/k soundness of free-choice workflow nets 1 of 10 Soundness of Free Choice Workflow Nets K.M. van Hee, M. Voorhoeve Eindhoven Univ. Tech.

/k soundness of free-choice workflow nets 2 of 10 Workflow (WF) net Connected net with unique source ( i ) and sink ( f ) place. R (X ) : markings reachable from marking X. i f WF net is k -sound if Free choice WF (FCWF) net has no multiple arcs and satisfies

/k soundness of free-choice workflow nets 3 of 10 FCWF net: k-sound if 1-sound if N:N: N* : A free choice WF (FCWF) net N is 1-sound iff closure N* marked with [i] is live and bounded. Hence: N* S-coverable. Therefore: 1-sound FCWF net is k – sound for any k. closure

/k soundness of free-choice workflow nets 4 of 10 FCWF net: 1 -sound if k -sound Suppose the above net N is k -sound for some k and Then The only recurrent marking in R [ i ] then is [f], so N must be 1 -sound! if ? Note that R [ i ] is finite!

/k soundness of free-choice workflow nets 5 of 10 Recurrent markings Definition: A marking X is recurrent if (non-recurrent = transient) Lemma 1: If X recurrent, then every marking in R ( X ) is recurrent. Note: so deadlock markings are recurrent! Lemma 2: If R ( X ) is finite, then it contains a recurrent marking.

/k soundness of free-choice workflow nets 6 of 10 Proof: 1 -sound if k -sound The above FCWF net N satisfies Consider a set of recurrent markings in R [ i ]: two cases: Contradiction, so the only recurrent marking in R [i] is [ f ]. if ? |I |=1, so R [ p ] contains deadlock marking X then kX is also deadlock ( N has no multiple edges!), so X = f. |I |>1, so then (FC property!) so these markings are recurrent and cannot reach k[f].

/k soundness of free-choice workflow nets 7 of 10 FC Generalization i f Multiple arcs allowed, plus Also for this class: 1-soundness implies k –soundness direct proof (no S-coverability) Converse does not hold!

/k soundness of free-choice workflow nets 8 of 10 Proof idea: id coloring i f k initial tokens with different ids; colored/colorable firing sequences. State after colorable sequence: superposition. N is k -sound if all firing sequences (FSs) colorable. Not true in depicted net. Given: 1 -sound FCWF net N ; to prove: k -sound. g h e d c b a colored adbeg not colorable aadadbadbe if ii – adbe→ X, then X = Y+Z, with i – ad→ Y and i – be→ Z.

/k soundness of free-choice workflow nets 9 of 10 Firing sequence extension Suppose  is colorable FS and  t not colorable. Then there exists a FS  such that  t  possible and  t colorable. Induction: every FS  can be extended to colorable FS . So N is indeed k -sound. if t  tt  t

/k soundness of free-choice workflow nets 10 of 10 Conclusions FCWF net: If k- sound for some k > 0 then k- sound for all k FCWF net with multiplicities: If 1- sound then k- sound for all k k- soundness with k > 1 : batchwise processing possible 1- soundness: individual order processing every subpart must be earmarked. Thank you!