V.Khomenko1, A.Kondratyev2, M.Koutny1 and W.Vogler3

1
Merged processes a new condensed
representation of Petri net behaviour

• V.Khomenko1, A.Kondratyev2, M.Koutny1 and
  W.Vogler3
• 1University of Newcastle upon Tyne
• 2Cadence Berkeley Labs
• 3University of Augsburg

2
Petri net unfoldings

• An acyclic net obtained through unfolding the PN
  by successive firings of transitions
• for each new firing a fresh transition (called an
  event) is generated
• for each newly produced token a fresh place
  (called a condition) is generated
• The full unfolding can be infinite
• If the PN has finitely many reachable states then
  the unfolding eventually starts to repeat itself
  and can be truncated (by identifying a set of
  cut-off events) without loss of essential
  information, yielding a finite prefix

3
Example Dining Philosophers
4
Example Dining Philosophers
T1
P1
P7
P8
P9
T6
5
Example Dining Philosophers
T2
P2
T1
P1
T3
P3
P7
P8
P9
T6
6
Example Dining Philosophers
T2
P2
T1
P1
T3
P3
P7
T7
P8
P10
P9
T8
T6
P11
7
Example Dining Philosophers
T2
P2
P4
T1
P1
T3
P3
P7
T7
P8
P10
P9
T8
T6
P11
8
Example Dining Philosophers
T2
P2
P4
T1
T4
P1
T3
P5
P3
P7
T7
P8
P10
P9
T8
T6
P11
9
Example Dining Philosophers
T2
P2
P4
T1
T4
P1
T3
P5
P3
P7
T7
P12
P8
P10
P9
T8
T6
P11
10
Example Dining Philosophers
T2
P2
P4
T1
T4
P1
T3
P5
P3
P7
T7
P12
P8
P10
T9
P9
T8
P13
T6
P11
11
Example Dining Philosophers
T2
P2
P4
T1
T4
T5
P6
P1
T3
P5
P3
P7
T7
P12
P8
P10
T9
P9
T8
P13
T6
P11
12
Example Dining Philosophers
T2
P2
P4
T1
T4
T5
P6
P1
T3
P5
P3
P7
T7
P12
P8
P10
T9
P14
T10
P9
T8
P13
T6
P11
13
Example Dining Philosophers
T2
P2
P4
T1
T4
T5
P6
P1
T3
P5
P3
P7
T7
P12
P8
P10
T9
P14
T10
P9
T8
P13
T6
P11
14
Example Dining Philosophers
T2
P2
P4
T1
T4
T5
P6
P1
T3
P5
P3
P7
T7
P12
P8
P10
T9
P14
T10
P9
T8
P13
T6
P11
15
Example Dining Philosophers
T2
P2
P4
P7
T1
T4
T5
P6
P1
P1
T3
P5
P8
P3
P7
T7
P12
P8
P10
P7
T9
P14
T10
P9
P9
T8
P13
T6
P8
P11
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Characteristics of unfoldings

• Alleviate the state space explosion problem for
  highly concurrent systems
• e.g. for Dining Philosophers the prefix size is
  linear in the number of philosophers even though
  the number of states is exponential
• Efficient model checking algorithms
• e.g. deadlock checking is PSPACE-complete for
  safe PNs but only NP-complete for prefixes
• Do not cope well with other than concurrency
  sources of state space explosion, e.g. with
  sequence of choices
• Do not cope well with non-safe PNs

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Example sequence of choices
No event is cut-off, the prefix is exponential
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Example non-safe PN
m
m
Tokens in the same place are distinguished in the
unfolding, the prefix is exponential
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Wanted A data structure coping not only with
concurrency but also with other sources of state
space explosion
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Example a Petri net
1
3
2
4
21
Example unfolding
3
1
4
3
2
4
Idea Fuse some of the nodes with the same label
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Occurrence-depth
1
1
1
3
2
1
2
1

• Merged Process
• Fuse conditions with the same label and
  occurrence-depth
• Delete duplicate events

23
Examples
m
m
Merged processes of these nets coincide with the
original nets, even though unfoldings are
exponential!
24
Experimental results

• Corbetts benchmarks were used
• Merged processes are often by orders of magnitude
  smaller than unfolding prefixes
• In many cases they are just slightly larger than
  the original PNs
• In some cases they are smaller than the original
  PNs due to removal of dead places

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Upper bounds on the size

• Trivial bound Merged processes are no larger
  than unfolding prefixes and hence no larger than
  the reachability graph
• too pessimistic in practice
• Merged processes of acyclic PN coincide with the
  original PNs with the dead nodes removed
• unfoldings can be exponential
• Merged processes of live and safe free-choice PNs
  with minor restrictions are polynomial in the
  size of the original PNs
• unfoldings can be exponential

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Model checking

• Merged processes are small, but are they of any
  use?
• Can model checking algorithms developed for
  unfoldings be lifted to merged prefixes?

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Problem cycles
A Petri net
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Problem cycles
1
1
2
Unfolding
Criss-cross fusion results in a cycle!
2
1
1
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Problem cycles
Merged process with a cycle
Still worse, the marking equation (ME) used for
unfolding-based verification can have spurious
solutions
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Problem cycles
Fire
Borrow a token
The borrowed token is returned
Fire
The current marking is unreachable
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Solution

• Add to the marking equation another constraint,
  ACYCLIC, requiring the run to be acyclic
• ME ACYCLIC

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Another problem spurious runs
2
Can visit this condition without first visiting
the other one! not possible in the unfolding
1
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Solution

• Add another constraint, NG (no-gap), conveying
  that if a condition with occurrence-depth kgt1 is
  visited then the condition with the same label
  and occurrence-depth k-1 is also visited
• ME ACYCLIC NG
• This is enough to lift unfolding-based model
  checking algorithms to merged processes!
• Deadlock checking is NP-complete in the size of
  merged process no worse than for unfoldings

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Experimental results

• Corbetts benchmarks were used
• Model checking is practical running times are
  comparable with those of an unfolding-based
  algorithm
• Still deteriorates on a couple of benchmarks
  but its early days of this approach and we keep
  improving it

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Open problems / future work

• Direct characterization of merged processes
• currently much is done via unfoldings
• Improve the efficiency of model checking
• A direct algorithm for building merged processes
• currently built by fusing nodes in the unfolding
  prefix
• significant progress has been made in this
  direction