Discovering Block-Structured Process Models from Incomplete Event Logs

Transcription

1 Disovrin Blok-Strutur Pross Mols rom Inomplt Evnt Los Snr J.J. Lmns, Dirk Fhln, n Wil M.P. vn r Alst Einhovn Univrsity o Thnoloy, th Nthrlns {s.j.j.lmns,.hln, w.m.p.v..lst}@tu.nl Astrt On o th min hllns in pross minin is to isovr pross mol sriin osrv hviour in th st possil mnnr. Sin vnt los only ontin xmpl hviour n on nnot ssum to hv sn ll possil pross xutions, pross isovry thniqus n to l to hnl inompltnss. In this ppr, w stuy th ts o suh inomplt los on pross isovry. W nlys th impt o inompltnss o los on hviourl rltions, whih r strtions otn us y pross isovry thniqus. W introu proilisti hviourl rltions tht r lss snsitiv to inompltnss, n xploit ths rltions to provi mor roust pross isovry lorithm. W prov this lorithm to l to risovr mol o th oriinl systm. Furthrmor, w show in xprimnts tht our pproh vn risovrs mols rom inomplt vnt los tht r muh smllr thn rquir y othr pross isovry lorithms. Kywors: pross isovry, lok-strutur pross mols, risovrility, pross trs Introution Ornistions nowys ollt n stor onsirl mounts o vnt t. For instn, worklow mnmnt systms lo uit trils, n ntrpris rsour plnnin systms stor trnstion los. From ths vnt los, pross minin ims to xtrt inormtion, suh s usinss pross mols, soil ntworks, ottlnks n omplin with rultions []. In this ppr w ous on th most hllnin prolm: isovrin pross mol rom xmpl trs. Lrnin pross mol (.., Ptri nt) rom xmpl trs in n vnt lo, ll pross isovry, is on o th irst n most hllnin stps o pross minin. Two prolms o los r prtiulrly hllnin or pross isovry lorithms. First, th lo my ontin inrqunt hviour, whih ors lorithms to ithr xlu this hviour or rturn omplit, unrl mols sriin ll hviour [8]. Son, th lo miht ontin insuiint inormtion to isovr pross mol tht rprsnts th systm wll: th lo miht inomplt. Inompltnss ors lorithms to ithr xlu th missin hviour, thry ruin th s yt unsn hviour th mol n prou, or inlu th missin, unknown, hviour, thry risk ussin wron. In this ppr, w ous on hnlin inomplt los.

2 2 Snr J.J. Lmns, Dirk Fhln, n Wil M.P. vn r Alst systm prous vnt lo pross isovry mol risovrility msur itnss, prision, nrlistion Fiur : Tritionl mol qulity ssssmnt (itnss, prision, nrlistion) n risovrility. A notion losly rlt to inompltnss is risovrility. I pross isovry thniqu hs risovrility, it is l to isovr mols tht hv th sm lnu s th rl-li pross y whih lo ws prou [3,5,7]. Fiur shows th ontxt o pross isovry, risovrility, n how isovr mols n vlut. Tritionlly, mols r vlut with rspt to th vnt lo: itnss msurs wht prt o th vnt lo is sri y th mol, prision is hih whn th mol os not llow too muh hviour tht ws not in th vnt lo, n nrlistion is hih whn th mol llows mor hviour thn just th hviour in th vnt lo. Althouh itnss, prision, n nrlistion r intuitivly lr, irnt orml initions r possil [3,24,25]. Msurin th qulity o isovr mol with rspt to its vnt lo miht usul, ut whthr th st mol or th vnt lo is th st mol or th systm is not ptur y ths msurs. Thror, to ompr pross isovry thniqus it is usul to stuy risovrility, s tht ivs thortil ouns to whn mol is lnu-quivlnt to its rl-li systm. Risovrility is usully provn usin ssumptions out oth lo n mol [3,5,7]. A mol must rom rtin lss, n lo must ontin suiint inormtion. Th notion wht inormtion is suiint, ompltnss, pns on th isovry lorithm. Gnrlly, th stronst ompltnss notion is lnu-ompltnss, i.., h tr throuh th pross must prsnt in th lo. Th wkst ompltnss notion is tht h pross stp must our t lst on in th lo: tivity-ompltnss [7]. Typilly, risovrility n only urnt i th lo is omplt. In this ppr, w invstit th prolm o risovrin pross mols rom vnt los, in prtiulr rom inomplt vnt los. Anothr sirl proprty o pross isovry lorithms is tht thy rturn simpl n soun mols. A simpl mol ns w onstruts to xprss its hviour, n soun mol is mol r o loks n othr nomlis. Whil n unsoun mol miht usul, it is, or instn, not wll suit or omplin vlution n ottlnk nlysis [8]. Thror, in this ppr w will ous on pross trs: strt hirrhil lok-strutur Ptri nts tht r urnt to soun. Th Inutiv Minr (IM) [7] is n xmpl o n lorithm tht isovrs pross trs n or whih risovrility hs n provn. IM pplis ivi-n-onqur pproh: it prtitions th tivitis, slts th most importnt pross onstrut, splits th lo n rurss until s s is nountr. In this ppr, w pt IM to hnl inomplt los: w kp th ivi-n-onqur pproh, ut rpl th tivity prtition stp y n optimistion prolm. W intro-

