On the Expressiveness of Infinite Behavior and Name Scoping in - - PowerPoint PPT Presentation

On the Expressiveness of Infinite Behavior and Name Scoping in Process Calculi

Pablo Giambiagi (KTH, Sweden) Gerardo Schneider (IRISA/INRIA) Speaker: Frank D. Valencia (Uppsala Univ., Sweden) FOSSACS’04, Barcelona, March 2004

FOSSACS’04: Inf. Behaviour and Scoping – p.1/34

Motivation: Process Calculi

CCS is an important representative of process calculi

FOSSACS’04: Inf. Behaviour and Scoping – p.2/34

Motivation: Process Calculi

CCS is an important representative of process calculi : Simplicity of its definition and techniques, Applicability to synchr. communication. Foundational ideas grown out from it.

FOSSACS’04: Inf. Behaviour and Scoping – p.2/34

Motivation: Process Calculi

CCS is an important representative of process calculi : Simplicity of its definition and techniques, Applicability to synchr. communication. Foundational ideas grown out from it. This talk: CCS Variants wrt infinite behavior and name scoping. Variants exhibits interesting connections wrt Relative Expressiveness. Verification.

FOSSACS’04: Inf. Behaviour and Scoping – p.2/34

Motivation: Process Calculi

CCS is an important representative of process calculi : Simplicity of its definition and techniques, Applicability to synchr. communication. Foundational ideas grown out from it. This talk: CCS Variants wrt infinite behavior and name scoping. Variants exhibits interesting connections wrt Relative Expressiveness. Verification.

FOSSACS’04: Inf. Behaviour and Scoping – p.2/34

Motivation: Scoping Variants

Consider CCS construct

• ✁✄✂

with rule: REC

• ✁✄✂

• ✁✄✂

FOSSACS’04: Inf. Behaviour and Scoping – p.3/34

Motivation: Scoping Variants

Consider CCS construct

• ✁✄✂

with rule: REC

• ✁✄✂

• ✁✄✂

Does

involve name

• conversion ?

FOSSACS’04: Inf. Behaviour and Scoping – p.3/34

Motivation: Scoping Variants

Consider CCS construct

• ✁✄✂

with rule: REC

• ✁✄✂

• ✁✄✂

Does

involve name

• conversion ?

Name Scoping: If yes, Static else Dynamic.

FOSSACS’04: Inf. Behaviour and Scoping – p.3/34

Motivation: Scoping Variants

Consider CCS construct

• ✁✄✂

with rule: REC

• ✁✄✂

• ✁✄✂

Does

involve name

• conversion ?

Name Scoping: If yes, Static else Dynamic. This affects not only semantics (e.g.,

• equivalence), also

expressiveness and verification.

FOSSACS’04: Inf. Behaviour and Scoping – p.3/34

Motivation: Infinite Behaviour

Ways of specifying infinite behaviour in process calculi: Parametric vs. Constant definitions

• 1. Finitely many constants
• ✁✄✂

• 2. Finitely many definitions
• ✝✟✞

FOSSACS’04: Inf. Behaviour and Scoping – p.4/34

Motivation: Infinite Behaviour

Ways of specifying infinite behaviour in process calculi: Parametric vs. Constant definitions

• 1. Finitely many constants
• ✁✄✂

• 2. Finitely many definitions
• ✝✟✞

Can we encode (2) into (1) ? (Without Relabelling or Infinite Sums).

FOSSACS’04: Inf. Behaviour and Scoping – p.4/34

Motivation: Infinite Behaviour

Ways of specifying infinite behaviour in process calculi: Parametric vs. Constant definitions

• 1. Finitely many constants
• ✁✄✂

• 2. Finitely many definitions
• ✝✟✞

Can we encode (2) into (1) ? (Without Relabelling or Infinite Sums). What about other constructions: Replication or Recursive Expressions ?

FOSSACS’04: Inf. Behaviour and Scoping – p.4/34

Motivation: Infinite Behaviour

Ways of specifying infinite behaviour in process calculi: Parametric vs. Constant definitions

• 1. Finitely many constants
• ✁✄✂

• 2. Finitely many definitions
• ✝✟✞

Can we encode (2) into (1) ? (Without Relabelling or Infinite Sums). What about other constructions: Replication or Recursive Expressions ? In

• calculus all forms coincide. What about less expressive cal-

culi?.

