From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
An introduction to Concurrency Theory
Communicating and Mobile Systems
Frank D. Valencia
French National Center for Scientic Research (CNRS)
Computer Science Laboratory of École Polytechnique de Paris (LIX)
Oct 2005/CLEI'05
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency is everywhere...
Concurrent Systems
Multiple
agents (processes) that interact among each other.
Some fundamental concurrent systems:
Reactive Systems.
Synchronous Communicating Systems
Mobile Systems.
Concurrent systems can combine:
Example
E.g.,
The Internet (a complex one!)
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency is everywhere...
Concurrent Systems
Multiple
agents (processes) that interact among each other.
Some fundamental concurrent systems:
Reactive Systems.
Synchronous Communicating Systems
Mobile Systems.
Concurrent systems can combine:
Example
E.g.,
The Internet (a complex one!)
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency is everywhere...
Concurrent Systems
Multiple
agents (processes) that interact among each other.
Some fundamental concurrent systems:
Reactive Systems.
Synchronous Communicating Systems
Mobile Systems.
Concurrent systems can combine:
Example
E.g.,
The Internet (a complex one!)
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency is everywhere...
Concurrent Systems
Multiple
agents (processes) that interact among each other.
Some fundamental concurrent systems:
Reactive Systems.
Synchronous Communicating Systems
Mobile Systems.
Concurrent systems can combine:
Example
E.g.,
The Internet (a complex one!)
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency is everywhere...
Concurrent Systems
Multiple
agents (processes) that interact among each other.
Some fundamental concurrent systems:
Reactive Systems.
Synchronous Communicating Systems
Mobile Systems.
Concurrent systems can combine:
Example
E.g.,
The Internet (a complex one!)
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency is everywhere...
Concurrent Systems
Multiple
agents (processes) that interact among each other.
Some fundamental concurrent systems:
Reactive Systems.
Synchronous Communicating Systems
Mobile Systems.
Concurrent systems can combine:
Example
E.g.,
The Internet (a complex one!)
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency is everywhere...
Concurrent Systems
Multiple
agents (processes) that interact among each other.
Some fundamental concurrent systems:
Reactive Systems.
Synchronous Communicating Systems
Mobile Systems.
Concurrent systems can combine:
Example
E.g.,
The Internet (a complex one!)
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency: A Serious Challenge...
Models of Concurrency
Formal Models to describe and analyze concurrent systems.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency: A Serious Challenge...
Models of Concurrency
Formal Models to describe and analyze concurrent systems.
As other models of reality, these models must be
be simple
be expressive,
be formal
provide reasoning techniques.
Just like
λ−calculus
model of sequential computation !
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency: A Serious Challenge...
Models of Concurrency
Formal Models to describe and analyze concurrent systems.
Models for sequential computation don't apply
putation is usually
. Concurrent com-
Non-Terminating
Reactive (or Interactive)
Nondeterministic (Unpredictable).
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency: A Serious Challenge...
Models of Concurrency
Formal Models to describe and analyze concurrent systems.
In concurrency theory
Each model focuses in a fundamental phenomenon: E.g.,
Synchrony and Mobility.
But there is no yet a cannonical model .
...Probably because concurrency is a very broad (young) area.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concurrency: A Serious Challenge...
Models of Concurrency
Formal Models to describe and analyze concurrent systems.
Some well established model of concurrency:
Process Calculi (Like
λ-calculi)
CCS (Synchronous communication)
π -calculus (CCS Extension to Mobility)
Petri Nets (Like Automata)
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Mobility
What kind of
mobility will we consider in this tutorial ?
Processes move.
√
Links move
The last one, because it is the
π -calculus'
choice; for exibility and
simplicity.
π -calculus
names).
has the ability of sending private and public
(
CLEI'05 Cali
Communicating and Mobile Systems
links
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Mobility
What kind of
mobility will we consider in this tutorial ?
Processes move.
√
Links move
The last one, because it is the
π -calculus'
choice; for exibility and
simplicity.
π -calculus
names).
has the ability of sending private and public
(
CLEI'05 Cali
Communicating and Mobile Systems
links
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Mobility
What kind of
mobility will we consider in this tutorial ?
Processes move.
√
Links move
The last one, because it is the
π -calculus'
choice; for exibility and
simplicity.
π -calculus
names).
has the ability of sending private and public
(
CLEI'05 Cali
Communicating and Mobile Systems
links
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Link Mobility
Mobile phones
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Link Mobility
Mobile phones
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
A avour of the pi-calculus
In the π calculus:
(ν a)(b̄a.S k a(x ).P ) k b(c ).c̄d .C
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
A avour of the pi-calculus
π the pi-calculus:
(ν a)(b̄a.S k a(x ).P ) k b(c ).c̄d .C −→ S k (ν a)(a(x ).P k ād .C )
In
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Outline
1
From Computability to Concurrency Theory
Basic Concepts from Automata Theory
Bisimilarity Equivalence
2
Calculus of Comunicating Systems CCS
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
3
Mobility and the pi calculus
Syntax and Semantics
Applications
Equivalences
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Outline
1
From Computability to Concurrency Theory
Basic Concepts from Automata Theory
Bisimilarity Equivalence
2
Calculus of Comunicating Systems CCS
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
3
Mobility and the pi calculus
Syntax and Semantics
Applications
Equivalences
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory
Automata
Denition
An automata
A over an alphabet Act
is a tuple
(Q , Q0 , Qf , T )
where
S (A) = Q = {q0 , q1 , . . .} is the set of states,
S0 (A) = Q0 ⊆ Q is the set of initial states,
Sf (A) = Qf ⊆ Q is the set of accepting (or nal) states,
T (A) = T ⊆ Q × Act × Q is the set of transitions.
Write
a 0
q −→
q
for
(q , a, q 0 ) ∈ T
CLEI'05 Cali
. Omit
Act
when it is understood.
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory
Automata
Example: A Typical Graph Representation
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory
Regular Sets and Expressions
Denition (Acceptance, Regularity)
a
a
an
1
2
A accepts a1 ...an i there is a path q0 −→
q1 −→·
· · −→
qn
in T (A) with q0 ∈ So (A) and qn ∈ Sf (A).
The Language of (or recognized by) A, L(A), is the set of
sequences accepted by A.
Regular sets are those recognized by nite-state automata
(FSA).
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory
Regular Sets and Expressions
Denition (Acceptance, Regularity)
a
a
an
1
2
A accepts a1 ...an i there is a path q0 −→
q1 −→·
· · −→
qn
in T (A) with q0 ∈ So (A) and qn ∈ Sf (A).
The Language of (or recognized by) A, L(A), is the set of
sequences accepted by A.
Regular sets are those recognized by nite-state automata
(FSA).
Fact
Regular Expressions (e.g. a.(b + c )∗ ) are as expressive as FSA.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory
Regular Sets and Expressions
Example
An string accepted: 10001.
The language accepted: All binary strings with an even
number of 1's.
∗ (10∗ 10∗ )∗
A Corresponding Regular Expression: 0
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory
Some Nice Properties and Exercises
Fact
1
2
Deterministic and NonDeterministic FSA are equally
expressive.
Regular sets are closed under (a) union, (b) complement, (c)
intersection.
Exercises
1
Prove 2.b and 2.c.
