Process Algebras and Petri Nets are Discrete Dynamical Systems

Process Algebras and Petri Nets are DiscreteDynamical Systems

Fernando Lopez Pelayo

Real Time and Concurrent Systems Group

Computing Systems Department of University of Castilla-La Mancha

Facultad de Informatica

Universidad Complutense

December 4th, 2014 Madrid, SPAIN

Process Algebras and Petri Nets are Discrete Dynamical Systems – p.1/55

Objectives

• To get a more usable Process Algebra

• To codify Petri Nets as Discrete DynamicalSystems

• To study the DDS

• To study the ODE


Outline

1. The Problem: LTS generation process• Roughly ROSA• Normal Formed ROSA• A Complete Metric Space• GENETIC ROSA

2. Petri Nets to Dynamical Systems• 1-safe PNs ⇒ Discrete Dynamical Systems I• PNs ⇒ DDSs II• TAPNs ∗ ⇒ DDSs III


Outline

1. The Problem: LTS generation process• Roughly ROSA• Normal Formed ROSA• A Complete Metric Space• GENETIC ROSA

2. Petri Nets to Dynamical Systems• 1-safe PNs ⇒ Discrete Dynamical Systems I• PNs ⇒ DDSs II• TAPNs ∗ ⇒ DDSs III


The Problem: LTS are unbroachable


Markovian Process Algebra ROSA

Three kinds of transitions:

• Non-Deterministic transitions: P −→ Q

Process P can evolve non-deterministically to proc. Q

• Probabilistic Transitions: P −→r Q

Process P can evolve, with probability r to process Q

• Action transitions: Pa,λ−→ Q

Process P can evolve by executing the action labelledby a, taking a time described by an Exponentialrandom distribution with parameter λ, to process Q





























Non-deterministic Transitions Rules

(ND-Def) P ⊕ Q −→ P (ND-Def) P ⊕ Q −→ Q

(ND-Ext) P −→ P ′

P + Q −→ P ′ + Q(ND-Ext) Q −→ Q′

P + Q −→ P + Q′

(ND-Pro) P −→ P ′

P ⊕r Q −→ P ′ ⊕r Q(ND-Pro) Q −→ Q′

P ⊕r Q −→ P ⊕r Q′

(ND-Par) P −→ P ′

P ||AQ −→ P ′||AQ(ND-Par) Q −→ Q′

P ||AQ −→ P ||AQ′

(ND-Rec)recX.P −→ P [recX.P/X]


Probabilistic Transition Rules

(P-Def) P ⊕r Q −→r P (P-Def) P ⊕r Q −→1−r Q

(P-Ext) P −→r P ′ ∧ Action[Q]P + Q −→r P ′ + Q

(P-Ext) Q −→t Q′ ∧ Action[P ]P + Q −→t P + Q′

(P-Par) P −→r P ′ ∧ Action[Q]P ||AQ −→r P ′||AQ

(P-Par) Q −→t Q′ ∧ Action[P ]P ||AQ −→t P ||AQ′

(P-BothExt) P −→r P ′ ∧Q −→t Q′

P + Q −→r·t P ′ + Q′ (P-BothPar) P −→r P ′ ∧Q −→t Q′

P ||AQ −→r·t P ′||AQ′

Provided that DS[P ] ∧DS[Q]


Action Transition Rules

(A-Def)a.P

a,∞−→ P

(A-Def)〈a, λ〉.P

a,λ−→ P

(A-Ext) Pa,λ−→ P ′ ∧ a /∈ Type[Available[Q]]

P + Qa,λ−→ P ′

(A-Ext) Qa,λ−→ Q′ ∧ a /∈ Type[Available[P ]]