FOSSACS’04: Inf. Behaviour and Scoping – p.4/34

Motivation and Contributions

We discuss: Static vs Dynamic Scoping. Parametric vs. Constant definitions. Recursion vs Replication.

FOSSACS’04: Inf. Behaviour and Scoping – p.5/34

Motivation and Contributions

We discuss: Static vs Dynamic Scoping. Parametric vs. Constant definitions. Recursion vs Replication. ...and show that these issues affect Expressiveness Decidability of Divergency.

FOSSACS’04: Inf. Behaviour and Scoping – p.5/34

Overview

The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks

FOSSACS’04: Inf. Behaviour and Scoping – p.6/34

Overview

The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks

FOSSACS’04: Inf. Behaviour and Scoping – p.6/34

The Finite Core: Syntax

Given: A set of names

• ✡

A set of co-names

• ✆
• ✄

A set of actions

FOSSACS’04: Inf. Behaviour and Scoping – p.7/34

The Finite Core: Syntax

Given: A set of names

• ✡

A set of co-names

• ✆
• ✄

A set of actions

Processes specifying finite behaviour:

• ✆

• ✆

FOSSACS’04: Inf. Behaviour and Scoping – p.7/34

The Finite Core: Semantics

SUM

• ✁

if

RES

• ✠

• ✠

• if

• ✡
• ✝

PAR

• ✠

• ✠

PAR

• ✠

• ✠

COM

FOSSACS’04: Inf. Behaviour and Scoping – p.8/34

Overview of the presentation

The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks

FOSSACS’04: Inf. Behaviour and Scoping – p.9/34

Scoping: Example

Consider

• ✁✄✂

with

• ✞

• ✂

• FOSSACS’04: Inf. Behaviour and Scoping – p.10/34

Scoping: Example

Consider

• ✁✄✂

with

• ✞

• ✂

• Consider the following rule:

REC

• ✁✄✂

• ✠

• ✁✄✂

• ✠

without name

• conversion.

FOSSACS’04: Inf. Behaviour and Scoping – p.10/34

Scoping: Example

Consider

• ✁✄✂

with

• ✞

• ✂

• Then, for the unfolding

• ✁✄✂

• ✞

• ✂

• ✁✄✂

• FOSSACS’04: Inf. Behaviour and Scoping – p.10/34

Scoping: Example

Consider

• ✁✄✂

with

• ✞

• ✂

• Then, for the unfolding

• ✁✄✂

• ✞

• ✂

• ✁✄✂

• ✆
• ✞

• ✂

• ✁✄✂

• ✞

• ✂

• ✌

• FOSSACS’04: Inf. Behaviour and Scoping – p.10/34

Scoping: Example

Consider

• ✁✄✂

with

• ✞

• ✂

• Then, for the unfolding

• ✁✄✂

• ✞

• ✂

• ✁✄✂

• ✆
• ✞

• ✂

• ✁✄✂

• ✞

• ✂

• ✌

• FOSSACS’04: Inf. Behaviour and Scoping – p.10/34

Scoping: Example

Consider

• ✁✄✂

with

• ✞

• ✂

• Then, for the unfolding

• ✁✄✂

• ✞

• ✂

• ✁✄✂

• ✆
• ✞

• ✂

• ✁✄✂

• ✞

• ✂

• ✌

• Then

may be executed.

FOSSACS’04: Inf. Behaviour and Scoping – p.10/34

Scoping: Example 2

Consider again

• ✁✄✂

with

• ✞

• ✂

• FOSSACS’04: Inf. Behaviour and Scoping – p.11/34

Scoping: Example 2

Consider again

• ✁✄✂

with

• ✞

• ✂

• Consider now the following rule:

