Canonical Prefixes of Petri Net Unfoldings Walter Vogler Universität Augsburg in cooperation with V. Khomenko, M. Koutny (CAV 2002, Acta Informatica 2003)

UFO 072 some concepts you know unfolding Unf of a Petri net (full branching process) C : set of (finite) configurations C loc : set of local configurations [e] with e  E is [e] without e if E’ is a suffix of C’, write C’  E’

UFO 073 McMillan: construct complete finite prefix of Unf stepwise; for cut-offs, compare configurations by size ERV:more abstract setting: compare with adequate order  ( e  f if [e]  [f] ) –ERV-algorithm constructs complete finite prefix

UFO 074 step: add  - minimal possible extension e declare cut-off if there is f in  with [f]  [e] & [e] and corresponding configuration [f] reach the same marking non-deterministic choice; different results? (maybe f exists in Unf, but not yet added to this  ) e f  f

UFO 075 it seems: prefix depends on algorithm and non-deterministic choices aim: define algorithm-independent canonical prefix application: (parallel) slicing algorithm [HKK 02] processes e.g. all  - minimal possible extensions together –original correctness proof: compares each run with ERV- algorithm –new: all runs (slicing, ERV) construct the canonical prefix

UFO 076 this approach in an even more general setting: cutting contexts canonical prefix finite and complete in ~ all such contexts

UFO 077 Overview cutting contexts canonical prefix completeness finiteness and König’s Lemma algorithmics a recent application (Khomenko ATPN 07)

UFO 078 cutting contexts – motivation parameter: adequate order parameter: corresponding configuration local or general or … (general configuration for minimised prefix) parameter: information to be represented –usually: marking, i.e. two configurations with the same marking are equivalent, one of them suffices –or: marking plus state vector (for asynchronous circuits, STGs) more information, finer equivalence –or: marking up to symmetries (smaller prefix) less information, coarser equivalence

UFO 079 cutting contexts – definition cutting context  = ( , , {C e } e  E ) where 1.for each event e  E of Unf, C e  C usually C e = C loc  is dense if C e  C loc 2.  is an equivalence on C 3.  is an adequate order, i.e. a well-founded strict partial order on C (no infinite decreasing sequence), refining  4.… preserved by finite extensions …

UFO 0710 cutting contexts – definition cutting context  = ( , , {C e } e  E ) where 4.  and  are preserved by finite extensions, i.e. whenever C’  C’’ and E’ is a suffix of C’, then there exists suffix E’’ such that a.C’’  E’’  C’  E’ (not needed in standard case) b.if C’’  C’, then C’’  E’’  C’  E’ (more general than standard case) standard: E’’ obtained from E’ by natural isomorphism

UFO 0711 canonical prefix simple idea: e cut-off if C  C e with C  [e] and C  [e] Does not work in standard setting! Such a local C can actually contain a cut-off; will never be constructed, not be in canonical prefix.  e not detected as cut-off Actually, some smaller C’ with the same marking and without cut-offs exists, but it might be non-local.

UFO 0712 canonical prefix Def.: feasible events fsbl  (events with no cut-off in past, will be in prefix) and static cut-offs cut  satisfy 1. e is feasible if  cut  =  ; ( is [e] without e) 2. feasible e is static cut-off, if there is C  C e with C  [e], C  [e] and C  fsbl  \ cut . Canonical prefix Unf  induced by fsbl .

UFO 0713 canonical prefix Def.: 1.e is feasible if  cut  =  ; ( is [e] without e) 2.feasible e is static cut-off, if there is C  C e with C  [e], C  [e] and C  fsbl  \ cut .  is well-founded on E; by Noetherian induction, status of predecessors already defined: 1.f   f < e  [f]  [e]  f  e 2.f  C  [e]  [f]  C  [e]  [f]  C  [e]

UFO 0714 completeness – general notion A branching process  is complete w.r.t. a set E’ of events if: –for every C  C there is a configuration C’ in  such that C  C’ and C’  E’ =  –if C is a configuration of  such that C  E’ =  and {e} is an extension of C in Unf, then C  {e} is also a configuration in  strictly stronger than standard notion: for every configuration, all possible firings are included; –important for deadlock detection

UFO 0715 completeness – result Theorem: Unf  is complete w.r.t. cut . Proof: essentially like in [ERV], using preservation under finite extensions

UFO 0716 finiteness and König’s Lemma König’s Lemma: An infinite, locally finite, rooted, directed graph has an infinite path. not applicable to branching processes: conditions may have infinitely many outgoing arcs! Theorem: An infinite branching process always contains an infinite chain of causally related events. The result holds also for unbounded nets!

UFO 0717 finiteness and König’s Lemma used this version of König’s Lemma to prove Theorem: 1. If  has finite index and  is dense, then Unf  is finite. 2. If  has infinite index, then Unf  is infinite. Remarks: 1. Finite index means bounded in standard case; 2. clear from completeness.

UFO 0718 algorithmics Theorem: If Unf  is finite, then the ERV- and the slicing algorithm generate Unf  – for any . essential fact: if e is a possible extension, then all feasible f  [e] have already been added.

UFO 0719 a recent application V. Khomenko: Behaviour-Preserving Transition-Insertions in Unfolding Prefixes, ATPN 2007 useful for designing asynchronous circuits

UFO 0720 a recent application setting: find place for insertion by studying a prefix Unf  desirable: prefix Unf’  of the new net can be obtained from Unf  by the “same” insertions – but cut-offs may be different! solution: the “same” insertions give the canonical prefix for a different cutting context  ’ !