P + Qa,λ−→ Q′

(A-Par) Pa,λ−→ P ′ ∧ a /∈ Type[Available[Q]] ∪A

P ||AQa,λ−→ P ′||AQ

(A-Par) Qa,λ−→ Q′ ∧ a /∈ Type[Available[P ]] ∪A

P ||AQa,λ−→ P ||AQ′

(A-RaceExt) Pa,∞−→ P ′ ∧Q

a,λ−→ Q′ ∧ λ 6= ∞

P + Qa,∞−→ P ′

(A-RaceExt) Pa,λ−→ P ′ ∧Q

a,∞−→ Q′ ∧ λ 6= ∞

P + Qa,∞−→ Q′

(A-RacePar) Pa,∞−→ P ′ ∧Q

a,λ−→ Q′ ∧ λ 6= ∞∧ a /∈ A

P ||AQa,∞−→ P ′||AQ

(A-RacePar) Pa,λ−→ P ′ ∧Q

a,∞−→ Q′ ∧ λ 6= ∞∧ a /∈ A

P ||AQa,∞−→ P ||AQ′

(A-RaceExtCoop) Pa,λ1−→ P ′ ∧Q

a,λ2−→ Q′ ∧ (λ1 = ∞ = λ2 ∨ λ1 6= ∞ 6= λ2)

P + Qa,λ1+λ2−→ P ′ ⊕ Q′

(A-Syn) Pa,λ1−→ P ′ ∧Q

a,λ2−→ Q′ ∧ a ∈ A

P ||AQa,min[{λ1,λ2}]

−→ P ′||AQ′

(A-RaceParCoop) Pa,λ1−→ P ′ ∧Q

a,λ2−→ Q′ ∧ (λ1 = ∞ = λ2 ∨ λ1 6= ∞ 6= λ2) ∧ a /∈ A

P ||AQa,λ1+λ2−→ P ′||AQ ⊕ P ||AQ′

Provided that DS[P ] ∧DS[Q]


ROSA Specification of MPEG (H.262)

I frame:

I ≡ 〈DCT_QI , α1〉.(〈V LCI , α2〉.out ||A 〈IQ_IDCTI , α3〉.〈FSI , α4〉.ESTP .COMPP .Θ)

P frame:

P ≡ 〈ESTP , γ1〉.〈COMPP , γ2〉.〈ERRORP , γ3〉.〈DCT_Q, γ4〉.(〈V LCP , γ5〉.out ||A 〈IQ_IDCT, γ6〉.〈FSP , γ7〉.Θ)

Bi frames:

Bi≡〈ESTBi, β1〉.〈COMPBi, β2〉.〈ERRORBi, β3〉.〈DCT_Q, β4〉.〈V LCBi, β5〉.out

Θ ≡ (ESTB1.COMPB1 ||A ESTB2.COMPB2)

MPEG Encoding Algorithm:

MPEG ≡ I ||A P ||A B1 ||A B2

A = {ESTP , ESTB1, ESTB2, COMPP , COMPB1, COMPB2}


LTS of MPEG specification

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Looking for solutions: Genetics?


Pseudocode - Description

1: P = generateInitialPopulation();

2: evaluate(P );

FITNESS FUNCTION

3: while not stoppingCondition()do4: P0 = selectParents(P );

5: P0 = applyV ariationOperators(P0);

6: evaluate(P0);

FITNESS FUNCTION

7: P1 = selectNewPopulation(P0);

8: end while;9: return the best found solution;


Pseudocode - Description

1: P = generateInitialPopulation();

2: evaluate(P ); FITNESS FUNCTION3: while not stoppingCondition()do4: P0 = selectParents(P );

5: P0 = applyV ariationOperators(P0);

6: evaluate(P0); FITNESS FUNCTION7: P1 = selectNewPopulation(P0);

8: end while;9: return the best found solution;


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VARIATIONS

• Crossover

• Operational Semantics• Non-deterministic Transition Rules• Probabilistic Transition Rules• Action Transition Rules

• Mutation

• Variation on FITNESS


Towards a Cheaper Semantics for ROSA


A Complete Metric Space

Given a pair of ROSA processes P and Q thedistance between them is d(P, Q)

d(P, Q) =1

2l(PuQ)−

1

2n

• l(T ) is the length of process T

• n = max{l(P ), l(Q)}

• P uQ is the longest common initial part ofprocesses P and Q


Consistent with the Semantics of ROSA

• ∀P, Q ∈ {ROSA}.d(P, Q) = 0 ⇔ P = Q

• Commutability and Associativity of ⊕ and + .Weighted Commutability and W. Assoc. of ⊕r