2
Prove that emptiness problem of a given FSA is
3
Prove that language equivalence of two given FSA is
decidable.
decidable.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory
Some Nice Properties and Exercises
Fact
1
2
Deterministic and NonDeterministic FSA are equally
expressive.
Regular sets are closed under (a) union, (b) complement, (c)
intersection.
Exercises
1
Prove 2.b and 2.c.
2
Prove that emptiness problem of a given FSA is
3
Prove that language equivalence of two given FSA is
decidable.
decidable.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory
What is good for ?
Classic Automata Theory is solid and foundational with several
applications in computer science. BUT...
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory
What is good for ?
Classic Automata Theory is solid and foundational with several
applications in computer science. BUT...
Example: A FSA Vending Machine Representation
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory
What is good for ?
Classic Automata Theory is solid and foundational with several
applications in computer science. BUT...
Example: A FSA Vending Machine Representation
•
L(B1 ) = L(B2 ) but B2
machine!
would represent an annoying vending
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory
What is good for ?
Classic Automata Theory is solid and foundational with several
applications in computer science. BUT...
Fact
Language equivalence may be too weak for interactive behavior.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Classic Automata Theory.
The Problem with Language Equivalence
In automata theory
a · (b + c ) = a · b + a · c .
I.e.,
p0
and
qo
are
equivalent.
•
We need stronger equivalences that does not validate the above.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Outline
1
From Computability to Concurrency Theory
Basic Concepts from Automata Theory
Bisimilarity Equivalence
2
Calculus of Comunicating Systems CCS
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
3
Mobility and the pi calculus
Syntax and Semantics
Applications
Equivalences
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Bisimilarity: The Equivalence of Reactive Systems.
Transition Systems
Transition systems are just automata in which initial and nal
states are irrelevant.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Bisimilarity: The Equivalence of Reactive Systems.
Simulation
T be transition system. A relation R ⊆ S (T ) × S (T ) is a
simulation i for each (p , q ) ∈ R :
a
a
. If p −→ p 0 then there is q 0 s.t. q −→ q 0 and (p 0 , q 0 ) ∈ R .
Let
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Bisimilarity: The Equivalence of Reactive Systems.
Bisimulation
A relation
R
is a
bisimulation i R
and its
converse R −1
are both
simulations.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Bisimilarity: The Equivalence of Reactive Systems.
Similaridad & Bisimilaridad
. p simulates q i there exists a simulation R s.t (p , q ) ∈ R .
. p and q are bisimilar, written p ∼ q , if there exists a
bisimulation R s.t. (p , q ) ∈ R .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Bisimilarity: The Equivalence of Reactive Systems.
Example: Trace Equivalence is not Bisimilarity
Here
p0
simulates
q0
but
q0
does not simulate
p0 , so p0
and
not bisimilar.
CLEI'05 Cali
Communicating and Mobile Systems
q0
are
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Bisimilarity: The Equivalence of Reactive Systems.
Example: Bisimilarity is not graph equivalence
Bisimilarity equates LTS with dierent graph structure: E.g.,
and
q0
p0
are bisimilar. But how do we prove they are bisimilar ?
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Bisimilarity: The Equivalence of Reactive Systems.
Example: Proving Bisimilarity
p0 and q0 are bisimilar simply exhibit a bisimulation
R = {(p0 , q0 ), (p0 , q2 ), (p1 , q1 ), (p2 , q1 )}.
To prove that
R.
E.g.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Bisimilarity: The Equivalence of Reactive Systems.
Example: Proving Bisimilarity
p0 and q0 are bisimilar simply exhibit a bisimulation
R = {(p0 , q0 ), (p0 , q2 ), (p1 , q1 ), (p2 , q1 )}.
To prove that
R.
E.g.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Bisimilarity: The Equivalence of Reactive Systems.
Example: Bisimilarity and simulation
Suppose that
p
simulates
q
and
q
simulates
p.
Are
p
and
q
necesarly bisimilar ?
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Concepts from Automata Theory
Bisimilarity Equivalence
Bisimilarity: The Equivalence of Reactive Systems.
Example: Bisimilarity and simulation
No! Here
p
Notice that
q and q simulates p but they are not bisimilar.
reliable than q ; i.e., it deadlocks after a.
simulates
p
is less
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Outline
1
From Computability to Concurrency Theory
Basic Concepts from Automata Theory
Bisimilarity Equivalence
2
Calculus of Comunicating Systems CCS
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
3
Mobility and the pi calculus
Syntax and Semantics
Applications
Equivalences
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Process Calculi
Process calculi treat processes much like λ-calculus treats
computable functions:
Structure of
terms represent the structure of processes.
E.g.:
PkQ
Operational Semantics to represent computational steps.
E.g.:
PAR
P −→ P 0
P k Q −→ P 0 k Q
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Process Calculi
Process calculi treat processes much like λ-calculus treats
computable functions:
Structure of
terms represent the structure of processes.
E.g.:
PkQ
Operational Semantics to represent computational steps.
E.g.:
PAR
P −→ P 0
P k Q −→ P 0 k Q
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Process Calculi
Process calculi treat processes much like λ-calculus treats
computable functions:
Structure of
terms represent the structure of processes.
E.g.:
PkQ
Operational Semantics to represent computational steps.
E.g.:
PAR
P −→ P 0
P k Q −→ P 0 k Q
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Typical Issues in Developing Process Calculi
Which process constructs t the intended phenomena?
E.g. atomic actions, parallelism, nondeterminism.
How should these constructs be given meaning ?
E.g. Operational Semantics, Denotational Semantics.
How should processes be compared ?
E.g. Observable Behavior, Process Equivalences.
How should process properties be specied and proved ?
E.g. Logic for expressing process specications (like in Hoare's
Logic).
How expressive is the calulus ?
E.g. Expressiveness of a nice fragment of the calculus.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Typical Issues in Developing Process Calculi
Which process constructs t the intended phenomena?
E.g. atomic actions, parallelism, nondeterminism.
How should these constructs be given meaning ?
E.g. Operational Semantics, Denotational Semantics.
How should processes be compared ?
E.g. Observable Behavior, Process Equivalences.
How should process properties be specied and proved ?
E.g. Logic for expressing process specications (like in Hoare's
Logic).
How expressive is the calulus ?
E.g. Expressiveness of a nice fragment of the calculus.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Typical Issues in Developing Process Calculi
Which process constructs t the intended phenomena?
E.g. atomic actions, parallelism, nondeterminism.
How should these constructs be given meaning ?
E.g. Operational Semantics, Denotational Semantics.
How should processes be compared ?
E.g. Observable Behavior, Process Equivalences.
How should process properties be specied and proved ?
E.g. Logic for expressing process specications (like in Hoare's
Logic).
How expressive is the calulus ?
E.g. Expressiveness of a nice fragment of the calculus.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Typical Issues in Developing Process Calculi
Which process constructs t the intended phenomena?
E.g. atomic actions, parallelism, nondeterminism.
How should these constructs be given meaning ?
E.g. Operational Semantics, Denotational Semantics.
How should processes be compared ?
E.g. Observable Behavior, Process Equivalences.
How should process properties be specied and proved ?
E.g. Logic for expressing process specications (like in Hoare's
Logic).
How expressive is the calulus ?
E.g. Expressiveness of a nice fragment of the calculus.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Typical Issues in Developing Process Calculi
Which process constructs t the intended phenomena?
