An Operational Petri Net Semantics for the Join-Calculus

An Operational Petri Net Semantics for theJoin-Calculus

Stephan Mennicke

EXPRESS/SOS 2012September 3, 2012

Motivation I

Distributed SystemsConsist of components on different locationsResource conflicts are solved locallyNo concurrency on componentsCommunication between components is asynchronous

Distributable SystemsGiven a specification. It is distributable if there exists an implementationthat is distributed.

van Glabbeek et al. characterize these notions and criteria in terms ofPetri nets

S. Mennicke (IPS) Petri Net Semantics for Join EXPRESS/SOS 2012 2 / 17

Motivation II

Interesting structure:p q

t1 t2 t3

The M is not distributedA net is distributable iff it is M-free [van Glabbeek et al.]

Starting point for this work: There is this interesting formalism which isdesigned to be distributable.


Motivation II

Interesting structure:p q

t1 t2 t3

The M is not distributedA net is distributable iff it is M-free [van Glabbeek et al.]

Starting point for this work: There is this interesting formalism which isdesigned to be distributable.


Meet the Reflexive Chemical Abstract Machine[Fournet and Gonthier 1996]

x〈k〉 | y〈v〉 . k〈v〉

x〈c〉 | a〈2〉

y〈3〉 | a〈1〉

y〈3〉x〈c〉 a〈1〉

a〈2〉x〈c〉 | y〈3〉

a〈1〉

a〈2〉 ⇀[Berry and Boudol 1990]

↽→

c〈3〉

a〈1〉

a〈2〉



x〈k〉 | y〈v〉 . k〈v〉

x〈c〉 | a〈2〉

y〈3〉 | a〈1〉

y〈3〉x〈c〉 a〈1〉

a〈2〉x〈c〉 | y〈3〉

a〈1〉

a〈2〉 ⇀[Berry and Boudol 1990]

↽→

c〈3〉

a〈1〉

a〈2〉



x〈k〉 | y〈v〉 . k〈v〉

x〈c〉 | a〈2〉

y〈3〉 | a〈1〉

y〈3〉x〈c〉 a〈1〉

a〈2〉x〈c〉 | y〈3〉

a〈1〉

a〈2〉

⇀[Berry and Boudol 1990]

↽→

c〈3〉

a〈1〉

a〈2〉



x〈k〉 | y〈v〉 . k〈v〉

x〈c〉 | a〈2〉

y〈3〉 | a〈1〉

y〈3〉x〈c〉 a〈1〉

a〈2〉

x〈c〉 | y〈3〉

a〈1〉

a〈2〉

⇀[Berry and Boudol 1990]

↽→

c〈3〉

a〈1〉

a〈2〉



x〈k〉 | y〈v〉 . k〈v〉

x〈c〉 | a〈2〉

y〈3〉 | a〈1〉

y〈3〉x〈c〉 a〈1〉

a〈2〉

x〈c〉 | y〈3〉

a〈1〉

a〈2〉 ⇀

[Berry and Boudol 1990]↽

→

c〈3〉

a〈1〉

a〈2〉



x〈k〉 | y〈v〉 . k〈v〉

x〈c〉 | a〈2〉

y〈3〉 | a〈1〉

y〈3〉x〈c〉 a〈1〉

a〈2〉

x〈c〉 | y〈3〉

a〈1〉

a〈2〉

⇀

[Berry and Boudol 1990]↽

→

c〈3〉

a〈1〉

a〈2〉



x〈k〉 | y〈v〉 . k〈v〉

x〈c〉 | a〈2〉

y〈3〉 | a〈1〉

y〈3〉x〈c〉 a〈1〉

a〈2〉

x〈c〉 | y〈3〉

a〈1〉

a〈2〉

⇀

[Berry and Boudol 1990]

↽

→

c〈3〉

a〈1〉

a〈2〉



x〈k〉 | y〈v〉 . k〈v〉

x〈c〉 | a〈2〉

y〈3〉 | a〈1〉

y〈3〉x〈c〉 a〈1〉

a〈2〉x〈c〉 | y〈3〉

a〈1〉

a〈2〉 ⇀

[Berry and Boudol 1990]

