Shortest%20Violation%20Traces%20in%20Model%20Checking%20Based%20on%20Petri%20Net%20Unfoldings%20and%20SAT

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Shortest Violation Traces in Model Checking Based
on Petri Net Unfoldings and SAT

• Victor Khomenko
• University of Newcastle upon Tyne

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Shortest violation traces

• Can be much shorter than the first computed trace
• Do not contain incidental system activity
  unrelated to the found error
• Facilitate debugging, saving the designers time

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Petri net unfolding prefixes

• Partial-order semantics of PNs
• Concurrency represented explicitly, using an
  acyclic PN
• Alleviate the state space explosion problem
• Efficient model checking algorithms

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Dining Philosophers
P13
P5
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Model checking on PN unfoldings

• A Boolean expression ?? is built using the
  prefix, such that
• ? is unsatisfiable iff the property holds
• Every satisfiable assignment of ? gives a
  violation trace
• ? has a form CONF?VIOL
• Some of the variables of ? are associated with
  the events of the prefix

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CONF Causality

• If an e is executed than its causal predecessors
  are also executed (its enough to require that
  the direct predecessors of e are executed)

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CONF Conflicts

• If an e is executed than events in conflict
  cannot be executed (its enough to require that
  the events in direct conflict with e are not
  executed)

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VIOL Deadlock

• For every e either
• some direct predecessor is not executed, or
• an event in direct conflict has fired, or
• e itself has fired

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Computing shortest traces

• input ? - a Boolean expression
• output T - a shortest violation trace or UNSAT
• A ? SAT_Assignment(?)
• if A UNSAT then T ? UNSAT stop
• T ? Extract_Trace(A)
• r ? T l ? 0
• while l lt r do
• t ? ?(l r)/2?
• A ? SAT_Assignment( ? ? Thresholdt )
• if A UNSAT then l t 1
• else
• T ? Extract_Trace(A)
• r ? T

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Threshold constraint

• First build a Boolean circuit and then translate
  it into a boolean expression (linear translation
  is possible by adding new variables)
• Try to minimize the changes in the circuit if the
  threshold changes good for incremental SAT

O(log n)
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Implementation of the counter
Size (if n is a power of 2) 4n 2 log2n 4
auxiliary variables 16n 10 log2n 16
clauses 52n 36 log2n 52 literals

• Linear translation
• Large multiplicative constants

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Exploiting conflicts

• Events in a conflict cluster are mutually
  exclusive
• An ?-gate can be used as a counter

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Implementation of the counter

• Significant gains if
• the number of
• clusters is much
• smaller than the
• number of events
• Need to partition
• the prefix into the
• minimum number
• of conflict clusters
• An NP-complete
• problem (reduction
• from partition into
• cliques)
• A greedy algorithm can be used in practice

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Exploiting causality

• If an event in a cluster has fired, some event in
  a preceding cluster has also fired
• If Cl1ltCl2ltltCln, then the outputs of the
  corresponding ?-gates are ordered

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Implementation of the counter

• A sort-adder is simpler than a conventional one!

n
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Partitioning into chains of clusters

• Gains if the number of ordered chains of clusters
  is small
• Need to partition the conflict clusters into the
  minimum number of ordered chains
• The problem can be reduced to maximum matching in
  bipartite graphs and solved in polynomial time,
  but this might be inefficient due to the need to
  work with an implicitly represented graph
• A greedy algorithm can be used in practice

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Experimental results

• The first computed violation trace can be much
  longer than a shortest one computing shortest
  violation traces can indeed greatly facilitate
  the debugging process
• The number of conflict clusters is by many orders
  of magnitude smaller than the number of events
  significant reductions in the size of threshold
  constraint

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The ideal case

• If the adder tree can be implemented as a single
  ?-gate
• 1 (rather than 4n 2 log2n 4) auxiliary
  variables
• n1 (rather than 16n 10 log2n 16) clauses
• 3n1 (rather than 52n 36 log2n 52) literals
• Improvement ratios for n??
• variables ?
• clauses 16
• literals 17?

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Experimental results variables
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Experimental results clauses
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Experimental results literals