REC

• ✁✄✂

• ✠

• ✁✄✂

• ✠

but applying name

• conversion when necessary

FOSSACS’04: Inf. Behaviour and Scoping – p.11/34

Scoping: Example 2

Consider again

• ✁✄✂

with

• ✞

• ✂

• Then,

• ✁✄✂

• ✞

• ✂

• ✁✄✂

• FOSSACS’04: Inf. Behaviour and Scoping – p.11/34

Scoping: Example 2

Consider again

• ✁✄✂

with

• ✞

• ✂

• Then,

• ✁✄✂

• ✞

• ✂

• ✁✄✂

• FOSSACS’04: Inf. Behaviour and Scoping – p.11/34

Scoping: Example 2

Consider again

• ✁✄✂

with

• ✞

• ✂

• Then,

• ✁✄✂

• ✞

• ✂

• ✁✄✂

• FOSSACS’04: Inf. Behaviour and Scoping – p.11/34

Scoping: Example 2

Consider again

• ✁✄✂

with

• ✞

• ✂

• Then,

• ✁✄✂

• ✞

• ✂

• ✁✄✂

• ✆
• ✞

• ✂

• ✁✄✂

• ✞

• ✂

• ✌

• FOSSACS’04: Inf. Behaviour and Scoping – p.11/34

Scoping: Example 2

Consider again

• ✁✄✂

with

• ✞

• ✂

• Then,

• ✁✄✂

• ✞

• ✂

• ✁✄✂

• ✆
• ✞

• ✂

• ✁✄✂

• ✞

• ✂

• ✌

• Then

will never be executed.

FOSSACS’04: Inf. Behaviour and Scoping – p.11/34

Static vs Dynamic Scoping

If name occurrences get dynamically (i.e. during execution) captured under the scope of a restriction, we talk about dynamic

• scoping. Otherwise, static scoping.

• conversion to avoid captures

• static scoping

Otherwise

• dynamic scoping

FOSSACS’04: Inf. Behaviour and Scoping – p.12/34

Overview of the presentation

The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks

FOSSACS’04: Inf. Behaviour and Scoping – p.13/34

Infinite Behaviour

At least four constructions specifying infinite behaviour in CCS-like calculi:

FOSSACS’04: Inf. Behaviour and Scoping – p.14/34

Infinite Behaviour

At least four constructions specifying infinite behaviour in CCS-like calculi: CCSp: Given by a finite set of parametric definitions

• ✝✟✞

. Basically, the variant in Milner’s 99 book on

• calculus.

FOSSACS’04: Inf. Behaviour and Scoping – p.14/34

Infinite Behaviour

At least four constructions specifying infinite behaviour in CCS-like calculi: CCSp:

• ✝✟✞

CCSk: Like CCSp but using constant definitions

• ✁✄✂

. Essentially CCS (Milner’89) without relabelling nor infinite summations.

FOSSACS’04: Inf. Behaviour and Scoping – p.14/34

Infinite Behaviour

At least four constructions specifying infinite behaviour in CCS-like calculi: CCSp:

• ✝✟✞

CCSk:

• ✁✄✂

CCS!: Given by replication !

. This variant is presented, e.g., in works by Busi, Gabbrielli and Zavattaro [2003,2004].

FOSSACS’04: Inf. Behaviour and Scoping – p.14/34

Infinite Behaviour

At least four constructions specifying infinite behaviour in CCS-like calculi: CCSp:

• ✝✟✞

CCSk:

• ✁✄✂

CCS!: !

CCS

• : Given by recursive expressions of the form
• ✁✄✂

as in Milner’89. We adopt, as in ECCS, static scoping of names.

FOSSACS’04: Inf. Behaviour and Scoping – p.14/34

Infinite Behaviour

At least four constructions specifying infinite behaviour in CCS-like calculi: CCSp:

• ✝✟✞

CCSk:

• ✁✄✂

CCS!: !

CCS

• :
• ✁✄✂

FOSSACS’04: Inf. Behaviour and Scoping – p.14/34

Parametric Definitions: CCSp

Syntax:

• ✆

• ✝

where

• ✝✟✞

,

• ✌

.

FOSSACS’04: Inf. Behaviour and Scoping – p.15/34

Parametric Definitions: CCSp

Syntax:

• ✆

• ✝

where

• ✝✟✞

,

• ✌

. Semantics: CALL

• ✠

• ✝

• ✠

if

• ✁

def

name

• conversion when necessary.

FOSSACS’04: Inf. Behaviour and Scoping – p.15/34

Constant Definitions: CCSk

Syntax:

• ✆

• where
• ✁✄✂

FOSSACS’04: Inf. Behaviour and Scoping – p.16/34

Constant Definitions: CCSk

Syntax:

• ✆

• where
• ✁✄✂

Semantics: CONS

• ✠

• ✠

if

• def

FOSSACS’04: Inf. Behaviour and Scoping – p.16/34

Constant Definitions: CCSk

Syntax:

• ✆

• where
• ✁✄✂

Semantics (alternative): Use

• ✁✄✂

instead and REC

• ✁✄✂

• ✠

• ✁✄✂

• ✠

without name

• conversion.