E.g. atomic actions, parallelism, nondeterminism.
How should these constructs be given meaning ?
E.g. Operational Semantics, Denotational Semantics.
How should processes be compared ?
E.g. Observable Behavior, Process Equivalences.
How should process properties be specied and proved ?
E.g. Logic for expressing process specications (like in Hoare's
Logic).
How expressive is the calulus ?
E.g. Expressiveness of a nice fragment of the calculus.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Outline
1
From Computability to Concurrency Theory
Basic Concepts from Automata Theory
Bisimilarity Equivalence
2
Calculus of Comunicating Systems CCS
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
3
Mobility and the pi calculus
Syntax and Semantics
Applications
Equivalences
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
CCS Syntax
Basic Entities of CCS:
A set
N = a, b , . . .
co-names.
of
names and N = {a | a ∈ N } of
labels (ranged over by l , l 0 , . . .).
A set Act = L ∪ {τ } of actions (ranged over by a, b, . . .).
Action τ is called the silent or unobservable action.
Actions a and a are viewed as being complementary. I.e.,
a = a.
A set
L=N ∪N
of
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
CCS Syntax
Denition (CCS Syntax of Processes)
P , Q , . . . := 0 | a.P | P k Q | P + Q | (ν a)P | A(a1 , ..., an )
Bound names of P , fn(P ): Those with a bound occurrence in
P.
Free names of P , bn(P ): Those with a not bound occurrence
in P .
For each (call) A(a1 , . . . , an ) there is a unique process
denition A(b1 . . . bn ) = P , with fv (P ) ⊆ {b1 , . . . , bn }.
The set of all processes is denoted by
CLEI'05 Cali
P.
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
CCS: Operational Semantics.
ACT
a
a.P −→
SUM1
P
a
−→ 0
a
+
−→ 0
P
P Q
P
a
−→
a
−→
P
COM1
P
PkQ
P0
P0 k Q
COM3
l P 0 Q −→
l Q0
P −→
P k Q −τ→ P 0 k Q 0
RES
P −a→ P 0
a
(ν a)P −→ (ν a)P 0
REC
PA [b1 , . . . , bn /a1 , . . . , an ] −a→ P 0
A hb1 , . . . , bn i −a→ P 0
SUM2
Q −a→ Q 0
P + Q −a→ Q 0
COM2
Q −a→ Q 0
P k Q −a→ P k Q 0
if a 6= a and a 6= a
CLEI'05 Cali
if
A(a1 , . . . , an ) def
= PA
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Outline
1
From Computability to Concurrency Theory
Basic Concepts from Automata Theory
Bisimilarity Equivalence
2
Calculus of Comunicating Systems CCS
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
3
Mobility and the pi calculus
Syntax and Semantics
Applications
Equivalences
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
CCS: Bisimilarity
Labeled Transition System of CCS
The LTS of CCS has P as its states and its transitions given by
the operational semantics. Dene P ∼ Q i the states
corresponding to P and Q are bisimilar.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
CCS: Bisimilarity
Labeled Transition System of CCS
The LTS of CCS has P as its states and its transitions given by
the operational semantics. Dene P ∼ Q i the states
corresponding to P and Q are bisimilar.
Example: Processes and LTS states
P = a.(b.0 + c .0) corresponds to p0
corresponds to q0 . Hence P 6∼ Q .
CLEI'05 Cali
and
Q = a.b.0 + a.c .0
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
CCS: Bisimilarity
Labeled Transition System of CCS
The LTS of CCS has P as its states and its transitions given by
the operational semantics. Dene P ∼ Q i the states
corresponding to P and Q are bisimilar.
Some Basic Equalities
.
.
.
.
.
.
PkQ∼QkP
Pk0∼P
(P k Q ) k R ∼ P k (Q k R )
(ν a)0 ∼ 0
P k (ν a)Q ∼ (ν a)(P k Q ) ?
(ν a)P ∼ (ν b)P [b/a] ?
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
CCS: The expansion law.
a.0 k b.0 is bisimilar to a.b.0 + b.a.0.
More generally, we have the expansion law:
(ν~a)(P1 k . . . k Pn ) ∼
Notice that
a
i
Σ{ai .(ν~a)(P1 k . . . k Pi0 k . . . k Pn | Pi −→
Pi0 , a, ai 6∈ ~a}
+
l
l
Σ{τ.(ν a)(P1 k . . . k Pi0 k . . . k Pj0 . . . k Pn | Pi −→ Pi0 , Pj −→ Pj0 }
So, every move in
(ν~a)(P1 k . . . k Pn ) is either
Pi and Pj .
one of the
or a communication between some
CLEI'05 Cali
Communicating and Mobile Systems
Pi
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
CCS: The expansion law.
a.0 k b.0 is bisimilar to a.b.0 + b.a.0.
More generally, we have the expansion law:
(ν~a)(P1 k . . . k Pn ) ∼
Notice that
a
i
Σ{ai .(ν~a)(P1 k . . . k Pi0 k . . . k Pn | Pi −→
Pi0 , a, ai 6∈ ~a}
+
l
l
Σ{τ.(ν a)(P1 k . . . k Pi0 k . . . k Pj0 . . . k Pn | Pi −→ Pi0 , Pj −→ Pj0 }
So, every move in
(ν~a)(P1 k . . . k Pn ) is either
Pi and Pj .
one of the
or a communication between some
CLEI'05 Cali
Communicating and Mobile Systems
Pi
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
CCS: The expansion law.
a.0 k b.0 is bisimilar to a.b.0 + b.a.0.
More generally, we have the expansion law:
(ν~a)(P1 k . . . k Pn ) ∼
Notice that
a
i
Σ{ai .(ν~a)(P1 k . . . k Pi0 k . . . k Pn | Pi −→
Pi0 , a, ai 6∈ ~a}
+
l
l
Σ{τ.(ν a)(P1 k . . . k Pi0 k . . . k Pj0 . . . k Pn | Pi −→ Pi0 , Pj −→ Pj0 }
So, every move in
(ν~a)(P1 k . . . k Pn ) is either
Pi and Pj .
one of the
or a communication between some
CLEI'05 Cali
Communicating and Mobile Systems
Pi
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Congruence issues
Suppose that
P ∼ Q.
We would like
P k R ∼ Q k R.
More generally we would like
C [P ] ∼ C [Q ]
C [.] is a process context.
congruence.
where
I.e., we want
The notion of congruence allows us to replace equals.
In fact,
∼
∼
to be a
equals with
is a congruence. How can we prove this ?.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Congruence issues
Suppose that
P ∼ Q.
We would like
P k R ∼ Q k R.
More generally we would like
C [P ] ∼ C [Q ]
C [.] is a process context.
congruence.
where
I.e., we want
The notion of congruence allows us to replace equals.
In fact,
∼
∼
to be a
equals with
is a congruence. How can we prove this ?.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Congruence issues
Suppose that
P ∼ Q.
We would like
P k R ∼ Q k R.
More generally we would like
C [P ] ∼ C [Q ]
C [.] is a process context.
congruence.
where
I.e., we want
The notion of congruence allows us to replace equals.
In fact,
∼
∼
to be a
equals with
is a congruence. How can we prove this ?.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Congruence issues
Suppose that
P ∼ Q.
We would like
P k R ∼ Q k R.
