Discovering Petri Nets From Event Logs

Transcription

1 Disovring Ptri Nts From Evnt Logs W.M.P. vn r Alst n B.F. vn Dongn Dprtmnt of Mthmtis n Computr Sin, Thnish Univrsitit Einhovn, Th Nthrlns. {W.M.P.v..Alst,B.F.v.Dongn}@tu.nl Astrt. As informtion systms r oming mor n mor intrtwin with th oprtionl prosss thy support, multitus of vnts r ror y toys informtion systms. Th gol of pross mining is to us suh vnt t to xtrt pross rlt informtion,.g., to utomtilly isovr pross mol y osrving vnts ror y som systm or to hk th onformn of givn mol y ompring it with rlity. In this rtil, w fous on pross isovry, i.., xtrting pross mol from n vnt log. W fous on Ptri nts s rprsnttion lngug, us of th onurrnt n unstrutur ntur of rl-lif prosss. Th gol is to introu svrl pprohs to isovr Ptri nts from vnt t (notly th α-lgorithm, stt-s rgions, n lngug-s rgions). Morovr, importnt rquirmnts for pross isovry r isuss. For xmpl, pross mining is only mningful if on n l with inompltnss (only frtion of ll possil hvior is osrv) n nois (on woul lik to strt from infrqunt rnom hvior). Ths rquirmnts rvl signifint hllngs for futur rsrh in this omin. Kywors: Pross mining, Pross isovry, Ptri nts, Thory of rgions 1 Introution Pross mining provis nw mns to improv prosss in vrity of pplition omins [2, 41]. Thr r two min rivrs for this nw thnology. On th on hn, mor n mor vnts r ing ror thus proviing til informtion out th history of prosss. Dspit th omniprsn of vnt t, most orgniztions ignos prolms s on fition rthr thn fts. On th othr hn, vnors of Businss Pross Mngmnt (BPM) n Businss Intllign (BI) softwr hv n promising mirls. Although BPM n BI thnologis riv lots of ttntion, thy i not liv up to th xpttions ris y mis, onsultnts, n softwr vnors. Pross mining is n mrging isiplin proviing omprhnsiv sts of tools to provi ft-s insights n to support pross improvmnts [2, 7]. This nw isiplin uils on pross mol-rivn pprohs n t mining. Howvr, pross mining is muh mor thn n mlgmtion of xisting pprohs. For xmpl, xisting t mining thniqus r too t-ntri to

2 provi omprhnsiv unrstning of th n-to-n prosss in n orgniztion. BI tools fous on simpl shors n rporting rthr thn lr-ut usinss pross insights. BPM suits hvily rly on xprts moling iliz to- prosss n o not hlp th stkholrs to unrstn th s-is prosss. Ovr th lst vnt t hs om rily vill n pross mining thniqus hv mtur. Morovr, pross mining lgorithms hv n implmnt in vrious mi n ommril systms. Exmpls of ommril systms tht support pross mining r: ARIS Pross Prformn Mngr y Softwr AG, Diso y Fluxion, Entrpris Visuliztion Suit y Businssp, Intrstg BPME y Fujitsu, Pross Disovry Fous y Ionts, Rflt on y Plls Athn, n Rflt y Futur Pross Intllign. Toy, thr is n tiv group of rsrhrs working on pross mining n it hs om on of th hot topis in BPM rsrh. Morovr, thr is hug intrst from inustry in pross mining. This is illustrt y th rntly rls Pross Mining Mnifsto [41]. Th mnifsto is support y 53 orgniztions n 77 pross mining xprts ontriut to it. Th mnifsto hs n trnslt into ozn lngugs ( Th tiv ontriutions from n-usrs, tool vnors, onsultnts, nlysts, n rsrhrs illustrt th growing rlvn of pross mining s rig twn t mining n usinss pross moling. Morovr, mor n mor softwr vnors strt ing pross mining funtionlity to thir tools. Th uthors hv n involv in th vlopmnt of th opn-sour pross mining tool ProM right from th strt [11, 56, 57]. ProM is wily us ll ovr th glo n provis n sy strting point for prtitionrs, stunts, n mis. Whrs it is sy to isovr squntil prosss, it is vry hllnging to isovr onurrnt prosss, spilly in th ontxt of noisy n inomplt vnt logs. Givn th onurrnt ntur of most rl-lif prosss, Ptri nts r n ovious nit to rprsnt isovr prosss. Morovr, most rl-lif prosss r not nily lok-strutur, thrfor, th grph s ntur of Ptri nts is mor suitl thn nottions tht nfor mor strutur. Th rtil is s on ltur givn t th Avn Cours on Ptri nts in Rostok, Grmny (Sptmr 2010). Th prtil rlvn of pross isovry n th suitility of Ptri nt s si rprsnttion for onurrnt prosss motivt us to writ this tutoril. Figur 1 illustrts th onpt of pross isovry using smll xmpl. Th figur shows n strtion of n vnt log. Thr r 1391 ss, i.., pross instns. Eh s is sri s squn of tivitis, i.., tr. In this prtiulr log thr r 21 iffrnt trs. For xmpl, tr,,,, h ours 455 tims, i.., thr r 455 ss for whih this squn of tivitis ws xut. Th hllng is to isovr Ptri nt givn suh n vnt log. A isovry lgorithm suh s th α-lgorithm [9] is l to isovr th Ptri nt shown in Figur 1. Pross isovry is hllnging prolm us on nnot ssum tht ll possil squns r in prsnt. Consir for xmpl th vnt log 2

3 # tr 455 h 191 g 177 h 144 h 111 g 82 g 56 h 47 fh 38 g 33 fh 14 fg # tr 11 fg 9 fh 8 fh 5 fg 3 ffg 2 fg 2 ffg 1 ffh 1 ffg 1 fffg xmin thoroughly g strt rgistr rqust xmin sully hk tikt i f rinitit rqust py ompnstion h rjt rqust n Fig. 1. A Ptri nt isovr from n vnt log ontining 1391 ss. shown in Figur 1. If w rnomly tk 500 ss from th st of 1391 ss, w woul lik to isovr mor or lss th sm mol. Not tht thr r svrl trs tht ppr only on in th log. Mny of ths will isppr whn onsiring log with only 500 ss. Also not tht th pross mol isovr y th α-lgorithm llows for mor trs thn th ons pit in Figur 1,.g.,,,,, f,,,, f,,,, h is possil oring to th pross mol ut os not our in th vnt log. This illustrts tht vnt logs tn to fr from omplt, i.., only smll sust of ll possil hvior n osrv us th numr of vritions is lrgr thn th numr of instns osrv. Th pross mol in Figur 1 is rthr simpl. Rl-lif prosss will onsist of ozns or vn hunrs of iffrnt tivitis. Morovr, som hviors will vry infrqunt ompr to othrs. Suh rr hviors n sn s 3

4 nois (.g., xptions). Typilly, it is unsirl n lso unfsil to ptur frqunt n infrqunt hvior in singl igrm. Pross isovry thniqus n to l to l with nois n inompltnss. This mks pross mining vry iffrnt from synthsis. Clssil synthsis thniqus im t rting mol tht pturs th givn hvior prisly. For xmpl, lssil lngug-s rgion thniqus [14, 17, 19, 28, 42, 43, 45] istill Ptri nt from (possily infinit) lngug, suh tht th hvior of th Ptri nt is only minimlly mor thn th givn lngug. In lssil stt-s rgion thory [13, 15, 23, 24, 26, 27, 35] on th othr hn, trnsition systm is us to synthsiz Ptri nt of whih th hvior is isimilr with th givn trnsition systm. Intuitivly two mols r isimilr if thy n mth h othr s movs, i.., thy nnot istinguish from on nothr y n osrvr [36]. In trms of mining this implis tht th nïvly synthsiz Ptri nt nnot gnrliz yon th xmpl trs sn. Pross isovry thniqus n to ln four ritri: fitnss (th isovr mol shoul llow for th hvior sn in th vnt log), prision (th isovr mol shoul not llow for hvior ompltly unrlt to wht ws sn in th vnt log), gnrliztion (th isovr mol shoul gnrliz th xmpl hvior sn in th vnt log), n simpliity (th isovr mol shoul s simpl s possil). This mks pross isovry hllnging n highly rlvnt topi. Th rminr of this rtil is orgniz s follows. Stion 2 introus th pross mining sptrum showing tht pross isovry is n ssntil ingrint for pross nlysis s on fts rthr thn fition. Stion 3 prsnts prliminris n formlizs th pross isovry tsk. Th α-lgorithm is prsnt in Stion 4. Stion 5 isusss th min hllngs rlt to pross mining. In Stion 6, w ompr pross isovry with rgion thory in mor til. This stion shows tht lssil pprohs nnot l with prtiulr rquirmnts ssntil for pross mining. Thn, in stions 7 n 8, w show how rgion thory n pt to l with ths rquirmnts. Both stt-s rgions n lngug-s rgions r onsir. All pprohs sri in this rtil r support y ProM, th ling opn-sour pross mining frmwork. ProM is sri in Stion 9. Stion 10 ns this rtil with som onlusions n hllngs tht rmin. 2 Pross Mining Pross mining is n importnt tool for morn orgniztions tht n to mng non-trivil oprtionl prosss. On th on hn, thr is n inril growth of vnt t [44]. On th othr hn, prosss n informtion n to lign prftly in orr to mt rquirmnts rlt to omplin, ffiiny, n ustomr srvi. Pross mining is muh ror thn just ontrolflow isovry, i.., isovring Ptri nt from multi-st of trs. Thrfor, w strt y proviing n ovrviw of th pross mining sptrum. 4

5 worl usinss prosss popl mhins omponnts orgniztions mols nlyzs supports/ ontrols spifis onfigurs implmnts nlyzs softwr systm rors vnts,.g., mssgs, trnstions, t. (pross) mol isovry onformn xtnsion vnt logs Fig. 2. Positioning of th thr min typs of pross mining: isovry, onformn, n nhnmnt. Evnt logs n us to onut thr typs of pross mining s shown in Figur 2 [2, 7]. Th first typ of pross mining is isovry. A isovry thniqu tks n vnt log n prous mol without using ny -priori informtion. An xmpl is th α-lgorithm [9] tht will sri in Stion 4. This lgorithm tks n vnt log n prous Ptri nt xplining th hvior ror in th log. For xmpl, givn suffiint xmpl xutions of th pross shown in Figur 1, th α-lgorithm is l to utomtilly onstrut th Ptri nt without using ny itionl knowlg. If th vnt log ontins informtion out rsours, on n lso isovr rsour-rlt mols,.g., soil ntwork showing how popl work togthr in n orgniztion. Th son typ of pross mining is onformn. Hr, n xisting pross mol is ompr with n vnt log of th sm pross. Conformn hking n us to hk if rlity, s ror in th log, onforms to th mol n vi vrs. For instn, thr my pross mol initing tht purhs orrs of mor thn on million Euro rquir two hks. Anlysis of th vnt log will show whthr this rul is follow or not. Anothr xmpl is th hking of th so-ll four-ys prinipl stting tht prtiulr tivitis shoul not xut y on n th sm prson. By snning th vnt log using mol spifying ths rquirmnts, on n isovr potntil ss of fru. Hn, onformn hking my us to tt, lot n xplin vitions, n to msur th svrity of ths vitions. An xmpl is th onformn hking lgorithm sri in [51]. Givn th mol shown in Figur 1 n orrsponing vnt log, this lgorithm n quntify n ignos vitions. In [4] nothr pproh s on rting lignmnts is prsnt. An lignmnt is optiml if it rlts th tr in th log to most similr pth in th mol. 5