• Distributivity:d((P ⊕ Q) + R, (P + R) ⊕ (Q + R)) = 0

• Derivative operators:d(a.0||∅b.0, a.b.0 + b.a.0) = 0

NORMAL FORMS:⊗

i

[qi]⊕

Aj∈Ai

+a∈Type(Aj)

〈a, λAj〉.nf(PAj ,a)


Consistent with the Semantics of ROSA

• ∀P, Q ∈ {ROSA}.d(P, Q) = 0 ⇔ P = Q

• Commutability and Associativity of ⊕ and + .Weighted Commutability and W. Assoc. of ⊕r

• Distributivity:d((P ⊕ Q) + R, (P + R) ⊕ (Q + R)) = 0

• Derivative operators:d(a.0||∅b.0, a.b.0 + b.a.0) = 0

NORMAL FORMS:⊗

i

[qi]⊕

Aj∈Ai

+a∈Type(Aj)

〈a, λAj〉.nf(PAj ,a)


Normal-Formed Genetic ROSA

∀P ∈ G−ROSA process :

P ::= 0 |⊗

i

[qi]Qi

Decreasing ordered qi and λA

Q ::=⊕

Aj∈A

Tj

Alphabetically ordered Tj and actions

T ::= +a∈Type(A)

〈a, λA〉.Va


The Complete Metric Space

Given a pair of ROSA processes P and Q thedistance between them is D(P, Q)

D(P, Q) =1

2l(‖P‖u‖Q‖)−

1

2N

• ‖P‖ is the ROSA normal form of process P

• l(T ) is the length of process T

• N = max{l(‖P‖), l(‖Q‖)}

• ‖P‖ u ‖Q‖ is the longest common initial part ofthe normal form of processes P and Q


Cheaper Semantics

According to a Means - Ends Policy:

p− f : {ROSA procs.} −→ (0,1]P 7→ 1−D(P, SF)

Higher values to the more promising states

p− f is a FITNESS FUNCTION

PROs: Several Symbolic Metrics could be valid

CONs: Harder specification process


PETRI NETS


Marked Ordinary Petri Nets

An Ordinary Petri Net (OPN) is a tuple

N = (P, T, F ):

P Set of places

T Set of transitions (P ∩ T = ∅)

F Flow relation , F ⊆ (P × T ) ∪ (T × P )

• Marking of N = (P, T, F ) is M : P −→ IN

• (P, T, F, M) is a Marked Ordinary Petri Net

• A PN is k-safe (k ∈ IN) if |M(p)| ≤ k,∀p ∈ P


Firing a transition

• X = P ∪ T

• ∀x ∈ X:

• •x = {y ∈ X | (y, x) ∈ F} (precondition set of x)

• x• = {y ∈ X | (x, y) ∈ F} (postcondition set of x)

• A transition t ∈ T of N = (P, T, F, M) isENABLED∗ AT MARKING M (M [t〉)iff ∀p ∈ •t . M(p) > 0

• M [t〉 can be fired ⇒ M ′ (M [t〉M ′):M ′(p) = M(p)−Wf(•t) + Wf(t•) ∀p ∈ P

• More than 1 transition can be fired at once


Non-determinism

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Examples and Analysis by Petri Nets