More generally we would like
C [P ] ∼ C [Q ]
C [.] is a process context.
congruence.
where
I.e., we want
The notion of congruence allows us to replace equals.
In fact,
∼
∼
to be a
equals with
is a congruence. How can we prove this ?.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Observable Behavior
equivalent i another process
(the environment, an observer) cannot observe any dierence
In principle,
P
and
Q
should be
in their behavior.
Notice
τ.P 6∼ P ,
could be too
although
strong:).
τ
is an
unobservable action.
So
We look for other notion of equivalence focused in terms of
l
observable actions (i.e., actions −→
, l ∈ L).
CLEI'05 Cali
Communicating and Mobile Systems
∼
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Observable Behavior
equivalent i another process
(the environment, an observer) cannot observe any dierence
In principle,
P
and
Q
should be
in their behavior.
Notice
τ.P 6∼ P ,
could be too
although
strong:).
τ
is an
unobservable action.
So
We look for other notion of equivalence focused in terms of
l
observable actions (i.e., actions −→
, l ∈ L).
CLEI'05 Cali
Communicating and Mobile Systems
∼
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Observable Behavior
equivalent i another process
(the environment, an observer) cannot observe any dierence
In principle,
P
and
Q
should be
in their behavior.
Notice
τ.P 6∼ P ,
could be too
although
strong:).
τ
is an
unobservable action.
So
We look for other notion of equivalence focused in terms of
l
observable actions (i.e., actions −→
, l ∈ L).
CLEI'05 Cali
Communicating and Mobile Systems
∼
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Outline
1
From Computability to Concurrency Theory
Basic Concepts from Automata Theory
Bisimilarity Equivalence
2
Calculus of Comunicating Systems CCS
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
3
Mobility and the pi calculus
Syntax and Semantics
Applications
Equivalences
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Observations
l
−→
l 6= τ as an observation.
Think of an experiment e as a sequence l1 .l2 . . . ln
Think of any
with
of
observable actions.
Notation:
If
s = a1 . . . an ∈ Act ∗
then dene
a1
s
an
τ
τ
τ
τ
=⇒= (−→)∗ −→
(−→)∗ . . . (−→)∗ −→
(−→)∗
e
=⇒
e = l1 .l2 . . . ln ∈ L denotes a sequence of
observable actions inter-spread with τ actions: The notion of
experiment.
Notice that
for
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Observations
l
−→
l 6= τ as an observation.
Think of an experiment e as a sequence l1 .l2 . . . ln
Think of any
with
of
observable actions.
Notation:
If
s = a1 . . . an ∈ Act ∗
then dene
a1
s
an
τ
τ
τ
τ
=⇒= (−→)∗ −→
(−→)∗ . . . (−→)∗ −→
(−→)∗
e
=⇒
e = l1 .l2 . . . ln ∈ L denotes a sequence of
observable actions inter-spread with τ actions: The notion of
experiment.
Notice that
for
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Observations
l
−→
l 6= τ as an observation.
Think of an experiment e as a sequence l1 .l2 . . . ln
Think of any
with
of
observable actions.
Notation:
If
s = a1 . . . an ∈ Act ∗
then dene
a1
s
an
τ
τ
τ
τ
=⇒= (−→)∗ −→
(−→)∗ . . . (−→)∗ −→
(−→)∗
e
=⇒
e = l1 .l2 . . . ln ∈ L denotes a sequence of
observable actions inter-spread with τ actions: The notion of
experiment.
Notice that
for
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Observations
l
−→
l 6= τ as an observation.
Think of an experiment e as a sequence l1 .l2 . . . ln
Think of any
with
of
observable actions.
Notation:
If
s = a1 . . . an ∈ Act ∗
then dene
a1
s
an
τ
τ
τ
τ
=⇒= (−→)∗ −→
(−→)∗ . . . (−→)∗ −→
(−→)∗
e
=⇒
e = l1 .l2 . . . ln ∈ L denotes a sequence of
observable actions inter-spread with τ actions: The notion of
experiment.
Notice that
for
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Trace Equivalence
Denition (Trace Equivalence)
P
and
Q
are
trace equivalent, written P ∼t Q , i for every
e = l1 . . . ln ∈ L∗
experiment (here called trace)
e
e
P =⇒
i Q =⇒
Observations:
τ.P ∼t P (nice!)
a.b.0 + a.c .0 ∼t a.(b.0 + c .0)
(not that nice:)
a.b.0 + a.0 ∼t a.b.0 (not sensitive to deadlocks)
a.0 + b.0 ∼t (ν c )(c .0 k c .a.0 k c .b.0)
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Trace Equivalence
Denition (Trace Equivalence)
P
and
Q
are
trace equivalent, written P ∼t Q , i for every
e = l1 . . . ln ∈ L∗
experiment (here called trace)
e
e
P =⇒
i Q =⇒
Observations:
τ.P ∼t P (nice!)
a.b.0 + a.c .0 ∼t a.(b.0 + c .0)
(not that nice:)
a.b.0 + a.0 ∼t a.b.0 (not sensitive to deadlocks)
a.0 + b.0 ∼t (ν c )(c .0 k c .a.0 k c .b.0)
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Failures Equivalence
Denition (Failures Equivalence)
A pair
(e , L),
where
0
i there is P s.t.:
e ∈ L∗
(a
trace) and L ⊂ L, is a failure for P
e
l
τ
(1)P =⇒ P 0 , (2) P 0 6−→ for all l ∈ L, and (3)P 0 6−→
P ∼f Q
i
P
and
Q
have the same failures.
Observations:
τ.P ∼f P .
a.b.0 + a.c .0 6∼f a.(b.0 + c .0) (Exercise: why?)
a.b.0 + a.0 6∼f a.b.0.
a.0 + b.0 6∼f (ν c )(c .0 k c .a.0 k c .b.0) (Exercise)
a.(b.c .0 + b.d .0) ∼f a.b.c .0 + a.b.d .0.
Let D = τ.D . We have
0 6∼f Communicating
D.
CLEI'05τ.
Cali
and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Failures Equivalence
Denition (Failures Equivalence)
A pair
(e , L),
where
0
i there is P s.t.:
e ∈ L∗
(a
trace) and L ⊂ L, is a failure for P
e
l
τ
(1)P =⇒ P 0 , (2) P 0 6−→ for all l ∈ L, and (3)P 0 6−→
P ∼f Q
i
P
and
Q
have the same failures.
Observations:
τ.P ∼f P .
a.b.0 + a.c .0 6∼f a.(b.0 + c .0) (Exercise: why?)
a.b.0 + a.0 6∼f a.b.0.
a.0 + b.0 6∼f (ν c )(c .0 k c .a.0 k c .b.0) (Exercise)
a.(b.c .0 + b.d .0) ∼f a.b.c .0 + a.b.d .0.
Let D = τ.D . We have
0 6∼f Communicating
D.
CLEI'05τ.
Cali
and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Weak Bisimilarity (Observational Equivalence)
Denition (Weak Bisimilarity)
A
symmetric relation R
is a
weak bisimulation i for every
(P , Q ) ∈ R :
e
e
. If P =⇒ P 0 then there is Q 0 s.t. Q =⇒ Q 0 and (P 0 , Q 0 ) ∈ R .
Dene P ≈ Q i (P , Q ) ∈ R for some weak bisimulation R .
