Implementing Petri Net Transformations

1
Implementing Petri Net Transformations using
Graph Transformation Tools

• Enrico Biermann, Claudia Ermel,Tony Modica and
  Peggy Sylopp
• 3rd Workshop on Petri Nets and Graph
  Transformations

2
Overview
Petri nets to typed attributed graphs
3
Category of Petri nets

• Common frameworkHigh-level Replacement Systems
  (DPO)
• category PTSys
• Petri systems PS (PN, M)
• algebraic with monoids
• net PN (P,T, pre,postT?P )
• (similar to graphs with hyper edges)
• marking M?P
• morphisms (fP,fT)PS1?PS2
• place mapping fPP1?P2
• transition mapping fTT1?T2
• with some structural conditions

?
?
4
Properties of Petri net morphisms

• conditions for (fP,fT)PS1?PS2
• preservation of transitions pre andpost domain
  (environment)
• preservation of markings per token
• Morphism called strict if (and injective)
• in general not
• conditions ensure preservation of firing behavior

5
Petri net transformation rules

• strict rule morphisms l, r
• rule is applicable if l and m satisfy gluing
  condition (ensuring existence of pushout
  complement D)
• Gluing Points are all LHS nodes in the image of
  l(matched by the rule but not deleted)
• Dangling Points are all LHS places that would
  leave a dangling edge after deletion
• Identification Points are all LHS nodes being
  matched non-injectively
• gluing condition (1) (2) m is strict on places
  to be deleted
• In short Dont delete places without their
  adjacent transitions!And dont match those
  places with less tokens!
• we only consider injective matches for now
• (no transitions as DP because P/T-morphisms
  preserve environments)

6
Simulating Petri net transformationwith graph
transformation

• rule application and match search
• obvious translation of P/T-systems and extension
  to (injective) P/T-morphisms
• problems because of P/T-morphism properties
  straightforward translation of P/T-rules yields
  graph rules with
• too many applications (graph matches do not
  preserve transition node environments)
• less applications (graph matching of token
  attribute is strict)
• we want match calculation, so we need a proper
  rule translation
• (translation is not a functor, non-strict
  P/T-morphisms do not have valid translations)

7
Problem 1 domain preservation

• graph morphisms are not restricted to preserve
  transition environments
• translated graph rule can have more applicable
  matches than original Petri net rule
• solution two negative application conditions
  for each transition in LHS
• forbid matching of transition if there are
  unmatched places in its environment

8
Problem 2 token matching

• graph attributes can be matched on same values
  only!
• non-strict P/T-morphisms can not be directly
  translated
• introduce token variable for each place in LHS
  that will not be deleted (i.e., has preimage in
  K)
• rule attributes assume values determined by match
• attribute condition to allow non-strict token
  attribute matching for graph morphisms
• (each xy equivalent to y NACs with less than y
  tokens on respective place)

9
Tool implementation

• Eclipse Plug-In based on EMF (Eclipse Modeling
  Framework) and GEF (Graphical Editor Framework)
• RON - Reconfigurable Object Nets (variant of
  Algebraic Higher-Order Nets)
• editing of P/T-systems, rules, controlling
  high-level net
• uses AGG as graph transformation engine for match
  search and rule application
• converts Petri nets and rules to AGG graphs and
  rules
• reflects changes on translated graph back to
  Petri net after rule application

10
Example Train loading

• trains that can be (un)loaded over a ramp on the
  last wagon
• each wagon can hold 3 pieces of load
• loads can be shifted to adjacent wagons
• if the loading wagon is full we can add an empty
  one after it
• we can remove an empty loading wagon, but only if
  theres still another wagon

11
Example Train loading (formal)

• Petri net for train with 2 wagons
• places for counting free spaces instead of
  capacities
• firing of transitions for (un)loading and
  shifting
• rule for extending a train
• applicable on loading wagons carrying 3 cargo
  units
• rule for reducing a train
• applicable on loading wagons with 3 free spaces
• the next wagon is preserved

12
Future work and outlook

• extended translation for non-injective matches
  (regarding arc weight sums)
• theory for P/T-rules with non-strict morphisms to
  change markings
• formal proofs for ourextended translation
• correctness for a Petri net rule each result of
  a translated possible application is equivalent
  to translationof the applications result
• completeness for a Petri net rule there are no
  other applications for its translation than the
  translated possible applications for itself

13

• Thank you!

14
Literature

• AGG AGG Homepage. http//tfs.cs.tu-berlin.de/agg
  
• BEHM07 E. Biermann, C. Ermel, F. Hermann, T.
  Modica. A Visual Editor for Recon?gurable Object
  Nets based on the ECLIPSE Graphical Editor
  Framework. Proc. Workshop on Algorithms and Tools
  for Petri Nets (AWPN07). 2007.http//tfs.cs.tu-b
  erlin.de/roneditor
• BM08 E. Biermann, T. Modica. Independence
  Analysis of Firing and Rule-based Net
  Transformations in Recon?gurable Object Nets.
  Proc. Workshop on Graph Transformation and Visual
  Modeling Techniques. (GT-VMT08). Vol. 10.
  EC-EASST, 2008.
• EHP08 H. Ehrig, K. Hoffmann, J. Padberg, C.
  Ermel, U. Prange, E. Biermann, T. Modica. Petri
  Net Transformations. In Petri Net Theory and
  Applications. Pp. 116. I-Tech Education and
  Publication, 2008.