Toward Efficient Reachability Analysis of

Algebraic Petri-NetsSteve Hostettler

Geneva University

SummaryMotivationState Space RepresentationAPN Firing homomorphismsBenchmarksWhy it does not perform.Future Work & Open issuesConclusion

MotivationAPN

Petri Net (Concurrency, atomicity)Algebraic Data Types (Data Types with User Defined Semantic)

Reachability AnalysisCompute the exhaustive set of reachable states.Check if a property always/never holds.Check whether a specific state exists.

State space blows up quicklyDomains are not limited to black token as in P/T nets.

Example : The Dining Philosophersp0, p1

goEat

<fk> <fk>

f0, f1

<fk> <fk>

goThink

takeL takeR

Think

WaitL WaitR

Forks

HasRHasL

p

l

l

r

r

ff

l, r

rl

l

rightF(p)leftF(p)

f = rf = l

next(l) = r

ADT Philossort phuse fkOps

p0 : -> php1 : -> phphilo : fk ->

phrightF : ph ->

fkleftF : ph ->

fkAxioms

rightF(p0) = f1

rightF(p1) = f0

leftF(p0) = f0leftF(p1) = f1philo(f0) = p0philo(f1) = p1

ADT Forkssort fkOps

f0 : -> fkf1 : -> fknext : fk -> fk

Axioms next(f0) = f1next(f1) = f0

State Space Representation (1)Decisions Diagrams

Limit the combinatorial explosion.Have a Canonicity of the representation.Powerful set operatorsUser Defined Operators : inductive homomorphisms.

Data Decisions Diagrams (DDD) – Couvreur02Values on arcs, reachability P/T nets : 100 philosophers.

Set Decisions Diagrams (SDD) – Mieg05Sets on arcs , reachability P/T nets : 10’000 philosophers.

State Space Representation (2)

Does not work with SDD!Breaks the one path = one state” paradigm!

T WL

M

1

1

WR FK HL HR

{p0, p1}M 1

{f0, f1}

Multiset of tokens

Cardinality

Place Empty multiset

Set of token with cardinality = 1

APN Firing homomorphism (1)

p0

t

q0 qm

pn

ci0’a0 cin’an

co0’b0 com’bm

…

…

H-(place, card, variable)1. For each pi, check whether there is

sufficient token on it.2. Create a new assignment

Check(left, right)3. Check whether ∧lk = rk holds

H+(place, card, term)4. For each qj, add coj’bj to it.

RemoveVars()5. Remove the assignments created in 2.

…

…

qj

coj’bj

pi

cii’ai

ai are variables

∧lk = rk

goEat = H-(T, 1, p) H+(WL, 1, leftF(p)) H+ (WR, rightF(p))

APN Firing homomorphism (2)

H-(T, 1, p)

T {p0}

P

T {p1}

{p0}

{p1}

T WL 1WR FK HL HR{p0, p1} {p0, p1}

APN Firing homomorphism (3)

T

P

T

WL 1WR FK HL HR{p0, p1}

{p1}{p0}

{p1}

H+(WL, 1, leftF(p))H+(WL, 1, leftF(p1))

H+(WL, 1, leftF(p0))

{p0}

P

{p0}

{p1} T

T

WL

{p1}

H+(WL, 1, p1)

H+(WL, 1, p0)

{p0}

WL

WL

WL

{p0}

{p1}

APN Firing homomorphism (4)Complex homomorphism are built from simple one.H+ : Term may be still open when we get to the place to

update.Structural clustering : substitutions may come from

other clusters.Need something to carry substitutions from one cluster to

another.Same problem as for local H+ .

Benchmarks

Poor performances (linear to #states)Due to Bad management of the concurrency.

