Monotone Precision and Recall Measures for Comparing Executions and Specifications of Dynamic Systems

Transcription

1 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER Monoton Prision n Rll Msurs for Compring Exutions n Spifitions of Dynmi Systms Artm Polyvynyy, Anrs Solti, Mtthis Wilih, Cluio Di Ciio, n Jn Mnling rxiv: v1 [s.fl] 18 D 2018 Astrt Th omprison of th hviours of softwr systms is n importnt onrn in softwr nginring rsrh. For xmpl, in th rs of spifition isovry n spifition mining, it is importnt to msur th onsistny twn olltion of xution trs n th progrm spifition tht ws utomtilly onstrut from ths trs. This prolm is lso tkl in pross mining, whr for lmost two s rsrhrs propos msurs to ssss th qulity of pross spifitions utomtilly isovr from xution logs of informtion systms. Though vrious msurs hv n propos, it ws rntly osrv tht non of thm fulfils ssntil proprtis, suh s monotoniity. To rss this rsrh prolm, w uil on th following osrvtion: If two hviours r not quivlnt, th xtnt of vition n quntifi y quotint of rtin spt of on hviour ovr th sm spt of th othr hviour. Howvr, thr is no systmti pproh for fining suh quotints n it is unlr whih spts shll onsir for mningful omprison of systms tht sri infinit hviours, whih is oftn th s for softwr systms. It is th ontriution of this ppr to introu frmwork to fin hviour quotints tht pply on systm s hviour is ptur y lngug ovr st of tions. W instntit th frmwork with msurs for th rinlity n ntropy s spifi spts of lngugs, thry hnling oth finit n infinit hviours. In ition, w prov importnt proprtis of ths quotints. W monstrt th pplition of th quotints to ptur prision n rll twn olltion of ror xutions of systm n systm spifition, i.., twn th ror n spifi hviours of systm. An xprimntl vlution of th quotints using our opn-sour implmnttion monstrts thir fsiility n inits tht thy nl monotoni ssssmnt. 1 INTRODUCTION Th nlysis of ynmi systms is fous of softwr nginring rsrh [1, 2], n othr rlt rs, for xmpl usinss pross mngmnt [3, 4], informtion systms [5, 6], soil sin [7, 8], n mngmnt sin [9]. Softwr nginring rsrh is primrily onrn with th nlysis of hviours ptur in softwr systms, progrm spifitions, n xution trs. This nlysis oftn tks th form of hviour omprison, with us ss rnging from spifition isovry [10, 11, 12, 13] n spifition mining [14, 15, 16, 17], through onformn of rquirmnts with spifitions [18], softwr volution [19], softwr tst ovrg [20, 21], n lk-ox softwr tsting [22, 23], to msurmnts of ury of th rvrsnginr spifitions [24, 25]. For xmpl, spifition isovry n spifition mining stuy wys to infr softwr spifitions from progrm xutions. Th qulity of suh infrn thniqus is oftn fin in trms of msurmnts of isrpnis twn th xution trs us s input n th rsulting progrm spifitions. Pross mining [26] hs intgrt ths prsptivs, s it rlts hviour of systm s spifi with th hviour A. Polyvynyy is with th Univrsity of Mlourn, Prkvill, VIC, 3010, Austrli. E-mil: s A. Solti, C. Di Ciio, n J. Mnling r with th Vinn Univrsity of Eonomis n Businss, Austri. Emil: {solti,iiio,mnling}@i.wu..t M. Wilih is with Humolt Univrsity of Brlin, Grmny. E-mil: mtthis.wilih@hu-rlin. ror uring th systm s xution, with pplitions in omputtionlly-intnsiv thory vlopmnt [27]. A ky hllng in th nlysis of ynmi systms is th finition of mningful msurs tht xprss th gr to whih iffrnt systm hviours r in lin with h othr. Thnilly, suh omprisons r formult in rltiv mnnr, fining quotint of som spt of on hviour ovr th sm spt of nothr hviour. For instn, th quotints of th hviours of systm t iffrnt points in tim rvl how th systm hs