FOSSACS’04: Inf. Behaviour and Scoping – p.16/34

Recursion Expressions: CCS

• Syntax:

• ✆

• ✁✄✂

FOSSACS’04: Inf. Behaviour and Scoping – p.17/34

Recursion Expressions: CCS

• Syntax:

• ✆

• ✁✄✂

Semantics: REC

• ✁✄✂

• ✠

• ✁✄✂

• ✠

with name

• conversion when necessary.

FOSSACS’04: Inf. Behaviour and Scoping – p.17/34

Replication: CCS!

Syntax:

• ✆

!

FOSSACS’04: Inf. Behaviour and Scoping – p.18/34

Replication: CCS!

Syntax:

• ✆

!

Semantics: REP

!

• ✠

!

• ✠

FOSSACS’04: Inf. Behaviour and Scoping – p.18/34

Overview of the presentation

The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks

FOSSACS’04: Inf. Behaviour and Scoping – p.19/34

Expressiveness and Classification Criteria

Let

• be weak bisimilarity.

Expressiveness: CCS

is as expressive as CCS

iff for every

, there exists

such that

• ✞

FOSSACS’04: Inf. Behaviour and Scoping – p.20/34

Expressiveness and Classification Criteria

Let

• be weak bisimilarity.

Expressiveness: CCS

is as expressive as CCS

iff for every

, there exists

such that

• ✞

Divergence:

FOSSACS’04: Inf. Behaviour and Scoping – p.20/34

Expressiveness and Classification Criteria

Let

• be weak bisimilarity.

Expressiveness: CCS

is as expressive as CCS

iff for every

, there exists

such that

• ✞

Divergence:

is divergent iff

• , i.e., there exists an infinite

sequence

.

FOSSACS’04: Inf. Behaviour and Scoping – p.20/34

Expressiveness and Classification Criteria

Let

• be weak bisimilarity.

Expressiveness: CCS

is as expressive as CCS

iff for every

, there exists

such that

• ✞

Divergence:

is divergent iff

• , i.e., there exists an infinite

sequence

. We’ll discuss:

• 1. The relative expressiveness w.r.t. weak bisimilarity
• 2. The decidability of divergence

FOSSACS’04: Inf. Behaviour and Scoping – p.20/34

Expressiveness Results

Encodings: (weak) bisimulation preserving mappings

• ✞

• ✄

FOSSACS’04: Inf. Behaviour and Scoping – p.21/34

Expressiveness Results

Encodings: (weak) bisimulation preserving mappings

• ✞

• ✄

Encoding CCSp into CCSk Encoding CCSk into CCSp Encoding CCS

• into CCS!

Encoding CCS! into CCS

• FOSSACS’04: Inf. Behaviour and Scoping – p.21/34

Expressiveness Results

Encodings: (weak) bisimulation preserving mappings

• ✞

• ✄

Encoding CCSp into CCSk Encoding CCSk into CCSp Encoding CCS

• into CCS!

Encoding CCS! into CCS

• FOSSACS’04: Inf. Behaviour and Scoping – p.21/34

Encoding CCSp into CCSk

• ✞

• ✄✆☎✝

• ✡

. Idea: Given

in CCSp,

FOSSACS’04: Inf. Behaviour and Scoping – p.22/34

Encoding CCSp into CCSk

• ✞

• ✄✆☎✝

• ✡

. Idea: Given

in CCSp, Suppose

uses

• ✝

def

. For each call

• ✝

that

may make, generate constant

def

and replace

• ✝

with

• .

FOSSACS’04: Inf. Behaviour and Scoping – p.22/34

Encoding CCSp into CCSk

• ✞

• ✄✆☎✝

• ✡

. Idea: Given

in CCSp, Suppose

uses

• ✝

def

. For each call

• ✝

that

may make, generate constant

def

and replace

• ✝

with

• .

Problem: Potentially infinite generation of constants!

FOSSACS’04: Inf. Behaviour and Scoping – p.22/34

Encoding CCSp into CCSk

• ✞

• ✄✆☎✝

• ✡

. Idea: Given

in CCSp, Suppose

uses

• ✝

def

. For each call

• ✝

that

may make, generate constant

def

and replace

• ✝

with

• .

Problem: Potentially infinite generation of constants!

• Due to name

• conversion in CCSp, an infinite number of fresh

names may be generated.