#Philo #States #DD %Cache Mem (MB) Time (ms)2 18 1122 31.01 1 223 76 3539 32.4 3 544 322 14201 40.05 7 1495 1364 61809 48.40 25 6976 5778 270820 56.19 113 29527 24476 1206461 62.97 531 150458 103682 / / / /

State of the ArtSoftware URL #Philo

Helena http://helena.cnam.fr 15

Maria http://www.tcs.hut.fi/Software/maria/ 12

CNAM Has started a new project 2007 to tackle this problem

/

PetriNetMC http://smv.unige.ch 7

Why Poor performances?Bad performances

Symptom : #DD grows as fast as #states.Power set effect is not handle efficiently

We don’t use the Cartesian product effect of DD.Concurrency induced by token multiplicity in not efficiently

managed.Several instances of the same net.

The P/T implementation is (far) more efficient We need to reproduce the P/T effect.

Cartesian product effect• Data Decision Diagrams• Represents 6 states• 9 arcs and 7 places• Unfolded : 30 arcs, 30 places

• Set Decision Diagrams• Represents 6*6 = 36 states• 11 arcs and 10 nodes are required (canonicity)• Unfolded : 360 arcs, 360 places

C1 C2

A

1

11

B C E F

F

2

5

2 2 2

33

5

A 1B C E F

F5 33

5

2222 2A 1B C E F

F 35

2222 2

3

Cluster (1)Clustering

If subnet x is not modified by transition t on subnet y we don’t touch it.

Structural clusteringEfficient to modularize the behaviorNot efficient to handle concurrency due to many instances of

the same token.Algebraical clustering

To tackle the problem of the concurrency induced by the token multiplicity.

Cluster (2)All input arcs and output arcs of a transition in the

same clusterLocal transitionAllow saturation : apply all local transitions till fixpoint and

then propagate.Else we try to synchronize only the clusters involved in

the transition.

Algebraic Cluster (1)The Clustering function Cl associates all pairs <Place, token>

to a cluster. Here token represents all the possible values of the domain.

Split the DD in clusters according to Cl.Transitions take and put token according to Cl.Enable a static analysis to build cluster application in advanceExample

∀pi {p∈ 0, ..., pn − 1} With n # of philosophers : CL(T, pi) = ci CL(F, fi) = ci CL(WL, fi) = ci CL(WR , fi) = succ(ci) CL(HL, fi) = ci CL(HR , fi) = succ (ci)

Algebraic Cluster (2)Express Cluster as an ADTCluster function may be tedious to defineRemain as symbolic as possiblePatterns (identity)

Algebraic Cluster (3)Example:

Pi = p0

CL(T, p0) = c0 CL(WL, leftF(p0)) = CL(WL , f0) = c0

CL(WR , rightF(p0)) = CL(WR , f1) = succ(c1) = c0 Pi = p1

CL(T, p1) = c1 CL(WL, leftF(p1)) = CL(WL , f1) = c1

CL(WR , rightF(p1)) = CL(WR , f0) = succ(cO) = c1 GoEat is Local

Benchmarks

General Homs but static Cluster application Still not a proof of conceptTime for static analysis not included.

#Philo #States #DD %Cache Mem (MB) Time (ms)2 18 401 13.83 2 254 322 1555 22.94 2 728 103682 13395 35.97 7 23116 1.075 E10 130731 47.85 50 165232 1.156 E20 1184692 56.13 410 13896

Future Work & Open issuesMore examples.Cost of the static analysis.How to build a dynamic homomorphism to avoid static

analysis.Example with both structural and algebraic saturation.Clustering and Homomorphism formalization.Complete eclipse integration, Complete software

platform (doc, web, …)

ConclusionWe (will) have a complete suite of tools from model

design to model validation.Good performances (vs. competition).User friendly.

RoadmapToward Efficient Reachability Analysis of Algebraic

Petri Nets -> ICATPN09Deadline : January 5, 2009ExperimentsFormalization

Set of Terms Rewriting -> TermGraph09Deadline : December 15, 2008Formalization1 Case study (Chemical Abstract Machine)