6 Aftr rting optiml lignmnts, ll hvior in th log n rlt to th mol. Th thir typ of pross mining is nhnmnt. Hr, th i is to xtn or improv n xisting pross mol using informtion out th tul pross ror in som vnt log. Whrs onformn hking msurs th lignmnt twn mol n rlity, this thir typ of pross mining ims t hnging or xtning th -priori mol. On typ of nhnmnt is rpir, i.., moifying th mol to ttr rflt rlity. For xmpl, if two tivitis r mol squntilly ut in rlity n hppn in ny orr, thn th mol my orrt to rflt this. Anothr typ of nhnmnt is xtnsion, i.., ing nw prsptiv to th pross mol y ross-orrlting it with th log. An xmpl is th xtnsion of pross mol with prformn t. For instn, Figur 1 n xtn with informtion out rsours, ision ruls, qulity mtris, t. Th Ptri nt in Figur 1 only shows th ontrol-flow. Howvr, whn xtning pross mols, itionl prsptivs n to. Morovr, isovry n onformn thniqus r not limit to ontrol-flow. For xmpl, on n isovr soil ntwork n hk th vliity of som orgniztionl mol using n vnt log. Hn, orthogonl to th thr typs of mining (isovry, onformn, n nhnmnt), iffrnt prsptivs n intifi. Th orgniztionl prsptiv fouss on informtion out rsours hin in th log, i.., whih tors (.g., popl, systms, rols, n prtmnts) r involv n how r thy rlt. Th gol is to ithr strutur th orgniztion y lssifying popl in trms of rols n orgniztionl units or to show th soil ntwork. Th tim prsptiv is onrn with th timing n frquny of vnts. Whn vnts r timstmps it is possil to isovr ottlnks, msur srvi lvls, monitor th utiliztion of rsours, n prit th rmining prossing tim of running ss. 3 Pross Disovry: Prliminris n Purpos In this stion, w sri th gol of pross isovry. In orr to o this, w prsnt prtiulr formt for logging vnts n prtiulr pross moling lngug (i.., Ptri nts). Bs on this w skth vrious pross isovry pprohs. 3.1 Evnt Logs Th gol of pross mining is to xtrt knowlg out prtiulr (oprtionl) pross from vnt logs, i.., pross mining sris fmily of - postriori nlysis thniqus xploiting th informtion ror in uit trils, trnstion logs, tss, t. Typilly, ths pprohs ssum tht it is possil to squntilly ror vnts suh tht h vnt rfrs to n tivity (i.., wll-fin stp in th pross) n is rlt to prtiulr s (i.., pross instn). Furthrmor, som mining thniqus us itionl 6

7 informtion suh s th prformr or origintor of th vnt (i.., th prson / rsour xuting or inititing th tivity), th timstmp of th vnt, or t lmnts ror with th vnt (.g., th siz of n orr). To lrify th notion of n vnt log onsir Tl 1 whih shows frgmnt of som vnt log. Only two trs r shown, oth ontining four vnts. Eh vnt hs uniqu i n svrl proprtis. For xmpl vnt longs to s x123 n is n instn of tivity tht ourr on Dmr 30th t 11.02, ws xut y John, n ost 300 uros. Th son tr (s x128) strts with vnt n lso rfrs to n instn of tivity. Th Tl 1. A frgmnt of som vnt log. s i vnt i proprtis timstmp tivity rsour ost... x :11.02 John x :11.06 John x :11.12 John x :11.18 John x :16.10 Ann x :16.14 John x :16.26 Pt x :16.36 Ann informtion pit in Tl 1 is th typil vnt t tht n xtrt from toy s systms. Systms stor vnts in vry iffrnt wys. Pross-wr informtion systms (.g., workflow mngmnt systms) provi it uit trils. In othr systms, this informtion is typilly sttr ovr svrl tls. For xmpl, in hospitl vnts rlt to prtiulr ptint my stor in iffrnt tls n vn iffrnt systms. For mny pplitions of pross mining, on ns to xtrt vnt t from iffrnt sours, mrg ths t, n onvrt th rsult into suitl formt. W vot th us of th so-ll XES (Xtnsil Evnt Strm) formt tht n r irtly y ProM ( [5,57]). XES is th sussor of MXML. Bs on mny prtil xprins with MXML, th XES formt hs n m lss rstritiv n truly xtnil. In Sptmr 2010, th formt ws opt y th IEEE Tsk For on Pross Mining. Th formt is support y tools suh s ProM (s of vrsion 6), Nitro, XESm, n OpnXES. S for til informtion out th stnr. XES is l to stor th informtion shown in Tl 1. Most of this informtion is optionl, i.., if it is thr, it n us for pross mining, ut it is not nssry for ontrol-flow isovry. 7

8 In this rtil, w fous on ontrol-flow isovry. Thrfor, w only onsir th tivity olumn in Tl 1. This mns tht n vnt is link to s (pross instn) n n tivity, n no furthr ttriuts r n. Evnts r orr (pr s), ut o not n to hv xpliit timstmps. This llows us to us th following simplifi finition of n vnt log. Dfinition 1 (Evnt, Tr, Evnt log). Lt A st of tivitis. σ A is tr, i.., squn of vnts. L IB(A ) is n vnt log, i.., multi-st of trs. Th first four vnts in Tl 1 form tr,,,. This tr rprsnts th pth follow y s x123. Th son s (x128) n rprsnt y th tr,,,. Not tht thr my multipl ss tht hv th sm tr. Thrfor, n vnt log is fin s multi-st of trs. A multi-st (lso rfrr to s g) is lik st whr h lmnt my our multipl tims. For xmpl, [hors, ow 5, uk 2 ] is th multi-st with ight lmnts: on hors, fiv ows n two uks. IB(X) is th st of multi-sts (gs) ovr X. W ssum th usul oprtors on multi-sts,.g., X Y is th union of X n Y, X \Y is th iffrn twn X n Y, x X tsts if x pprs in X, n X Y vluts to tru if X is ontin in Y. For xmpl, [hors, ow 2 ] [hors 2, uk 2 ] = [hors 3, ow 2, uk 2 ], [hors 3, ow 4 ] \ [ow 2 ] = [hors 3, ow 2 ], [hors, ow 2 ] [hors 2, ow 3 ], n [hors 3, ow 1 ] [hors 2, ow 2 ]. Not tht sts n onsir s gs hving only on instn of vry lmnt. Hn, w n mix sts n gs,.g., {hors, ow} [hors 2, ow 3 ] = [hors 3, ow 4 ]. For prtil pplitions of pross mining it is ssntil to iffrntit twn trs tht r infrqunt or vn uniqu (multipliity of 1) n trs tht r frqunt. Thrfor, n vnt log is multi-st of trs rthr thn n orinry st. Howvr, in this rtil w fous on th fountions of pross isovry thry oftn strting from nois n frqunis. S [2] for thniqus tht tk frqunis into ount. This ook lso sris vrious s stuis showing th importn of multipliitis. In th rminr, w will us th following xmpl log: L 1 = [,,, 5,,,, 8,,, 9 ]. L 1 ontins informtion out 22 ss; fiv ss following tr,,,, ight ss following tr,,,, n nin ss following tr,,. Not tht suh simpl rprsnttion n xtrt from sours suh s Tl 1, MXML, XES, or ny othr formt tht links vnts to ss n tivitis. 3.2 Ptri Nts Th gol of pross isovry is to istil pross mol from som vnt log. Hr w us Ptri nts [50] to rprsnt suh mols. In ft, w xtrt sulss of Ptri nts known s workflow nts (WF-nts) [1]. Dfinition 2. An Ptri nt is tupl (P, T, F ) whr: 1. P is finit st of pls, 8

9 2. T is finit st of trnsitions suh tht P T =, n 3. F (P T ) (T P ) is st of irt rs, ll th flow rltion. An xmpl Ptri nt is shown in Figur 3. This Ptri nt hs six pls rprsnt y irls n four trnsitions rprsnt y squrs. Pls my ontin tokns. For xmpl, in Figur 3 oth p1 n p6 ontin on tokn, p3 ontins two tokns, n th othr pls r mpty. Th stt, lso ll mrking, is th istriution of tokns ovr pls. A mrk Ptri nt is pir (N, M), whr N = (P, T, F ) is Ptri nt n whr M IB(P ) is g ovr P noting th mrking of th nt. Th initil mrking of th Ptri nt shown in Figur 3 is [p1, p3 2, p6]. Th st of ll mrk Ptri nts is not N. t1 t3 p3 p1 p2 p5 p6 p4 t2 t4 Fig. 3. A Ptri nt with six pls (p1, p2, p3, p4, p5, n p6) n four trnsitions (t1, t2, t3, n t4). Lt N = (P, T, F ) Ptri nt. Elmnts of P T r ll nos. A no x is n input no of nothr no y iff thr is irt r from x to y (i.., (x, y) F ). No x is n output no of y iff (y, x) F. For ny x P T, x = {y (y, x) F } n x = {y (x, y) F }. In Figur 3, t3 = {p3, p6} n t3 = {p5}. Th ynmi hvior of suh mrk Ptri nt is fin y th so-ll firing rul. A trnsition is nl if h of its input pls ontins tokn. An nl trnsition n fir thry onsuming on tokn from h input pl n prouing on tokn for h output pl. Dfinition 3 (Firing rul). Lt (N, M) mrk Ptri nt with N = (P, T, F ). Trnsition t T is nl, not (N, M)[t, iff t M. Th firing rul [ N T N is th smllst rltion stisfying for ny (N, M) N n ny t T, (N, M)[t (N, M) [t (N, (M \ t) t ). In th mrking shown in Figur 3, oth t1 n t3 r nl. Th othr two trnsitions r not nl us t lst on of th input pls is mpty. If t1 firs, on tokn is onsum (from p1) n two tokns r prou (on for p2 n on for p3). Formlly, (N, [p1, p3 2, p6]) [t1 (N, [p2, p3 3, p6]). So th rsulting mrking is [p2, p3 3, p6]. If t3 firs in th initil stt, two tokns r 9

10 onsum (on from p3 n on from p6) n on tokn is prou (for p5). Formlly, (N, [p1, p3 2, p6]) [t3 (N, [p1, p3, p5]). Lt (N, M 0 ) with N = (P, T, F ) mrk P/T nt. A squn σ T is ll firing squn of (N, M 0 ) iff, for som nturl numr n IN, thr xist mrkings M 1,..., M n n trnsitions t 1,..., t n T suh tht σ = t 1... t n n, for ll i with 0 i < n, (N, M i )[t i+1 n (N, M i ) [t i+1 (N, M i+1 ). Lt (N, M 0 ) th mrk Ptri nt shown in Figur 3, i.., M 0 = [p1, p3 2, p6]. Th mpty squn σ = is nl in (N, M 0 ). Th squn σ = t1, t3 is lso nl n rsults in mrking [p2, p3 2, p5]. Anothr possil firing squn is σ = t3, t4, t3, t1, t4, t3, t2, t1. A mrking M is rhl from th initil mrking M 0 iff thr xists squn of nl trnsitions whos firing ls from M 0 to M. Th st of rhl mrkings of (N, M 0 ) is not [N, M 0. [p1,p3 2,p6] t1 [p2,p3 3,p6] t2 [p2,p3 2,p4,p6] t3 t4 t3 [p2,p3 2,p5] t1 [p1,p3,p4,p6] t3 t4 t1 [p1,p3,p5] t2 [p2,p3,p4 2,p6] t3 t4 t2 [p2,p3,p4,p5] t1 [p1,p4 2,p6] t2 [p2,p4 3,p6] t3 t4 t4 t1 [p1,p4,p5] t2 [p2,p4 2,p5] Fig. 4. Th rhility grph of th mrk Ptri nt shown in Figur 3. For th mrk Ptri nt shown in Figur 3 thr r 12 rhl stts. Ths stts n omput using th so-ll rhility grph shown in Figur 4. All nos orrspon to rhl mrkings n h r orrspons to th firing of prtiulr trnsition. Any pth in th rhility grph orrspons to possil firing squn. For xmpl, using Figur 4 is is sy to s tht t3, t4, t3, t1, t4, t3, t2, t1 is in possil n rsults in [p2, p3, p4, p5]. A mrk nt my unoun, i.., hv n infinit numr or rhl stts. In this s, th rhility grph is infinitly lrg, ut on n still onstrut th so-ll ovrility grph [50]. 10

11 3.3 Workflow Nts For pross isovry, w look t prosss tht r instntit multipl tims, i.., th sm pross is xut for multipl ss. For xmpl, th pross of hnling insurn lims my xut for thousns or vn millions of lims. Suh prosss hv lr strting point n lr ning point. Thrfor, th following sulss of Ptri nts (WF-nts) is most rlvnt for pross isovry. Dfinition 4 (Workflow nts). Lt N = (P, T, F ) Ptri nt n t frsh intifir not in P T. N is workflow nt (WF-nt) iff: 1. ojt rtion: P ontins n input pl i (lso ll sour pl) suh tht i =, 2. ojt ompltion: P ontins n output pl o (lso ll sink pl) suh tht o =, 3. onntnss: N = (P, T { t}, F {(o, t), ( t, i)}) is strongly onnt, i.., thr is irt pth twn ny pir of nos in N. Clrly, Figur 3 is not WF-nt us sour n sink pl r missing. Figur 5 shows n xmpl of WF-nt: strt =, n =, n vry no is on pth from strt to n. p1 p3 strt n p2 p4 Fig. 5. A workflow nt with sour pl i = strt n sink pl o = n. Th Ptri nt pit in Figur 1 is nothr xmpl of WF-nt. Not vry WF-nt rprsnts orrt pross. For xmpl, pross rprsnt y WF-nt my xhiit rrors suh s loks, tsks whih n nvr om tiv, livloks, grg ing lft in th pross ftr trmintion, t. Thrfor, w fin th following orrtnss ritrion. Dfinition 5 (Sounnss). Lt N = (P, T, F ) WF-nt with input pl i n output pl o. N is soun iff: 1. sfnss: (N, [i]) is sf, i.., pls nnot hol multipl tokns t th sm tim, 11