FOSSACS’04: Inf. Behaviour and Scoping – p.22/34

Encoding CCSp into CCSk

...nevertheless, Theorem: Let

CCSp with set of definitions

• ✠

. One can effectively construct

and its set of constants

• ✠

s.t.,

FOSSACS’04: Inf. Behaviour and Scoping – p.23/34

Encoding CCSp into CCSk

...nevertheless, Theorem: Let

CCSp with set of definitions

• ✠

. One can effectively construct

and its set of constants

• ✠

s.t.,

But CCSp can encode the perhaps most used kind of relabellings: The injective ones.

FOSSACS’04: Inf. Behaviour and Scoping – p.23/34

Encoding CCSp into CCSk

...nevertheless, Theorem: Let

CCSp with set of definitions

• ✠

. One can effectively construct

and its set of constants

• ✠

s.t.,

But CCSp can encode the perhaps most used kind of relabellings: The injective ones. Corollary: Injective relabellings are derivable in CCSk

FOSSACS’04: Inf. Behaviour and Scoping – p.23/34

Encoding CCS

• into CCS!

• ✞

• ✄✆☎✝

• ✡

! Idea:

• ✆

• ✁

!

• ✌

FOSSACS’04: Inf. Behaviour and Scoping – p.24/34

Encoding CCS

• into CCS!

• ✞

• ✄✆☎✝

• ✡

! Idea:

• ✆

• ✁

!

• ✌

... But the encoding does not work if we hadn’t assumed

• conversion

in CCS

• .

FOSSACS’04: Inf. Behaviour and Scoping – p.24/34

Encoding CCS

• into CCS!: Example

Let be the following CCS

• process:

• ✁✄✂

• ✂

FOSSACS’04: Inf. Behaviour and Scoping – p.25/34

Encoding CCS

• into CCS!: Example

Let be the following CCS

• process:

• ✁✄✂

• ✂

Then the corresponding encoding is:

!

• ✂
• ✞

• ✞

FOSSACS’04: Inf. Behaviour and Scoping – p.25/34

Encoding CCS

• into CCS!: Example

Let be the following CCS

• process:

• ✁✄✂

• ✂

Then the corresponding encoding is:

!

• ✂
• ✞

• ✞

Clearly

and

are not strongly bisimilar. However, Theorem: For

• ,

• ✆

. Moreover,

diverges iff

diverges.

FOSSACS’04: Inf. Behaviour and Scoping – p.25/34

Overview of the presentation

The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks

FOSSACS’04: Inf. Behaviour and Scoping – p.26/34

Conclusions

CCSp

CCSk CCS

• CCS!

Divergence: Undecidable Divergence: Decidable Slight variations of

• ✁✄✂

lead to:

• 1. Static Scoping

Replication.

• 2. Dynamic Scoping

• Param. Definitions.

FOSSACS’04: Inf. Behaviour and Scoping – p.27/34

Conclusions

CCSp

CCSk CCS

• CCS!

Divergence: Undecidable Divergence: Decidable Slight variations of

• ✁✄✂

lead to:

• 1. Static Scoping

Replication.

• 2. Dynamic Scoping

• Param. Definitions.

Injective relabellings are redundant in CCS with constants.

FOSSACS’04: Inf. Behaviour and Scoping – p.27/34

Conclusions

CCSp

CCSk CCS

• CCS!

Divergence: Undecidable Divergence: Decidable Slight variations of

• ✁✄✂

lead to:

• 1. Static Scoping

Replication.

• 2. Dynamic Scoping

• Param. Definitions.

Injective relabellings are redundant in CCS with constants. CCS with constants does not preserve

• conversion.

FOSSACS’04: Inf. Behaviour and Scoping – p.27/34

Conclusions

CCSp

CCSk CCS

• CCS!

Divergence: Undecidable Divergence: Decidable Slight variations of

• ✁✄✂

lead to:

• 1. Static Scoping

Replication.

• 2. Dynamic Scoping

• Param. Definitions.

Injective relabellings are redundant in CCS with constants. CCS with constants does not preserve

• conversion.

FOSSACS’04: Inf. Behaviour and Scoping – p.27/34

Final Remarks

CCS variant in Milner’99 uses parametric definitions with static scoping. CWB uses parametric definitions with dynamic scoping.

FOSSACS’04: Inf. Behaviour and Scoping – p.28/34

Final Remarks

CCS variant in Milner’99 uses parametric definitions with static scoping. CWB uses parametric definitions with dynamic scoping. ECCS advocates the static scoping of names. CHOCS advocates for dynamic name scoping—in the context of higher-order CCS.

