:

• Receptionists

• External actors

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Synchronous vs. AsynchronousComminucation

• π-Calculus (and other process algebras such as CSS, CSP) take synchronous communication as a primitive

• Actors assume asynchronous communication is more primitive

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Communication Medium

• In π-Calculus, channels are explicitly modelled.Multiple processes can share a channel, potentially causing intereference.

• In the actor model, the communication medium is not explicit. Actors (active objects) are first-class, history-sensitive entities with an explicit identity used for communication.

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Fairness

• The actor model theory assumes fair computations:1. Message delivery is guaranteed.

2. Individual actor computations are guaranteed to progress.

Fairness is very useful for reasoning about equivalences of actor programs but can be hard/expensive to guarantee; in particular when distribution and failures are considered.

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Programming Languages Influenced by П-Calculus and

Actors• Scheme ‘75

• Act1 ‘87

• Acore ‘87

• Rosette ‘89

• Obliq ‘94

• Erlang ‘93

• ABCL ‘90

• SALSA ‘99

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• Amber ’86

• Facile ’89• CML ’91

• Pict ’94

• Nomadic Pict ’99• JOCAML ‘99

Actor (Agent) Model

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Agha, Mason, Smith & Talcott

1. Extend a functional language (λ-calculus + ifs and pairs) with actor primitives.

2. Define an operational semantics for actor configurations.

3. Study various notions of equivalence of actor expressions and configurations.

4. Assume fairness:– Guaranteed message delivery.

– Individual actor progress.

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λ-Calculus

Syntax

e ::= v value

| λv.e functional abstraction

| ( e e ) application

Example

( λx.x2 2 )

x2{2/x} (in π-calculus)

[2/x]x2 (in λ-calculus)

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λ-Calculus

• pr(x,y) returns a pair containing x & y

• ispr(x) returns t if x is a pair; f otherwise

• 1st(pr(x,y)) = x returns the first value of a pair

• 2nd(pr(x,y)) = y returns the second value of a pair

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Actor Primitives

• send(a,v)– Sends value v to actor a.

• new(b)– Creates a new actor with behavior b and returns the identity/name

of the newly created actor.

• ready(b)– Becomes ready to receive a new message with behavior b.

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Actor Language Example

b5 = rec(λy. λx.seq(send(x,5), ready(y)))

Receives an actor name x and sends the number 5 to that actor, then it becomes ready to process new messages with the same behavior y.

Sample usage:

send(new(b5), a)

A Sink:

sink = rec(λb. λm.ready(b))

An actor that disregards all messages.C. Varela 13

Reference Cell

Cell = rec(λb.λc.λm.

if ( get?(m),

seq( send(cust(m), c),

ready(b(c)))

if ( set?(m),

ready(b(contents(m))),

ready(b(c)))))

Using the cell:Let a = new(cell(0)) in seq( send(a, mkset(7)),

send(a, mkset(2)),

send(a, mkget(c)))

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Exercises

1. Write get?custset?contentsmksetmkget

to complete the reference cell example in the AMST actor language.

2. Modify Bcell to notify a customer when the cell value is updated (such as in the П-calculus cell example).

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Dining Philosophers

phil = rec(λb.λl.λr.λself.λsticks.λm.

if (eq?(sticks, 0),

ready(b(l,r,self,1)),

seq( send(l,mkrelease(self)),

send(r,mkrelease(self)),

send(l,mkpickup(self)),

send(r,mkpickup(self)),

ready(b(l,r,self,0)))))

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Dining Philosophers

chopstick = rec(λb.λh.λw.λm.

if(pickup?(m),

if(eq?(h,nil),

seq( send(getphil(m), nil),

ready(b(getphil(m),nil))),

ready(b(h,getphil(m))),

if(release?(m),

if(eq?(w,nil),

ready(b(nil, nil)),

seq( send(w,nil),

ready(b(w,nil)))),

ready(b(h,w)))))