↽

→

c〈3〉

a〈1〉

a〈2〉


The (Core) Join-CalculusSyntax and Semantics [Fournet and Gonthier 1996]

Syntax

P ::= 0 x〈v〉 P |P def x〈u〉|y〈v〉 . P inP

Semantics` P |Q ` P,Q` def D inP Dσdv ` Pσdv

J . P ` Jσrv → J . P ` Pσrv

In the paper: LTS-Semantics


The (Core) Join-CalculusSyntax and Semantics [Fournet and Gonthier 1996]

Syntax

P ::= 0 x〈v〉 P |P def x〈u〉|y〈v〉 . P inP

Semantics` P |Q ` P,Q` def D inP Dσdv ` Pσdv

J . P ` Jσrv → J . P ` Pσrv

In the paper: LTS-Semantics


Petri Nets


Net Semantics and the Join-Calculus

Why Net Semantics?Induce a step semantics on top of the natural oneMake Petri net theory/tools available for JoinTransfer phenomena between Petri nets and Join

Existing Net SemanticsBuscemi and Sassone 2001Bruni et al. 2006

Shared ProblemConcurrency!


Net Semantics and the Join-Calculus

Why Net Semantics?Induce a step semantics on top of the natural oneMake Petri net theory/tools available for JoinTransfer phenomena between Petri nets and Join

Existing Net SemanticsBuscemi and Sassone 2001Bruni et al. 2006

Shared ProblemConcurrency!


Concurrent Petri Net SemanticsOverview

Ingredients1 A universe of places P

2 A decomposition function mapping Join terms to subsets of P3 A transition rule

transition rule:

dec :



Ingredients1 A universe of places P2 A decomposition function mapping Join terms to subsets of P

3 A transition rule

transition rule:

dec :



Ingredients1 A universe of places P2 A decomposition function mapping Join terms to subsets of P

3 A transition rule

. . .dec:

transition rule:

dec :



Ingredients1 A universe of places P2 A decomposition function mapping Join terms to subsets of P3 A transition rule

. . .dec:

transition rule:

dec :



Ingredients1 A universe of places P2 A decomposition function mapping Join terms to subsets of P3 A transition rule

. . .

D

dec:

transition rule:

dec :


Example

def c〈v〉 . v〈〉︸︷︷︸D1

in def a〈k〉 | b〈v〉 . k〈v〉︸︷︷︸D2

in a〈c〉 | b〈42〉

a〈c〉 b〈42〉

c〈42〉42〈〉

D2

D1


Example

def c〈v〉 . v〈〉︸︷︷︸D1

in def a〈k〉 | b〈v〉 . k〈v〉︸︷︷︸D2

in a〈c〉 | b〈42〉

a〈c〉 b〈42〉

c〈42〉42〈〉

D2

D1


Example

def c〈v〉 . v〈〉︸︷︷︸D1

in def a〈k〉 | b〈v〉 . k〈v〉︸︷︷︸D2

in a〈c〉 | b〈42〉

a〈c〉 b〈42〉

c〈42〉

42〈〉

D2

D1


Example

def c〈v〉 . v〈〉︸︷︷︸D1

in def a〈k〉 | b〈v〉 . k〈v〉︸︷︷︸D2

in a〈c〉 | b〈42〉

a〈c〉 b〈42〉

c〈42〉42〈〉

D2

D1


Examples with Concurrency

1 def x〈v〉 . P inx〈EXPRESS〉 | def y〈w〉 . Q in y〈SOS〉

2 def x〈v〉 . P in def y〈w〉 . Q inx〈EXPRESS〉 | y〈SOS〉3 def x〈v〉 . P inx〈a〉 |x〈b〉4 def k1〈〉 . P1 in . . . def kn〈〉 . Pn in def cons〈〉 .k1〈〉 | . . . | k2〈〉 in cons〈〉

x〈EXPRESS〉

y〈SOS〉

P

Q

x〈v〉 . P

y〈w〉 . Q



1 def x〈v〉 . P inx〈EXPRESS〉 | def y〈w〉 . Q in y〈SOS〉

2 def x〈v〉 . P in def y〈w〉 . Q inx〈EXPRESS〉 | y〈SOS〉3 def x〈v〉 . P inx〈a〉 |x〈b〉4 def k1〈〉 . P1 in . . . def kn〈〉 . Pn in def cons〈〉 .k1〈〉 | . . . | k2〈〉 in cons〈〉

x〈EXPRESS〉

y〈SOS〉

P

Q

x〈v〉 . P

y〈w〉 . Q



1 def x〈v〉 . P inx〈EXPRESS〉 | def y〈w〉 . Q in y〈SOS〉2 def x〈v〉 . P in def y〈w〉 . Q inx〈EXPRESS〉 | y〈SOS〉

