Parallel LTLX Model Checking of HighLevel Petri Nets Based on Unfoldings

1
Parallel LTL-X Model Checking of High-Level Petri
Nets Based on Unfoldings
UNIVERSITY OF STUTTGART

• Claus Schröter and Victor Khomenko
• University of Stuttgart, Germany
• University of Newcastle upon Tyne, UK

2
Basis for our work
Esparza and Heljanko (ICALP 2000, SPIN 2001) A
New Unfolding Approach to LTL Model-Checking

• Net system is constructed as the product of
• the original net system and
• an Büchi automaton accepting ??

• Model-checking problem is reduced to detection
  of
• illegal ?-traces and
• illegal livelocks
• by exploiting finite complete prefixes

3
Basis for our work

• Simplicity of this approach
• Partial order semantics of Petri nets
• Alleviates the state space explosion problem

? Input are low level Petri nets ? Low level
Petri nets are not convenient for modelling
4
Coloured PNs
a good intermediate formalism

• Low-level PNs
• Can be efficiently verified
• Not convenient for modelling

• High-level descriptions
• Verification is hard
• ? Convenient for modelling

Gap
5
Coloured PNs
1,2
1,2
1
2
v
u
wltuv
w
1..4
6
Coloured PNs
1,2
1,2
1
2
v
u
wltuv
w
1..4
7
Coloured PNs
1,2
1,2
v
u
wltuv
w
1..4
1
8
Coloured PNs
1,2
1,2
v
u
wltuv
w
1..4
2
9
Expansion
10
Expansion
?
?
11
Expansion
?
12
Expansion
?
?
13
Expansion
?
14
Expansion
1,2
1,2
1
2
u
v
wltuv
w
1..4
? Blow up in size

• The expansion faithfully models the original net

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Finite complete prefix

• Introduced by McMillan in 1992
• Relies on the partial order view of concurrent
  computation
• Represents system states implicitly, using an
  acyclic net
• Satisfies two key properties

• Completeness Each reachable marking of the
  original net is represented by at least one
  reachable marking in the prefix

• Finiteness The prefix is finite and thus can be
  used as an input to model-checking algorithms

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Relationship diagram
expansion
Low-level PNs
Coloured PNs
unfolding
unfolding
?
Low-level prefix
Coloured prefix
17
Relationship diagram
expansion
Low-level PNs
Coloured PNs
unfolding
unfolding

Low-level prefix
Coloured prefix
Khomenko and Koutny proved isomorphism (TACAS03)
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Relationship diagram
1,2
1,2
1
2
u
v
wltuv
w
1..4
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Example Buffer of capacity 2
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Example Buffer of capacity 2
p10,1
p30,1
0
1
a
a
a
a
t1
t2
t3
a
a
b
b
p20,1
p40,1
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Example Buffer of capacity 2
p10,1
p30,1
0
a
a
a
a
t1
t2
t3
a
a
b
b
1
p20,1
p40,1
22
Example Buffer of capacity 2
p10,1
p30,1
0
a
a
a
a
t1
t2
t3
a
a
b
b
1
p20,1
p40,1
23
Example Buffer of capacity 2
p10,1
p30,1
0
1
a
a
a
a
t1
t2
t3
a
a
b
b
p20,1
p40,1
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Example Buffer of capacity 2
p10,1
p30,1
0
a
a
a
a
t1
t2
t3
a
a
b
b
1
p20,1
p40,1
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Example Buffer of capacity 2
p10,1
p30,1
a
a
a
a
t1
t2
t3
a
a
b
b
0
1
p20,1
p40,1
26
Example Buffer of capacity 2
Property f ??(p2?0)
Büchi automaton A?f
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Synchronisation

• Standard technique Synchronisation on all
  transitions
• ? Synchronisation sequentialises the system
• ? Not suitable for unfolding based
  verification

• Solution Synchronisation just on those
  transitions which touch the atomic
  propositions of the formula
• ? Concurrency can be exploited

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Synchronisation
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Synchronisation
p2
p2
30
Synchronisation
S?
B?
p2
p2
31
Synchronisation
B?
p2
p2
32
Synchronisation
B?
p2
p2
33
Synchronisation
S?
p2
p2
34
Synchronisation
S?
p2
p2
35
Synchronisation
p2
p2
36
Illegal ?-traces

• Infinite transition sequence that touches q1
  infinitely often violates f

• To detect such runs we introduce a set I off all
  transitions putting a token into an accepting
  Büchi place

• An infinite transition sequence of the
  synchronised net which is fireable from the
  initial marking and contains infinitely many
  occurrences of I-transitions violates f (illegal
  ?-trace)

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Synchronisation
p2
p2
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Synchronisation
p2
L1
p2
39
Synchronisation
p2
p2
L2
p2
?(p2?0)
40
Synchronisation
p2
p2
p2
?(p2?0)
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Prefix
p10
p31
p31
t3
p41
S
q0
I0
S
u0
p10
q0
q0
B
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Prefix
p10
p31
p31
t3
p41
S
q0
I0
S
u0
p10
q0
q0
B
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Prefix
p10
p31
p31
t3
p41
S
q0
I0
S
u0
p10
q0
q0
B
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Prefix
p10
p31
p31
t3
p41
S
q0
I0
S
u0
p10
q0
q0
B
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Prefix
p10
p31
p31
t3
p41
S
q0
I0
S
u0
p10
q0
q0
B
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Experimental Results
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More Results
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More Results
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Results for Parallel Mode
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Conclusions

• Efficient parallel LTL-X model-checker for high
  level Petri nets
• Based on partial order techniques (unfoldings)
• Alleviates the state space explosion problem
• Experimental results showed a good performance of
  our checker for several examples