3 Disovrin Pross Mols rom Inomplt Evnt Los 3 u rltions twn tivitis, stimt proilitis o ths rltions n srh or prtition o tivitis tht is optiml with rspt to ths proilitis. Risovrility is provn ssumin lo ompltnss n suiintly lr lo; w iv lowr oun or suiiny. In th rminr o this ppr, w irst xplor rlt work. In Stion 3, w introu los, Ptri nts, pross trs n ompltnss notions. W stuy ts o inompltnss on hviourl rltions in Stion 4 n sri hviourl proilistions. Stion 5 sris th lorithm, Stion 6 provs risovrility or suiintly lr los, n illustrts how inompltnss is hnl y th nw pproh, ompr with othr pprohs. Stion 7 onlus th ppr. 2 Rlt Work Ptri nt synthsis ims to uil n quivlnt Ptri nt rom trnsition systm or lnu. Rion thory, tht hrtriss pls in Ptri nt, ws introu in [5], n svrl synthsis mthos wr propos, or instn in [,2,6,2]. Pross isovry irs rom Ptri nt synthsis in th ssumption rrin ompltnss. Synthsis ssums tht th omplt lnu o th systm is sri in som orm. For pross isovry w nnot ssum th lo to lnu-omplt, s typilly only rtion o th possil hviour n osrv in th vnt lo, mkin lnu-ompltnss otn impossil or insil. For xmpl, th lnu o mol with loop in it ontins ininitly mny trs, n th lnu o mol sriin th prlll xution o 0 tivitis ontins t lst 0! = irnt trs []. In ontrst, typil lo only ontins rtion o tht. Mny pross isovry thniqus hv n propos. For instn, tr trnsition systm hs n onstrut rom th lo, stt-s rion minr thniqus onstrut Ptri nt y olin rions o stts into pls [4,30]. Typilly, stts rion thniqus provi risovrility urnts [0], ut hv prolms lin with inompltnss (onurrny is only isovr i suiint/ll intrlvins r prsnt). Pross trs, or lok struturs in nrl, hv n us in pross isovry, oth insi th sop o Ptri nts [8,2,22], s outsi [26,27] th sop o Ptri nts. Thy provi nturl, strutur, wll-in wy o sriin prosss tht r otn sily trnsltl to Ptri nts. Th pross tr ormlisms us in [8,7,8] urnt sounnss s wll. Pross tr isovry thniqus hv lso n propos or. For instn, th pproh us y [28] onstruts pross tr rom lo y numrtin ll trs, tr whih th pross tr is simplii. Th Evolutionry Tr Minr (ETM) [8] uss nti pproh to isovr pross tr, i.., rnom popultion is mutt until rtin stop ritrion is mt, ut s it is str y lo-s mtris, itnss, prision, nrlistion n simpliity, n y its rnom ntur, it is unl to urnt risovrility. A nturl strty whn usin lok struturs is to pply ivi-n-onqur strty, whih hs n ppli to pross isovry in or instn [9,38,7,8]. In istinuishin lnus o lsss o Ptri nts, hviourl rltions hv prov thir worth [3], n thy hv n us to rin or orsn mols, i.., mkin thm

4 4 Snr J.J. Lmns, Dirk Fhln, n Wil M.P. vn r Alst mor or lss strt [29,6], to ompr pross mols [32], n to prorm pross isovry. For instn, th hviourl rltion us in th α lorithm [3], its rivtivs [35,36], n in [7,8], th irtly-ollows rltion, hols or two tivitis i on tivity n onsutivly ollow th othr tivity. A notion los to th irtly-ollows rltion is th vntully-ollows rltion, whih hols i on tivity n vntully ollow y nothr. This vntully-ollows rltion hs n us in th ontxt o pross isovry [28,3,8]. To th st o our knowl, th inlun o inompltnss hs not n systmtilly stui ithr on hviourl rltions or pross isovry. 3 Trs, Evnt Los, Ptri Nts n Compltnss Trs, Evnt Los. A tr is squn o tivitis: x,, y nots tr in whih irst ourr, thn in n inlly. Trs n ontnt: x, y xy x,, y. An vnt lo is multist o trs. For instn, rx,, y 3, x, y 2 s nots n vnt lo in whih th tr x,, y hppn 3 tims n x, y hppn 2 tims. Th untion st trnsorms multist into st: stplq tt t P Lu; th untion Σ ivs th lpht o th lo, i.., th tivitis us in it. Ptri Nts, Worklow Nts n Blok-Strutur Worklow Nts. A Ptri nt is iprtit irt rph o intronnt pls n trnsitions, in whih tokns on pls mol th systm stt n trnsitions mol pross stp xution. W us th stnr smntis o Ptri nts, s [23]. A worklow nt is Ptri nt hvin sinl input n sinl output pl, mollin th initil n inl stts o th systm. Morovr, h lmnt is on pth rom input to output [3]. A onsutiv squn o pross xutions tht rins th systm rom th initil stt into th inl stt, orrspons to tr. Th st o trs tht n prou y mol M, th lnu o M, is not y LpMq. A lok-strutur worklow nt is hirrhil worklow nt: it n ivi rursivly into worklow nts. An xmpl is shown in Fiur 2. Fiur 2: A lok-strutur worklow nt M E ; ill rions not th lokstrutur; pross tr Ñp p^p, q, q, pöpñp, q, q, qq orrspons to this nt. Pross Trs. A pross tr is n strt hirrhil rprsnttion o lokstrutur worklow nt. Th lvs o th tr r tivitis, rprsntin trnsitions. Th nos o th tr, oprtors, sri how thir hilrn r omin. This ppr uss our oprtors:, Ñ, ^ n ö. Th oprtor sris th xlusiv hoi twn its hilrn, Ñ th squntil omposition n ^ th prlll omposition. Th

5 Disovrin Pross Mols rom Inomplt Evnt Los 5 irst hil o ö tr is th loop oy, th non-irst hilrn r ro prts. For instn, öp, q is th omposition o tr o th oy, thn zro-or-mor tims tr rom ro prt n oy in: pq. Eh pross tr is sily trnsltl to soun worklow nt. For xmpl, Fiur 2 shows th lok-strutur worklow nt orrsponin to th pross tr M E Ñp p^p, q, q, pöpñp, q, q, qq. To in th smntis o pross trs, w ssum init st o tivitis Σ to ivn. Th lnu o n tivity is th xution o tht tivity ( pross stp). Th lnu o th silnt tivity τ ontins only th mpty tr: xutin τ s nothin to th lo. Th lnu o n oprtor is omintion o th lnus o its hilrn. In th ollowin inition, w us th stnr lnu nottions, n [20]. To hrtris ^, w us th shul prout S... S n, whih tks sts o trs rom S... S n n intrlvs thir trs t P S,..., t n P S n whil mintinin th prtil orr within h t i [7]. For instn, tx, yu tx, yu tx,,, y, x,,, y, x,,, y, x,,, y, x,,, y, x,,, yu Usin this nottion, w in th smntis o pross trs: Lpτq tx yu Lpq txyu or P Σ Lp pm,..., M n qq LpM q LpM 2 q... LpM n q LpÑpM,..., M n qq LpM q LpM 2 q LpM n q Lp^pM,..., M n qq LpM q LpM 2 q... LpM n q LpöpM,..., M n qq LpM qplp pm 2,..., M n qqlpm qq As n xmpl, th lnu o M E is p qppq q. Th untion Σ ivs th lpht o pross tr: ΣpM E q t,,,,,, u. W us À to not th st o oprtors, n otn ` to not pross tr oprtor: ` P À, À t, Ñ, ^, öu. Oviously, th orr o hilrn or n ^ n th orr o non-irst hilrn o ö is ritrry. Dirtly-Follows Rltion, Trnsitiv Closur n Grphs. Th irtly-ollows rltion ÞÑ hs n propos in [3] s n strtion o th hviour sri y mol or lo. From mol M, tk two tivitis n. I n ollow irtly in M, x...,,,...y P LpMq, thn ÞÑ M. For lo L, ÞÑ L is in similrly. For los, ÞÑ is monotoni: or pir o tivitis, ÞÑ nnot s to hol y in mor trs to th lo. A ÞÑ-pth is squn... k o tivitis suh tht k 2 i k i ÞÑ i. Th trnsitiv losur o ÞÑ is not y ÞÑ : or tivitis n, th rltion ÞÑ hols i thr xists ÞÑ-pth rom to. For mol M (rsp. lo L), StrtpMq W i not hoos th vntully-ollows/wk-orr rltion [8,3], s its ompltnss os not surviv lo splittin; Lmm os not hol or it.