3 def x〈v〉 . P inx〈a〉 |x〈b〉4 def k1〈〉 . P1 in . . . def kn〈〉 . Pn in def cons〈〉 .k1〈〉 | . . . | k2〈〉 in cons〈〉

x〈EXPRESS〉

y〈SOS〉

P

Q

x〈v〉 . P

y〈w〉 . Q



1 def x〈v〉 . P inx〈EXPRESS〉 | def y〈w〉 . Q in y〈SOS〉2 def x〈v〉 . P in def y〈w〉 . Q inx〈EXPRESS〉 | y〈SOS〉

3 def x〈v〉 . P inx〈a〉 |x〈b〉4 def k1〈〉 . P1 in . . . def kn〈〉 . Pn in def cons〈〉 .k1〈〉 | . . . | k2〈〉 in cons〈〉

x〈EXPRESS〉

y〈SOS〉

P

Q

x〈v〉 . P

y〈w〉 . Q



1 def x〈v〉 . P inx〈EXPRESS〉 | def y〈w〉 . Q in y〈SOS〉2 def x〈v〉 . P in def y〈w〉 . Q inx〈EXPRESS〉 | y〈SOS〉3 def x〈v〉 . P inx〈a〉 |x〈b〉

4 def k1〈〉 . P1 in . . . def kn〈〉 . Pn in def cons〈〉 .k1〈〉 | . . . | k2〈〉 in cons〈〉

x〈a〉

x〈b〉

P [a/v]

P [b/v]

x〈v〉 . P

x〈v〉 . P



1 def x〈v〉 . P inx〈EXPRESS〉 | def y〈w〉 . Q in y〈SOS〉2 def x〈v〉 . P in def y〈w〉 . Q inx〈EXPRESS〉 | y〈SOS〉3 def x〈v〉 . P inx〈a〉 |x〈b〉4 def k1〈〉 . P1 in . . . def kn〈〉 . Pn in def cons〈〉 .k1〈〉 | . . . | k2〈〉 in cons〈〉

x〈a〉

x〈b〉

P [a/v]

P [b/v]

x〈v〉 . P

x〈v〉 . P


Correctness of the Semantics

TheoremLet P be a (core) Join term. Then its Petri net semantics N(P ) isbisimilar to P .

We prove that R = {(J, dec(J)) | J is a Join term} is a bisimulation byinduction over the structure of P .

LemmaIf P ≡ Q for P,Q (core) Join terms. Then their Petri net representationsN(P ), N(Q) are isomorphic.

1 Our semantics is correct2 It yields a reasonable degree of concurrency


Revisiting Distributability

P = def x〈u〉 | y〈v〉 . u〈v〉︸︷︷︸D

inx〈a〉 | y〈1〉 |x〈b〉 | y〈2〉

x〈a〉

y〈1〉 x〈b〉 y〈2〉

a〈1〉 b〈1〉 b〈2〉 a〈2〉

D D D D



P = def x〈u〉 | y〈v〉 . u〈v〉︸︷︷︸D

inx〈a〉 | y〈1〉 |x〈b〉 | y〈2〉

x〈a〉

y〈1〉 x〈b〉 y〈2〉

a〈1〉 b〈1〉 b〈2〉 a〈2〉

D D D D



P = def x〈u〉 | y〈v〉 . u〈v〉︸︷︷︸D

inx〈a〉 | y〈1〉 |x〈b〉 | y〈2〉

x〈a〉

y〈1〉 x〈b〉 y〈2〉

a〈1〉

b〈1〉 b〈2〉 a〈2〉

D

D D D



P = def x〈u〉 | y〈v〉 . u〈v〉︸︷︷︸D

inx〈a〉 | y〈1〉 |x〈b〉 | y〈2〉

x〈a〉

y〈1〉 x〈b〉 y〈2〉

a〈1〉 b〈1〉 b〈2〉 a〈2〉

D D D D


What can we learn from Join?