MTAPN for generic GoP

0

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out

<0,0>

np

BBP


MTAPN Modelling MPEG-2 I

0

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out


MTAPN Modelling MPEG-2 II

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out00

Time = 595ms


MTAPN Modelling MPEG-2 III

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out

Time = 895ms

300

0


MTAPN Modelling MPEG-2 IV

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out

Time = 895ms

300

0


MTAPN Modelling MPEG-2 V

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out

Time = 1268ms

0

373


MTAPN Modelling MPEG-2 VI

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out

Time = 1268ms

373

0

000

0

0


MTAPN Modelling MPEG-2 VII

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out

Time = 1863ms

595595 968

0 0

595

595

595


MTAPN Modelling MPEG-2 VIII

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out

Time = 2163ms

895895 1268895

895

895

0

300


MTAPN Modelling MPEG-2 IX

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out

Time = 2163ms

895895 895

895

895

300

12680


MTAPN Modelling MPEG-2 X

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out

Time = 2448ms

1180

1180

585

1553285

1180

0


Demanding processors

DCT_Q I

IQ_IDCTI VLC I

FS I

Iout

I

Perror P

ESTP

COMPP

OUTI

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

B1out

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<0,0>

<48,48>

MV−B1 MV−B2

ERROR

DCT_Q

P

<521,521>

<0,0>

VLC P

<0,0>Pout

OUTP

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

B1

B2

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

<0,0>

B2error

B1

VLCB1

DCT_Q

ERRORB1

COMP B1

ESTB1

OUT B2

VLC B2

DCT_Q

ERRORB2

COMPB2

EST B2

OUT

MV−P

B2out

0

PROCESSING FINISHEDPROCESSING STARTED


Analysis on Petri Nets


Behavioural properties

• Reachability• Can ever a marking be reached from another

one?• Reversibility

• Is M0 reachable from each reachable marking?• Boundness / Safeness

• Is there an upper limit on the number of tokensin a/all given place/s?

• Vivacity / Liveliness / Deadlock freeness• Can the system become blocked?


Analysis Means

• Reachability / Reversibility• Reachability or coverability tree• Explosion on the number of markings / states!

• Boundness / Safeness• Incidence Matrix and State Equations

• Vivacity / Liveliness / Deadlock freeness• Structural Analysis based on net structures


Codification to Discrete Dynamical Systems


DDS associated to a PN

The DDS which encodes the MOPN N = (P, T, F, M) is the triple (X, T,Φ), where:

• the Phase Space X = P(INn) is the set of all subsets of INn, being n the numberof places of the MOPN.

• T is the monoid N ∪ {0}

• Φ : T ×X → X is the evolution operator Φ which holds:

1. Φ(0, x) = x ∀x ∈ X, i.e., Φ0(x) = idX

2. Φ(1, x) = y x, y ∈ X where:• x = {x1, . . . , xk} where xi ∈ INn encodes Markings of the MOPN N• y = ∪k

i=1yi• yi = ∪t

j=1yij, i.e. the union of all (t) possible reachable markings from

xi, defined by xi[Ri〉yijbeing Ri the set of transitions of the net enabled

at marking xi

3. Φ(t,Φ(s, x)) = Φ(t + s, x) ∀t, s ∈ T , ∀x ∈ X


Distance on 1-safe PNs

Given a pair x, y ∈ {0,1}n (corresponding to a pairof markings) the distance between them is given by:

d(x, y) =1

2l(xuy)−

1

2n, x, y ∈ {0,1}n

• l(x u y) is the length of the longest common initialpart of the vectors x and y

• n ∈ IN is the number of places of the PN• Metric well defined but blind to big differences in

both k-safe and non-bounded cases


Problem of binary case metrics I

p1

p2

p3

p4

p5

p6

t1

t2

t3

t4

p1

p2

p3

p4

p5

p6

t1

t2

t3

t4

p1

p2

p3

p4

p5

p6

t1

t2

t3

t4

p1

p2

p3

p4

p5

p6

t1

t2

t3

t4


Problem of binary case metrics I

p1

p2

p3

p4

p5

p6

t1

t2

t3

t4

p1

p2

p3

p4

p5

p6

t1

t2

t3

t4

p1

p2

p3

p4

p5

p6

t1

t2

t3

t4

p1

p2

p3

p4

p5

p6

t1

t2

t3

t4


Problem of binary case metrics II

• In the first pair of MOPNs the states are(0,1,1,4,1,0) and (0,1,0,5,1,0) which meansdistance 1

4− 1

64but only the first can fire t2

• In the second pair of MOPNs the states are(0,1,1,4,1,0) and (0,1,2,3,1,0) which means thesame distance 1

4− 1

64but both can fire t2

• To solve this problem we will use the (ENABLED)TRANSITIONS VECTOR


Distance on k-safe and non-bounded PNs

Given a pair x, y ∈ INm the distance between them isgiven by:

d(x, y) =1

2l(tv(x)utv(y))−

1

2m, x, y ∈ INm

• tv(x)i = 1 ⇔ transitioni is enabled, 0 o.w.

