Process Algebra C alculus of C ommunicating S ystems

Process AlgebraCalculus of Communicating Systems

Daniel ChoiProvable Software Lab.

KAIST

Content

• Introduction

• Calculus of Communicating Systems

• Equivalence for CCS

• Discussions

23年 4月 21日 2/59Provable Software Laboratory Seminar

Why are we going to study Process Algebra?

• Need– Mathematical models for

concurrent communicating processes?

• How– How can we define a mathematical

models for concurrent communicating process?


Why are we going to study Process Algebra?






The reason why we study Process Algebra

23年 4月 21日 Provable Software Laboratory Seminar

5 /59



– Process Algebra, Petri Net, etc.



– By defining structured operational semantics

Families of Algebraic Approaches

• Milner’s Calculus of Communicating Systems (CCS)

• Hoare’s theory of Communicating Sequential Processes

• The Algebra of Communicating Processes (ACP) of Bergstra & Klop

23年 4月 21日 6 /59Provable Software Laboratory Seminar

Content

• Introduction – Why are we going to study Process Algebra?

• Calculus of Communicating Systems– Definitions– Operational Semantic of CCS Terms– Examples

• Equivalence for CCS

• Discussions


DefinitionsTheoretical definitions

• Assume a non-empty set S of states, together with a finite, non-empty set of transition labels A and a finite set of predicate symbols

• Signature– Consist of a finite set of function symbols f, g, … where each function symbol f has an arity ar(f), being its number of

arguments.– Symbol of arity zero : constant (a, b, c, …)– Countably infinite set of variables (x, y, z, …)

• Finite non-empty set A of (atomic) actions– Each atomic action a is a constant that can execute itself, after which it terminates successfully.

• Term– Set T(∑) of open terms, s, t, u, … over ∑ is defined as the least set satisfying

• Each variable is T(∑);• If f ∈ ∑ and t1, …, tar(f) ∈ T(∑), then f(t1, …, tar(f)) ∈ T(∑)

– A term is closed if it does not contain variables. The set of closed terms is denoted by T(∑)

• Labeled transition system – A transition is a triple (s, a, s’) with a ∈ A, or a pair (s, P) with P a predicate, where s, s’ ∈ S. A labeled transition system (LTS)

is a possibly infinite set of transitions. An LTS is finitely branching if each of its states has only finitely many outgoing transitions

– The states of an LTS are always the closed terms over a signature ∑. – In view of the syntactic structure of closed terms over a signature, such transitions can be derived by means of inductive proof

rules, where the validity of a number of transitions (the premises) may imply the validity of some other transition (the conclusion)

• Process graph – A Process (graph) p is an LTS in which one state s is elected to be the root. If the LTS contains a transition s – a -> s’, then p –

a -> p’ where p’ has root state s’. Moreover, if the LTS contains a transition sP, then pP.


DefinitionsTheoretical definitions

• Assume a non-empty set S of states, together with a finite, non-empty set of transition labels A and a finite set of predicate symbols

• Signature– Consist of a finite set of function symbols f, g, … where each function symbol f has an arity ar(f), being its number of

arguments.– Symbol of arity zero : constant (a, b, c, …)– Countably infinite set of variables (x, y, z, …)

• Finite non-empty set A of (atomic) actions– Each atomic action a is a constant that can execute itself, after which it terminates succefully.

• Term– Set T(∑) of open terms, s, t, u, … over ∑ is defined as the least set satisfying

• Each variable is T(∑);• If f ∈ ∑ and t1, …, tar(f) ∈ T(∑), then f(t1, …, tar(f)) ∈ T(∑)

– A term is closed if it does not contain variables. The set of closed terms is denoted by T(∑)

• Labeled transition system – A transition is a triple (s, a, s’) with a ∈ A, or a pair (s, P) with P a predicate, where s, s’ ∈ S. A labeled transition system (LTS)

is a possibly infinite set of transitions. An LTS is finitely branching if each of its states has only finitely many outgoing transitions

– The states of an LTS are always the closed terms over a signature ∑. – In view of the syntactic structure of closed terms over a signature, such transitions can be derived by means of inductive proof

rules, where the validity of a number of transitions (the premises) may imply the validity of some other transition (the conclusion)

• Process graph – A Process (graph) p is an LTS in which one state s is elected to be the root. If the LTS contains a transition s – a -> s’, then p –

a -> p’ where p’ has root state s’. Moreover, if the LTS contains a transition sP, then pP.