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Dining Philosophers

letrec c1 = new(chopstick(nil,nil)),

c2 = new(chopstick(nil,nil)),

p1 = new(phil(c1,c2,p1,0)),

p2 = new(phil(c2,c1,p2,0))

in seq( send(c1,mkpickup(p1)),

send(c2,mkpickup(p1)),

send(c1,mkpickup(p2)),

send(c2,mkpickup(p2)))

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Dining Philosophers

Auxiliary definitions:

mkpickup = λp.p

mkrelease = nil

pickup? = λm.not(eq?(m,nil))

release? = λm.eq?(m,nil)

getphil = λm.m

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Actor Garbage Collection

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2

8

34

5

7

9

11

10

12

6

1

Blocked (idle)

Unblocked (non-idle)

Root

Join Continuations

Consider:

treeprod = rec(λf.λtree.

if(isnat(tree),

tree,

f(left(tree))*f(right(tree))

))

Which multiplies all leaves of a tree, which are numbers.

You can do the “left” and “right” computations concurrently.

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Tree Product Behavior

Btreeprod =

rec(λb.λself.λm.

seq(become(b(self)),

if(isnat(tree(m)),

send(cust(m),tree(m)),

letactor{newcust := Bjoincont

(cust(m),0,nil)}

seq(send(self,

pr(left(tree(m)),newcust)),

send(self,

pr(right(tree(m)),newcust)))

)

)

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Tree Product (continued)

Bjoincont =

rec(λb.λcust.λnargs.λfirstnum.λnum.

if(eq(nargs,0),

become(b(cust,1,num)),

seq(become(sink),

send(cust,firstnum*num))))

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Sample Execution

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Cust

f(tree,cust)

JC JC

Cust Cust JC

(a) (b)

Sample Execution

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Cust

JC’ JC’

JC JC

CustJC

firstnum

(c)

JC'

JCJC

firstnum

firstnum

JC'

Cust Cust

firstnum

JC

(d)

Sample Execution

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num

Cust

firstnum

Cust

JC

(e)

firstnum * num

Cust

(f)

Operational Semantics for Actor Configurations

k = << α | µ >>

α is a function mapping actor names (represented as variables) to actor states.

µ is a multi-set of messages “en-route.”

ρ is a set of receptionists.

χ is a set of external actors

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ρχ

Operational Semantics

Given A = Dom(α):

P is a subset of A, A ∩ χ is empty.

• If α(a) = (?a’), then a’ is in A.

• If a in A, then fv(α(a)) is a subset of A U χ.

• If <v0 in v1> in µ, then fv(v0,v1) is a subset of A U χ.

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Operational Semantics

α in X As

As = (?x) U (V) U [E]

µ in Mw [M]

M = <V <= V>

ρ,χ in Pw[X] <a <= 5>

V = At U X U λX.E U pr(V,V)

E = V U app(E,E) U Fn(En)

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├

Labeled Transition Relation

<fun: a>

e e'

<<α, [e]a | µ>> << α, [e'] | µ>>

<newactor: a, a’>

<<α, [R[newactor(e)]]a | µ>> <<α, [R[a']]a, [e]a' | µ>>

<send: a, v0, v1>

<<α, [R[send(v0, v1)]]a | µ>> << α, [R[nil]]a | µ<v0 <= v1> >>

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Dom(α)U{a}ρχ

ρχ

ρχ

ρχ

(where a’ is fresh)

ρχ

ρχ

Labeled Transition Relation

<receive: v0, v1>

<<α, [ready(v)]v0| <v0 <= v1>, µ>> <<α, [app(v,v1)]v0 | µ>>

<out: v0, v1>

<<α | µ, <v0 <= v1> >> <<α | µ>>

<in: v0, v1>

<< α | µ>> << α | µ, <v0 <= v1> >>

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ρχ

ρχ

ρχ

ρ'χ

If v0 is in χ and ρ'= ρ U (fv(v1) ∩ Dom(α))

ρχ

ρχ'

If v0 is in ρ and fv(v1) ∩ Dom(α) is a subset of ρ and χ' =χ U (fv(v1) – Dom(α)

Computation Sequences and Paths

• If k is a configuration, then the computation tree T(k) is the set of all finite sequences of labeled transitions

[ki ki+1 | i < n ]

for some n in the natural numbers with k = k0. Such sequences are called computation sequences.

