BehaviourPreserving Transition Insertions in Unfolding Prefixes

1
Behaviour-Preserving Transition Insertions in
Unfolding Prefixes

• Victor Khomenko
• University of Newcastle upon Tyne

2
Motivation

• Some design methods based on Petri nets
  repeatedly execute the following steps
• Analyze the original PN spec
• Modify the PN by behaviour-preserving transition
  insertion

3
Example VME Bus Controller
4
Example Encoding Conflict
5
State Graphs vs. Unfoldings

• State Graphs
• Relatively easy theory
• Many efficient algorithms
• Not visual
• State space explosion problem

6
State Graphs vs. Unfoldings

• Unfoldings
• Alleviate the state space explosion problem
• More visual than state graphs
• Proven efficient for model checking
• Quite complicated theory
• Not sufficiently investigated
• Relatively few algorithms

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Example Encoding Conflict
e10
e8
dtack-
dsr
e1
e2
e3
e4
e5
e6
e7
e12
lds
ldtack
dtack
dsr
lds
d-
dsr-
d
Code(conf)10110
Code(conf)10110
lds-
ldtack-
e9
e11
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Example Resolving the conflict
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Example Resolving the conflict
dtack-
dsr
csc
001000
000000
100000
100001
lds
ldtack-
ldtack-
ldtack-
dtack-
dsr
011000
100101
010000
110000
ldtack
lds-
lds-
lds-
dtack-
dsr
110101
011100
110100
010100
d
d-
dtack
dsr-
csc-
011111
111111
110111
011110
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Example Resulting Circuit
Data Transceiver
Device
Bus
d
lds
dtack
dsr
csc
ldtack
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Motivation validity

• Need to check the validity of the transformation
• safeness
• bisimulation
• The validity should be checked before the
  transformation is performed, i.e. on the original
  prefix (to avoid backtracking)

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Motivation avoid re-unfolding

• Perform the transformation directly on the prefix
  to avoid re-unfolding
• Re-unfolding is time-consuming
• Good for visualization (re-unfolding can
  dramatically change the look of the prefix)
• Can transfer information (e.g. encoding
  conflicts) between the iterations of the algorithm

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Example Re-unfolding
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Sequential pre-insertion

• Preserves safeness
• Preserves traces
• Can introduce deadlocks need to check that the
  new transition never steals tokens from any
  other enabled transition
• simple state property
• can be checked on the original prefix

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Sequential post-insertion

• Preserves safeness
• Yields a bisimular PN
• Nothing to check!

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Concurrent insertion

• Can introduce unsafeness
• Can introduce deadlocks

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Place insertion token
t
t
p

• If the place insertion is valid and t or t is
  not dead then p contains token iff there is a
  t-labelled event in the prefix which does not
  have t-labelled predecessor

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Place insertion validity
t
t
n
p

• Tokens(C)n tC tC
• The transformation is valid if
• for all instances e of t and t of the prefix,
  Tokens(e)??0,1, and
• for all cut-offs e with a corresponding
  configuration C, Tokens(e)Tokens(C)
• If a valid transformation is rejected by this
  criterion then t and t are not live

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Pre-insertion in the prefix

• Naïve splitting can yield an incomplete prefix!

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Pre-insertion in the prefix

• Naïve splitting can yield an object which is not
  a branching process!

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Pre-insertion in the prefix

• Find all possible extensions of the prefix by the
  new transition
• Amend the instances of the split transitions
• Amend the cut-off corresponding configurations

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Post-insertion in the prefix

• Naïve splitting can yield an incomplete prefix!

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Post-insertion in the prefix

• Definition a configuration is extendible if in
  the modified prefix it can be extended by an
  instance of the new transition
• If there is a cut-off event e with a
  corresponding configuration C such that e is
  extendible and C is not extendible then terminate
  unsuccessfully
• Amend the instances of the split transition
• Amend the cut-off corresponding configurations

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Place insertion in the prefix

• Assumption the place insertion has passed the
  validity check
• If n 1 then create a new (causally minimal)
  instance cmin of p
• For each instance e of t' (including cut-offs),
  create a new instance of p and connect it to e
• For each instance e of t'' (including cut-offs)
  connect e to cmin if e has no t'-labelled
  predecessor and to the instance of p in the
  postset of the (unique) maximal t'-labelled
  predecessor of e otherwise

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Concurrent insertion in the prefix

• Perform the corresponding place insertion
• Perform the sequential pre-insertion
• This two steps can easily be combined

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Equivalent insertions

• Equivalence is easy to check
• Fewer transformations to consider
• Can convert to canonical form, e.g.
  pre-insertions good for unfolding
• No need to check validity post-insertions are
  always valid

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Commutative insertions

• Definition two transition insertions commute if
  they can be performed in any order
• concurrent insertions commute with any other
  insertions
• pre-insertions commute with post-insertions
• two pre/post-insertions commute iff they split
  different transitions or the sets of split off
  places do not overlap
• A valid insertion remains valid if another valid
  commutative insertion is applied first, i.e. the
  validity needs to be checked only once

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Summary

• Rigorous validity criteria developed
• can be checked on the original prefix no
  backtracking
• Algorithms for performing transformations
  directly on the prefix
• avoids re-unfolding, good for performance and
  visualization
• proofs of correctness
• Optimisation
• equivalent transformations
• commutative transformations

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• Thank you!
• Any questions?