DefinitionsDefinitions in CCS

• Actions– Atomic – uninterruptible execution steps

(with some other internal computation steps(τ))– Representing potential interactions with its environment

(inputs/outputs on ports)


10 /59

action not vaild is α, α

}{} α|α{ A

step.n computatio internalan represents

αport on signal a emitting ofact therepresents , α where,α

αport on signal a receiving ofact therepresents , α whereα,

CCS

DefinitionsOperator of CCS

• nil – terminated process that has finished execution

• a.p– Capable first of a and then behaves like p

• + – Choice construct– p1 + p2 offers the potential of behaving like either p1 or p2, depending on the interactions offered by the

environment

• | – parallel composition– p1 | p2 offers interleaves the execution of p1 and p2 – Permitting complementary actions of p1 and p2 to synchronize (τ)

• Restriction operator– Permits actions to be localized within a system

• [f]– Actions in a process to be renamed– P[f] behaves exactly like p except that f is applied to each action that p wishes to engage in

• Defining equation– C represents a valid system


DefinitionsLabeled transition Systems

• Labeled transition system (LTS) – Triple <Q, A,→>

• Q : a set of states • A : a set of actions• → : transition relation →⊆Qⅹ A ⅹ Q

– B = ((a.(b.B + c.0) + b.0)|a’.0 )\a


B

((b.B + c.0)|0)\a

(0|a’.0)\a

(0|0)\a

(B|0)\a …

τ

b

c

b

Temporal Structure

Operation Semantics of CCS Terms


Referenced from lecture note of Prof. Kim

ExamplesLovers

• Assume that there is a man and a woman in the society• Man and Woman can manifest their emotion independently

(concurrently)

• M = ‘man.(acc.M‘+ rej.M) • W = man.(‘acc.W’ + ‘rej.W)

• M’ = lov.M’ + ‘lov.M’ + ‘neg_man.M• W’ = lov.W’ + ‘lov.W’ + neg_man.W

• Does L = (M|W) is a model of happy lovers?


ExamplesLTS of Unhappy lovers


M|W

(acc.M‘+ rej.M) |W M|(’acc.W’ + ‘rej.W)

‘man man

(acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W)

M’|W M|W’ M’|(’acc.W’ + ‘rej.W)

acc‘rej

‘acc

acc‘acc

rej

‘rej

rej

(acc.M‘+ rej.M)|W’

M‘|W’

τ

τ

τ

man ‘man

‘acc

acc

τ,lov,’lov

τ

neg_man‘neg_man

lov,’lov

lov,’lov

‘neg_man

neg_man

ExamplesLTS of Unhappy lovers


M|W

(acc.M‘+ rej.M) |W M|(’acc.W’ + ‘rej.W)

‘man man


M’|W M|W’ M’|(’acc.W’ + ‘rej.W)

acc‘rej

‘acc

acc‘acc

rej

‘rej

rej

(acc.M‘+ rej.M)|W’

M‘|W’

τ

τ

τ

man ‘man

‘acc

acc

τ,lov,’lov

τ

neg_man‘neg_man

lov,’lov

lov,’lov

‘neg_man

neg_man

One sided Love

ExamplesLTS of Happy lovers


HL = (M|W) \{man, lov, acc, rej}

M|W


M‘|W’

τ

τ

τ

ExamplesLTS of Happy lovers


HL = (M|W) \{man, lov, acc, rej}

M|W


M‘|W’

τ

τ

τ

proc HL = (M|W)\{manifest,love,neg_manifest,accept,reject}proc UHL = (M|W)

proc M = 'manifest.(accept.M1 + reject.M) proc W = manifest.('accept.W1 + 'reject.W) proc M1 = love.M1 + 'love.M1 + 'neg_manifest.Mproc W1 = 'love.W1 + love.W1 + neg_manifest.W

ExamplesProof

• Proof of (M|W)\{man, lov, acc, rej} => (M|W)\{man, lov, acc, rej}


(M|W)\{man, lov, acc, rej} -τ->((acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W)) \{man, lov, acc, rej}

ExamplesProof




Res

ExamplesProof




‘man.(acc.M‘+ rej.M) | man.(‘acc.W’ + ‘rej.W) -τ->(acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W)