• A computation path from k is a maximal linearly ordered set of computation sequences in the computation tree,T(k).

• T∞(k) denotes the set of all paths from k.

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li

A path π = [ki ki+1 | i < ] in the computation tree T∞(k) is fair if each enabled transition eventually happens or becomes permanently disabled.

For a configuration k we define F(k) to be the subset of T∞(k) that are fair.

Fairness

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li

Composition of Actor Configurations

k0 = <<α0 | µ0>>

k1 = <<α1 | µ1>>

k0 || k1 = <<α0 U α1 | µ0 U µ1>>

Configurations are “composable” if

Dom(α0) ∩ Dom(α1) is empty and

χ0 ∩ Dom(α1) is a subset of ρ1

χ1 ∩ Dom(α0) is a subset of ρ0

k0 = << 0 | 0>>

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ρ0

χ0ρ1

χ1

ρ0 U ρ1

(χ0 U χ1) – (ρ0 U ρ1)

00

Composition of Actor Configurations

(AC)

k0 || k1 = k1 || k0

k0 || k = k0

(k0 || k1) || k2 = k0 || (k1 || k2)

Closed Configurations: A configuration in which both the receptionist and external actor sets are empty is said to be closed

Composition preserves computations paths:

T(k0 || k1) = T(k0) || T(k1)

F(k0|| k1) = F(k0) || F(k1)

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0

Equivalence of Expressions

Operational equivalence

Testing equivalence

Two program expressions are said to be equivalent if they behave the same when placed in any observing context.

An observing context is a complete program with a hole, such that all free variables in expressions being evaluated become captured when placed in the hole.

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Observational equivalence

Events and Observing Contexts

A new event primitive operator is introduced.

The reduction relation is extended:

<e: a>

<<α, [R[event()]]a | µ>> <<α, [R[nil]]a | µ>>

An observing configuration is on of the form:

<< α, [C]a | µ>>

where C is a hole-containing expression or context.

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ρχ

ρχ

Observations

Let k be a configuration of the extended language and let π = [ki ki+1 | i < ] be a fair path, i.e., π is in F(k).

Define:

obs(π) =

obs(k) =

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li

S if there exists a, i < such that li = <e: a>F otherwise

S if, for all π in F(k), obs(π) = s SF otherwiseF if, for all π in F(k), obs(π) = s

Three Equivalences

The natural equivalence is equal are made in all closing configuration contexts.

Other two equivalences (weaker) arise if SF observations are considered as good as S observations; or if SF observations are considered as bad as F observations.

1. Testing or Convex or Plotkin or Egli-Milner

e0 =1 e1 iff Obs(O[e0]) = Obs(O[e1]).

1. Must or Upper or Smythe0 =2 e1 iff Obs(O[e0]) = S Obs(O[e1]) = S.

2. May or Lower or Hoaree0 =3 e1 iff Obs(O[e0]) = F Obs(O[e1]) = F.

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~

~

~

Congruence

e0 =j e1 c[e0] =j c[e1] for j = 1,2,3,…

By construction, all equivalences defined are congruencies.

Partial Collapse

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~ ~

=1 S SF FS

SF

F

X X

XX

X X

=2 S SF FS

SF

F

=3 S SF FS

SF

F

(1 = 2): e0 =1 e1 iff e0 =2 e1 (due to fairness)(1 => 3): e0 =1 e1 implies e0 =3 e1

~~~ ~

X X

X

X X X

X

X

*

*

e0

e1