12 2. propr ompltion: for ny mrking M [N, [i], o M implis M = [o], 3. option to omplt: for ny mrking M [N, [i], [o] [N, M, n 4. sn of tsks: (N, [i]) ontins no trnsitions (i.., for ny t T, thr is firing squn nling t). Th WF-nts shown in figurs 5 n 1 r soun. Sounnss n vrifi using stnr Ptri-nt-s nlysis thniqus. In ft sounnss orrspons to livnss n sfnss of th orrsponing short-iruit nt [1]. This wy ffiint lgorithms n tools n ppli. An xmpl of tool tilor towrs th nlysis of WF-nts is Wofln [55]. This funtionlity is lso m in our pross mining tool ProM [5]. 3.4 Prolm Dfinition n Approhs Aftr introuing vnts logs n WF-nts, w n fin th min gol of pross isovry. Dfinition 6 (Pross isovry). Lt L n vnt log ovr A, i.., L IB(A ). A pross isovry lgorithm is funtion γ tht mps ny log L onto Ptri nt γ(l) = (N, M). Illy, N is soun WF-nt n ll trs in L orrspon to possil firing squns of (N, M). Th gol is to fin pross mol tht n rply ll ss ror in th log, i.., ll trs in th log r possil firing squns of th isovr WF-nt. Assum tht L 1 = [,,, 5,,,, 8,,, 9 ]. In this s th WF-nt shown in Figur 5 is goo solution. All trs in L 1 orrspon to firing squns of th WF-nt n vi vrs. Throughout this rtil, w us L 1 s n xmpl log. Not tht it my possil tht som of th firing squns of th isovr WF-nt o not ppr in th log. This is ptl s on nnot ssum tht ll possil squns hv n osrv. For xmpl, if thr is loop, th numr of possil firing squns is infinit. Evn if th mol is yli, th numr of possil squns my normous u to hois n prlllism. Ltr in this rtil, w will isuss th qulity of isovr mols in mor til. Sin th mi-nintis svrl groups hv n working on thniqus for pross mining [7, 9, 10, 25, 29, 32, 33, 58], i.., isovring pross mols s on osrv vnts. In [6] n ovrviw is givn of th rly work in this omin. Th i to pply pross mining in th ontxt of workflow mngmnt systms ws introu in [10]. In prlll, Dtt [29] look t th isovry of usinss pross mols. Cook t l. invstigt similr issus in th ontxt of softwr nginring prosss [25]. Hrst [40] ws on of th first to tkl mor omplit prosss,.g., prosss ontining uplit tsks. Most of th lssil pprohs hv prolms ling with onurrny. Th α-lgorithm [9] is n xmpl of simpl thniqu tht tks onurrny s strting point. Howvr, this simpl lgorithm hs prolms ling with omplit routing onstruts n nois (lik most of th othr pprohs 12

13 sri in litrtur). In [32, 33] mor roust ut lss pris pproh is prsnt. Rntly, popl strt using th thory of rgions to pross isovry. Thr r two pprohs: stt-s rgions n lngug-s rgions. Stts rgions n us to onvrt trnsition systm into Ptri nt [13,15, 23,24,26,27,35]. Lngug-s rgions pls s long s it is still possil to rply th log [14, 17, 19, 28, 42, 43]. Mor from thortil point of viw, th pross isovry prolm is rlt to th work isuss in [12, 37, 38, 49]. In ths pprs th limits of inutiv infrn r xplor. For xmpl, in [38] it is shown tht th omputtionl prolm of fining minimum finit-stt ptor omptil with givn t is NP-hr. Svrl of th mor gnri onpts isuss in ths pprs n trnslt to th omin of pross mining. It is possil to intrprt th prolm sri in this rtil s n inutiv infrn prolm spifi in trms of ruls, hypothsis sp, xmpls, n ritri for sussful infrn. Th omprison with litrtur in this omin riss intrsting qustions for pross mining,.g., how to l with ngtiv xmpls (i.., suppos tht sis log L thr is log L of trs tht r not possil,.g., y omin xprt). Howvr, spit th rltions with th work sri in [12,37,38,49] thr r lso mny iffrns,.g., w r mining t th nt lvl rthr thn squntil or lowr lvl rprsnttions (.g., Mrkov hins, finit stt mhins, or rgulr xprssions), tkl onurrny, n o not ssum ngtiv xmpls or omplt logs. Th ov pprohs ssum tht thr is no nois or infrqunt hvior. For pprohs ling with ths prolms w rfr to th work on y Christin Günthr [39], Ton Wijtrs [58], n An Krl Alvs Miros [47]. 4 α-algorithm Aftr introuing th pross isovry prolm n proviing n ovrviw of pprohs sri in litrtur, w fous on th α-lgorithm [9]. Th α- lgorithm is not intn s prtil mining thniqu s it hs prolms with nois, infrqunt/inomplt hvior, n omplx routing onstruts. Nvrthlss, it provis goo introution into th topi. Th α-lgorithm is vry simpl n mny of its is hv n m in mor omplx n roust thniqus. Morovr, it ws th first lgorithm to rlly rss th isovry of onurrny. 4.1 Bsi I Th α-lgorithm sns th vnt log for prtiulr pttrns. For xmpl, if tivity is follow y ut is nvr follow y, thn it is ssum tht thr is usl pnny twn n. To rflt this pnny, th orrsponing Ptri nt shoul hv pl onnting to. W istinguish four log-s orring rltions tht im to ptur rlvnt pttrns in th log. 13

14 Dfinition 7 (Log-s orring rltions). Lt L n vnt log ovr A, i.., L IB(A ). Lt, A: > L iff thr is tr σ = t 1, t 2, t 3,... t n n i {1,..., n 1} suh tht σ L n t i = n t i+1 =, L iff > L n L, # L iff L n L, n L iff > L n > L. Consir for xmpl L 1 = [,,, 5,,,, 8,,, 9 ]. > L1 us irtly follows in tr,,,. Howvr, L1 us nvr irtly follows in ny tr in th log. > L1 = {(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )} ontins ll pirs of tivitis in irtly follows rltion. L1 us somtims irtly follows n nvr th othr wy roun ( > L1 n L1 ). L1 = {(, ), (, ), (, ), (, ), (, ), (, )} ontins ll pirs of tivitis in uslity rltion. L1 us > L1 n > L1, i., somtims follows n somtims th othr wy roun. L1 = {(, ), (, )}. # L1 us L1 n L1. # L1 = {(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )}. Not tht for ny log L ovr A n x, y A: x L y, y L x, x# L y, or x L y. () squn pttrn: () XOR-split pttrn:,, n # () XOR-join pttrn:,, n # () AND-split pttrn:,, n () AND-join pttrn:,, n Fig. 6. Typil pross pttrns n th footprints thy lv in th vnt log. 14

15 Th log-s orring rltions n us to isovr pttrns in th orrsponing pross mol s is illustrt in Figur 6. If n r in squn, th log will show > L. If ftr thr is hoi twn n, th log will show L, L, n # L us n follow y n, ut will not follow y n vi vrs. Th logil ountrprt of this so-ll XOR-split pttrn is th XOR-join pttrn s shown in Figur 6(-). If L, L, n # L, thn this suggsts tht ftr th ourrn of ithr or, shoul hppn. Figur 6(-) shows th so-ll AND-split n ANDjoin pttrns. If L, L, n L, thn it pprs tht ftr oth n n xut in prlll (AND-split pttrn). If L, L, n L, thn it pprs tht ns to synhroniz n (AND-join pttrn). Figur 6 only shows simpl pttrns n os not prsnt th itionl onitions n to xtrt th pttrns. Howvr, th figur nily illustrts th si i. f p ({},{}) p ({},{}) g i L p ({,f},{}) p ({},{f,g}) o L p ({},{}) p ({},{}) Fig. 7. WF-nt N 2 riv from L 2 = [,,,,, f,,,,, g,,,,,, g,,,,,, f,,,,, f,,,,, g ]. Consir for xmpl WF-nt N 2 pit in Figur 7 n th log vnt log L 2 = [,,,,, f,,,,, g,,,,,, g,,,,,, f,,,,, f,,,,, g ]. Th α-lgorithm onstruts WF-nt N 2 s on L 2. Not tht th pttrns in th mol in mth th log-s orring rltions xtrt from th vnt log. Consir for xmpl th pross frgmnt involving,,, n. Oviously, this frgmnt n onstrut s on L2, L2, L2, L2, n L2. Th hoi following is rvl y L2 f, L2 g, n f# L2 g. Et. Anothr xmpl is shown in Figur 8. WF-nt N 3 n riv from L 3 = [,, 45,,, 42,,, 38,,, 22 ]. Not tht hr thr r two strt n two n tivitis. Ths n foun sily y looking for th first n lst tivitis in trs. 4.2 Algorithm Aftr showing th si i n som xmpls, w sri th α-lgorithm. 15

16 i L p ({,},{}) p ({},{,}) ol Fig. 8. WF-nt N 3 riv from L 3 = [,, 45,,, 42,,, 38,,, 22 ]. Dfinition 8 (α-lgorithm). Lt L n vnt log ovr T. α(l) is fin s follows. 1. T L = {t T σ L t σ}, 2. T I = {t T σ L t = first(σ)}, 3. T O = {t T σ L t = lst(σ)}, 4. X L = {(A, B) A T L A B T L B A B L 1, 2 A 1 # L 2 1, 2 B 1 # L 2 }, 5. Y L = {(A, B) X L (A,B ) X L A A B B = (A, B) = (A, B )}, 6. P L = {p (A,B) (A, B) Y L } {i L, o L }, 7. F L = {(, p (A,B) ) (A, B) Y L A} {(p (A,B), ) (A, B) Y L B} {(i L, t) t T I } {(t, o L ) t T O }, n 8. α(l) = (P L, T L, F L ). L is n vnt log ovr som st T of tivitis. In Stp 1 it is hk whih tivitis o ppr in th log (T L ). Ths will orrspon to th trnsitions of th gnrt WF-nt. T I is th st of strt tivitis, i.., ll tivitis tht ppr first in som tr (Stp 2). T O is th st of n tivitis, i.., ll tivitis tht ppr lst in som tr (Stp 3). Stps 4 n 5 form th or of th α-lgorithm. Th hllng is to fin th pls of th WF-nt n thir onntions. W im t onstruting pls nm p (A,B) suh tht A is th st of input trnsitions ( p (A,B) = A) n B is th st of output trnsitions (p (A,B) = B). Th si i for fining p (A,B) is shown in Figur 9. All lmnts of A shoul hv usl pnnis with ll lmnts of B, i.., for ny (, ) A B: L. Morovr, th lmnts of A shoul nvr follow ny of th othr lmnts, i.., for ny 1, 2 A: 1 # L 2. A similr rquirmnt hols for B. Lt us onsir L 1 = [,,, 5,,,, 8,,, 9 ]. Clrly A = {} n B = {, } mt th rquirmnts stt in Stp 4. Also not tht A = {} n B = {} mt th sm rquirmnts. X L is th st of ll suh pirs tht mt th rquirmnts just mntion. In this s, X L1 = {({}, {}), ({}, {}), ({}, {}), ({}, {, }), ({}, {, }), ({}, {}), ({}, {}), ({}, {}), ({, }, {}), ({, }, {})}. If on woul insrt pl for ny lmnt in X L1 thr woul too mny pls. Thrfor, only th mximl pirs (A, B) shoul inlu. Not tht for ny pir (A, B) X L, non-mpty st A A, n non-mpty st B B, it is impli tht (A, B ) X L. In Stp 5 ll nonmximl pirs r rmov. So Y L1 = {({}, {, }), ({}, {, }), ({, }, {}), ({, }, {})}. 16

17 p (A,B)... m n A={ 1, 2, m } B={ 1, 2, n } Fig. 9. Pl p (A,B) onnts th trnsitions in st A to th trnsitions in st B. Evry lmnt of (A, B) Y L orrspons to pl p (A,B) onnting trnsitions A to trnsitions B. In ition P L lso ontins uniqu sour pl i L n uniqu sink pl o L (f. Stp 6). In Stp 7 th rs r gnrt. All strt trnsitions in T I hv i L s n input pl n ll n trnsitions T O hv o L s output pl. All pls p (A,B) hv A s input nos n B s output nos. Th rsult is Ptri nt α(l) = (P L, T L, F L ) tht sris th hvior sn in vnt log L. Thus fr w prsnt thr logs n thr WF-nts. Clrly α(l 2 ) = N 2, n α(l 3 ) = N 3. In figurs 7 n 8 th pls r nm s on th sts Y L2 n Y L3. Morovr, α(l 1 ) = N 1 moulo rnming of pls (us iffrnt pl nms r us in Figur 5). Ths xmpls show tht th α-lgorithm is in l to isovr WF-nts s vnt logs. i L o L p ({},{}) p ({,},{}) p ({},{,f}) f p ({},{}) p ({},{f}) Fig. 10. WF-nt N 4 riv from L 4 = [,,, f 2,,,,,,, f 3,,,,,,, f 2,,,,,,, f 4,,,,,,, f 3 ]. 17