6 6 Snr J.J. Lmns, Dirk Fhln, n Wil M.P. vn r Alst () ÞÑ-rph o M E () ÞÑ -rph o M E Fiur 3: Grphs o M E showin its irtly-ollows rltion ÞÑ n its trnsitiv losur ÞÑ. (rsp. StrtpLq) nots th strt tivitis, oun t th innin o tr, n EnpMq (rsp. EnpLq) th n tivitis, tht n onlu tr. Fiur 3 shows th irtly-ollows rltion o M E in rph nottion: irtlyollows rph. In this rph, n is rwn twn pir o tivitis px, yq i x ÞÑ y. Similrly, Fiur 3 shows th rph o ÞÑ or M E. Compltnss. Usin ths rltions, w introu two ompltnss notions, twn mol M n lo L: L is tivity omplt to M (L Σ M), i h tivity o M is prsnt in L t lst on: ΣpMq ΣpLq. L is irtly-ollows omplt to M (L ÞÑ M), i L is tivity-omplt to M, its irtly-ollows rltion is omplt, n oth strt n n tivitis r omplt: L Σ M, ÞÑ M ÞÑ L, StrtpMq StrtpLq n EnpMq EnpLq. Prtitions n Cuts. A prtition is istriution o n tivity st Σ into isjoint nonmpty susts Σ... Σ n, with n. A pir o tivitis p, q is prtition y prtition Σ,..., Σ n i n r not oth in th sm Σ i. A ut is prtition omin with pross tr oprtor. I pir o tivitis is prtition y th prtition in ut, th pir rosss th ut. For xmpl, pñ, tu, t,,,, uq is ut, tivity pir p, q rosss it n tivity pir p, q os not. Oviously, ny pross tr n rwrittn to lnu-quivlnt inry pross tr. Thror, without loss o nrlity, in this ppr w onsir only inry prtitions n uts. 4 Bhviourl Rltions In mny Ptri nt isovry lorithms, suh s [3,7,8,35,36], two-st pproh is us: irst, n strtion o th lo is riv, n son, rom this strtion mol is nrt. Th irtly-ollows rltion ÞÑ is otn us s hviourl rltion. In this stion, w irst sri th inlun o inompltnss on hviourl rltions. To this n, w lssiy pirs o tivitis inspir y th pross tr oprtors, y usin th ÞÑ rltion, tr whih w show th t inompltnss hs on

7 Disovrin Pross Mols rom Inomplt Evnt Los 7 this lssiition. Son, w introu proilisti vrsion o th lssiition tht hlps isovry thniqus l with inompltnss. Fiur 4 intiis nin ss or ÞÑ n ÞÑ twn two ivn tivitis n, n orniss ths ss in ltti. Th strutur o th ltti ollows rom ÞÑ n ÞÑ : n in th ltti orrspons to n xtnsion o th ÞÑ-rltion with on pir o tivitis. Th ltti yils iv rltions twn tivitis: th ommuttiv, ^ n ö i, n th non-ommuttiv Ñ n ö s. For instn, i ÞÑ n ÞÑ, thn Ñp, q, n i ÞÑ, ÞÑ, ÞÑ n ÞÑ, thn ö i p, q. Inormlly, p, q nots tht n r in n xlusiv hoi rltion, Ñp, q nots tht n r in squn rltion, n ^p, q nots tht n r in prlll rltion. Ths r similr to th α-rltions # W, Ñ W n W [3], ut t lolly inst o lolly. Both ö i p, q (loop inirt) n ö s p, q (loop sinl) not tht n r in loop rltion. I w omin thm into sinl rltion, this sinl rltion woul not iv suiint inormtion to prtition th tivitis. Usin th two rltions ö s n ö i s ivn y th ltti os, s will provn in Stion 6. W onsir th ommuttiv ss, or instn ^p, q n ^p, q, to quivlnt. + + (, ) (, ) s (, ) s (, ) (, ) (, ) (, ) + + i (, ) i(, ) + + (, ) + + (, ) (, ) Fiur 4: Ativity rltions; th rrows in ltti. Consir in Ptri nt M E shown in Fiur 2. Fiur 5 shows th tivity rltions o M E s rphs. Consir th lo L E rx,,,,,,,, y, x,,, y, x,,,,,, y, x, ys, whih w prou usin M E, ut L E is not irtly-ollows omplt to M E, s ÞÑ, ÞÑ, ÞÑ n ÞÑ hol in M E ut not in L E. Thror, p, q n p, q hol in L E ; Fiur 6 shows how n Ñ hn. For L E, pross isovry lorithm will rr n to xlusiv to, whil M E puts thm in squn, n thus unl to risovr M E. Th prolm illustrt with ths tivity rltions is inhrnt to ny pross isovry lorithm usin hviourl rltions; ny thniqu tht just uss hviourl rltions is likly unl to risovr mol i th hviourl rltions o th lo r not omplt.

8 8 Snr J.J. Lmns, Dirk Fhln, n Wil M.P. vn r Alst () -rph () Ñ-rph () ö i -rph () ö s -rph Fiur 5: Ativity rltions o M E () ^-rph s rphs. In th Ñ-rph irt is rwn rom to i Ñp, q hols, n similr or ö s. For, ^ n ö i, whih r ommuttiv, unirt s r rwn. () -rph () Ñ-rph Fiur 6: Two tivity rltions o L E s rphs. Noti tht Ñp, q n Ñp, q o not hol nymor, whil p, q n p, q now o. In th ollowin, w xplor wys to us inormtion rom inomplt los tht oul hlp to risovr th oriinl mol. Thror, in th rminr o this ppr w ssum tht th lo only ontins hviour rom its systm, i.., no nois is prsnt. First, som inormtion in th lo my llow us to onlu tht prtiulr rltion twn two tivitis nnot hol. For instn, i th lo ontins tr x, y, thn Ñp, q nnot hol. Ths violtions ollow rom Fiur 4: i th lo ontins inormtion tht rltion ` hols, thn ny wkr rltion, i.., not rhl rom `, nnot hol; on n only mov up in th ltti. Son, th i is, inst o usin inry hoi, to rthr us n stimt proility tht rltion hols, n i lso us in or instn th Huristis minr [33,34]. For h o th tivity rltions `, w introu proilisti vrsion p`: or tivitis n, p`p, q nots n rtiiilly stimt proility tht p, q r in `-rlt. Usin th proilisti vrsions mks it sir or thniqus to hnl inompltnss: in our xmpl, inst o inry hoi whthr Ñp, q n Ñp, q hol or not, w n ompr th proilitis p Ñ n p to mk hoi.