• l(tv(x)u tv(y)) = length of the longest commoninitial part of the transition vectors of x and y

• m ∈ IN is the number of transitions of the PN


New distance

p1

p2

p3

p4

p5

p6

t1

t2

t3

t4

p1

p2

p3

p4

p5

p6

t1

t2

t3

t4

tv = (0,1,1,0) tv = (0,0,1,0)p1

p2

p3

p4

p5

p6

t1

t2

t3

t4

p1

p2

p3

p4

p5

p6

t1

t2

t3

t4

tv = (0,1,1,0) tv = (0,1,1,0)


Complete Metric Space X

• From this metric d, we can define the distancebetween a vector x ∈ INn and a subset B ofvectors of INn, i.e., an element in P(INn):

d(x, B) = min{d(x, y) : y ∈ B}

• Distance between A, B in P(INn):

D(A, B) = max{d(x, B) : x ∈ A}


Marked Timed-Arcs∗ Petri Net

��

��

��

��

��

��

?

?

?

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?

?

�

aaa��

��

��

��>

PPPPPPPi

��*

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......

��-

.......................... ..............

......

.......................�

t1

t2

t3

t4

p1

p2

p3

p4

p5

p6

00 0

< 0 >< 0 >

< 0 >

< 4 >

< 5 >

< 0 >

t = 0


MTAPNs∗ t = 4

& t = 5


MTAPNs∗ t = 4 & t = 5


Urgent TAPNs to DDSs

• X = P({IN, ∅}n). n = #(places) of the MTAPN.

• τ is the monoid N ∪ {0}

• Φ : τ ×X → X is the evolution operator Φ verifying:

1. Φ(0, A) = A ∀A ∈ X, i.e., Φ0 = idX

2. Φ(1, A) = B A, B ∈ X where:• A = {z1, . . . , zk}. zi ∈ {IN, ∅}n is a Marking.• ∀t ∈ {1 . . . k} . xt = 1 + zt where ∀i ∈ {1 . . . n}:

xti =

∅ when zti = ∅

zti + 1 otherwise• B = ∪k

i=1Bi

• Bi = ∪tj=1{y

ji }/xi[RRi〉y

ji being RRi maximal.

3. Φ(t,Φ(s, A)) = Φ(t + s, A) ∀t, s ∈ τ , ∀A ∈ XProcess Algebras and Petri Nets are Discrete Dynamical Systems – p.51/55

Conclusions and Future Work


Conclusions

• A Topology over ROSA processes consistent withROSA semantics has been provided

• A cheaper P.A. according to a Means - Ends Policy

• Discrete and Stochastic Petri Nets encoded asDynamical Systems

• A Complete Metric Space for the DDSs associatedto 1-safe, k-safe and unbounded PNs


Future Work

• If P.A. heuristics does not work (chosen path iswrong) → Mutation means different FITNESS . . .

• Improving expressive power of nf − P.A.

• Key points in DSs Theory:• Periodic Points• Semi-periodic points• Orbits from M0• In DDSs are sequences of elements of X• In CDSs are curves in X

• α-limit and ω-limit sets

• Comparison over PNs


Future Work







Future Work







Future Work




• α-limit and ω-limit sets• Comparison over PNs


Process Algebras and Petri Nets are DiscreteDynamical Systems

Fernando Lopez Pelayo

Real Time and Concurrent Systems Group

Computing Systems Department of University of Castilla-La Mancha

Facultad de Informatica

Universidad Complutense

December 4th, 2014 Madrid, SPAIN


Education

Process Algebras and Petri Nets are Discrete Dynamical Systems