An introduction to process calculi:Calculus of Communicating Systems

(CCS)Lecture 2 of Modelli Matematici dei Processi

Concorrenti

Paweł Sobocinski

University of Southampton, UK

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Introduction

In the first lecture we discussed behavioural preordersand equivalences, in particular bisimilarity.

This lecture is an introduction to a well-knownprocess-calculus, the Calculus of CommunicatingSystems (CCS) introduced by Robin Milner in the early1980’s.

Process calculi:a pseudo-programming language which usuallyfocuses on a small language feature;usually try to eliminate syntactic sugar and extraprogrammer-friendly features;idea is to isolate basic principles and reasoningtechniques.

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CCS

focuses on a very simple paradigm of synchronoushandshakes.

processes a?P and a!Q, when executing in parallel(a?P ‖ a!Q);

can synchronise their execution by synchronisingon channel a;after the synchronisation continue as P and Q,respectively;a?P ‖ a!Q → P ‖ Q;

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CCS syntax

Assume that we have a set of names A and a countable setof process variables.

P ::= 0 | a?P | a!P | P ‖ P | P + P | νaP | X | µX.P

NB. sometimes a?P is written aP and a!P is written aP . Inearly texts νaP is written P\a.

a?P : input on a and proceed as P ;

a!P : output on a and proceed as P ;

τP : perform an internal reduction and proceed as P ;

P1 ‖ P2 : put P1 and P2 in parallel;

P1 + P2 : act either as P1 or as P2;

νaP : treat a as a local channel visible only in P ;Intro to process calculi: CCS – p.4/25

Recursion

Infinite behaviour can be added to CCS in several(nonequivalent) ways.

P ::= . . . | X | µX.P

The expression µX.P stands for treats the X as a recursivevariable in P .

Idea:

µX.a?X “ ≡ ” a?a? . . . ;

µX.P ‖ X “ ≡ ” P ‖ P ‖ . . . ;

µX.a!(b?X + c?X) “ ≡ ” ?

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Contexts and Congruences

Definition 1 (Contexts).

C ::= − | P ‖ C | C ‖ P | C + P | P + C | νaC |a?C | a!C

Definition 2 (Substitution). Given a term P and a context C , let C[P ]denote the term obtained by substituting P for −. Free names may becaptured!

Definition 3. A relation R on the terms of CCS is said to be acongruence when

PRQ ⇒ σ(P )Rσ(Q)

for all operations σ of CCS. More formally, if PRQ then C[P ]RC[Q] forall contexts C .

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Structural congruence

P ‖ Q denotes the same system as Q ‖ P ;

We quotient the “raw” syntax via a relation which is acongruence with respect to the operations of CCS.

Definition 4 (SC). Let ≡ be the smallest congruence which includes thefollowing:

(P ‖ Q) ‖ R ≡ P ‖ (Q ‖ R) P ‖ Q ≡ Q ‖ P P ‖ 0 ≡ P

(P + Q) + R ≡ P + (Q + R) P + Q ≡ Q + P P + 0 ≡ P

νa(P ‖ Q) ≡ (νaP ) ‖ Q (a /∈ Q) νaP ≡ νbP [b/a] (b /∈ P )

µX.P ≡ P [µX.P/X]

Remark 5. Contexts are not quotiented by SC. This is because we wantthe contexts to have the power to bind.

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Reduction semantics 1

Idea, basic reduction:

a?P1 + P2 ‖ a!Q1 + Q2 → P1 ‖ Q1

Letting

la,P1,P2,Q1,Q2

def= a?P1 + P2 ‖ a!Q1 + Q2

ra,P1,P2,Q1,Q2

def= P1 ‖ Q1

we want, for all a, P1, P2, Q1, Q2

la,P1,P2,Q1,Q2→ ra,P1,P2,Q1,Q2

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Reduction semantics 2

But parallel composition shouldn’t inhibit reduction, so thatif P → P ′ then for all Q we should also haveP ‖ Q → P ′ ‖ Q. Similarly, νaP → νaP ′.

Evaluation contexts:

E ::= − ‖ P | νa−

Definition 6. P → P ′ iff ∃a, P1, P2, Q1, Q2 and evaluation context Esuch that P ≡ E[la,P1,P2,Q1,Q2

] and P ′ ≡ E[ra,P1,P2,Q1,Q2].

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SOS

There is a useful way of presenting the reduction semanticsusing structural operational semantics (SOS).

a?P1+P2‖a!Q1+Q2→P1‖Q1

P→P ′

P‖Q→P ′‖Q

P→P ′

νaP→νaP ′

The rules above generate a transition system (which wecan think of as an LTS with only one label);

we will refer to this as the reduction transition system.

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Examples

c t

s

c t

s s

M1def= s?(c! + t!) M2

def= s?c! + s?t!

Coffee drinker C1def= s!c?0;

Simulation 1: C1 ‖ M1 → c?0 ‖ (c!0 + t!0) → 0

Simulation 2: C1 ‖ M2 → c?0 ‖ t!0

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Process equivalence?

What is a good notion of process equivalence?we can try bisimilarity on the reduction LTS...but then, for instance, a!P ∼ b!P . In fact, allprocesses which do not reduce are bisimilar. Clearly,this is not sufficient.

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Contextual equivalence

A canonical process equivalence for a process language iscontextual equivalence . The idea originally arose in thetheory of the λ-calculus.