9 Disovrin Pross Mols rom Inomplt Evnt Los 9 Our hoi or ths p` is shown in Tl. Lt M mol n L lo o M. Thn, usin Fiur 4, w istinuish thr ss n hoos p`p, q s ollows: i `p, q hols in L, it mks sns to hoos p`p, q s th hihst o ll rltions or th pir p, q. Th mor rqunt tivitis n our in L, th mor onint w r tht `p, q hols or M, n not som stronr rltion. W hoos p`p, q s ollows: lt zp, q 2 not th vr numr o ourrns o n, thn w in p`p, q zp,q, yilin numr twn 2 n. i som rltion p, q, hols in L rom whih `p, q is unrhl, thn L ontins violtion to p`p, q, s w ssum L to nois-r n th hviourl rltions nnot s to hol y in osrvtions. Thror, w hoos p`p, q low: 0. i som rltion p, q hols in L rom whih `p, q n rh, i.., p`p, q oul hol y in mor trs to L, w hoos to ivi th rminin zp,q vnly ovr ll rminin ntris, suh tht th proilitis or h pir p, q sum up to. For xmpl, in s o L E, w t p p, q 0.6 n p Ñ p, q Tl : Our proposl or proilisti tivity rltions or tivitis n, with zp, q p q{2. Ntions o rltions r omitt rom th irst olumn. p p, q pñp, q pñp, q pö i p, q pö s p, q pö s p, q p^p, q 6 z (nothin) z ÞÑ 0 z 0 6 z ÞÑ 0 0 z 6 z 4 z 4 z ÞÑ ^ ÞÑ z ÞÑ z ÞÑ ^ ÞÑ z 6 z 4 z 4 z 4 z 4 z 3 z 3 z 0 2 z z 0 ÞÑ z 2 z ÞÑ ^ ÞÑ z ÞÑ ^ ÞÑ z 4 z 4 z 3 z 2 z z 2 z z In th nxt stion, w monstrt how to us ny systm o proilisti rltions in onrt lorithm; on oul in Tl irntly, s lon s or h pir o tivitis p, q n h rltion `, proility p`p, q is vill. In Stion 6, w will show tht our hois or p` l to orrt lorithm. W xpt tht th proos ivn in Stion 6 sily xtn to othr hois, ut th pris lss o ptl p` ns urthr rsrh. 5 Alorithm In this stion, w monstrt how th proilisti tivity rltions in in Stion 4 n us to isovr pross trs. W us ivi-n-onqur pproh n pt is rom IM [7] to introu nw isovry lorithm tht w ll Inutiv Minr - inompltnss (IMin). IMin

10 0 Snr J.J. Lmns, Dirk Fhln, n Wil M.P. vn r Alst onsists o thr stps tht r ppli rursivly: irst, th ÞÑ-rph o th lo n its trnsitiv losur ÞÑ r omput. Son, ut is hosn suh tht th rltions twn pirs rossin th ut hv th hihst proility orin to Tl. Th oprtor o th hosn ut is ror. Thir, usin th ut, th lo is split into sulo or h prt n on h sulo, IMin rurss. Th rursion ns whn s s, lo ontinin just sinl tivity, is nountr. Th hirrhy o ror oprtors is pross tr. W irst sri how to umult th proilitis o Tl to ssss th proility o ut. Son, w iv th lorithm, n xmpl n sription o our implmnttion. 5. Aumult Estimt Proilitis or Cuts Givn tivity rltion proilitis, suh s th ons in in Tl, w omput n umult proility or ut. Inormlly, or ` P t, Ñ, ^u, th umult proility p` is th vr p` ovr ll prtition pirs o tivitis. Dinition (umult proility or, Ñ n ^). Lt p`, Σ, Σ 2 q ut, with ` P t, Ñ, ^u. Thn p`pσ, Σ 2 q nots th umult proility o : PΣ p`pσ, Σ 2 q,pσ 2 p`p, q Σ Σ 2 Not tht Ñ,, or ^ ut rquirs ll pirs o tivitis to in th sm rltion suiintly otn. For loop ut, this is not suiint, s ll rossin pirs o tivitis in loop r in loop rltion (ö s Y ö i ). This loop rltion suis to sri th proility whthr ll tivitis r in in loop, ut on its own nnot istinuish th oy o loop rom its ro prts. For this, w hv to xpliitly pik th strt n n tivitis o th ro prts, suh tht ro strt tivity ollows oy n tivity, n ro n tivity is ollow y oy strt tivity. This irt sussion in loop is xprss in ö s. Hn, w otin th ollowin proility tht pö, Σ, Σ 2 q is loop ut or th hosn ro strt tivitis S 2 n loop ro n tivitis E 2 ; th strt n n tivitis o th oy r th strt n n tivitis o th lo. In th nxt stion, w show how S 2 n E 2 oul hosn. Dinition 2 (umult proility or ö). Lt pö, Σ, Σ 2 q ut, L lo, n S 2, E 2 sts o tivitis. W rt ovr thr prts: strt o ro prt, n o ro prt n vrythin ls: ro strt ro n inirt p,qpenplq S 2 pö s p, q p,qpe 2 StrtpLq pö s p, q PΣ,PΣ 2 p,qrpenplq S 2qYpE 2 StrtpLqq pö i p, q