18 Figur 10 shows nothr xmpl. WF-nt N 4 n riv from L 4 = [,,, f 2,,,,,,, f 3,,,,,,, f 2,,,,,,, f 4,,,,,,, f 3 ], i.., α(l 4 ) = N 4. Th WF-nt in Figur 1 is isovr whn pplying th α-lgorithm to th vnt log in th sm figur. 4.3 Limittions In [9] it ws shown tht th α-lgorithm is l to isovr lrg lss of WF-nts if on ssums tht th log is omplt with rspt to th log-s orring rltion > L. This ssumption implis tht, for ny vnt log L, > L if n irtly follow y. W rvisit th notion of ompltnss ltr in this rtil. f p 1 p 2 g Fig. 11. WF-nt N 5 riv from L 5 = [,,, g 2,,,, g 3,,, f, g 2,, f,, g 4 ]. Evn if w ssum tht th log is omplt, th α-lgorithm hs som prolms. Thr r mny iffrnt WF-nts tht hv th sm possil hvior, i.., two mols n struturlly iffrnt ut tr quivlnt. Consir for xmpl L 5 = [,,, g 2,,,, g 3,,, f, g 2,, f,, g 4 ]. α(l 5 ) is shown in Figur 11. Although th mol is l to gnrt th osrv hvior, th rsulting WF-nt is nlssly omplx. Two of th input pls of g r runnt, i.., thy n rmov without hnging th hvior. Th pls not s p 1 n p 2 r so-ll impliit pls n n rmov without llowing for mor trs. In ft, Figur 11 shows only on of mny possil tr quivlnt WF-nts. Th originl α-lgorithm hs prolms ling with short loops, i.., loops of lngth 1 or 2. This is illustrt y WF-nt N 6 in Figur 12 whih shows th rsult of pplying th si lgorithm to L 6 = [, 2,,, 3,,,, 2 ]. It is sy to s tht th mol os not llow for, n,,,. In ft, in N 6, trnsition ns to xut prisly on n thr is n impliit pl onnting n. This prolm n rss sily s shown in [46]. Using n improv vrsion of th α-lgorithm on n isovr th WF-nt shown in Figur

19 Fig. 12. Inorrt WF-nt N 6 riv from L 6 = [, 2,,, 3,,,, 2 ]. Fig. 13. WF-nt N 7 hving so-ll short-loop. A mor iffiult prolm is th isovry of so-ll non-lol pnnis rsulting from non-fr hoi pross onstruts. An xmpl is shown in Figur 14. This nt woul goo nit ftr osrving for xmpl L 8 = [,, 45,,, 42 ]. Howvr, th α-lgorithm will riv th WF-nt without th pl ll p 1 n p 2. Hn, α(l 8 ) = N 3 shown in Figur 8 lthough th trs,, n,, o not ppr in L 8. Suh prolms n (prtilly) rsolv using rfin vrsions of th α-lgorithm suh s th on prsnt in [59]. p 1 p 2 Fig. 14. WF-nt N 8 hving non-lol pnny. Th ov xmpls show tht th α-lgorithm is l to isovr lrg lss of mols. Th si 8-lin lgorithm hs som limittions whn it oms to prtiulr pross pttrns (.g., short-loops n non-lol pnnis). Som of ths prolms n solv using vrious rfinmnts. Howvr, svrl mor funmntl prolms rmin s shown nxt. 19

20 5 Chllngs Th α-lgorithm ws on of th first pross isovry lgorithms to qutly ptur onurrny. Toy thr r muh ttr lgorithms tht ovrom th wknsss of th α-lgorithm. Ths r ithr vrints of th α-lgorithm or lgorithms tht us ompltly iffrnt pproh,.g., gnti mining or synthsis s on rgions [34]. Ltr w will sri som of ths pprohs. Howvr, first w isuss th min rquirmnts for goo pross isovry lgorithm. To isovr suitl pross mol it is ssum tht th vnt log ontins rprsnttiv smpl of hvior. Thr r two rlt phnomn tht my mk n vnt log lss rprsnttiv for th pross ing stui: Nois: th vnt log ontins rr n infrqunt hvior not rprsnttiv for th typil hvior of th pross. Inompltnss: th vnt log ontins too fw vnts to l to isovr som of th unrlying ontrol-flow struturs. Oftn w woul lik to strt from nois whn isovring pross. This os not mn tht nois is not rlvnt. In ft, th gol of onformn hking is to intify xptions n vitions. Howvr, for pross isovry it mks no sns to inlu noisy hvior in th mol s this will luttr th igrm n hs littl pritiv vlu. Whrs nois rfrs to th prolm of hving too muh t (sriing rr hvior), ompltnss rfrs to th prolm of hving too littl t. To illustrt th rlvn of ompltnss, onsir pross onsisting of 10 tivitis tht n xut in prlll n orrsponing log tht ontins informtion out 10,000 ss. Th totl numr of possil intrlvings in th mol with 10 onurrnt tivitis is 10! = 3,628,800. Hn, it is impossil tht h intrlving is prsnt in th log s thr r fwr ss (10,000) thn potntil trs (3,628,800). Evn if thr r 3,628,800 ss in th log, it is xtrmly unlikly tht ll possil vritions r prsnt. For th pross in whih 10 tivitis n xut in prlll, lol notion of ompltnss n ru th rquir numr of osrvtions rmtilly. For xmpl, for th α-lgorithm only 10 (10 1) = 90 rthr thn 3,628,800 iffrnt osrvtions r n to onstrut th mol. Compltnss n nois rfr to qulitis of th vnt log n o not sy muh out th qulity of th isovr mol. Dtrmining th qulity of pross mining rsult is iffiult n is hrtriz y mny imnsions. As shown in Figur 15, w intify four min qulity imnsions: fitnss, simpliity, prision, n gnrliztion [2, 4, 51]. A mol with goo fitnss llows for th hvior sn in th vnt log. A mol hs prft fitnss if ll trs in th log n rply y th mol from ginning to n. Thr r vrious wys of fining fitnss. It n fin t th s lvl,.g., th frtion of trs in th log tht n fully rply. It n lso fin t th vnt lvl,.g., th frtion of vnts in th log tht r in possil oring to th mol [2, 4, 51]. Not tht w fin n vnt log to multi-st of trs rthr thn n orinry st: 20

21 l to rply vnt log fitnss Om s rzor simpliity pross isovry gnrliztion not ovrfitting th log prision not unrfitting th log Fig. 15. Blning th four qulity imnsions: fitnss, simpliity, prision, n gnrliztion [2]. th frqunis of trs r importnt for trmining fitnss. If tr nnot rply y th mol, thn th signifin of this prolm pns on th rltiv frquny. Th simpliity imnsion in Figur 15 rfrs to Om s Rzor, th prinipl tht stts tht on shoul not inrs, yon wht is nssry, th numr of ntitis rquir to xplin nything. Following this prinipl, w look for th simplst pross mol tht n xplin wht is osrv in th vnt log. Th omplxity of th mol oul fin y th numr of nos n rs in th unrlying grph. Also mor sophistit mtris n us,.g., mtris tht tk th struturnss or ntropy of th mol into ount. Fitnss n simpliity r ovious ritri. Howvr, this is not suffiint s will illustrt using Figur 16. Assum tht th four mols tht r shown r isovr s on th vnt log lso pit in th figur. (Not tht this vnt log ws lry shown in Stion 1.) Thr r 1391 ss. Of ths 1391 ss, 455 follow th tr,,,, h. Th son most frqunt tr is,,,, g whih ws follow y 191 ss. If w pply th α-lgorithm to this vnt log, w otin mol N 1 shown in Figur 16. A omprison of th WF-nt N 1 n th log shows tht this mol is quit goo; it is simpl n hs goo fitnss. WF-nt N 2 mols only th most frqunt tr, i.., it only llows for th squn,,,, h. Hn, non of th othr = 936 ss fits. WF-nt N 2 is simpl ut hs poor fitnss. Lt us now onsir WF-nt N 3, this is vrint of th so-ll flowr mol [2, 51], i.., mol tht llows for ll known tivitis t ny point in tim. Not tht Ptri nt without ny pls n rply ny log n hs hvior similr to th flowr mol (ut is not WF-nt). Figur 16 os not show pur flowr mol, ut still llows for ivrsity of hviors. N 3 pturs th strt n n tivitis wll. Howvr, th mol os not put ny onstrints on th othr tivitis. For xmpl tr,,,,,,, f, f, f, f, f, g 21

22 strt strt strt strt rgistr rqust rgistr rqust rgistr rqust xmin sully xmin sully xmin thoroughly xmin thoroughly xmin sully hk tikt i hk tikt f i hk tikt rinitit rqust i rinitit rqust py ompnstion py ompnstion rjt rqust rjt rqust N 1 : fitnss = +, prision = +, gnrliztion = +, simpliity = + rjt rqust N 2 : fitnss = -, prision = +, gnrliztion = -, simpliity = + N 3 : fitnss = +, prision = -, gnrliztion = +, simpliity = + rgistr rqust rgistr rqust rgistr rqust rgistr rqust hk tikt xmin sully hk tikt xmin sully (ll xmin sully hk tikt xmin sully hk tikt f i i i i 21 vrints sn in th log) g h g h h g g h rjt rqust h rjt rqust n n n py ompnstion py ompnstion n # tr 455 h 191 g 177 h 144 h 111 g 82 g 56 h 47 fh 38 g 33 fh 14 fg 11 fg 9 fh 8 fh 5 fg 3 ffg 2 fg 2 ffg 1 ffh 1 ffg 1 fffg 1391 rgistr rqust rgistr rqust rgistr rqust xmin thoroughly hk tikt xmin thoroughly hk tikt xmin thoroughly hk tikt i i i rjt rqust N 4 : fitnss = +, prision = +, gnrliztion = -, simpliity = - g py ompnstion h rjt rqust h Fig. 16. Diffrnt Ptri nts isovr for n vnt log ontining 1391 ss. 22

23 is possil, whrs it sms unlikly tht this tr is possil whn looking t th vnt log, i.., th hvior is vry iffrnt from ny of th trs in th log. Extrm mols suh s th flowr mol (nything is possil) show th n for n itionl imnsion: prision. A mol is pris if it os not llow for too muh hvior. Clrly, th flowr mol lks prision. A mol tht is not pris is unrfitting. Unrfitting is th prolm tht th mol ovr-gnrlizs th xmpl hvior in th log, i.., th mol llows for hviors vry iffrnt from wht ws sn in th log. WF-nt N 4 in Figur 16 rvls nothr potntil prolm. This mol simply numrts th 21 iffrnt trs sn in th vnt log. Not tht N 4 is so-ll ll Ptri nt, i.., multipl trnsitions n hv th sm ll (thr r 21 trnsition with ll ). Th WF-nt in Figur 16 is pris n hs goo fitnss. Howvr, N 4 is lso ovrly omplx n is ovrfitting. WFnt N 4 illustrts th n to gnrliz; on shoul not rstrit hvior to th trs sn in th log s ths r just xmpls. Ovrfitting is th prolm tht vry spifi mol is gnrt whrs it is ovious tht th log only hols xmpl hvior, i.., th mol xplins th prtiulr smpl log, ut nxt smpl log of th sm pross my prou ompltly iffrnt pross mol. Rll tht logs r typilly fr from omplt. Morovr, gnrliztion n us to simplify mols. WF-nt N 1 shown in Figur 16 llows for hvior not sn in th log,.g.,,,,, f,,,, f,,,, h. Any WF-nt tht rstrits th hvior to only sn ss will muh mor omplx n xlu hvior whih sms likly s on similr trs in th vnt log. For rl-lif vnt logs it is hllnging to ln th four qulity imnsions shown in Figur 15. For instn, n ovrsimplifi mol is likly to hv low fitnss or lk of prision. Morovr, thr is n ovious tr-off twn unrfitting n ovrfitting [2, 4, 48, 51]. 6 Pross Disovry n th Thory of Rgions Prolms similr to pross isovry ris in othr rs rnging from hrwr sign n to ontrollr synthsis of mnufturing systms. Oftn th so ll thory of rgions is us to onstrut Ptri nt from hviorl spifition (.g., lngug or stt sp), suh tht th hvior of this nt orrspons to th spifi hvior (if suh nt xists). Th gnrl qustion nswr y th thory of rgions is: Givn th spifi hvior of systm, wht is th Ptri nt tht rprsnts this hvior?. Two min typs of rgion thory n istinguish, nmly stt-s rgion thory n lngug-s rgion thory. Th stt-s thory of rgions fousss on th synthsis of Ptri nts from stt-s mols, whr th stt sp of th Ptri nt is isimilr to th givn stt-s mol. Th lngugs rgion thory, onsirs lngug ovr finit lpht s hviorl spifition. Using th notion of rgions, Ptri nt is onstrut, suh tht ll wors in th lngug r firing squns in tht Ptri nt. 23