hng. In pross mining, in turn, th quotint of th hviour of systm s ror in log ovr th hviour s spifi n us to nlys th trustworthinss of th lttr. Yt, fining suh quotints is hllnging: A rnt ommntry on msurs in pross mining intifis st of intuitiv proprtis n shows tht non of th vill msurs fulfils thm [28]. W pproh th ov prolm s on th notion of forml lngug. This is suitl point of prtur us th squntil (stt-s) hviour of ynmi systm,.g., softwr systm or informtion systm, is oftn moll s stt mhin or n utomton [29, 30]. An tion rprsnts n tomi unit of work, whih, pning on th typ of systm, my, for xmpl, progrm instrution, W srvi ll or mnul tivity xut y humn gnt. Th hviour of systm, thrfor, n rprsnt y lngug tht fins st of wors ovr its tions. Thn, h wor is on possil xution (lso known s run, tr, squn, or pross) of th systm. Bhviourl omprison s on quotints of lngugs fs two mjor hllngs. First n formost, in orr to llow for rsonl intrprttion, quotints shll stisfy

2 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER ssntil proprtis. On suh proprty is monotoniity: Whn inrsing th mount of hviour in th numrtor of quotint whil lving th mount of hviour in th nomintor unhng, th quotint shll inrs s wll. Existing quotints s propos,.g., in th fil of pross mining [26] to ompr ror n spifi hviour, o not stisfy suh wll-motivt proprty [28, 31]. Th son hllng rlts to th finition of quotints in th prsn of systms tht sri infinit hviours, i.., th hviours tht onsist of infinitly mny wors. In tht s, quotints fin ovr stnr spts of lngugs, suh s thir rinlity, r not mningful for hviourl omprison. In pross mining, this issu hs n voi y using hviourl strtions tht ptur lngug y mns of pirwis rltions ovr its tions [32]. Yt, suh n strtion os not ptur th omplt lngug smntis of systm [33] n, thus, introus is into th hviourl omprison, rfr to Stion 7 for tils. In softwr nginring, this issu is voi y sustituting th hviour of progrm spifition with finit olltion of its simult xution trs [24, 25]. Still, ths pprohs suffr from th prolm of smpling th suitl finit portion of possily infinit hviour [34]. Aginst this kgroun, w rss th prolm of how to fin mningful quotints for hviourl omprison of finit n infinit lngugs? W nswr this qustion y fining msurs tht quntify th rltion twn th spifi n ror hviours. Conrtly, this rtil ontriuts: (1) A frmwork for th finition of hviourl quotints tht gurnt svrl sir proprtis. (2) Dfinitions of two quotints s instntitions of th frmwork tht r groun in th rinlity of lngug (for finit lngugs) n th ntropy of n utomton (for finit n infinit lngugs). (3) Applition of th propos quotints to fin monoton prision n rll msurs twn th hviour s ror in n xution log of systm n th hviour ptur in spifition of th systm. (4) A pulily vill implmnttion of th propos prision n rll quotints. (5) An vlution tht monstrts th fsiility of omputing th propos prision n rll quotints, omprs thm with th stt-of-th-rt msurs in pross mining, n shows tht, unlik th propos prision msur, ll th vlut prision msurs r not monoton. Th rminr of this rtil is strutur s follows. Stion 2 sris th kgroun of th rsrh prolm w rss. Stion 3 introus forml prliminris in trms of lngugs n utomt. Th frmwork for th finition of quotints is introu in Stion 4. This stion lso inlus two instntitions of th frmwork n isussion of forml proprtis of th quotints. In Stion 5, w prsnt notions of prision n rll for omprisons of olltion of ror systm xutions with systm spifition. Ths notions r vlut in sris of xprimnts using rl-worl t in Stion 6. Finlly, Stion 7 isusss our ontriutions in th light of rlt work, for w onlu th ppr in Stion 8. 