11 Disovrin Pross Mols rom Inomplt Evnt Los Thn, pöpσ, Σ 2, S 2, E 2 q nots th umult proility o : pöpσ, Σ 2, S 2, E 2 q ro strt ro n inirt Σ Σ 2 In this inition, ro strt n ro n ptur th strnth o S 2 n E 2 rlly in th strt n n o th ro prts; inirt pturs th strnth tht ll othr pirs o tivitis tht ross Σ, Σ 2 r in loop rltion. For rility rsons, in th ollowin, w will omit th prmtrs S 2 n E Th Alorithm: Inutiv Minr - inompltnss (IMin) Nxt, w introu pross isovry lorithm tht uss th umult stimtions o initions n 2 in ivi-n-onqur pproh. For this, w introu prmtr tht inluns thrshol o ptl inompltnss. By ult, ut with hihst p` is to slt t ll tims. Howvr, low p` miht init tht th hviour in th lo nnot sri wll y lok-strutur Ptri nt. Thror, prmtr h is inlu: i thr is no ut with p` h, lowr mol öpτ,,..., m q with t,..., m u ΣpLq, llowin or ny tr ovr ΣpLq [7], is rturn. untion IMIN(L) i L rxy x s with P Σ n x thn rturn n i p`, Σ, Σ 2 q Ð ut o ΣpLq with hihst p`pσ, Σ 2 q; ` P À i p`pσ, Σ 2 q h thn L, L 2 Ð SPLITpL, p`, Σ, Σ 2 qq rturn `piminpl q, IMinpL 2 qq ls rturn öpτ,,..., m q whr t,..., m u ΣpLq n i n untion IMin ontins two non-trivil oprtions: sltin ut with hihst p` n th SPLIT untion. To slt ut with hihst p`, n in s o ö to hoos S 2 n E 2, our implmnttion uss n SMT-solvr. For mor tils o th trnsltion to SMT, pls rr to [9]. Th untion SPLIT splits lo L into sulos L n L 2, orin to ivn ut p`, Σ, Σ 2 q, y projtin th trs o L on Σ n Σ 2. For xmpl, SPLIT ppli to squn ut pñ, tu, tuq n tr x,,, y yils x, y n x, y. In ition, or ö, trs r split on th points whr th tr lvs Σ n ntrs Σ 2. For xmpl: SPLITprx,,,,, ys, pö, tu, tuqq yils rxy 2, x, ys n rxy 2 s. For mor til orml sription, pls rr to [7]. IMin hs n implmnt s prt o th Inutiv Minr plu-in o th ProM rmwork [4], vill t Exmpl 3. As n xmpl, onsir in th lo L E rx,,,,,,,, y, x,,, y, x,,,,,, y, x, ys. I IMin is ppli to L E with h 0, th irst

12 2 Snr J.J. Lmns, Dirk Fhln, n Wil M.P. vn r Alst most likly ut is pñ, t,, u, t,,, uq, with p Ñ o out Th hoi or Ñ is ror, n L E is split into rxy 2, x, y, x, ys n rx,,,,,,, y, x, y, x,,,, y, xys. Thn, IMin rurss on oth ths sulos. Fiur 7 shows th rursiv stps tht r tkn y IMin. Th inl rsult is Ñp p^p, q, q, pöpñp, q, q, qq, whih is qul to M E (, {,, }{,,, }) 0.74 (, {, }, {}) 0.74 (, {,, }, {}).00 (, {}, {}) 0.82 (, {, }, {}) 0.86 (, {}, {}) Fiur 7: Runnin xmpl: IMinpL E q. As irst stp, th ut with hihst p` is pñ, t,, u, t,,, uq, with p` Thn, IMin rurss s shown. 6 Risovrility In this stion, w rport on th risovrility o IMin. W irst sri lss o pross trs, or whih w thn prov tht IMin hs risovrility, ivn irtlyollows omplt lo in whih h tivity ours suiintly otn. Atr tht, w rport on xprimnts showin tht IMin mns to risovr ths pross trs, vn rom smllr los thn thos n y othr isovry lorithms. 6. Clss o Risovrl Pross Trs; Norml Form Th lss o pross trs C R or whih w will prov risovrility is s ollows: Dinition 4 (Clss C R ). Lt M pross tr. Thn M lons to C R i or h (su)tr M t ny position in M, it hols tht Th sutr is not silnt tivity: M τ I M `pm... M nq, with ` P À, thn no tivity pprs mor thn i j n ΣpM i q X ΣpM j q H I M öpm... M nq, thn M is rquir to hv isjoint strt n n tivitis: StrtpM q X EnpM q H In orr to prov lnu-risovrility, w us lnu-uniqu norml orm. Eh pross tr n onvrt into this norml orm usin th ollowin lnuprsrvin rution ruls. I no rul n ppli to tr, th tr is in lnuuniqu norml orm [7]. Not tht th orr o hilrn o n ^, n ro hilrn o ö, is ritrry.

13 Disovrin Pross Mols rom Inomplt Evnt Los 3 Dinition 5 (Norml Form). Lt M pross tr. Thn pplyin th ollowin rution ruls xhustivly on sutrs o M yils lnu-uniqu norml orm, in whih ` nots pross tr oprtor: `pm q Ñ M p, p 2 q, 3 q Ñ p, 2, 3 q Ñp, Ñp 2 q, 3 q Ñ Ñp, 2, 3 q ^p, ^p 2 q, 3 q Ñ ^p, 2, 3 q öpöpm, q, 2 q Ñ öpm,, 2 q öpm,, p 2 q, 3 q Ñ öpm,, 2, 3 q Usin this norml orm, IMin n isovr th lnu o ny tr y srhin or only inry uts. For xmpl, i M ÑpM, M 2, M 3 q, it is prtly in to isovr ithr ÑpM, ÑpM 2, M 3 qq or ÑpÑpM, M 2 q, M 3 q. W sy tht ut onorms to mol M in norml orm i sltin os not isl isovry o tr quivlnt to M: Dinition 6. Lt p`, Σ, Σ 2 q ut n lt M `pm... M n q mol in norml orm. Thn onorms to M i no ΣpM i q is i D j ΣpM i q Σ j. Morovr, or non-ommuttiv oprtors, orr must mintin. 6.2 Forml Risovrility Th min thorm stts tht ny mol rom lss C R n risovr rom irtly-ollows omplt lo whos tivitis our t lst rtin numr o tims. Lt lstplq not th numr o tims th lst ourrin tivity ours in lo L. Thorm 7. Assum mol M tht is o lss C R. Thn thr xists k P N suh tht or ll los L with stplq LpMq, L ÞÑ M n lstplq k, it hols tht LpIMinpLqq LpM q. W prov th thorm s ollows: w irst show tht IMin slts th orrt root oprtor (Lmm 9), thn tht IMin slts prtition orrsponin to M (Lmm 0), n inlly tht lo splittin yils orrt irtly-ollows omplt sulos (Lmm ), on whih IMin rurss. In ths lmms, w will us vry nrl proprty o prtitions: ny two prtitions shr t lst on pir o tivitis tht rosss oth prtitions. Lmm 8. Tk two inry prtitions Σ, Σ 2 n Σ, Σ 2, oth o th sm Σ. Thn thr is pir o tivitis tht is prtition y oth prtitions. Proo. Towrs ontrition, ssum thr is no pir tht is prtition y oth Σ, Σ 2 n Σ, Σ 2. Tk, P Σ, 2 P Σ 2. Pirs p, 2 q n p, 2 q r prtition y Σ, Σ 2, so y ssumption thy r not prtition y Σ, Σ 2. Thus, thr is n i 2 suh tht,, 2 P Σ i. As w pos no rstritions on n, or som i 2, Σ Σ i. By similr rsonin, Σ 2 Σ i, so Σ Y Σ 2 Σ i. Thror, Σ i Σ n hn Σ, Σ 2 is not prtition. [\