Res

ExamplesProof





Parτ

Res

ExamplesProof





‘man.(acc.M‘+ rej.M) – ‘man-> (acc.M‘+ rej.M) man.(‘acc.W’ + ‘rej.W) – man-> (‘acc.W’ + ‘rej.W)

Act Act

Parτ

Res

ExamplesProof



((acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W)) \{man, lov, acc, rej} -τ-> (M’|W’) \{man, lov, acc, rej}

(acc.M‘+ rej.M) | (‘acc.W’ + ‘rej.W) -τ-> (M’|W’)

(acc.M‘+ rej.M) - acc-> M’ (‘acc.W’ + ‘rej.W) – ‘acc -> W’

acc.M‘ – acc -> M’ ‘acc.W‘ – ‘acc -> W’

Act Act

ChoiceL ChoiceL

Parτ

Res

Content• Introduction

– Why are we going to study Process Algebra?


• Equivalence for CCS– Trace Equivalence – Strong Bisimulation Equivalence– Weak Bisimulation Equivalence

• Discussions


Trace EquivalenceDefinition

• Language Equivalence – Two machines are equivalent if they accept the same sequences

of symbol

• Can we directly apply language equivalence to rooted LTS? No– Identify every state in a rooted LTS as being accepting

• Definition Let <Q, A,→> be a labeled transition system– Let A* consists of the set of finite sequences of elements of A– Let s = a1 … an ∈A* be a sequence of actions. Then q – s-> q’ if there are

states q0, ..., qn such that q = q0, qi –ai-> qi+1 and q’ = qn – s is a strong trace of q if there exists q’ such that q – s -> q’. We use S(q) to

represent the set of all strong traces of q– p ≈s q exactly when S(p) = S(q)

(strong traces do not distinguish between internal and external actions)

• Can we use trace equivalence to decide whether two system are behavioral congruent? No


Trace Equivalence Definition


of symbol




represent the set of all strong traces of q– p ≈s q exactly when S(p) = S(q)




Trace EquivalenceDefinition


of symbol




represent the smallest set of all strong traces of q (prefix-closed)– p ≈s q exactly when S(p) = S(q)




Trace Equivalence Definition


of symbol




represent the smallest set of all strong traces of q (prefix-closed)– p ≈s q exactly when S(p) = S(q)




ExampleTrace Equivalence


q0

q1

q1

’

q2 q3

p0

p2 p3

p1

P = a.(b.nil + c.nil)S(P) = {ε,a,ab,ac}

a

cb

a

cb

a

Q = a.b.nil + a.c.nilS(Q) = {ε,a,ab,ac}



q0

q1

q1

’

q2 q3

p0

p2 p3

p1


a

cb

a

cb

a

Q = a.b.nil + a.c.nilS(Q) = {ε,a,ab,ac}

S(P) = S(Q)S(P) = S(Q)



q0

q1

q1

’

q2 q3

p0

p2 p3

p1


a

cb

a

cb

a

Q = a.b.nil + a.c.nilS(Q) = {ε,a,ab,ac}Trace EquivalentTrace Equivalent

S(P) = S(Q)S(P) = S(Q)



q0

q1

q1

’

q2 q3

p0

p2 p3

p1


a

cb

a

cb

a

Q = a.b.nil + a.c.nilS(Q) = {ε,a,ab,ac}Trace EquivalentTrace Equivalent

S(P) = S(Q)S(P) = S(Q)

It is not behavioral congruentIt is not behavioral congruent

Strong Bisimulation Equivalence

Definition• Execution sequences for equivalent systems ought to pass

through equivalent states

• Definition Let <Q, A,→> be an LTS. A relation R ⊆ Q x Q is a bisimulation if whenever <p, q> ∈R, then the following conditions hold for any a, p’ and q’

– If p –a-> p’ then q – a -> q’ for some q’ such that <p’, q’> ∈R– If q –a-> q’ then p – a -> p’ for some p’ such that <p’, q’> ∈R

• Definition System p and q are bisimulation equivalent, or bisimilar, if there exists a bisimulation R containing <p, q>. We write p ~ q whenever p and q are bisimilar