24 Th im of th thory of rgions is to synthsiz pris mol, with miniml gnrliztion, whil kping mximl fitnss. Th lssil pprohs sri in this stion (i.., onvntionl stt-s rgion thory n lngug s rgion thory) o not put muh mphsis on simpliity. Unlik lgorithms suh s th huristi minr [58], th gnti minr [47], n th fuzzy minr [39], onvntionl rgion-s mthos o not ompromis on prision in fvor of simpliity or gnrliztion. In th rminr of this stion, w introu th min rgion thory onpts n isuss th iffrns twn synthsis n pross isovry. In Stion 7 n Stion 8 w show how rgion thory n us in th ontxt of pross isovry. 6.1 Stt Bs Rgion Thory Th stt-s rgion thory [13, 15, 23, 24, 26, 27, 35] uss trnsition systm s input, i.., it ttmpts to onstrut Ptri nt tht is isimilr to th trnsition systm. Hn oth r hviorlly quivlnt n if th systm xhiits onurrny, th Ptri nt my muh smllr thn th trnsition systm. Dfinition 9 (Trnsition systm). T S = (S, E, T ) fins ll trnsition systm whr S is th st of stts, A is th st of visil tivitis (i.., tivitis ror in vnt log), τ A is us to rprsnt silnt stps (i.., tions not ror in vnt log), E = A {τ} is th st of trnsition lls, n T S E S is th trnsition rltion. W us s 1 s2 to not trnsition from stt s 1 to s 2 ll with. Furthrmor, w sy tht S s = {s S s S, E s s} S is th st of initil stts, n S = {s S s S, E s s } S is th st of finl stts. In th trnsition systm, rgion orrspons to st of stts suh tht ll stts hv similrly ll input n output gs. Figur 17 shows n xmpl of trnsition systm. In ft, this figur pits th rhility grph of th Ptri nt in Figur 5, whr th stts r nonymous, i.., thy o not ontin informtion out how mny tokns r in pl. Dfinition 10 (Stt rgion). Lt T S = (S, E, T ) trnsition systm n S S st of stts. W sy S is rgion, if n only if for ll E on of th following onitions hols: 1. ll th trnsitions s 1 s2 ntr S, i.., s 1 / S n s 2 S, 2. ll th trnsitions s 1 s2 xit S, i.., s 1 S n s 2 / S, 3. ll th trnsitions s 1 s2 o not ross S, i.., s 1, s 2 S or s 1, s 2 / S Any trnsition systm T S = (S, E, T ) hs two trivil rgions: (th mpty rgion) n S (th rgion onsisting of ll stts). Typilly, only non-trivil rgions r onsir. A rgion r is si to surgion of nothr rgion r if r r. A rgion r is miniml if thr is no othr rgion r whih is surgion of r. Rgion r is prrgion of if thr is trnsition ll with whih 24

25 s5 s6 s7 s1 s2 s3 s4 s8 Fig. 17. A trnsition systm with 8 stts, 5 lls, 1 initil stt n 2 finl stts. s5 s6 s7 s1 s2 s3 s4 s8 p2 p4 p1 p6 p3 p5 Fig. 18. Th trnsition systm of Figur 17 is onvrt into Ptri nt using th stt rgions. Th six rgions orrspon to pls in th Ptri nt. 25

26 s3 s1 s2 s4 (lssil rgion thory) (using ll splitting) p3 2 1 p1 p2 p3 p1 p2 p4 Fig. 19. Th trnsition systm is not lmntry. Thrfor, th gnrt Ptri nt using lssil rgion thory is not quivlnt (moulo isimilrity). Howvr, using ll-splitting n quivlnt Ptri nt n otin. xits r. Rgion r is postrgion of if thr is trnsition ll with whih ntrs r. For Ptri nt synthsis, rgion orrspons to Ptri nt pl n n vnt orrspons to Ptri nt trnsition. Thus, th min i of th synthsis lgorithm is th following: for h vnt in th trnsition systm, trnsition ll with is gnrt in th Ptri nt. For h miniml rgion r i pl p i is gnrt. Th flow rltion of th Ptri nt is uilt th following wy: p i if r i is prrgion of n p i if r i is postrgion of. Figur 18 shows th miniml rgions of th trnsition systm of Figur 17 n th orrsponing Ptri nt. Th first pulitions on th thory of rgions only lt with spil lss of trnsition systms ll lmntry trnsition systms. S [13, 15, 30] for tils. Th lss of lmntry trnsition systms is vry rstrit. In prti, most of th tim, popl n to l with ritrry trnsition systms tht only y oinin fll into th lss of lmntry trnsition systms. In th pprs of Cortll t l. [26, 27], mtho for hnling ritrry trnsition systms ws prsnt. This pproh uss ll Ptri nts, i.., iffrnt trnsitions n rfr to th sm vnt. (WF-nt N 4 in Figur 16 is n xmpl of ll Ptri nt,.g., thr r 21 trnsitions ll.) For this pproh it hs n shown tht th hvior (f. rhility grph) of th synthsiz Ptri nt is isimilr to th initil trnsition systm vn if th trnsition systm is non-lmntry. 26

27 Mor rntly, in [23,24], n pproh ws prsnt whr th onstrut Ptri nt is not nssrily sf, ut oun 1. Agin, th rhility grph of th synthsiz Ptri nt is isimilr to th givn trnsition systm. To illustrt th prolm of non-lmntry trnsition systms, onsir Figur 19. This trnsition systm is not lmntry. Th prolm is tht thr r two stts s2 n s3 tht r intil in trms of rgions, i.., thr is no rgion suh tht on is prt of it n th othr is not. As rsult, th onstrut Ptri nt on th lft hn si of Figur 19 fils to onstrut isimilr Ptri nt. Howvr, using ll-splitting s prsnt in [26, 27], th Ptri nt on th right hn si n otin. This Ptri nt hs two trnsitions 1 n 2 orrsponing to tivity in th log. Th splitting is s on th so-ll notions of xittion n gnrliz xittion rgion, s [26]. As shown in [26, 27] it is lwys possil to onstrut n quivlnt Ptri nt. Howvr, ll-splitting my l to lrgr Ptri nts. In [21] th uthors show how to otin th most pris mol whn ll splitting is not llow. In stt-s rgion thory, th im is to onstrut Ptri nt, suh tht its hvior is isimilr to th givn trnsition systm. In pross isovry howvr, w hv log s input, i.., w hv informtion out squns of trnsitions, ut not out stts. In Stion 7, w show how w n intify stt informtion from vnt logs n thn us stt-s rgion thory for pross isovry. Howvr, w first introu lngug-s rgion thory. 6.2 Lngug Bs Rgion Thory In ition to stt-s rgion thory, w lso onsir lngug-s rgion thory [14,17,19,28,42,43]. In thir survy ppr [45], Musr n Lornz show how for iffrnt lsss of lngugs (stp lngugs, rgulr lngugs n (infinit) prtil lngugs) Ptri nt n riv suh tht th rsulting nt is th Ptri nt with th smllst hvior in whih th wors in th lngug r possil firing squns. Givn prfix-los lngug A ovr som non-mpty, finit st of tivitis A, th lngug-s thory of rgions tris to fin finit Ptri nt N(A) in whih th trnsitions orrspon to th lmnts in th st A n of whih ll squns in th lngug r firing squns (fitnss ritrion). Furthrmor, th Ptri nt shoul minimiz th numr of firing squns not in th lngug (prision ritrion). Th Ptri nt N(A) = (, A, ) is finit Ptri nt with infinitly mny firing squns llowing for ny squn involving tivitis A. Suh mol is typilly unrfitting, i.., llowing for mor hvior thn suggst y th vnt log. Thrfor, th hvior of this Ptri nt ns to ru so tht th Ptri nt still llows to rprou ll squns in th lngug, ut os not llow for hvior unrlt to th xmpls sn in th vnt log. This is hiv y ing pls to th Ptri nt. Th thory of rgions provis mtho to intify ths pls, using lngug rgions. 1 A Ptri nt is sf if thr n nvr mor thn 1 tokn in ny pl. Bounnss implis tht thr xists n uppr oun for th numr of tokns in ny pl. 27

28 Dfinition 11 (Lngug Rgion). Lt A st of tivitis. A rgion of prfix-los lngug L ovr A is tripl ( x, y, ) with x, y {0, 1} A n {0, 1}, suh tht for h non-mpty squn w = w L, w L, A: + ( ) w (t) x(t) w(t) y(t) 0 t A This n rwrittn into th inqution systm: 1 + M x M y 0 whr M n M r two L A mtris with M(w, t) = w(t), n M (w, t) = w (t), with w = w. Th st of ll rgions of lngug is not y R(L) n th rgion ( 0, 0, 0) is ll th trivil rgion. 2 Consir rgion r = ( x, y, ) orrsponing to som pl p r. For ny prfix w = w in L, rgion r stisfis + ( ) t A w (t) x(t) w(t) y(t) 0 whr is th initil numr of tokns in pl p r, t A w (t) x(t) is th numr of tokns prou for pl p r just for firing (not tht w is th prfix without inluing th lst ), n t A w(t) y(t) is th numr of tokns onsum from pl p r ftr firing (w is th ontntion of w n ). w is th Prikh vtor of w, i.., w(t) is th numr of tims t pprs in squn w. x(t) is th numr of tokns t prous for pl p r. Trnsition t onsums y(t) tokns from pl p r pr firing. So, w(t) y(t) is th totl numr of tokns t onsums from pl p r whn xuting w. t 1 t 2 y 1 x 2 x 1 y 2 x 4 y 3 y 2 t 3 x 3 t 4 Fig. 20. Rgion for lngug with four lttrs: t 1, t 2, t 3, n t 4. Figur 20 illustrts th lngug-s rgion onpt using for lngug ovr four tivitis ( A =4), i.., h solution ( x, y, ) of th inqution systm n rgr in th ontxt of Ptri nt, whr th rgion orrspons to 2 To ru lultion tim, th inqution systm n rwrittn to th form [ 1; M ; M] r 0 whih n simplifi y liminting uplit rows. 28

29 fsil pl with prst {t t T, x(t) 1} n postst {t t T, y(t) 1}, n initilly mrk with tokns. In this ppr, w ssum r-wights to 0 or 1 s w im t unrstnl mols (i.., x, y {0, 1} A ). As shown in [14, 16, 28, 43] it is possil to gnrliz th ov notions to ritrry rwights. A pl rprsnt y rgion n to Ptri nt, without limiting its hvior with rspt to trs sn in th vnt log. Thrfor, w ll suh pl fsil. Dfinition 12 (Fsil pl). Lt L prfix-los lngug ovr A n lt N = ((P, T, F ), M) mrk Ptri nt with T = A n M IB(P ). A pl p P is ll fsil if n only if thr xists orrsponing rgion ( x, y, ) R(L) suh tht M(p) =, n x(t) = 1 if n only if t p, n y(t) = 1 if n only if t p. In [16,43] it ws shown tht ny solution of th inqution systm of Dfinition 11 n to Ptri nt without influning th ility of tht Ptri nt to rply th log. Howvr, sin thr r infinitly mny solutions of tht inqution systm (ssuming r wights), thr r infinit mny fsil pls n th uthors of [16, 43] prsnt two wys of finitly rprsnting ths pls, nmly sis rprsnttion [43] n sprting rprsnttion [16, 43]. 6.3 Pross Disovry vs. Rgion Thory Whn ompring rgion thory stt-s or lngug s with pross isovry, som importnt iffrns shoul not. First of ll, in rgion thory, th strting point is full hviorl spifition, ithr in th form of (possily infinit) trnsition systm, or (possily infinit) lngug. Hn, th unrlying ssumption is tht th input is omplt n nois fr n thrfor mximl fitnss is ssur. Son, th im of rgion thory is to provi ompt, xt rprsnttion of tht hvior in th form of Ptri nt. If th nt llows for mor hvior thn spifi, thn this itionl hvior n provn to miniml, hn rgion thory provis pris rsults. Finlly, whn rgion thory is irtly ppli in th ontxt of pross isovry [16, 21, 53], th rsulting Ptri nts typilly prform poorly with rspt to two of th four imnsions shown in Figur 15. Th rsulting mols r typilly ovrfitting (lk of gnrliztion) n r too iffiult to omprhn (simpliity ritrion). Thrfor, in stions 7 n 8, w show how rgion thory n moifi for pross isovry. Th ky i is to llow th lgorithms to gnrliz n rlx on prisnss, with th im of otining simplr mols. 7 Pross Disovry Using Stt-Bs Rgion Thory In Stion 2 w introu th onpt of ontrol-flow isovry n isuss th prolms of xisting pprohs. In Stion 6, w introu rgion thory 29

