Process Algebras and Petri Nets are Discrete Dynamical Systems

transcript

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;

FITNESS - given 2 target states

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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)−

• 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:⊗

[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:⊗

[qi]⊕

Aj∈Ai

+a∈Type(Aj)

〈a, λAj〉.nf(PAj ,a)

Normal-Formed Genetic ROSA

∀P ∈ G−ROSA process :

P ::= 0 |⊗

[qi]Qi

Decreasing ordered qi and λA

Q ::=⊕

Aj∈A

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‖)−

• ‖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

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

MTAPN Modelling MPEG-2 I

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

MTAPN Modelling MPEG-2 II

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

B2out00

Time = 595ms

MTAPN Modelling MPEG-2 III

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

Time = 895ms

MTAPN Modelling MPEG-2 IV

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

Time = 895ms

MTAPN Modelling MPEG-2 V

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

Time = 1268ms

MTAPN Modelling MPEG-2 VI

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

Time = 1268ms

MTAPN Modelling MPEG-2 VII

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

Time = 1863ms

595595 968

MTAPN Modelling MPEG-2 VIII

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

Time = 2163ms

895895 1268895

MTAPN Modelling MPEG-2 IX

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

Time = 2163ms

895895 895

MTAPN Modelling MPEG-2 X

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

Time = 2448ms

1553285

Demanding processors

DCT_Q I

IQ_IDCTI VLC I

Perror P

<595,595>

<298,358><673,673>

<0,0> <0,0>

<1180,1181>

<32,32>

<48,48>

<189,454>

<0,0> <0,0>

<189,454>

<521,521><521,521>

<48,48>

MV−B1 MV−B2

<521,521>

<0,0>Pout

<378,389>

IQ_IDCT FSP

<0,0><515,515>

I−B2

I−B1

<2398,2422>

P−B1

P−B2

<2398,2422>

B1error

B2error

ERRORB1

COMP B1

OUT B2

VLC B2

ERRORB2

COMPB2

EST B2

MV−P

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)−

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

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))−

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

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

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

��

aaa��

��

��>

PPPPPPPi

��*

��..............

......

��-

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

......

.......................�

< 0 >< 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}:

∅ when zti = ∅

zti + 1 otherwise• B = ∪k

• 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

• 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

Process Algebras and Petri Nets are Discrete Dynamical Systems

Education