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Action Planning for Directed Model Checking of Petri Nets

Action Planning for Directed Model Checking of Petri Nets. Stefan Edelkamp Shahid Jabbar Computer Science Department University of Dortmund, Dortmund, Germany. Motivation. Can reachability analysis in Petri nets be accelerated by exploiting heuristic estimates ?. Part I.

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Action Planning for Directed Model Checking of Petri Nets

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  1. Action Planning for Directed Model Checking of Petri Nets Stefan Edelkamp Shahid Jabbar Computer Science Department University of Dortmund, Dortmund, Germany

  2. Motivation • Can reachability analysis in Petri nets be accelerated by exploiting heuristic estimates ? Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  3. Part I Heuristics for the Analysis of Petri nets

  4. Petri Nets • A Bipartite Directed Graph with two nodes set – places and transitions. • Formally, a 4-tuple (P, T, I-, I+)where, • P : Places • T : Transitions • I- :P x T  N –Backward incidence matrix • I+: Tx P  N –Forward incidence matrix Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  5. Execution • Marking – Tokens • Firing Enabled/Live Transition Tokens p1 M0 = <1,1,0> p3 t1 p2 Markings p1 M1 = <0,0,1> p3 t1 Disabled Transition p2 Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  6. Goal Condition • Specific Goal Condition: An explicit marking. • General Goal Condition: A set of different markings satisfying a particular property. E.g., A deadlock in the system – no transition is enabled. Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  7. Example: Dinning Philosphers thinking thinking Pick right fork Pick right fork waiting waiting Pick left fork Pick left fork eating eating Put forks down Put forks down Philosopher 1 Philosopher 2 Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  8. Deadlock in Dinning Philosphers No transition enabled thinking thinking Pick right fork Pick right fork waiting waiting Pick left fork Pick left fork eating eating Put forks down Put forks down Philosopher 1 Philosopher 2 Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  9. Distance Heuristics for Petri Nets – Basics • Heuristics:Evaluation functions that estimate the number of transitions necessary to achieve a goal condition. • Goal condition: • Shortest path between two markings M and M’ is the minimum number of firings necessary to reach M’ from M. • Shortest path to the goal: (M,) = min{(M,M’) | M’ |= } • Admissible: if h(M) ≤ (M,) • Monotone: if h(M) – h(M’) ≤ 1 for MM’ Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  10. 1. Hamming Distance Heuristics • where [M(p) ≠M’(p)]evaluates to{0,1} • Admissible Since a transition can add/delete more than one tokens • Consistent Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  11. 2. Subnet Distance Heuristics (Abstraction) 4-Philosophers 2-Philosophers φ  • Admissible Abstraction preserves triangular property • Consistent Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  12. 3. Activeness Heuristic • Specialized heuristic for Deadlock detection. • Deadlock => No enabled transition. • Prioritize the marking that has the minimum number of diabled transitions. • Admissible Since a single firing can effect the enableness of two or more transitions • Consistent Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  13. Part II:Planning as Directed Model Checking Motivation: Can we utilize planning heuristics by modeling Petri nets analysis problem as a planning problem ?

  14. PLANNER domain.pddl PLAN problem.pddl PDDL Modeling for Petri Nets • PDDL provides a modeling formalism for planning domains and problems. • PDDL Level 2 allows numerical predicates => Necessary to model number of tokens in a Petri net Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  15. Modeling of Topology: • Places :(?p – place) • Transitions :(?t – transition) • Incoming arcs to transitions • (incoming ?p – place ?t – transition) • Outgoing arcs from transitions • (outgoing ?t – transition ?p – place ) • Number of tokens • (number-of-tokens ?p – place) Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  16. Modeling of Goal conditions • Blocked Transition (:derivedblock (?t - transition) (exists (?p - place) (and (incoming ?p ?t) (= (number-of-tokens ?p) 0)))) • Deadlock: (:deriveddeadlock (forall (?t - transition) (blocked ?t))) Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  17. Propositional / ADL Encoding • ADL provides a flexible planning formalism providing support for • Negation • Disjunctive preconditions • Conditional effects • Universal/existensial quantification of objects • Transformation of Petri net model to ADL • Unary encoding of tokens (?n – number) • zero, one, two, three, .. Etc. • Predicates for their manipulations • (is-not-zero ?n – number) • (inc ?n1 ?n2 – number) Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  18. Propositional Planning Operator for Transition Firing (:actionfire-transition :parameters (?t - transition) :precondition (forall (?p - place) (or (not (incoming ?p ?t)) (exists (?n - number) (and (number-of-tokens ?p ?n) (is-not-zero ?n))))) :effect (and (forall (?p - place ?n1 ?n2 - number) (when (and (incoming ?p ?t) (inc ?n1 ?n2) (number-of-tokens ?p ?n2)) (and (not (number-of-tokens ?p ?n2)) (number-of-tokens ?p ?n1)))) (forall (?p - place ?n1 ?n2 - number) (when (and (outgoing ?t ?p) (inc ?n1 ?n2) (number-of-tokens ?p ?n1)) (and (not (number-of-tokens ?p ?n1)) (number-of-tokens ?p ?n2))))))) If all incoming places to t have tokens ? Delete tokens from input places Add tokens at output places Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  19. Planning Heuristic Post-condition droped • Action a = (pre(a), add(a), del(a)) • Relaxed action a+ = (pre(a), add(a), Ø) • Heuristic = length of the shortest plan that solves the relaxed problem. Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  20. Experiments • Used FF Planner developed byHoffmann. • Relaxed Planning Heuristic. • Extensive testing on deadlock checking benchmarks by Corbett. • 1-safe Petri nets models. • A net is called 1-safe, if M(p)≤ 1 for allp • Compared with the results by Heljanko and Niemelä on Bounded Model Checking. Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  21. Experimental Results: Analysis of 1-safe petri nets with FF vs. Bounded Model Checking 450 MHz 3.2 GHz Further results in the paper … Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

  22. Conclusions • Heuristics for analyzing Petri nets. • Hamming distance, abstraction, activeness. • For the first time, modeling of a model checking problem as a planning problem. • Enable us to utilize planning heuristics for analysis of Petri nets. • Experimental results show the strong potential of the approach. • Can incorporate more complex goal conditions like assertions. • (<= (number-of-tokens ?p) ?m) Action Planning for Directed Model Checking of Petri Nets - Edelkamp, Jabbar

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