30 n show th min iffrns with ontrol flow isovry. In this stion, w introu two-stp pproh to omin pross isovry with stt-s rgion thory [8]. In th rminr, w lort on ths two stps n isuss hllngs. 7.1 From Evnt Logs to Trnsition Systms In th first stp, w onstrut trnsition systm from th log, whr w gnrliz from th osrv hvior. Furthrmor, w mssg th output, suh tht th rgion thory us in th son stp is mor likly to prou simpl mol. In th son stp, w us lssil stt-s rgion thory to otin Ptri nt. This stion sris th first n most importnt stp. Dpning on th sir proprtis of th mol n th hrtristis of th log, th lgorithm n tun to provi mor simpl n/or gnri mol. Th most importnt spt of pross isovry is uing th stts of th oprtionl pross in th log. Most mining lgorithms hv n impliit notion of stt, i.., tivitis r glu togthr in som pross moling lngug s on n nlysis of th log n th rsulting mol hs hvior tht n rprsnt s trnsition systm. In this stion, w propos to fin stts xpliitly n strt with th finition of trnsition systm. urrnt stt tr: pst f g h h h i futur pst n futur Fig. 21. Thr si ingrints n onsir s sis for lulting th pross stt : (1) pst, (2) futur, n (3) pst n futur. In som ss, th stt n riv irtly,.g., h vnt nos th omplt stt y proviing vlus for ll rlvnt t ttriuts. Howvr, in th vnt log w typilly only s tivitis n not stts. Hn, w n to u th stt informtion from th tivitis xut for n ftr givn stt. Bs on this, thr r silly thr pprohs to fining th stt of prtilly xut s in log: pst, i.., th stt is onstrut s on th history of s, futur, i.., th stt of s is s on its futur, or pst n futur, i.., omintion of th prvious two. 30

31 <,> <,,> <,,,> <> <> <,> <,,>... <,,,> <,> <,,> () trnsition systm s on prfix <,,> <,> <,,,> <,,> <,> <> <> <,,,> <,,> <,> () trnsition systm s on postfix <> <,,,> <> <,,> <,> <,> <,,> <> <,,,> <> <> <,,> <> <,> <,> <> <,,> <> <> <,,,> <> <,,> <,> <,> <,,> <> <,,,> <> () trnsition systm s on prfix n postfix Fig. 22. Thr trnsition systms riv from th log L 1 = [,,, 5,,,, 8,,, 9 ]. Figur 21 shows n xmpl of tr n th thr iffrnt ingrints tht n us to lult stt informtion. Givn onrt tr, i.., th xution of s from ginning to n, w n look t th stt ftr xuting th first nin tivitis. This stt n rprsnt y th prfix, th postfix, or oth. To xplin th si i of onstruting trnsition systm from n vnt log, onsir Figur 22. If w just onsir th prfix (i.., th pst), w gt th trnsition systm shown in Figur 22(). Not tht th initil stt is not, i.., th mpty squn. Strting from this initil stt th first tivity is lwys in h of th trs. Hn, thr is on outgoing r ll, n th susqunt stt is ll. From this stt, thr trnsitions r possil 31

32 ll rsulting in iffrnt stts,.g., xuting tivity rsults in stt,, t. Not tht in Figur 22() thr is on initil stt n thr finl stts. Figur 22() shows th trnsition systm s on postfixs. Hr th stt of s is trmin y its futur. This futur is known us pross mining looks t th vnt log ontining omplt ss. Now thr r thr initil stts n on finl stt. Initil stt,, inits tht th nxt tivity will, follow y n. Not tht th finl stt hs ll initing tht no tivitis n to xut. Figur 22() shows trnsition systm s on oth pst n futur. Th no with ll,,, nots th stt whr n hv hppn n n still n to our. Not tht now thr r thr initil stts n thr finl stts. Th pst of s is prfix of th omplt tr. Similrly, th futur of s is postfix of th omplt tr. This my tkn into ount ompltly, whih ls to mny iffrnt stts n pross mols tht my too spifi (i.., ovrfitting mols). It is lso possil to tk lss informtion into ount (.g., just th lst stp in th pross). This my rsult in unrfitting mols. Th hllng is to slt n strtion tht lns twn ovrfitting n unrfitting. Mny strtions r possil; s for xmpl th systmti trtmnt of strtions in [8]. Hr, w only highlight som of thm. Mximl horizon (h) Th sis of th stt lultion n th omplt prfix (postfix) or prtil prfix (postfix). Filtr (F ) Th son strtion is to filtr th (prtil) prfix n/or postfix, i.., tivitis in F A r kpt whil tivitis A \ F r rmov. Mximum numr of filtr vnts (m) Th squn rsulting ftr filtring my ontin vril numr of lmnts. Agin on n trmin kin of horizon for this filtr squn. Squn, g, or st (q) Th first thr strtions yil squn. Th fourth strtion mhnism optionlly rmovs th orr or frquny from th rsulting tr. Visil tivitis (V ) Th fifth strtion is onrn with th trnsition lls. Ativitis in V A r shown xpliitly on th rs whil th tivitis in A \ V r not shown. Figur 23 illustrts th strtions. In Figur 23() only th st strtion is us q = st. Th rsult is tht svrl stts r mrg (ompr with Figur 22()). In Figur 23() tivitis n r filtr out (i.., F = {,, } n V = {,, }). Morovr, only th lst non-filtr vnt is onsir for onstruting th stt (i.., m = 1). Not tht th stts in Figur 23() rfr to th lst vnt in {,, }. Thrfor, thr r four stts:,,, n. It is intrsting to onsir th rol of n. First of ll, thy r not onsir for uiling th stt (F = {,, }). Son, thy r lso not visuliz (V = {,, }), i.., th lls r supprss. Th orrsponing trnsitions r ollps into th unll r from to. If V woul hv inlu n, thr woul hv n two suh rs ll rsptivly. 32

33 {,} {,,} {,,,} {} {} {,}... <> {,} {,,} () trnsition systm s on sts <> <> <> () trnsition systm strting from n Fig. 23. Two trnsition systms uilt on L 1 using th following prfix strtions: () h =, F = A (i.., ll tivitis), m =, q = st, n V = A, n () h =, F = {,, }, m = 1, q = sq, n V = {,, }. Th first four strtions n ppli to th prfix, th postfix, or oth. In ft, iffrnt strtions n ppli to th prfix n postfix. As rsult of ths hois mny iffrnt trnsitions systms n gnrt. If mor rigorous strtions r us, th numr of stts will smllr n th ngr of unrfitting is prsnt. If, on th othr hn, fwr strtions r us, th numr of stts my lrgr rsulting in n ovrfitting mol. An xtrm s of ovrfitting ws shown in Figur 22(). At first this my sm onfusing; howvr, s init in th introution it is importnt to provi rprtoir of pross isovry pprohs. Dpning on th sir gr of gnrliztion, suitl strtions r slt n in this wy th nlyst n ln twn ovrfitting n unrfitting, i.., twn gnrliztion n prision in ontroll wy. Using lssil rgion thory, w n trnsform th trnsition systm into pross mol. Howvr, whil w n now ln prision n gnrliztion, w i not fous on simpliity yt. Thrfor, w mk us of th innr workings of stt-s rgion thory to mssg th trnsition systm. This is intn to pv th pth for rgion thory. For xmpl, on my rmov ll slfloops, i.., trnsitions of th form s s (f. Figur 24()). Th rson my tht on is not intrst in vnts tht o not hng th stt or tht th synthsis lgorithm in th son stp nnot hnl this. Anothr xmpl 33

34 woul to los ll imons s shown in Figur 24(). If s 1 1 s2, s 2 1 s3, n s 2 2 s4, thn s 1 3 s4 is. Th rson for oing so my tht us (1) oth 1 n 2 r nl in s 1 n (2) ftr oing 1, tivity 2 is still nl, it is ssum tht 1 n 2 n xut in prlll. Although th squn 2, 1 ws not osrv, it is ssum tht this is possil n hn th trnsition systm is xtn y ing s 1 3 s4. s1 s s s s2 2 s3 s2 2 1 s3 () rmoving slf-loops s4 s4 () losing th imon Fig. 24. Two xmpls of moifitions of th trnsition systm to i th onstrution of th pross mol. 7.2 From Trnsition Systms to Ptri Nts In th son stp, th trnsition systm is trnsform into Ptri nt using th thniqus sri in [13,15,23,24,26,27,35]. In Stion 6.1, w introu th si i of stt-s rgions. Thrfor, w o not lort on this hr. Th importnt thing to not is tht thr is rng of thniqus to onvrt trnsition systm into Ptri nt. Ths thniqus typilly only rss two of th four qulity imnsions mntion in Figur 15: fitnss n prision. Th othr two imnsions simpliity n gnrliztion n to rss whn onstruting th trnsition systm or y imposing itionl onstrints on th Ptri nt. Th gol of pross mining is to prsnt mol tht n intrprt sily y pross nlysts n n-usrs. Thrfor, omplx pttrns shoul voi. Rgion-s pprohs hv tnny to introu smrt pls, i.., pls tht omptly srv multipl purposs. Suh pls hv mny onntions n my hv non-lol ffts (i.., th sm pl is us for iffrnt purposs in iffrnt phss of th pross). Thrfor, it my usful to gui th gnrtion of pls suh tht thy r sir to unrstn. This is firly strightforwr in oth stt-s rgion thory n lngug-s rgion thory. In [26, 27] it is shown tht itionl rquirmnts n with rspt to th proprtis of th rsulting nt. For xmpl, th nt n for to fr-hoi, pur, t. S [8] for xmpls. 34

35 () () () () Fig. 25. Vrious Ptri nts riv for th trnsitions systms in figurs 22 n 23 using stt-s rgions. All mols r s on vnt log L 1 = [,,, 5,,,, 8,,, 9 ]. Th pproh ws lry illustrt using Figur 18. Figur 25 shows som mor xmpls s on th trnsition systms in figurs 22 n 23. Ths mols whr omput using th lssil synthsis pproh prsnt in [26, 27]. This pproh pplis ll-splitting if n. Not tht ll trnsition systms wr riv from vnt log L 1 = [,,, 5,,,, 8,,, 9 ]. Th Ptri nt in Figur 25() is otin y pplying stt-s rgion thory to th trnsition systm in Figur 22(). Th sm mol is otin whn omputing th rgions for th trnsition systm in Figur 23(). Th Ptri nt in Figur 25() is otin whn pplying stt-s rgion thory to th trnsition systm in Figur 22(). Two things n not: (1) th multipl initil stts in Figur 22() rsult in mny initil tokns n sour pls, n (2) ll splitting is us (.g., thr r two trnsitions) to llow for th multipl strting points. Th rgion-s pproh synthsizs th mol in Figur 25() for th trnsition systm in Figur 22(). Also this mol suffrs from th prolm tht thr r multipl initil stts. In gnrl, w suggst to voi hving multipl initil stts in th trnsition systm to synthsiz. It is trivil to mrg th initil stts or nw rtifiil initil stt for pplying rgion-s synthsis. Figur 25() ws otin from th trnsition systm in Figur 23(). Th Ptri nt shows tht if w strt from n, w otin n unll 35

36 trnsition initing th stt in whih n woul hv ourr. This silnt trnsition is u to th slf-loop in th trnsition systm of Figur 23(). Eliminting th slf-loop using th strtgy prsnt in Figur 24() woul rmov th unll trnsition in Figur 25(). 7.3 Chllngs In this stion, w hv shown tht y omining strtion thniqus n rgion thory, powrful pross mining lgorithm n otin. Through svrl strtions, w n otin th sir lvl of prision n gnrliztion, whil y mssging th trnsition systm, w n try to otin simpl mols. Howvr, thr r lso som rwks of this pproh. Fig. 26. Ptri nt otin using rgion thory ppli to log L 6 = [, 2,,, 3,,,, 2 ] using th following sttings: h =, F = A (i.., ll tivitis), m =, q = st, n V = A n post-prossing stp in whih stts with intil inflow or outflow r mrg. It is fr from trivil to slt th right prmtrs for th strtions. Existing thniqus n tools r snsitiv to hngs of prmtr vlus, n th rsult is oftn unpritl. Hn, otining suitl pross mol is mttr of tril-n-rror. Figur 26 shows, for xmpl, th sttings with whih w n otin th sir mol for log L 6, i.., th Ptri nt with slf-loop on trnsition. Howvr, th mol shown in Figur 27 illustrts tht th wrong sttings my l to n ovrfitting mol. Nonthlss, th stt-s rgion pproh is on of th fw tht n tt long-trm pnnis, s shown y Figur 28, whih rsult from pplying th thniqu to log L 8. Furthrmor, th mjor rwk of th pproh outlin hr is th omputtionl omplxity. For lrgr logs, th rsulting trnsition systm my not fit in min mmory n son, th rgion thory us to otin Ptri nt hs tim omplxity whih is xponntil in th siz of th trnsition systm. 8 Pross Disovry Using Intgr Linr Progrmming In Stion 7, w hv prsnt two-stp pproh to pply rgion thory in th ontxt of pross mining. W fouss on otining trnsition systm 36