14 4 Snr J.J. Lmns, Dirk Fhln, n Wil M.P. vn r Alst In th ollowin lmm, w prov tht or h lo or whih lst is suiintly lr, IMin slts th orrt root oprtor. Lmm 9. Assum ru mol M `pm,..., M n q. Thn thr xists k P N suh tht or ll los L with stplq LpMq, L ÞÑ M n lstplq k, it hols tht IMinpLq slts `. Proo. IMin slts inry uts, whil M n hv n ritrry numr o hilrn. Without loss o nrlity, ssum tht p`, Σ, Σ 2 q is inry ut onormin to M. Lt p, Σ, Σ2q n ritrry ut o M, with `. W n to prov tht p`pσ, Σ 2 q p pσ, Σ2q, whih w o y omputin lowr oun or p`pσ, Σ 2 q n n uppr oun or p pσ, Σ2q n thn omprin ths two ouns. Apply s istintion on whthr ` ö: Cs ` ö. W strt with th lowr oun or p`pσ, Σ 2 q. By Dinition, PΣ p`pσ, Σ 2 q,pσ 2 p`p, q Σ Σ 2 By smntis o pross trs, Fiur 4, stplq LpMq n L ÞÑ M, or h tivity pir p, q tht rosss, `p, q hols. For h suh pir, w hos p`p, q zp,q (not tht this woul n qulity, sv or p^p, q, whih is ). Thus, p`pσ, Σ 2 q PΣ,PΣ 2 zp,q Σ Σ 2 For ll n, zp, q 2 minp, q lstplq. Thus, p`pσ, Σ 2 q lstplq () Nxt, w prov n uppr oun or p pσ, Σ 2q. By Dinition, PΣ,PΣ 2 p p, q Σ Σ 2 p pσ, Σ 2q Lt pu, vq pir prtition y oth Σ, Σ 2 n Σ, Σ 2. By Lmm 8, suh pir xists. For ll othr p, q pu, vq, it hols tht p p, q (usin nottion it y ominin ö i n ö s ), n thr r Σ Σ 2 o thos pirs. p Σ Σ2 q p pu, vq Σ Σ 2 p pσ, Σ2q As pu, vq rosss, `pu, vq hols. Thn y insption o Tl, p pu, vq zpu,vq. Din y to Σ Σ 2. py q zpu,vq y p pσ, Σ 2q

15 Disovrin Pross Mols rom Inomplt Evnt Los 5 From zp, q 2 ollows tht zpu,vq 2. Thus, py q 2 y Usin th two ouns () n (2), w n to prov tht p pσ, Σ 2q (2) lstplq py q 2 y (3) Not tht y is t most tσpmq{2u rσpmq{2s, whih llows us to hoos k suh tht k 2y. By initil ssumption lstplq k, n thror (3) hols. Hn, p`pσ, Σ 2 q p pσ, Σ 2q. Cs ` ö. Usin rsonin similr to th ` ö s, w riv (). W irtly rus (2) to rriv t (3) n onlu tht p`pσ, Σ 2 q p pσ, Σ 2q. Thus, p`pσ, Σ 2 q p pσ, Σ 2q hols or ll `. As IMin slts th ut with hihst p`, IMin slts `. [\ Nxt, w prov tht or lo L, i lstplq is suiintly lr, thn IMin will slt prtition onormin to M. Lmm 0. Assum mol M `pm,..., M n q in norml orm. Lt p`, Σ, Σ 2 q ut onormin to M, n lt p`, Σ, Σ 2q ut not onormin to M. Thn thr xists k P N suh tht or ll los L with stplq LpMq, L ÞÑ M n lstplq k, hols tht p`pσ, Σ 2 q p`pσ, Σ 2q. Th proo strty or this lmm is similr to th proo o Lmm 9: w prov tht t lst on mislssii tivity pir pu, vq ontriuts to th vr p`pσ, Σ 2q. Pls rr to [9] or til proo. As lst lmm, w show tht lo splittin prous orrt n irtly-ollows omplt sulos. Lmm. Assum mol M in norml orm n lo L suh tht stplq LpMq n L ÞÑ M. Lt p`, Σ, Σ 2 q ut orrsponin to M, n lt L, L 2 th rsult o SPLITpL, q. Thn, thr xist pross trs M n M 2, suh tht Σ ΣpM q, Σ 2 ΣpM 2 q, th norml orm o `pm, M 2 q is M, stpl q LpM q, L ÞÑ M, stpl 2 q LpM 2 q n L 2 ÞÑ M 2. For this lmm, w us tht M n onvrt into inry tr y usin th rution ruls o Dinition 5 rvrs. As onorms to M, it is possil to onvrt M to `pm, M 2 q suh tht Σ ΣpM q n Σ 2 ΣpM 2 q. Th strty or th rminin prt o th proo is to show or h oprtor tht SPLIT rturns sulos L n L 2 tht r vli or M n M 2 (@i : stpl i q LpM i q). W thn prov tht L n L 2 r irtly-ollows omplt to M n M 2 (@i : L i ÞÑ M i ). Pls rr to [9] or tils. Usin ths lmms, w n prov risovrility or suiintly lr los. Proo (o Thorm 7). W prov th thorm y inution on mol sizs, in ΣpM q.

16 6 Snr J.J. Lmns, Dirk Fhln, n Wil M.P. vn r Alst Bs s: M. As stplq LpMq, L rxy x s or som x. By o insption, LpIMinpLqq LpM q. Inution stp: ssum tht th thorm hols or ll mols smllr thn M. By Lmm 9 n 0, IMin slts ut p`, Σ, Σ 2 q onormin to M. Nxt SPLIT(L, ) rturns n L n L 2. By Lmm, thr xists pross trs M, M 2 suh tht Lp`pM, M 2 qq LpMq. By Lmm, stpl q LpM q, L ÞÑ M, stpl 2 q LpM 2 q n L 2 ÞÑ M 2. As o th inution hypothsis n th t tht L n L 2 r suiintly lr y onstrution, Lp`pIMinpL q, IMinpL 2 qqq Lp`pM, M 2 qq LpMq. Bus IMinpLq `piminpl q, IMinpL 2 qq, thr xists k P N suh tht i lstplq k, thn LpIMinpLqq LpMq. [\ In th proos o Lmms 9 n 0, w hos k 2 tσpmq{2u rσpmq{2s. This ivs n uppr oun or th minimum lstplq rquir, n hrtristion o suiiny: Corollry 2. A oun or k n lstplq s us in Thorm 7 is trmin y th siz o th lpht: lstplq k 2 t ΣpM q {2u r ΣpM q {2s. Lst, th unsolv qustion rminin is whthr irtly-ollows ompltnss o lo implis tht th lo is suiintly lr, n tht nrlis vrsion o Thorm 7 hols: Conjtur 3. Assum mol M n lo L suh tht stplq LpMq n L ÞÑ M. Thn LpIMinpLqq LpMq. Th xprimntl rsults rport in th rminr o this ppr support this onjtur. 6.3 Exprimntl Rsult In this stion, w show tht IMin n risovr mols rom smll los. In ition, w invstit how vrious pross isovry lorithms, inluin IMin, hnl inompltnss. Exprimnt. In th xprimnt, w im to nswr thr qustions: ) Cn IMin risovr th lnu o mols? 2) How os IMin hnl inomplt los? 3) How o othr lorithms hnl inomplt los? To nswr qustions n 2, w invstit how lr th lo o ivn mol M hs to to risovr th lnu o M, y nrtin los o vrious sizs n tryin to risovr M rom ths los. For qustion 3, w invstit how lr los n to or othr lorithms, suh tht in mor trs to th lo woul not hn th rsult o th lorithm. Stup. For nswrin qustions n 2, w nrt 25 rnom pross trs with 5 tivitis rom lss C R. For h tr M, 20 rnom, suiintly lr, irtlyollows omplt los wr nrt. For h lo L, w vrii tht LpMq ws risovr rom it: LpIMinpLqq LpMq. Thn w prorm inry srh on L to in th smllst sulo o L rom whih, in norml orm, M ws risovr.