Idea: Processes P and Q can be distinguished if in somecontext C they “behave differently”.

we can try taking the largest bisimulation which is alsoa congruence ;

this almost works, the problem is that processes whichcan always reduce are equated;

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Reduction barbed congruence

One sensible definition of a contextual equivalence for CCSis reduction barbed congruence .Definition 7. A barb is a basic observation. In CCS it makes sense totake the instantaneous ability to input or output on a name. We say that

P ↓a iff P ≡ νk(a!P1 + P2 ‖ P3) or P ≡ νk(a?P1 + P2 ‖ P3) anda /∈ k.

Definition 8. Reduction barb congruence Let ∼= be the largestequivalence relation which is

a congruence;

barb-closed: if P ∼= Q and P ↓a then Q↓a;

reduction-closed: if P ∼= Q and P → P ′ then ∃Q′, Q → Q′ andQ ∼= Q′ (for all contexts C , C[P ] ∼= C[Q], bisimulation on thereduction LTS)

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Examples

Claim 9 (Firewall). νa.a! ∼= 0.

Proof. ?

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Problems with contextual equivalence

∼= is a priori defined to be a congruence;

thus, in principle, to check that P ∼= Q we have to have aproof which takes into account an infinite number ofarbitrarily complex contexts C

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LTS characterisation

We will give a labelled transition system on whichbisimulation characterises contextual equivalence, ie

it is sound : ∼⊆∼=;

it is complete : ∼=⊆∼.

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LTS for CCS

(IN)a?P

a?

−→P(OUT)

a!Pa!

−→P

Pa!

−→P ′ Qa?

−→Q′

(TAU)P‖Q

τ

−→P ′‖Q′

Pα

−→P ′

(PAR)P‖Q

α

−→P ′‖Q

Pα

−→P ′ (a/∈α)(NU)

νaPα

−→νaP ′

P≡P ′ P ′α

−→Q′ Q′≡Q(STRCONG)

Pα

−→Q

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Example

Lemma 10 (Firewall). νa.a! ∼ 0

Proof. Obvious, since both sides do not have any transitions.

Lemma 11 (Expansion). If a 6= b then a? ‖ b! ∼ a?b! + b!a?.

Proof. {(a? ‖ b!, a?b! + b!a?), (b!, b!), (a?, a?), (0, 0)} is abisimulation.

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Proof of soundness

Theorem 12. ∼ is a congruence.

Proof. We’ll do the case − ‖ Q. We will prove thatR = {(P1 ‖ R,P2 ‖ R), P1 ∼ P2} is a bisimulation.

Suppose that P1 ‖ Qα−→ R. We’ll do the case α = τ . If P1

τ−→ P ′

1 and

R ≡ P ′1 ‖ Q then also P2

τ−→ P ′

2 such that P ′1 ∼ P ′

2. Then

P2 ‖ Qτ−→ P ′

2 ‖ Q, but (R,P ′2 ‖ Q) ∈ R.

The case Qτ−→ Q′ such that R ≡ P1 ‖ Q′ is similar.

The other possibility is, P1a!−→ P ′

1 and Qa?−→ Q′ and R ≡ P ′

1 ‖ Q′. But

then P2a!−→ P ′

2 such that P ′1 ∼ P ′

2 and so P2 ‖ Qτ−→ P ′

2 ‖ Q′.

(note the case P1a?−→ P ′

1, Qa!−→ Q′ is symmetric)

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Congruent bisimilarities

Having a congruent bisimilarity is quite useful becauseit allows the use of a familiar algebraic principle –substituting “equal for equal”.

It can also reduce the burden of constructingbisimulations since bisimulations can be deconstructed:

Example 13. If P1 ∼ Q1 and P2 ∼ Q2 then P1 ‖ P2 ∼ Q1 ‖ Q2.

Proof. P1 ‖ P2 ∼ Q1 ‖ P2 ∼ Q1 ‖ Q2.

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Proof of soundness

Lemma 14. P ↓a iff Pa!−→ or P

a?−→.

Lemma 15. Pτ−→ P ′ iff P → P ′.

Theorem 16. ∼⊆∼=.

Proof. Recall that ∼= is defined to be the largest barb and reductionclosed congruence. ∼ is reduction and barbed closed (the two lemmas)and is a congruence. Hence ∼⊆∼=.

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Proof of completeness

Theorem 17. ∼=⊆∼.

We need to make sure that the label transitions do not ob-

serve too much about the processes. Hence for each label

α, we’ll find a context which “observes” the label.

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Weak equivalences

“Weak” in the jargon of process calculists means thatinternal reductions are not observable.

Definition 18 (Weak bisimulation). A relation R is a weak bisimulationwhen

PRQ and Pτ−→ P ′ then Q

τ−→

∗Q′

PRQ and Pα−→ P ′ (α 6= τ) then Q

τ−→

∗ α−→

τ−→

∗Q′;

the symmetric versions of the above hold.

Definition 19. We will write τP for νa(a!‖a?P ) where a is fresh for P .

Example 20.

τa! ≈ a!.

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Weak bisimilarity not a congruence

The counterexample is very simple, we have τa! ≈ a!, but

τa!+b! 6 ≈a!+b!. Weak bisimilarity behaves better with respect

to other operators.

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