How to find out P and Q are bisimular?• Strong Simulation

– Let <Q, A,→> be an LTS, and let S be a binary relation over Q. Then S is called a strong simulation over <Q, A,→> if, whenever pSq, if p – a -> p’ then there exists q’ ∈ Q such that q – a -> q’ and p’ S q’

• q strongly simulates p if there exists a strong simulation S such that pSq



How to find out P and Q are bisimular? : Example


q0

q1

q1

’

q2 q3

p0

p2 p3

p1

a

cb

a

cb

a

Suppose, (p0, q0)∈ S




Suppose p0 strongly simulates q0, (q0, p0)∈ S or q0Sp0

q1 p1

q0 S p0

a a





q1 p1

q0 S p0

a a

q1 S p1

q1' p1

q0 S p0

a a

q1' S p1





q1 p1

q0 S p0

a a

q1 S p1

q1' p1

q0 S p0

a a

q1' S p1

q2 p2

q1 S p1

b b

q2 S p2





q1 p1

q0 S p0

a a

q1 S p1

q1' p1

q0 S p0

a a

q1' S p1

q2 p2

q1 S p1

b b

q2 S p2

q3 p3

q1' S p1

c c

q3 S p3





q1 p1

q0 S p0

a a

q1 S p1

q1' p1

q0 S p0

a a

q1' S p1

q2 p2

q1 S p1

b b

q2 S p2

q3 p3

q1' S p1

c c

q3 S p3

Therefore S = {(q0, p0), (q1, p1), (q1’, p1), (q2, p2), (q3, p3)}




Suppose q0 strongly simulates p0, (p0, q0)∈ S or p0Sq0

p1 q1

p0 S q0

a a

p1 S q1

p2 q2

p1 S q1

b b

q1' S p1

p3

p1 S q1

c




Suppose q0 strongly simulates p0, (p0, q0)∈ S or p0Sq0

p1 q1’

p0 S q0

a a

p1 S q1’

p3 q3

p1 S q1’

c c

p3 S q3

p2

p1 S q1’

b





• S-1 is the set of pairs (y, x) such that (x, y) ∈ S

• Strong bisimulation– A binary relation S over Q is said to be a strong bisimulation over the LTS if both

S and its converse are simulations







• Strong bisimulation– A binary relation S over Q is said to be a strong bisimulation over the LTS if both

S and its converse are simulations





p0

p2

p1

a

b

S = {(p0, q0), (p1, q1), (p2, q1), (p0, q2)}

a

a

a

b

q0q1

q2

a

a

a

b

S’ = {(q0, p0), (q1, p1), (q1, p2), (q2, p0)}




p0

p2

p1

a

b

S = {(p0, q0), (p1, q1), (p2, q1), (p0, q2)}

a

a

a

b

q0q1

q2

a

a

a

b

S’ = {(q0, p0), (q1, p1), (q1, p2), (q2, p0)}Strong BisimulationStrong Bisimulation




p0

p2

p1

P strongly simulates QS = {(q0, p0), (q1, p2), (q2, p3)}

a

a

b

q0q1

q2

a

b

Q strongly simulates PS’ = {(p0, q0), (p1, q1), (p2, q1), (p3, q2)}

p3

It is not Strong BisimulationIt is not Strong Bisimulation






• Strong bisimulation (P ~ Q)– A binary relation S over Q is said to be a strong bisimulation over the LTS if both S and

its converse are simulations– Strong bisimulation equivalence : reflexive, symmetric, transitive

– P ~ Q implies P ≈s Q

• What about internal computation τ ? – Weak bisimulation







• Strong bisimulation (P ~ Q)– A binary relation S over Q is said to be a strong bisimulation over the LTS if both S and

its converse are simulations– Strong bisimulation equivalence : reflexive, symmetric, transitive

– P ~ Q implies P ≈s Q

• What about internal computation τ ? – Weak bisimulation


Weak Bisimulation Equivalence

Definition• How are we going to treat internal computation?

– We cannot ignore τ.

• Definition S is a weak simulation (observational simulation) if and only if, whenever PSQ, if P → P’ then there exists Q’∈ P such that Q ⇒ Q’ and P’SQ’if P -λ-> then there exists Q’ ∈ P such that Q = λ => Q’ and P’SQ’

• → : unobservable reactions (like τ) λ : observable actions ⇒ : zero or more reactions= λ => : observation – λ -> accompanied (before and after) by any number of reactions = λ => →* – λ -> →*= τ => : = at least one reaction



Definition• How are we going to treat internal computation?