FOSSACS’04: Inf. Behaviour and Scoping – p.28/34

Thanks!

FOSSACS’04: Inf. Behaviour and Scoping – p.29/34

Bisimilarity

A relation

• ✆

is a (strong) simulation if for all

• :

• ✠

• ✞

FOSSACS’04: Inf. Behaviour and Scoping – p.30/34

Bisimilarity

A relation

• ✆

is a (strong) simulation if for all

• :

• ✠

• ✞
• ✠

FOSSACS’04: Inf. Behaviour and Scoping – p.30/34

Bisimilarity

A relation

• ✆

is a (strong) simulation if for all

• :

• ✠

• ✞
• ✠

• is a (strong) bisimulation if both
• and its converse are

(strong) simulations:

.

FOSSACS’04: Inf. Behaviour and Scoping – p.30/34

Bisimilarity

A relation

• ✆

is a weak simulation if for all

• :

• ✆
• ☎

• ✞
• ✆
• ✞

• is a weak bisimulation if both
• and its converse are weak

simulations:

• ✞

.

• “
• ✆
• ” (where
• ✟
• ✂

• ✁

) is

• ✂☎✄

• ✂

• ✂

• ✂

FOSSACS’04: Inf. Behaviour and Scoping – p.30/34

Encoding CCSp into CCSk

• ✞

• ✄✆☎✝

• ✡

Idea: For each

CCSp, let

• ☎

CCSk replacing in

each

• ccurrence of

with

• For each definition
• ✝✟✞

def

• , generate a constant

definition

• ✁

def

• FOSSACS’04: Inf. Behaviour and Scoping – p.31/34

Encoding CCSk into CCSp

The encoding of

into CCSp is a process

with associated set of definitions

• ✁

The encoding

• ✞

• ✄✆☎✝

• :

It satisfies

• ✞

• ✝

where

• ✌

It is an homomorphism over all other operators

FOSSACS’04: Inf. Behaviour and Scoping – p.32/34

Encoding CCS! into CCS

• The encoding

• ✞

• ✄✆☎✝

!

• :

It satisfies

!

• ✁✄✂

It is an homomorphism over all other operators

FOSSACS’04: Inf. Behaviour and Scoping – p.33/34

Encoding CCSp into CCSk: Example

Let

• ✝

• ✞

• ✞
• ✝✁

FOSSACS’04: Inf. Behaviour and Scoping – p.34/34

Encoding CCSp into CCSk: Example

Let

• ✝

• ✞

• ✞
• ✝✁

1.

• ✁

• ✂

• ✞

• ✞

• FOSSACS’04: Inf. Behaviour and Scoping – p.34/34

Encoding CCSp into CCSk: Example

Let

• ✝

• ✞

• ✞
• ✝✁

1.

• ✁

• ✂

• ✞

• ✞

• 2.

• ✂
• ✞
• ✂
• ✞

FOSSACS’04: Inf. Behaviour and Scoping – p.34/34

Encoding CCSp into CCSk: Example

Let

• ✝

• ✞

• ✞
• ✝✁

1.

• ✁

• ✂

• ✞

• ✞

• 2.

• ✂
• ✞
• ✂
• ✞

FOSSACS’04: Inf. Behaviour and Scoping – p.34/34

Encoding CCSp into CCSk: Example

Let

• ✝

• ✞

• ✞
• ✝✁

1.

• ✁

• ✂

• ✞

• ✞

• 2.

• ✠

• ✂
• ✞
• ✂
• ✞
• ✟

• ✠

FOSSACS’04: Inf. Behaviour and Scoping – p.34/34

Encoding CCSp into CCSk: Example

Let

• ✝

• ✞

• ✞
• ✝✁

1.

• ✁

• ✂

• ✞

• ✞

• 2.

• ✠

• ✂
• ✞
• ✂
• ✞
• ✟

• ✠

3.

• ✟

• ✠

• ✞
• ✠

• ✞
• ✌

FOSSACS’04: Inf. Behaviour and Scoping – p.34/34

Encoding CCSp into CCSk: Example

Let

• ✝

• ✞

• ✞
• ✝✁

1.

• ✁

• ✂

• ✞

• ✞

• 2.

• ✠

• ✂
• ✞
• ✂
• ✞
• ✟

• ✠

3.

• ✟

• ✠

• ✞
• ✠

• ✞
• ✌

Remark: The generation of fresh names could continue forever!

FOSSACS’04: Inf. Behaviour and Scoping – p.34/34