Observations:
τ.P ≈ P
a.τ.P ≈ a.P
However, a.0 + b .0 6≈ a.0 + τ.b .0
a.(b.c .0 + b.d .0) 6≈ a.b.c .0 + a.b.d .0.(Exercise)
Let D = τ.D . We have τ.0 ≈ D .
a.0 + b.0 6≈ (ν c )(c .0 k c .a.0 k c .b.0) (Exercise)
,
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Weak Bisimilarity (Observational Equivalence)
Denition (Weak Bisimilarity)
A
symmetric relation R
is a
weak bisimulation i for every
(P , Q ) ∈ R :
e
e
. If P =⇒ P 0 then there is Q 0 s.t. Q =⇒ Q 0 and (P 0 , Q 0 ) ∈ R .
Dene P ≈ Q i (P , Q ) ∈ R for some weak bisimulation R .
Observations:
τ.P ≈ P
a.τ.P ≈ a.P
However, a.0 + b .0 6≈ a.0 + τ.b .0
a.(b.c .0 + b.d .0) 6≈ a.b.c .0 + a.b.d .0.(Exercise)
Let D = τ.D . We have τ.0 ≈ D .
a.0 + b.0 6≈ (ν c )(c .0 k c .a.0 k c .b.0) (Exercise)
,
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Process Logic: Verication and Specication
Processes can be used to
specify and verify behavioural
properties:
E.g., 2
p .tea.0 + 2p .coe .0 species a machine which does not
p .(tea.0 + coe .0).
satisfy the behaviour specied by a 2
Alternatively, a logic can be used. An example of a property is
P
will never not execute a bad action or P
eventually
executes execute a good action.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Process Logic: Verication and Specication
Processes can be used to
specify and verify behavioural
properties:
E.g., 2
p .tea.0 + 2p .coe .0 species a machine which does not
p .(tea.0 + coe .0).
satisfy the behaviour specied by a 2
Alternatively, a logic can be used. An example of a property is
P
will never not execute a bad action or P
eventually
executes execute a good action.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Outline
1
From Computability to Concurrency Theory
Basic Concepts from Automata Theory
Bisimilarity Equivalence
2
Calculus of Comunicating Systems CCS
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
3
Mobility and the pi calculus
Syntax and Semantics
Applications
Equivalences
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Hennessy&Milner Logic
Denition (Logic Syntax)
F := true | false | F1 ∧ F2 | F1 ∨ F2 | hK i F | [K ]F
where
K
is a set of actions
The boolean operators are interpreted as in propositional logic.
hK i F (possibility) asserts (of a given P ): It is possible
to do a a ∈ K and then evolve into a Q that satisfy F .
[K ]F (necessity)
then it
asserts (of a given
must evolve into a Q
CLEI'05 Cali
P ):
If
which satises
for
P
P can do a a ∈ K
F.
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Hennessy&Milner Logic
Denition (Logic Syntax)
F := true | false | F1 ∧ F2 | F1 ∨ F2 | hK i F | [K ]F
where
K
is a set of actions
The boolean operators are interpreted as in propositional logic.
hK i F (possibility) asserts (of a given P ): It is possible
to do a a ∈ K and then evolve into a Q that satisfy F .
[K ]F (necessity)
then it
asserts (of a given
must evolve into a Q
CLEI'05 Cali
P ):
If
which satises
for
P
P can do a a ∈ K
F.
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Hennessy&Milner Logic
Denition (Logic Syntax)
F := true | false | F1 ∧ F2 | F1 ∨ F2 | hK i F | [K ]F
where
K
is a set of actions
The boolean operators are interpreted as in propositional logic.
hK i F (possibility) asserts (of a given P ): It is possible
to do a a ∈ K and then evolve into a Q that satisfy F .
[K ]F (necessity)
then it
asserts (of a given
must evolve into a Q
CLEI'05 Cali
P ):
If
which satises
for
P
P can do a a ∈ K
F.
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Hennessy&Milner Logic
Denition (Logic Syntax)
F := true | false | F1 ∧ F2 | F1 ∨ F2 | hK i F | [K ]F
where
K
is a set of actions
The boolean operators are interpreted as in propositional logic.
hK i F (possibility) asserts (of a given P ): It is possible
to do a a ∈ K and then evolve into a Q that satisfy F .
[K ]F (necessity)
then it
asserts (of a given
must evolve into a Q
CLEI'05 Cali
P ):
If
which satises
for
P
P can do a a ∈ K
F.
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Semantics of Hennessy-Milner Logic
Denition
The compliance of
P
with the specication
F , written P |= F , is
given by:
P
P
P
P
P
P
6|= false
|= true
|= F1 ∧ F2
|= F1 ∨ F2
|= hK i F
|= [K ]F
i
i
i
i
P |= F1
P |= F1
P |= F2
P |= F2
a
for some Q , P −→ Q , a ∈ K and Q |= F
a
if P −→ Q and a ∈ K then Q |= F
and
or
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Hennessy&Milner Logic and Bisimilarity
Example
Let
P1 = a.(b.0 + c .0), P2 = a.b.0 + a.c .0.
Also let
F = h{a}i (h{b}i true ∧ h{c }i true).
Notice that
P1 |= F
but
P2 6|= F .
Theorem
P ∼ Q if and only, for every F , P |= F i Q |= F .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Hennessy&Milner Logic and Bisimilarity
Example
Let
P1 = a.(b.0 + c .0), P2 = a.b.0 + a.c .0.
Also let
F = h{a}i (h{b}i true ∧ h{c }i true).
Notice that
P1 |= F
but
P2 6|= F .
Theorem
P ∼ Q if and only, for every F , P |= F i Q |= F .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
A Linear Temporal Logic.
Denition (Logic Syntax)
F := true | false | L | F1 ∨ F2 | F1 ∧ F2 | ♦F | F
where
L is a set of non-silent actions.
Formulae assert properties of traces.
Boolean operators are interpreted as usual
L asserts (of a given trace s ) that the rst action of s must be
in L ∪ {τ }
♦F asserts (of a given trace s ) that at some point in s , F
holds.
F
asserts (of a given trace
s ) that at every point in s , F
holds.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
A Linear Temporal Logic.
Denition (Logic Syntax)
F := true | false | L | F1 ∨ F2 | F1 ∧ F2 | ♦F | F
where
L is a set of non-silent actions.
Formulae assert properties of traces.
Boolean operators are interpreted as usual
L asserts (of a given trace s ) that the rst action of s must be
in L ∪ {τ }
♦F asserts (of a given trace s ) that at some point in s , F
holds.
F
asserts (of a given trace
s ) that at every point in s , F
holds.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
A Linear Temporal Logic.
Denition (Logic Syntax)
F := true | false | L | F1 ∨ F2 | F1 ∧ F2 | ♦F | F
where
L is a set of non-silent actions.
Formulae assert properties of traces.
Boolean operators are interpreted as usual
L asserts (of a given trace s ) that the rst action of s must be
in L ∪ {τ }
♦F asserts (of a given trace s ) that at some point in s , F
holds.
F
asserts (of a given trace
s ) that at every point in s , F
holds.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
A Linear Temporal Logic.
Denition (Logic Syntax)
F := true | false | L | F1 ∨ F2 | F1 ∧ F2 | ♦F | F
where
L is a set of non-silent actions.
Formulae assert properties of traces.