37 Fig. 27. Ptri nt otin using rgion thory ppli to log L 6 using th following prmtrs: h =, F = A (i.., ll tivitis), m =, q = multist, n V = A. Fig. 28. Ptri nt otin using rgion thory ppli to log L 8 = [,, 45,,, 42 ] using th following sttings: h =, F = A (i.., ll tivitis), m =, q = st, n V = A. from n vnt log n us lssil rgion thory to otin Ptri nt. In this stion, w o not onsir th rgion thory s lk ox, ut inst, w xtn xisting pprohs to mk thm mor pplil in th ontxt of pross isovry, minly y llowing th thniqus to gnrliz from th log n to prou simplr mols. Both sis n th sprting rprsnttions of rgions prsnt in [16,43] r s on th sm prinipl, nmly tht finit rprsnttion is provi of th infinit st of pls stisfying Dfinition 11. By oing so, th lngugs rgion thory nsurs mximl prisnss n fitnss, with littl to no gnrliztion n no im for simpl mols. Hn, only two of th four qulity imnsions of Figur 15 r onsir. For pross isovry, w r iming t simpl, gnrlizing mols. Hn, w prsnt n pproh [60], whr w only rprsnt thos pls stisfying Dfinition 11 tht: h pl xprsss usl pnny lrly visil in th log, no impliit pls r inlu in th nt, n pls whih r mor xprssiv thn othrs r fvor, i.., pls with miniml input trnsitions n mximl output trnsitions r fvor. In ontrst to th stt-s rgion pproh, w o not try to influn gnrliztion n prision irtly. Inst, w fous on mol simpliity, y 37

38 limiting th numr of pls in th mol (n llowing for vrying this numr). As with th stt-s pproh, mximl fitnss is gurnt. In orr to slt pls stisfying Dfinition 11, w onvrt this qution into Intgr Linr Progrmming (ILP) prolm. 8.1 Intgr Linr Progrmming Rprsnttion W quntify th xprssivnss of pls, in orr to provi trgt funtion, nssry to trnslt th inqution systm of Dfinition 11 into n Intgr Linr Progrmming (ILP) prolm. In Stion 8.2, w thn us th rsult to gnrt Ptri nt in stp-y-stp fshion. In stion 8.3, w provi insights into th usl pnnis foun in log n how ths n us for fining pls. To pply th lngug-s thory of rgions in th fil of pross isovry, w n to rprsnt th vnt log s prfix-los lngug, i.., y ll th trs prsnt in th vnt log, n thir prfixs. Rll from Dfinition 1 tht n vnt log is finit g of trs. Dfinition 13 (Lngug of n vnt log). Lt A st of tivitis. Lt L IB(A ) n vnt log ovr this st of tivitis. Th lngug L tht rprsnts this vnt log, uss lpht A, n is fin y: L = {σ A σ L: σ σ } A trivil Ptri nt pl of rprouing lngug is nt with only trnsitions. This nt is simpl, n rprsnt ll trs, n hn hs mximl fitnss. It lso gnrlizs wll, ut th Ptri nt with only trnsitions is vry impris us nything is possil oring to th mol. To rstrit th hvior llow y th Ptri nt, ut not osrv in th log, w strt ing pls to tht Ptri nt. As stt for, th pls w to th Ptri nt shoul s xprssiv s possil, whih is th sm s sying tht suh pls hv mximl postst n miniml prst, i.., it shoul not possil to n output trnsition to or to rmov n input trnsition from pl without ruing th fitnss of th rsulting nt. Bsis srhing for rgions tht l to pls with mximum xprssivnss, w lso wnt to voi ing impliit pls to mol. Thrfor, w will srh for miniml rgions s introu in [30]. Using th inqution systm of Dfinition 11 n th xprssivnss of pl, w n fin trgt funtion for our ILP prolm to onstrut th pls of Ptri nt in logil orr [52]. Th following trgt funtion is shown to suh tht it fvors miniml rgions whih r mximlly xprssiv [60]: Dfinition 14 (Trgt funtion). Lt A st of tivitis. Lt L IB(A ) n vnt log n L th orrsponing lngug. Furthrmor, lt M th mtrix fin in Dfinition 11. W fin th funtion τ : R(L) IN y τ(( x, y, )) = + 1 T ( 1 + M ( x y)) 38

39 Comining this trgt funtion with th inqution systm of Dfinition 11 yils th following ILP prolm: Dfinition 15 (ILP formultion). Lt A st of tivitis, lt L IB(A ) n vnt log, n lt M n M th mtris s fin in Dfinition 11. W fin th ILP ILP L for vnt log L s: Min + 1 T ( 1 + M ( x y)) Dfinition 14 s.t. 1 + M x M y 0 Dfinition 11 1 T x + 1 T y 1 Thr shoul t lst on g 0 x 1 x {0, 1} T 0 y 1 y {0, 1} T 0 1 {0, 1} This ILP prolm provis th sis for our pross isovry prolm. Howvr, n optiml solution to this ILP only provis singl fsil pl with miniml vlu for th trgt funtion. Thrfor, in th nxt stion, w show how this ILP prolm n us s sis for onstruting Ptri nt from log. 8.2 Construting Ptri Nts Using ILP In th prvious sustion, w provi th sis for ing pls to Ptri nt s on knowlg xtrt from log. In ft, th trgt funtion of Dfinition 14 provis prtil orr on ll lmnts of th st R(L), i.., th st of ll rgions of lngug. In this sustion, w show how to gnrt th first n pls of Ptri nt, tht is (1) l to rprou log unr onsirtion n (2) of whih th pls r s xprssiv s possil. A trivil pproh woul to h foun solution s ngtiv xmpl to th ILP prolm, i.., xpliitly foriing this solution. Howvr, it is lr tht on rgion r hs n foun n th orrsponing fsil pl is to th Ptri nt, w r no longr intrst in rgions r for whih th orrsponing fsil pl hs mor tokns, lss outgoing rs or mor inoming rs, i.., w r only intrst in inpnnt rgions. Dfinition 16 (Rfining th ILP ftr h solution). Lt A st of tivitis, lt L IB(A ) n vnt log, lt M n M th mtris s fin in Dfinition 11 n lt ILP 0 L th orrsponing ILP. Furthrmor, for i 0 lt rgion r i = ( x i, y i, i ) miniml solution of ILP i L. W fin th rfin ILP s ILP i L, with th xtr onstrint spifying tht: i + y T ( 1 y i ) x T x i i T x i Not tht for ny solution r = ( x, y, ) of ILP i L: < i or thr is t A suh tht x(t) < x i (t) or y(t) > y i (t). If this is not th s (i.., i n x(t) x i (t) n y(t) y i (t) for ny t), thn i = i, x(t) x i (t) = x i (t), n y(t) y i (t) = 0 1. Hn, w fin ontrition with rspt to th xtr 39

40 onstrint. As rsult th nw rgion r is for to suffiintly iffrnt from r i. Th rfinmnt oprtor prsnt ov, silly fins n lgorithm for onstruting th pls of Ptri nt tht is pl of rprouing givn log. Th pls r gnrt in n orr whih nsurs tht th most xprssiv pls r foun first n tht only pls r tht hv lss tokns, lss outgoing rs, or mor inoming rs. Furthrmor, h solution of rfin ILP is lso solution of th originl ILP, sin th nw solution stisfis ll onstrints of th initil ILP formultion, n som xtr onstrints. Hn, ll pls onstrut using this prour r fsil pls. This prour, n us to ontinu ing pls, thus mking th mol mor pris, whil ompromising on mol omplxity s shown y Figur 29. Th Ptri nt in Figur 29 llows for mor hvior thn th log L 1 ontins, so in thory mor pls oul still. Nonthlss, ny nw pl woul suh tht it hs fwr output rs, or mor input rs thn th ons inlu in this mol. In th worst s, th totl numr of pls introu is xponntil in th numr of trnsitions. Sin thr is no wy to provi insights into n upproun for th numr of pls to gnrt, w propos mor suitl pproh, not using th rfinmnt stp of Dfinition 16. Inst, w propos to gui th srh for solutions (i.. for pls) using onpts from th α- lgorithm [9]. 8.3 Using Log-Bs Proprtis Rll from th ginning of this stion, tht w r spifilly intrst in pls xprssing xpliit usl pnnis twn trnsitions. In this sustion, w us th usl rltions L fin rlir in Dfinition 7 in omintion with th ILP of Dfinition 15 to onstrut Ptri nt. Fig. 29. Ptri nt otin using lngug-s rgion thory nivly ppli to log L 1. 40

41 Cusl pnnis twn trnsitions r us y mny pross isovry lgorithms [6,9,31,58] n gnrlly provi goo inition s to whih trnsitions shoul onnt through pls. Furthrmor, xtnsiv thniqus r vill to riv usl pnnis twn trnsitions using huristi pprohs [9, 31]. Howvr, it is not known whthr th log is omplt n whthr w ovr ll usl pnnis. Thrfor, w rstrit ourslvs to srh for Ptri nt suh tht if usl pnny is not in th log, it is lso not in th nt. In orr to fin pl xprssing spifi usl pnny, w xtn th ILP prsnt in Dfinition 15. Dfinition 17 (ILP for usl pnny). Lt A st of tivitis, lt L IB(A ) n vnt log, lt M n M th mtris s fin in Dfinition 11 n lt ILP L th orrsponing ILP. Furthrmor, lt t 1, t 2 A n ssum t 1 L t 2. W fin th rfin ILP, ILP t1 t2 L s ILP L, with two xtr ouns spifying tht: x(t 1 ) = y(t 2 ) = 1 A solution of th optimiztion prolm xprsss th usl pnny t 1 L t 2, n rstrits th hvior s muh s possil. Howvr, suh solution os not hv to xist, i.., th ILP might infsil, in whih s no pl is to th Ptri nt ing onstrut. Nonthlss, y onsiring sprt ILP for h usl pnny in th log, Ptri nt n onstrut, in whih h pl is s xprssiv s possil n xprsss t lst on pnny riv from th log. With this pproh, t most on pl is gnrt for h pnny n thus th uppr oun of pls in N(L) is th numr of usl pnnis, whih is worst-s qurti in th numr of trnsitions. Th rsult of pplying this log-s thniqu to our log L 1 is shown in Figur 30. This mol is vry los to th sir mol, xpt tht it os not ontin finl pl. This is gnrl rwk of lngug-s rgion thory: th fous is on th ility to rprou prfixs of log trs rthr thn trmintion in wll-fin finl stt. Up to now, w i not impos ny rstrition on th strutur of th rsulting Ptri nt. By ing onstrints, svrl Ptri nt proprtis n xprss, thus rsulting in lmntry nts, pur nts, (xtn) fr-hoi nts, stt mhins n mrk grphs [60]. This llows us to furthr simplify th rsulting Ptri nt. Not tht this is similr to th rfinmnt sri in Stion 7.2 for stt-s rgions. 8.4 Chllngs In stions 7 n 8 w prsnt svrl wys to us rgion thory in th ontxt of pross isovry in orr to llvit som of th prolms of th α-lgorithm. First, w hv shown how to w n ln prision n gnrliztion whil onstruting trnsition systm from log. Thn, y mssging th trnsition 41

42 Fig. 30. Ptri nt otin using lngug-s rgion thory using log-s proprtis ppli to log L 1. Not tht ompr to rlir solutions th sink pl noting trmintion is missing. systm, w n somwht improv th simpliity of th rsulting mols. Whn using lngug-s rgion thory, w hv shown tht w n fous on th simpliity of th rsulting mol. By inrmntlly introuing pls, w n mk th rsulting mol mor pris in stp-y-stp fshion. Figurs 31 n 32 show tht w n isovr mols for th logs L 6 n L 8, ut th long-trm pnny in L 8 is not intifi, u to th rlin on th usl pnnis us in th α-lgorithm. Furthrmor, s isuss for, lngug-s rgions hv prolms mking th finl stt xpliit (i.., sink pls r missing in figurs 31 n 32). Fig. 31. Ptri nt isovr for vnt log L 6. Th mol ws otin using lngugs rgion thory gui y log-s proprtis. Fig. 32. Ptri nt otin using lngug-s rgion thory gui y log-s proprtis ppli to log L 8 = [,, 45,,, 42 ]. No sink pl is rt n th long-trm pnnis r not isovr us only short-trm pnnis r us to gui th isovry of pls. 42