17 Disovrin Pross Mols rom Inomplt Evnt Los 7 Ths sulos wr otin y rmovin trs rom L, n on h smllst sulo oun, w msur th numr o trs n ompltnss o ÞÑ. To nswr qustion 3, omprin IMin to othr lorithms, w us similr prour: or h isovry lorithm D, w us th sm rnomly nrt pross trs to in, or h tr, th smllst los L D suh tht in mor trs to L D woul lwys rturn mol D DpL D q (up to isomorphism). W ll th mol DpL D q or suh smllst lo L D top mol M T. For this xprimnt, w onsir th ollowin isovry lorithms: Inutiv Minr (IM) [7], Intr Linr Prormmin minr (ILP) [37], α-lorithm (α) [3], Rion minr (RM) [30,4] n lowr mol, ll plu-ins o th ProM rmwork [4]. Th lowr mol ws inlu s slin, s it will rh its top mol i L Σ M: it only pns on th prsn o tivitis in th lo. All minrs wr ppli usin thir ult sttins, n or IMin h ws st to 0. For oth prours, w xprimntlly osrv tht vnt los with 6000 trs wr irtly-ollows omplt n suiintly lr to risovr th oriinl mol (in s o IMin) or to in th top mol (or othr lorithms). Rsults. Tl 2 shows th rsults. For xmpl, IM on vr rquir 97% o th ÞÑ-pirs o th mol to prsnt in th lo to isovr its top mol M T. For som mols, th ILP implmnttion w us i not rturn n nswr. Avrs r ivn without ths mols n r mrk with prin *. Tl 2: Rsults o th xprimnts. Column 2: or how mny mols M ws its lnu risovr in M T, vr ovr los. Column 3: vr numr o trs in th smllst sulos. Column 4: vr rtio o ÞÑ-pirs prsnt in smllst sulos ompr to th mols M. minr LpMq LpM T q numr o trs ÞÑ-ompltnss α 0% ILP 2% * *0.980 RM 4% IM 00% IMin 00% Flowr 0% Fiur 8: Ptri nt rprsnttion o M F : ÑpöpÑp 0, p, 2 qq, 3, 4 q, öp^pñp 5, 6 q, Ñp 7, 8 q, Ñp 9, 0 qq, Ñp, 2 q, 3, 4 q On o th rnomly nrt mols is shown in Fiur 8. To illustrt hnlin o inompltnss, w us this mol to in th smllst sulo or whih IMin r-

18 8 Snr J.J. Lmns, Dirk Fhln, n Wil M.P. vn r Alst isovr M F, n ppli othr isovry lorithms to tht sulo. Th rsults r shown in Fiur () Exrpt o α; 0 unsoun. nnot ir; () Exrpt o RM; lls hv n rmov; lots o pls nssry to rprsnt prlllism () Exrpt o ILP; 0 n ir t ny () IM; lls hv n rmov; misss th ntrl tim. prlllism. Fiur 9: Mols rsultin rom isovry o smllst sulo o IMin. Disussion. Answrin qustion, whthr IMin n risovr th lnu o mols, or ll mols n los, IMin isovr th oriinl mol or lnu-quivlnt on, n vn i not rquir th lo to irtly-ollows omplt, whih supports Conjtur 3. IMin rquir on vr 87.5% o th ÞÑ-rltion pirs to prsnt in th lo to isovr its top mol. This susts tht IMin is l to hnl irtlyollows inomplt los, nswrin qustion 2. Th lowr mol provis slin: it isovrs mol s on th tivitis tht r prsnt in lo; no pross isovry thniqu n xpt to rh its top mol without ll tivitis in prsnt in th lo. For ll mols, IMin rquir wr or qully mny trs thn ny othr isovry lorithm, xpt or th lowr mol, to rh its top mol. Rmrkly, lso IM i not rquir th ÞÑ rltion to omplt t ll tims. A possil xplntion is tht lo splittin miht hlp t tims. For instn, ^p,, q oul risovr s ^p, ^p, qq. I lo lks ÞÑp, q, it oul introu urin lo splittin: y splittin x,, y with tu n t, u yils th tr x, y or whih ÞÑ hols, nlin th risovry o ^p, q. Fiur 9 illustrts how othr isovry lorithms hnl mols within th rprsnttionl is o IM n IMin, or whih IMin risovrs its lnu. It woul intrstin to s how ths lorithms prorm on pross trs not riv rom lss C R, n on nrl Ptri nts. 7 Conlusion In this ppr, w stui th ts o inompltnss on pross isovry. W nlys th impt o inompltnss o los on hviourl rltions. W introu