Boolean operators are interpreted as usual
L asserts (of a given trace s ) that the rst action of s must be
in L ∪ {τ }
♦F asserts (of a given trace s ) that at some point in s , F
holds.
F
asserts (of a given trace
s ) that at every point in s , F
holds.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
A Linear Temporal Logic.
Denition (Logic Syntax)
F := true | false | L | F1 ∨ F2 | F1 ∧ F2 | ♦F | F
where
L is a set of non-silent actions.
Formulae assert properties of traces.
Boolean operators are interpreted as usual
L asserts (of a given trace s ) that the rst action of s must be
in L ∪ {τ }
♦F asserts (of a given trace s ) that at some point in s , F
holds.
F
asserts (of a given trace
s ) that at every point in s , F
holds.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Semantics of the temporal logic.
Denition
s = a1 .a2 . . . satises (or is a model
F , written s |= F , i hs , 1i |= F , where
An innite sequence of actions
of )
hs , i i |= true
hs , i i 6|= false
hs , i i |= L
hs , i i |= F1 ∨ F2
hs , i i |= F1 ∧ F2
hs , i i |= F
hs , i i |= ♦F
P |= F
i whenever
i
i
i
i
i
ai ∈ L ∪ τ
hs , i i |= F1 or hs , i i |= F2
hs , i i |= F1 and hs , i i |= F2
for all j ≥ i hs , j i |= F
there is a j ≥ i s.t. hs , j i |= F
s
P =⇒
then ŝ |= F , where ŝ = s .τ.τ. . . .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Temporal Logic and Trace Equivalence
Example
Consider the trace equivalent processes
A(a, b, c ) def
= a.(b.A(a, b, c ) + c .A(a, b, c )) and
def
B (a, b, c ) = a.b.B (a, b, c ) + a.c .B (a, b, c ), and the property
F = ♦(b ∨ c ).
We have A(a, b , c ) |= F and B (a, b , c ) |= F .
Theorem
P ∼t Q then for every linear temporal formula F ,
P |= F i Q |= F
Question. Does the other direction of the theorem hold ?
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Temporal Logic and Trace Equivalence
Example
Consider the trace equivalent processes
A(a, b, c ) def
= a.(b.A(a, b, c ) + c .A(a, b, c )) and
def
B (a, b, c ) = a.b.B (a, b, c ) + a.c .B (a, b, c ), and the property
F = ♦(b ∨ c ).
We have A(a, b , c ) |= F and B (a, b , c ) |= F .
Theorem
P ∼t Q then for every linear temporal formula F ,
P |= F i Q |= F
Question. Does the other direction of the theorem hold ?
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
Temporal Logic and Trace Equivalence
Example
Consider the trace equivalent processes
A(a, b, c ) def
= a.(b.A(a, b, c ) + c .A(a, b, c )) and
def
B (a, b, c ) = a.b.B (a, b, c ) + a.c .B (a, b, c ), and the property
F = ♦(b ∨ c ).
We have A(a, b , c ) |= F and B (a, b , c ) |= F .
Theorem
P ∼t Q then for every linear temporal formula F ,
P |= F i Q |= F
Question. Does the other direction of the theorem hold ?
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Outline
1
From Computability to Concurrency Theory
Basic Concepts from Automata Theory
Bisimilarity Equivalence
2
Calculus of Comunicating Systems CCS
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
3
Mobility and the pi calculus
Syntax and Semantics
Applications
Equivalences
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Mobility and the pi calculus
What kind of
mobility are we talking about?
Processes move.
√
Links move
The last one, because it is the
π -calculus'
choice; for exibility and
simplicity.
π -calculus
names).
has the ability of sending private and public
(
CLEI'05 Cali
Communicating and Mobile Systems
links
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Mobility and the pi calculus
What kind of
mobility are we talking about?
Processes move.
√
Links move
The last one, because it is the
π -calculus'
choice; for exibility and
simplicity.
π -calculus
names).
has the ability of sending private and public
(
CLEI'05 Cali
Communicating and Mobile Systems
links
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Mobility and the pi calculus
What kind of
mobility are we talking about?
Processes move.
√
Links move
The last one, because it is the
π -calculus'
choice; for exibility and
simplicity.
π -calculus
names).
has the ability of sending private and public
(
CLEI'05 Cali
Communicating and Mobile Systems
links
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Link Mobility
Mobile phones
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Link Mobility
Mobile phones
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
π−Calculus: Syntax
Denition (Syntax)
P , Q , . . . := P k Q | Σi ∈I α.P | (ν a)P | !P | if a = b then P
where
α := τ | ab | a(x ).
Names=Channels=Ports=Links.
ab.P : send b on channel a and then activate P a(x ).P : receive a name on channel a (if any), and replace x
with it in P (ν a)P : create a fresh name a private to P (ν a)P and a(x ).Q are the only binders.
!P : replicate P i.e., !P represents P k P k P k . . .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
π−Calculus: Syntax
Denition (Syntax)
P , Q , . . . := P k Q | Σi ∈I α.P | (ν a)P | !P | if a = b then P
where
α := τ | ab | a(x ).
Names=Channels=Ports=Links.
ab.P : send b on channel a and then activate P a(x ).P : receive a name on channel a (if any), and replace x
with it in P (ν a)P : create a fresh name a private to P (ν a)P and a(x ).Q are the only binders.
!P : replicate P i.e., !P represents P k P k P k . . .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
π−Calculus: Syntax
Denition (Syntax)
P , Q , . . . := P k Q | Σi ∈I α.P | (ν a)P | !P | if a = b then P
where
α := τ | ab | a(x ).
Names=Channels=Ports=Links.
ab.P : send b on channel a and then activate P a(x ).P : receive a name on channel a (if any), and replace x
with it in P (ν a)P : create a fresh name a private to P (ν a)P and a(x ).Q are the only binders.
!P : replicate P i.e., !P represents P k P k P k . . .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
π−Calculus: Syntax
Denition (Syntax)
P , Q , . . . := P k Q | Σi ∈I α.P | (ν a)P | !P | if a = b then P
where
α := τ | ab | a(x ).
Names=Channels=Ports=Links.
ab.P : send b on channel a and then activate P a(x ).P : receive a name on channel a (if any), and replace x
with it in P (ν a)P : create a fresh name a private to P (ν a)P and a(x ).Q are the only binders.
!P : replicate P i.e., !P represents P k P k P k . . .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
π−Calculus: Syntax
Denition (Syntax)
P , Q , . . . := P k Q | Σi ∈I α.P | (ν a)P | !P | if a = b then P
where
α := τ | ab | a(x ).
Names=Channels=Ports=Links.
ab.P : send b on channel a and then activate P a(x ).P : receive a name on channel a (if any), and replace x
with it in P (ν a)P : create a fresh name a private to P (ν a)P and a(x ).Q are the only binders.
!P : replicate P i.e., !P represents P k P k P k . . .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
π−Calculus: Syntax
Denition (Syntax)
P , Q , . . . := P k Q | Σi ∈I α.P | (ν a)P | !P | if a = b then P
where
α := τ | ab | a(x ).
Names=Channels=Ports=Links.
ab.P : send b on channel a and then activate P a(x ).P : receive a name on channel a (if any), and replace x
with it in P (ν a)P : create a fresh name a private to P (ν a)P and a(x ).Q are the only binders.