In Join, the Ms above must reside on one locationvan Glabbeek et al.’s theory is not directly applicableAre there any other Ms in the Join-calculus semantics?

PropositionIf N(P ) contains an M, then all transitions belonging to this Mcorrespond to the same definition.

It is worth looking at notions for distributability using a priorly givenlocation function [Best and Darondeau 2011]


Concluding Remarks

Making Petri Net Tools AvailableIn general, the proposed semantics is infiniteStructural properties:

1 •m0 = ∅2 For all p ∈ P : |•p| ≤ 13 F+ is irreflexive

So, the semantics may be used as a starting point for Petri netunfolding based techniquesStructural properties above + transition rule ensure 1-safety

Occurrence NetsThe conflict relation is not irreflexive

=⇒ We have transitions in the resulting nets which may never fire


Concluding Remarks

Making Petri Net Tools AvailableIn general, the proposed semantics is infiniteStructural properties:

1 •m0 = ∅2 For all p ∈ P : |•p| ≤ 13 F+ is irreflexive

So, the semantics may be used as a starting point for Petri netunfolding based techniquesStructural properties above + transition rule ensure 1-safety

Occurrence NetsThe conflict relation is not irreflexive

=⇒ We have transitions in the resulting nets which may never fire


Thank You!

AppendixCounter Example I

a〈b〉 | def a〈v〉 . v〈〉 in a〈c〉

and set X = {a〈v〉 . v〈〉}:

a〈b〉 a〈c〉

b〈〉 c〈〉

a〈v〉 . v〈〉 a〈v〉 . v〈〉

a〈b〉 a〈c〉

c〈〉

a〈v〉 . v〈〉





a〈b〉 a〈c〉

b〈〉 c〈〉

a〈v〉 . v〈〉 a〈v〉 . v〈〉

a〈b〉 a〈c〉

c〈〉

a〈v〉 . v〈〉





a〈b〉 a〈c〉

b〈〉 c〈〉

a〈v〉 . v〈〉 a〈v〉 . v〈〉

a〈b〉 a〈c〉

c〈〉

a〈v〉 . v〈〉





a〈b〉 a〈c〉

b〈〉 c〈〉

a〈v〉 . v〈〉 a〈v〉 . v〈〉

a〈b〉 a〈c〉

c〈〉

a〈v〉 . v〈〉


AppendixCounter Example II

def c〈v〉 . v〈〉︸︷︷︸D1

in def a〈k〉 | b〈v〉 . k〈v〉︸︷︷︸D2

in a〈c〉 | b〈42〉

a〈c〉 D2 b〈42〉 D2

c〈42〉

D2

a〈c〉 D2D1b〈42〉 D2D1

c〈42〉 D1

42〈〉

D2

D1



def c〈v〉 . v〈〉︸︷︷︸D1

in def a〈k〉 | b〈v〉 . k〈v〉︸︷︷︸D2

in a〈c〉 | b〈42〉

a〈c〉 D2 b〈42〉 D2

c〈42〉

D2

a〈c〉 D2D1b〈42〉 D2D1

c〈42〉 D1

42〈〉

D2

D1



def c〈v〉 . v〈〉︸︷︷︸D1

in def a〈k〉 | b〈v〉 . k〈v〉︸︷︷︸D2

in a〈c〉 | b〈42〉

a〈c〉 D2 b〈42〉 D2

c〈42〉

D2

a〈c〉 D2D1b〈42〉 D2D1

c〈42〉 D1

42〈〉

D2

D1



def c〈v〉 . v〈〉︸︷︷︸D1

in def a〈k〉 | b〈v〉 . k〈v〉︸︷︷︸D2

in a〈c〉 | b〈42〉

a〈c〉 D2 b〈42〉 D2

c〈42〉

D2

a〈c〉 D2D1b〈42〉 D2D1

c〈42〉 D1

42〈〉

D2

D1


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An Operational Petri Net Semantics for the Join-Calculus