43 Unfortuntly, ll rgion-s pprohs r omputtionlly hllnging. In th s of th lngug-s rgions, fining solution for h inrmntl ILP prolm is of worst-s xponntil tim omplxity. Furthrmor, th ommon proprty of ll rgion-s thniqus is tht th fitnss of th isovr nt is gurnt to 100%, rgrlss of th log. This mks ths pprohs vry roust, ut lso snsitiv to nois. Thus fr w only us toy xmpls to illustrt th iffrnt onpts. All funtionlity hs n m in th pross isovry frmwork ProM, whih is pl of onstruting nts for logs with thousns of ss rfrring to ozns of trnsitions. Th thniqus hv n tst on mny rl-lif n synthti vnt logs. Howvr, isussion of ths xprimntl rsults is outsi th sop of this rtil. For this w rfr to [2, 7, 39, 47, 53]. 9 Tool Support Both for pross mining n rgion thory, it is ssntil tht lgorithms n put to th tst in rl lif nvironmnts. Thrfor, lmost ll work prsnt in this rtil is implmnt in frly vill tools. For xmpl, lssil stts rgion thory is implmnt in Ptrify n Gnt [22], whil Rminr [54] pplis this in pross isovry ontxt. Som of th lngug-s rgion thory is implmnt in VIPTool [18]. Th pross mining lgorithms prsnt in stions 4, 7 n 8 hv ll n implmnt in th ProM frmwork [11, 56, 57]. All lgorithms isuss in this rtil n foun th most rnt vrsion of ProM (vrsion 6.0 n ltr). ProM is gnri opn-sour frmwork for implmnting pross mining lgorithms in stnr nvironmnt. Figur 33 shows th strtup srn of ProM. Hr, log ws opn for nlysis whih is shown in th worksp. Whn slting th log n liking on th tion utton, th usr is tkn to th tion rowsr, whr in Figur 34, th α-minr is slt. Th α-minr is n implmnttion of th work in Stion 4. In rlir vrsions of ProM, th tul pross mining lgorithms implmnt y plug-ins ssum th prsn of GUI. Most lgorithms rquir prmtrs, n th plug-in woul sk th usr for ths prmtrs using som GUI-s ilog. Furthrmor, som plug-ins isply sttus informtion using progrss rs n suh. Thus, th tul pross mining lgorithm n th us of th GUI wr intrtwin. As rsult, th lgorithm oul only run in GUI-wr ontxt, sy on lol worksttion. This wy, it ws impossil to fftivly run pross mining xprimnts using istriut infrstrutur n/or in th. In ProM 6, th pross mining lgorithm n th GUI hv n rfully sprt, n th onpts of ontxts hs n introu. For plug-in, th ontxt is th proxy for its nvironmnt, n th ontxt trmins wht th plug-in n o in its nvironmnt. A plug-in n only isply ilog or progrss r on th isply if th ontxt is GUI-wr. Typilly, in ProM 6, th implmnttion of n lgorithm is split into numr of plug-ins: A plug-in 43

44 Fig. 33. ProM 6 Worksp; opning srn ftr loing fil. Fig. 34. ProM 6 Ation Browsr; slting th lph-minr to isovr pross mol from th lo vnt log. 44

45 Fig. 35. ProM 6 Pkg mngr showing th pkgs rlvnt for th thniqus prsnt in this rtil. for vry ontxt. Th tul pross mining lgorithm will implmnt in gnri wy, suh tht it n run in gnrl (GUI-unwr) ontxt. This llows th lgorithm to run in ny ontxt, vn in istriut ontxt [20]. Th ilog for stting th rquir prmtrs is typilly implmnt in GUIwr vrint of th plug-in. Typilly, this GUI-wr plug-in first isplys th prmtr ilog, n whn th usr hs provi th prmtrs n hs los th ilog, it will simply run th gnri plug-in using th provi prmtrs. Th mjor vntg of this is tht th ProM frmwork my i to hv th gnri plug-in run on iffrnt omputr thn th lol worksttion. Som plug-ins my rquir lots of systm rsours (.g., omputing powr, mmory, n isk sp), lik for xmpl th gnti minr. Bsilly, th gnti minr tks mol n log, n thn gnrts numr of ltrntiv mols for th givn log. Th st of ths ltrntiv mols r thn tkn s nw strting points for th gnti minr. Th gnti minr rpts this until som stop ritrion hs n rh, ftr whih it rturns th st mol foun so fr. Clrly, this minr might tk onsirl tim (it my tk hunrs of itrtions for it stops n th fitnss lultion is vry tim-onsuming for lrg logs), n it my tk onsirl mmory (th numr of ltrntiv mols my grow rpily). For suh n lgorithm, it might prfrl to hv it run on srvr whih is mor powrful thn th lol worksttion. Morovr, gnti mining n istriut in svrl wys [20]. For xmpl, th pop- 45

46 ultion n prtition ovr vrious nos. Eh supopultion on no volvs inpnntly for som tim ftr whih th nos xhng iniviuls. Similrly, th vnt logs my portion ovr nos thus sping up th fitnss lultions. Bsis sprting th funtionlity from th usr intrf, ProM 6 rquirs funtionlity to provi in pkgs. Ths pkgs h ontin olltion of rlt lgorithms, typilly implmnt y on rsrh group. Whn ProM is strt for th first tim, th pkg mngr is opn s shown in Figur 35. Hr, for h known pkg, ProM shows who th uthor is, wht th urrnt vrsion is n whthr or not this vrsion is instll. Th work prsnt in this rtil, rquirs th following pkgs to instll: AlphMinr, TrnsitionSystms n ILPMinr. Th othr pkgs shown r utomtilly instll u to pnnis. Furthrmor, th pkg Ptrify provis import n xport funtionlity to n from th stt-s rgion tool Ptrify. Fig. 36. Rsult of α-minr: th α-lgorithm hs prolms ling with th multipl hotl ookings intrlv with othr ooking tivitis. Th vnt log opn in Figur 33 is log onsisting of 1000 ss of trvl gny. A ustomr rgistrs, thn purhss us tikt or pln tikt whil t th sm tim h ooks on or mor hotls. Aftr th ooking phs, th trip osts r omput n th ustomr hs to hoos twn two typs of insurn. Aftr tht, th totl osts r lult n th pymnt is omplt. This is rthr simpl xmpl us to show th rsults of th thr lgorithms. Th rsulting Ptri nt ftr pplying th α-lgorithm to this log is shown in Figur 36. Th rsult ftr xuting th trnsition systm minr is shown in Figur 37 n th rsult of th ILP minr is shown in 38. All thr lgorithms provi mol tht in mols th givn sitution. Th iffiulty hr is th ft tht th hotl ooking is xut on or mor tims. Th α-lgorithm os 46

47 Fig. 37. Rsult of TS Minr. Not tht thr r now two trnsitions rfrring to hotl ookings (ll splitting). Fig. 38. Rsult of ILP Minr. Th mol is l to rply th vnt log. Howvr, tokns my rmin in th pl following th hotl ooking n ookings n tk pl for th rgistrtion stp. 47

48 not onnt this trnsition (thus nling it ontinuously n stroying th WF-nt strutur), whil th trnsition systm minr introus two trnsitions for this stp, ut it nfors tht th son hotl n only ook ftr th us or pln tikt is ook. Th ILP minr llows for th hotl ooking to our ritrrily oftn, ut t lst on for th trip osts r lult. 10 Conlusion Pross mining n sn s th missing link twn t mining n tritionl mol-rivn BPM. Th sptulr growth of vnt t is n importnt nlr for pross nlysis s on rl osrvtions rthr thn hn-m mols only. W hv ppli ProM in ovr 100 orgniztions rnging from muniiplitis n hospitls to finnil institutions n mnufturrs of high-th systms. This illustrts th ppliility of th thniqus sri in this rtil. Pross mining n us to ignos th tul prosss. This is vlul us in mny orgniztions most stkholrs lk orrt, ojtiv, n urt viw on importnt oprtionl prosss. Howvr, pross mining is not limit to th pross isovry thniqus mntion in this rtil (s for xmpl [2]). Pross mining n lso us to improv th isovr prosss. Conformn hking n us for uiting n omplin. By rplying th vnt log on pross mol it is possil to quntify n visuliz vitions. Similr thniqus n us to tt ottlnks n uil pritiv mols. Givn th ppliility of pross mining, w nourg th rr to simply pply th thniqus isuss. Th vnt t n to onut suh xprimnts n foun in ny non-trivil orgniztion. Th frly vill opn-sour pross mining tool ProM n ownlo from n supports ll of th pross mining thniqus mntion. In this rtil w mphsiz tht four qulity imnsions fitnss, simpliity, prision, n gnrliztion n to ln [2]. Morovr, w zoom in on rgion-s pprohs. As shown, onvntionl stt-s rgions n lngug-s rgions fous on fitnss n prision, whil nglting simpliity n gnrliztion. Fortuntly, it is possil to moify ths thniqus to lso l with th othr two qulity imnsions. Stt-s rgions n us for pross isovry tsks provi tht th right strtion is us whn onstruting th trnsition systm. Lngug-s rgions n mpp onto n ILP prolm whr th trgt funtion n itionl onstrints r us to otin simpl n mor gnrl mol. Dspit th ppliility of pross mining thr r mny intrsting hllngs; ths illustrt tht pross mining is young isiplin. As isuss, it is fr from trivil to onstrut pross mol s on vnt logs tht r inomplt n noisy. Unfortuntly, thr r still rsrhrs n tool vnors tht ssum logs to omplt n fr of nois. Although huristi mining, gnti mining, n fuzzy mining provi s-hrn pross isovry th- 48

49 niqus, mny improvmnts r n to onstrut truly intuitiv mols tht r l to xplin th most likly/ommon hvior. Anothr hllng is to l with vr-growing tsts, i.., it is not unommon to hv vnt logs with millions of ss, illions of vnts, n thousns of tivitis [44]. In som ss it is impossil to stor ll vnts n pross mols n to isovr onth-fly. In othr ss, thr is n to istriut pross mining prolms ovr multipl omputrs. As isuss in [3] this n on in vrious wys. Thrfor, thr r mny intrsting prolms for rsrhrs with kgroun in Ptri nts n gr to nlyz prosss s on rl vnt t rthr thn unrlisti toy mols. Aknowlgmnts. Th uthors woul lik to thnk ll th popl tht ontriut to th vlopmnt of ProM ( Rfrns 1. W.M.P. vn r Alst. Th Applition of Ptri Nts to Workflow Mngmnt. Th Journl of Ciruits, Systms n Computrs, 8(1):21 66, W.M.P. vn r Alst. Pross Mining: Disovry, Conformn n Enhnmnt of Businss Prosss. Springr-Vrlg, Brlin, W.M.P. vn r Alst. Distriut Pross Disovry n Conformn Chking. In J. Lr n A. Zismn, itors, Intrntionl Confrn on Funmntl Approhs to Softwr Enginring (FASE 2012), volum 7212 of Ltur Nots in Computr Sin, pgs Springr-Vrlg, Brlin, W.M.P. vn r Alst, A. Arinsyh, n B. vn Dongn. Rplying History on Pross Mols for Conformn Chking n Prformn Anlysis. WIREs Dt Mining n Knowlg Disovry, 2(2): , W.M.P. vn r Alst, B.F. vn Dongn, C.W. Günthr, R.S. Mns, A.K. Alvs Miros, A. Rozint, V. Ruin, M. Song, H.M.W. Vrk, n A.J.M.M. Wijtrs. ProM 4.0: Comprhnsiv Support for Rl Pross Anlysis. In J. Klijn n A. Ykovlv, itors, Applition n Thory of Ptri Nts n Othr Mols of Conurrny (ICATPN 2007), volum 4546 of Ltur Nots in Computr Sin, pgs Springr-Vrlg, Brlin, W.M.P. vn r Alst, B.F. vn Dongn, J. Hrst, L. Mrustr, G. Shimm, n A.J.M.M. Wijtrs. Workflow Mining: A Survy of Issus n Approhs. Dt n Knowlg Enginring, 47(2): , W.M.P. vn r Alst, H.A. Rijrs, A.J.M.M. Wijtrs, B.F. vn Dongn, A.K. Alvs Miros, M. Song, n H.M.W. Vrk. Businss Pross Mining: An Inustril Applition. Informtion Systms, 32(5): , W.M.P. vn r Alst, V. Ruin, H.M.W. Vrk, B.F. vn Dongn, E. Kinlr, n C.W. Günthr. Pross Mining: A Two-Stp Approh to Bln Btwn Unrfitting n Ovrfitting. Softwr n Systms Moling, 9(1):87 111, W.M.P. vn r Alst, A.J.M.M. Wijtrs, n L. Mrustr. Workflow Mining: Disovring Pross Mols from Evnt Logs. IEEE Trnstions on Knowlg n Dt Enginring, 16(9): , R. Agrwl, D. Gunopulos, n F. Lymnn. Mining Pross Mols from Workflow Logs. In Sixth Intrntionl Confrn on Extning Dts Thnology, 49

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