!P : replicate P i.e., !P represents P k P k P k . . .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
π−Calculus: Syntax
Denition (Syntax)
P , Q , . . . := P k Q | Σi ∈I α.P | (ν a)P | !P | if a = b then P
where
α := τ | ab | a(x ).
Names=Channels=Ports=Links.
ab.P : send b on channel a and then activate P a(x ).P : receive a name on channel a (if any), and replace x
with it in P (ν a)P : create a fresh name a private to P (ν a)P and a(x ).Q are the only binders.
!P : replicate P i.e., !P represents P k P k P k . . .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
A avour of the pi-calculus
In the π calculus:
(ν a)(b̄a.S k a(x ).P ) k b(c ).c̄d .C
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
A avour of the pi-calculus
π the pi-calculus:
(ν a)(b̄a.S k a(x ).P ) k b(c ).c̄d .C −→ S k (ν a)(a(x ).P k ād .C )
In
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Mobile Exercises
Write an agent that:
reads something from port
generates
innitely many
a.
a and sends it twice along port b
dierent namesand send them
along channel
x̄z1 z2 .P which outputs both y1
x , and x (y1 , y2 ).Q which inputs two names
encodes biadic-comunication:
and
y2
on channel
sent on channel
x.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Mobile Exercises
Write an agent that:
reads something from port
generates
innitely many
a.
a and sends it twice along port b
dierent namesand send them
along channel
x̄z1 z2 .P which outputs both y1
x , and x (y1 , y2 ).Q which inputs two names
encodes biadic-comunication:
and
y2
on channel
sent on channel
x.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Mobile Exercises
Write an agent that:
reads something from port
generates
innitely many
a.
a and sends it twice along port b
dierent namesand send them
along channel
x̄z1 z2 .P which outputs both y1
x , and x (y1 , y2 ).Q which inputs two names
encodes biadic-comunication:
and
y2
on channel
sent on channel
x.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Reaction Semantics of π .
The reactive semantics of
≡
and the
reactive rules.
π
The relation≡ describes
consists of an
structural congruence
irrelevant syntactic aspects
The reactive semantics describes evolutions due to
synchronous communication.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Reaction Semantics of π .
The reactive semantics of
≡
and the
reactive rules.
π
The relation≡ describes
consists of an
structural congruence
irrelevant syntactic aspects
The reactive semantics describes evolutions due to
synchronous communication.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Reaction Semantics of π .
The reactive semantics of
≡
and the
reactive rules.
π
The relation≡ describes
consists of an
structural congruence
irrelevant syntactic aspects
The reactive semantics describes evolutions due to
synchronous communication.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Structural Equivalence
Structural Congruence
Structural Congruence ≡ is the congruence given by:
P ≡ Q if P can be α-converted into Q .
P k 0 ≡ P , P k Q ≡ Q k P , (P k Q ) k R = P k (Q k R ).
(ν a)0 = 0,(ν a)(ν b)P ≡ (ν b)(ν a)P ,
(ν a)(P k Q ) ≡ P k (ν a)Q if a 6∈ fn(P ).
!P ≡ P k!P .
Example
(1)
(2)
(3)
(ν a)P ≡ P if a 6∈ fn(P )?
x (y ).y (z ) ≡ x (z ).y (y ) ?
x k y ≡ x .y + y .x ?
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Structural Equivalence
Structural Congruence
Structural Congruence ≡ is the congruence given by:
P ≡ Q if P can be α-converted into Q .
P k 0 ≡ P , P k Q ≡ Q k P , (P k Q ) k R = P k (Q k R ).
(ν a)0 = 0,(ν a)(ν b)P ≡ (ν b)(ν a)P ,
(ν a)(P k Q ) ≡ P k (ν a)Q if a 6∈ fn(P ).
!P ≡ P k!P .
Example
(1)
(2)
(3)
(ν a)P ≡ P if a 6∈ fn(P )?
x (y ).y (z ) ≡ x (z ).y (y ) ?
x k y ≡ x .y + y .x ?
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Reactive Rules
TAU
τ.P + M −→
REACT
PAR
P
(a(x ).P + M ) k (ab .Q + N ) −→
P [b/x ] k Q
P −→ P 0
P k Q −→ P 0 k P
P −→ P 0
(ν a)P −→ (ν a)P 0
P ≡ P 0 −→ Q 0 ≡ Q
STRUCT
P −→ Q
RES
Exercise.
Assume
a 6∈ fn(S ) ∪ fn(C ).
Write a derivation for our
Server example:
(ν a)(b̄a.S k a(x ).P ) k b (c ).c̄d .C −→ S k (ν a)(a(x ).P k ād .C )
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Outline
1
From Computability to Concurrency Theory
Basic Concepts from Automata Theory
Bisimilarity Equivalence
2
Calculus of Comunicating Systems CCS
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
3
Mobility and the pi calculus
Syntax and Semantics
Applications
Equivalences
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Applications: Persistent and Mutable Data Structures
Denition
A
cell
is basis for dening mutable and persistent data structures.
Cell (u , r , z ) def
= u (z 0 ).Cell (u , r , z 0 ) + r (x ).x̄z .Cell (u , r , z )
The cell current contents
z
can be updated via
But we do not have recursive denitions in the
above cell be rewritten in with the
CLEI'05 Cali
π -calculus
u
or read via
π -calculus.
r.
Can the
primitives ?
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Applications: Persistent and Mutable Data Structures
Denition
A
cell
is basis for dening mutable and persistent data structures.
Cell (u , r , z ) def
= u (z 0 ).Cell (u , r , z 0 ) + r (x ).x̄z .Cell (u , r , z )
The cell current contents
z
can be updated via
But we do not have recursive denitions in the
above cell be rewritten in with the
CLEI'05 Cali
π -calculus
u
or read via
π -calculus.
r.
Can the
primitives ?
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Recursive Denitions
We wish invocations
def
A(x ) = P .
Do we need to extend the calculus ?
No! Jus take an
1
A(z ) to recursive denition (as in CCS)
Replace in
a 6∈ fn(P ):
P
every invocation
A(u ) with āu .
Call
P̂
the
resulting processes.
1
Encode
A(x ) = P
1
Encode
A(z ) as (ν a)(ā(z ) k!a(x ).P̂ ).
as
!a(x ).P̂ .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Recursive Denitions
We wish invocations
def
A(x ) = P .
Do we need to extend the calculus ?
No! Jus take an
1
A(z ) to recursive denition (as in CCS)
Replace in
a 6∈ fn(P ):
P
every invocation
A(u ) with āu .
Call
P̂
the
resulting processes.
1
Encode
A(x ) = P
1
Encode
A(z ) as (ν a)(ā(z ) k!a(x ).P̂ ).
as
!a(x ).P̂ .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Recursive Denitions
We wish invocations
def
A(x ) = P .
Do we need to extend the calculus ?
No! Jus take an
1
A(z ) to recursive denition (as in CCS)
Replace in
a 6∈ fn(P ):
P
every invocation
A(u ) with āu .
Call
P̂
the
resulting processes.
1
Encode
A(x ) = P
1
Encode
A(z ) as (ν a)(ā(z ) k!a(x ).P̂ ).
as
!a(x ).P̂ .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Recursive Denitions
We wish invocations
def
A(x ) = P .
Do we need to extend the calculus ?