– We cannot ignore τ.

• Definition S is a weak simulation (observational simulation) if and only if, whenever PSQ, if P → P’ then there exists Q’∈ P such that Q ⇒ Q’ and P’SQ’if P -λ-> then there exists Q’ ∈ P such that Q = λ => Q’ and P’SQ’

• → : unobservable reactions (like τ) • λ : observable actions • ⇒ : zero or more reactions• = λ => : observation – λ -> accompanied (before and after) by

any number of reactions– = λ => →* – λ -> →*

• = τ => : = at least one reaction



How to find out weak bisimulation equivalence?• Similar to strong bisimulation

• Definition A binary relation S over P is said to be a weak bisimulation if both S and its converse are weak simulations. We say that P and Q are weakly bisimilar, weakly equivalent, or observation equivalent, written P ≈ Q, if there exists a weak bisimulation S such that P S Q

• q weakly simulates p if there exists a strong simulation S such that pSq



How to find out weak bisimulation equivalence? : example


A = a.A’ B = b.B’A’ = ‘b.A B’ = ‘c.B

p0 = (A|B)\{b}p1 = (A’|B)\{b}p2 = (A|B’)\{b}p3 = (A’|B’)\{b}

p0

p1p2

a

a‘c

p3

‘c

τ

E = a.E’E’ = a.E’’ + ‘c.EE’’ = ‘c.E

q0 = Eq1 = E’q2 = E’’

q0q0

q1q1

a‘c

q2q2

a

‘c




A = a.A’ B = b.B’A’ = ‘b.A B’ = ‘c.B


p0

p1p2

a

a‘c

p3

‘c

τ

E = a.E’E’ = a.E’’ + ‘c.EE’’ = ‘c.E

q0 = Eq1 = E’q2 = E’’

q0q0

q1q1

a‘c

q2q2

a

‘c

S = {(p0 ,q0), (p1 ,q1), (p2 ,q1), (p3 ,q2)}




A = a.A’ B = b.B’A’ = ‘b.A B’ = ‘c.B


p0

p1p2

a

a‘c

p3

‘c

τ

E = a.E’E’ = a.E’’ + ‘c.EE’’ = ‘c.E

q0 = Eq1 = E’q2 = E’’

q0q0

q1q1

a‘c

q2q2

a

‘c

S = {(p0 ,q0), (p1 ,q1), (p2 ,q1), (p3 ,q2)}

Observational BisimulationObservational Bisimulation

Content• Introduction

– Why are we going to study Process Algebra?


• Equivalence for CCS– Trace Equivalence – Strong Bisimulation Equivalence– Weak Bisimulation Equivalence

• Discussions


Discussions


58 /59

Reference• Communicating and mobile systems: the pi-calculus

by Robin Milner, Cambridge,1999

• Communication and Concurrencyby Robin Milner, Prentice Hall, 1989

• Fundamentals of software engineeringby C. Chezzi, M. Jazayeri, D. Mandrioli, Prentice Hall, 2003

• Lecture Notes of Professor Bae, http://se.kaist.ac.kr/~course/DrBae/cs550_2006/

• Lecture Notes of Professor Kim, http://cs.kaist.ac.kr/~moonzoo/cs750b

• Notes on the methodology of CCS and CSPby R.J. van Glabbeek, TCS 177(2), pp. 329-349. Originally appeared as Report CS-R8624, CWI, Amsterdam, 1986

• Operational and algebraic semantics of concurrent processesby R. Milner, in J. van Leeuwen, editor: Handbook of Theoretical Computer Science, Chapter 19, Elsevier Science Publishers B.V. (North-Holland), pp. 1201-1242. (1990)

• Process Algebraby R. Cleaveland and S. Smolka, in J.G. Webster, editor, Encyclopedia of Electrical Engineering, John Wiley & Sons, 1999 (Chap. 1 ~ 3)


59 /59

http://se.kaist.ac.kr/~course/DrBae/cs550_2006/

http://cs.kaist.ac.kr/~moonzoo/cs750b

Process Algebra C alculus of C ommunicating S ystems

Documents

set s of states

process graph p

state s

set of closed terms

process algebracalculus

triple s

pair s

set of transition labels