19 Disovrin Pross Mols rom Inomplt Evnt Los 9 proilisti hviourl rltions to mk thm mor stl whn lin with inompltnss, n in n lorithm s on ths proilisti rltions. This lorithm ws provn to l to risovr th lnu o mols, ivn suiintly lr irtly-ollows omplt los. Morovr, in xprimnts it ws shown to l to risovr th lnu o mols, vn whn ivn smll inomplt los, n to n lss inormtion in th lo to onvr thn othr pross isovry lorithms. An opn qustion rminin is whthr risovrility hols or IMin (Conjtur 3). Othr points o utur rsrh oul wht hrtriss ptl hois o proilisti tivity rltions (Tl ), (tht oul vn l to hnl nois), n, i irtly-ollows ompltnss is n uppr oun or risovrility, n i tivity-ompltnss is lowr oun or it, whthr ths ouns r tiht. Th xprimnts w onut sust tht thr is tihtr uppr oun thn irtly-ollows ompltnss. Rrns. vn r Alst, W.: Pross Minin: Disovry, Conormn n Enhnmnt o Businss Prosss. Sprinr (20) 2. vn r Alst, W., Buijs, J., vn Donn, B.: Towrs improvin th rprsnttionl is o pross minin. In: SIMPDA. Ltur Nots in Businss Inormtion Prossin, vol. 6, pp Sprinr (20) 3. vn r Alst, W., Wijtrs, A., Mrustr, L.: Worklow minin: Disovrin pross mols rom vnt los. IEEE Trns. Knowl. Dt En. 6(9), (2004) 4. Boul, E., Dronu, P.: Thory o Rions. In: Lturs on Ptri Nts I: Bsi Mols. vol. 49, pp (998) 5. Boul, E.: On th α-ronstrutiility o worklow nts. In: Ptri Nts 2. LNCS, vol. 7347, pp Sprinr (202) 6. Brnthum, R., Dsl, J., Musr, S., Lornz, R.: Synthsis o Ptri nts rom trm s rprsnttions o ininit prtil lnus. Funm. Inorm. 95(), (2009) 7. Bloom, S.L., Ésik, Z.: Fr shul lrs in lnu vritis. Thor. Comput. Si. 63(&2), (996) 8. Buijs, J., vn Donn, B., vn r Alst, W.: A nti lorithm or isovrin pross trs. In: IEEE Conrss on Evolutionry Computtion. pp. 8. IEEE (202) 9. Crmon, J.: Projtion pprohs to pross minin usin rion-s thniqus. Dt Minin n Knowl Disovry 24(), (202) 0. Cortll, J., Kishinvsky, M., Lvno, L., Ykovlv, A.: Drivin Ptri nts or init trnsition systms. IEEE Trns. Computrs 47(8), (998). Dronu, P.: Rion s synthsis o p/t-nts n its potntil pplitions. In: ICATPN. pp (2000) 2. Dronu, P.: Unoun Ptri nt synthsis. In: Lturs on Conurrny n Ptri Nts. LNCS, vol. 3098, pp Sprinr (2003) 3. D Wrt, J., D Bkr, M., Vnthinn, J., Bsns, B.: A multi-imnsionl qulity ssssmnt o stt-o-th-rt pross isovry lorithms usin rl-li vnt los. Inormtion Systms 37, (202) 4. vn Donn, B., Miros, A., Vrk, H., Wijtrs, A., vn r Alst, W.: Th prom rmwork: A nw r in pross minin tool support. ICATPN 3536, (2005) 5. Ehrnuht, A., Roznr, G.: Prtil (st) 2-struturs. At Inormti 27(4), (990)

20 20 Snr J.J. Lmns, Dirk Fhln, n Wil M.P. vn r Alst 6. Günthr, C., vn r Alst, W.: Fuzzy minin ptiv pross simpliition s on multi-prsptiv mtris. Businss Pross Mnmnt pp (2007) 7. Lmns, S., Fhln, D., vn r Alst, W.: Disovrin lok-strutur pross mols rom vnt los - onstrutiv pproh. In: Ptri Nts 203. LNCS, vol. 7927, pp Sprinr (203) 8. Lmns, S., Fhln, D., vn r Alst, W.: Disovrin lok-strutur pross mols rom vnt los ontinin inrqunt hviour. In: Businss Pross Mnmnt Workshops. Sprinr (203), to ppr 9. Lmns, S., Fhln, D., vn r Alst, W.: Disovrin lok-strutur pross mols rom inomplt vnt los. Th. Rp. BPM-4-05, Einhovn Univrsity o Thnoloy (Mrh 204) 20. Linz, P.: An introution to orml lnus n utomt. Jons & Brtltt Lrnin (20) 2. Lornz, R., Musr, S., Juhás, G.: How to synthsiz nts rom lnus: survy. In: Wintr Simultion Conrn. pp WSC (2007) 22. Polyvynyy, A., Vnhtlo, J., Völzr, H.: Simplii omputtion n nrliztion o th rin pross strutur tr. In: WS-FM 0. LNCS, vol. 655, pp Sprinr (200) 23. Risi, W., Shnupp, P., Muhnik, S.: Primr in Ptri Nt Dsin. Sprinr (992) 24. Rozint, A., Miros, A., Günthr, C., Wijtrs, A., vn r Alst, W.: Th n or pross minin vlution rmwork in rsrh n prti. In: Businss Pross Mnmnt Workshops. pp Sprinr (2008) 25. Rozint, A., Vloso, M., vn r Alst, W.: Evlutin th qulity o isovr pross mols. In: 2n Int. Workshop on th Inution o Pross Mols. pp (2008) 26. Shimm, G.: Gnri linr usinss pross molin. In: ER (Workshops). LNCS, vol. 92, pp Sprinr (2000) 27. Shimm, G.: Pross minr - tool or minin pross shms rom vnt-s t. In: JELIA. LNCS, vol. 2424, pp Sprinr (2002) 28. Shimm, G.: Minin most spii worklow mols rom vnt-s t. In: Businss Pross Mnmnt. LNCS, vol. 2678, pp Sprinr (2003) 29. Smirnov, S., Wilih, M., Mnlin, J.: Businss pross mol strtion s on synthsis rom wll-strutur hviorl proils. Int. J. Cooprtiv In. Syst. 2(), (202) 30. Solé, M., Crmon, J.: Pross minin rom sis o stt rions. In: Ptri Nts. LNCS, vol. 628, pp Sprinr (200) 3. Wilih, M., vn r Wr, J.: On proils n ootprints - rltionl smntis or Ptri nts. In: Ptri Nts. LNCS, vol. 7347, pp Sprinr (202) 32. Wilih, M., Polyvynyy, A., Mnlin, J., Wsk, M.: Cusl hviourl proils - iint omputtion, pplitions, n vlution. Funm. Inorm. 3(3-4), (20) 33. Wijtrs, A., vn r Alst, W., Miros, A.: Pross minin with th huristis minrlorithm. BETA Workin Ppr sris 66, Einhovn Univrsity o Thnoloy (2006) 34. Wijtrs, A., Riiro, J.: Flxil Huristis Minr. In: CIDM. pp IEEE (20) 35. Wn, L., vn r Alst, W., Wn, J., Sun, J.: Minin pross mols with non-r-hoi onstruts. Dt Minin n Knowl Disovry 5(2), (2007) 36. Wn, L., Wn, J., Sun, J.: Minin invisil tsks rom vnt los. Avns in Dt n W Mnmnt pp (2007) 37. vn r Wr, J., vn Donn, B., Hurkns, C., Srrnik, A.: Pross isovry usin intr linr prormmin. Funm. Inorm. 94(3-4), (2009) 38. Yzquiro-Hrrr, R., Silvrio-Cstro, R., Lzo-Cortés, M.: Su-pross isovry: Opportunitis or pross inostis. In: EIS o th Futur, pp Sprinr (203)