No! Jus take an
1
A(z ) to recursive denition (as in CCS)
Replace in
a 6∈ fn(P ):
P
every invocation
A(u ) with āu .
Call
P̂
the
resulting processes.
1
Encode
A(x ) = P
1
Encode
A(z ) as (ν a)(ā(z ) k!a(x ).P̂ ).
as
!a(x ).P̂ .
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
π Equivalences
Should
P = x (z ) k ȳu
π like
in CCS?
Take
R = a(x ).P
Hence,
āy
be equivalent to
and
can tell
R
Q = x (z ).ȳu + ȳu .x (z ) in
S = a(x ).Q .
and
S
apart!. I.e.,
āy k R .
Passing links around complicates mattersnot unduly, though:).
Need to consider substitutions, contexts, etc!
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
π Equivalences
Should
P = x (z ) k ȳu
π like
in CCS?
Take
R = a(x ).P
Hence,
āy
be equivalent to
and
can tell
R
Q = x (z ).ȳu + ȳu .x (z ) in
S = a(x ).Q .
and
S
apart!. I.e.,
āy k R .
Passing links around complicates mattersnot unduly, though:).
Need to consider substitutions, contexts, etc!
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
π Equivalences
Should
P = x (z ) k ȳu
π like
in CCS?
Take
R = a(x ).P
Hence,
āy
be equivalent to
and
can tell
R
Q = x (z ).ȳu + ȳu .x (z ) in
S = a(x ).Q .
and
S
apart!. I.e.,
āy k R .
Passing links around complicates mattersnot unduly, though:).
Need to consider substitutions, contexts, etc!
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Outline
1
From Computability to Concurrency Theory
Basic Concepts from Automata Theory
Bisimilarity Equivalence
2
Calculus of Comunicating Systems CCS
General Aspects of Process Calculi
Syntax and Semantics
Bisimilarity
Observable Behaviour and other Equivalences
Verication and Specication
3
Mobility and the pi calculus
Syntax and Semantics
Applications
Equivalences
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Barbed Equivalence
Intuitively we write
channel
x.
P ↓l
Formally,
P ↓x (P ↓x̄ ) i P
where
can receive (send) a link on
l ∈ L is given by:
x (y ).P ↓x .
x̄z .P ↓x̄ .
Σi ∈I αi .Pi ↓l i for some i ∈ I , αi .Pi ↓l
P k Q ↓l i either P ↓l or Q ↓l
(ν z )P ↓l i l 6= z , l̄ 6= z and P ↓l
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Barbed Equivalence
Denition
barbed simulation i for every (P , Q ) ∈ R :
P −→ P 0 then ∃Q 0 : Q −→ Q 0 ∧ (P 0 , Q 0 ) ∈ R .
(2) If P ↓l then Q ↓l .
R
is a
(1) If
Denition
Barbed bisimilarity: P ∼B Q i there is R
are barbed simulations and (P , Q ) ∈ R .
such that
R
and
R −1
Example
P = x (z ) k ȳu and Q = x (z ).ȳu + ȳu .x (z ).
anda(x ).P ∼B a(x ).Q .
Let
CLEI'05 Cali
Then
P ∼B Q
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Barbed Equivalence
Denition
barbed simulation i for every (P , Q ) ∈ R :
P −→ P 0 then ∃Q 0 : Q −→ Q 0 ∧ (P 0 , Q 0 ) ∈ R .
(2) If P ↓l then Q ↓l .
R
is a
(1) If
Denition
Barbed bisimilarity: P ∼B Q i there is R
are barbed simulations and (P , Q ) ∈ R .
such that
R
and
R −1
Example
P = x (z ) k ȳu and Q = x (z ).ȳu + ȳu .x (z ).
anda(x ).P ∼B a(x ).Q .
Let
CLEI'05 Cali
Then
P ∼B Q
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Barbed Equivalence
Denition
barbed simulation i for every (P , Q ) ∈ R :
P −→ P 0 then ∃Q 0 : Q −→ Q 0 ∧ (P 0 , Q 0 ) ∈ R .
(2) If P ↓l then Q ↓l .
R
is a
(1) If
Denition
Barbed bisimilarity: P ∼B Q i there is R
are barbed simulations and (P , Q ) ∈ R .
such that
R
and
R −1
Example
P = x (z ) k ȳu and Q = x (z ).ȳu + ȳu .x (z ).
anda(x ).P ∼B a(x ).Q .
Let
CLEI'05 Cali
Then
P ∼B Q
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Barbed Congruence.
Let
≈B
be the congruence
induced by ∼B .
I.e:
Denition
P ≈B Q
i
for every context
C [.], C [P ] ∼B C [Q ]
Example
Note that
x (z ) k ȳu 6≈B x (z ).ȳu + ȳu .x (z ).
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Barbed Congruence.
Let
≈B
be the congruence
induced by ∼B .
I.e:
Denition
P ≈B Q
i
for every context
C [.], C [P ] ∼B C [Q ]
Example
Note that
x (z ) k ȳu 6≈B x (z ).ȳu + ȳu .x (z ).
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
A Labelled Semantics for Pi.
Transitions
α
P −→
Q
where
α ∈ {āb, a(x ), τ, ā ν u }
CLEI'05 Cali
are given by:
Communicating and Mobile Systems
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Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Early Bisimilarity for Pi
Denition
is a simulation i for ∀(P , Q ) ∈ R :
α
P −→
P 0 then
0
(1) If α is an input a(x ) then ∀u , ∃Q :
α
0
0
0
Q −→ Q ∧ (P [u /x ], Q [u /x ]) ∈ R
α
0 ∧ (P 0 , Q 0 ) ∈ R
(2) If α is not an input then ∃Q : Q −→ Q
R
If
.
Denition
(Early) Bisimilarity: P ∼E Q
simulations and (P , Q ) ∈ R .
Example: Let
i there is
R
such
R
and
R −1
P = x (z ) k ȳu and Q = x (z ).ȳu + ȳu .x (z ).
a(x ).P 6∼E a(x ).Q but P ∼E Q .
Then
CLEI'05 Cali
Communicating and Mobile Systems
are
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Early Congruence for Pi.
Denition
Dene
P ≈E Q
i
P σ ∼E Q σ
Example: Note that
The relation
≈E
for all substitutions
σ.
x (z ) k ȳu ≈
6 E x (z ).ȳu + ȳu .x (z ).
is a congruence. Furthermore
CLEI'05 Cali
≈E =≈B
In fact,
!.
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Thanks
Example
On rst slide.
Example
On second slide.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Thanks
Example
On rst slide.
Example
On second slide.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Make Titles Informative.
Theorem
On rst slide.
Corollary
On second slide.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Make Titles Informative.
Theorem
On rst slide.
Corollary
On second slide.
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Make Titles Informative.
Theorem
Corollary
In left column.
In right column.
New line
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Syntax and Semantics
Applications
Equivalences
Make Titles Informative.
Theorem
Corollary
In left column.
In right column.
New line
CLEI'05 Cali
Communicating and Mobile Systems
From Computability to Concurrency Theory
Calculus of Comunicating Systems CCS
Mobility and the pi calculus
Summary
Summary
The rst main message of your talk in one or two lines.
The second main message of your talk in one or two lines.
Perhaps a third message, but not more than that.
Outlook
What we have not done yet.
Even more stu.
CLEI'05 Cali
Communicating and Mobile Systems