2 BACKGROUND ON BEHAVIOURAL COMPARISON On th hviour of ynmi systms is ptur y lngugs ovr thir tions, insights into thir iffrns n ommonlitis r otin y ompring th rsptiv lngugs. This is ommonly on s on msur tht quntifis n spt of lngug, suh s its rinlity, i.., th numr of wors fin y th lngug. Bs thron, rtio of ths spts nls rltiv omprison of two lngugs n, thus, hviours of on or mor systms: On hviour is put into prsptiv (or normlis) rltiv to som s hviour. W rfr to suh rtio s lngug quotint: lngug quotint = msur(lngug 1) msur(lngug 2 ). For illustrtion purposs, onsir th snrio of usr logging into som pplition, whih is s on th tions list in Fig. 1, suh s rting login sssion or onuting th tul uthntition. Spifi rlistions of this snrio r givn s finit utomt in Figs. 1 n 1. Alit similr, th systms, S 1, S 2, n S 3, fin iffrnt lngug ovr th tions, not y L(S 1 ), L(S 2 ), n L(S 3 ), rsptivly. In ft, th lngugs r in sust rltion, L(S 3 ) L(S 2 ) L(S 1 ). Furthrmor, Fig. 1 pits thr logs, L 1, L 2, n L 3, whih rprsnt ror xutions of tul login prosss. Eh log L is multist of squns ovr tions n, thus, lso inus lngug L(L). Th lttr ontins ll wors tht our t lst on in th log. Th thr utomt my rprsnt (i) iffrnt systms, (ii) iffrnt vrsions of th sm systm, or (iii) systm spifitions n thir implmnttions. In ny s, it usful to quntify to whih xtnt th utomt sri th sm hviour this nswrs th qustion in how fr (i) iffrnt systms provi th sm funtionlity; (ii) th funtionlity of systm hs hng ovr svrl vrsions; n (iii) spifition hs n implmnt lry. Consiring lso th logs, w not tht similr qustions mrg in th fil of pross mining [26], whih trgts th nlysis of informtion systms s on ror xutions of pross. Givn spifition n log, pross mining strivs for quntifying th shr of ror hviour tht is in lin with th spifition (fitnss or rll of th log) or th shr of spifi hviour tht is tully ror (prision of th spifition). To rss th ov us ss, w my onsir th qustion y how muh on systm xtns th hviour of nothr systm? For systms S x n S y, suh tht L(S y ) L(S x ), w my nswr this qustion with quotint fin using lngug rinlity s msurmnt funtion: lngug xtnsion(s x, S y ) = L(S x) L(S y ). A slightly iffrnt wy to ssss th rltion twn ths systms, howvr, is th qustion of how muh of th hviour of on systm is ovr y nothr systm? To this n, st-lgri oprtions ovr lngugs my inorport in th finition of quotint, s in th following finition: lngug ovrg(s x, S y ) = L(S x) L(S y ) L(S x ). Th ov quotints of lngug xtnsion n ovrg provi stright-forwr mns for hviourl omprison of systms, spifitions of systms, n logs. Yt, thy r usful only if th ppli msurmnt funtion provis mningful mpping of lngug into numril omin. For th rinlity funtion us ov, w rgu tht this is th s solly for finit lngugs. For lngugs

3 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER τ Ations: C C 3 35 Ations: Ations: : Crt login sssion, : Ent r u sr n m 1 2,, ExCution 1: : Crt : login Crt sssion login sssion 3 I II f III (,,,,,,, ) A B C D E F G H : Ent r u sr n m : Entr psswor I II f III A f 4 Ations: : Ent r u sr n m I II f III A B, B D D : Crt login sssion : Entr : : psswor Entr r psswor u sr n m : Auth ntit I II f III A B ExDution 2: : Auth ntit : : Auth Entr ntit psswor : Rst login sssion 4 IV (,,,,,, ) J K L M : Rst : login Rst sssion login sssion IV IV E : Auth ntit E 10 f: Aort uthntition f: Aort f: uthntition Aort uthntition : Rst login sssion Systm IV s1 Systm E s2 Log l Systm sl of log l f: Aort uthntition () Ations () Systm s0 S 1 Systm () Systm s1 S 2 Systm s2() Systm Systm s2 3 S 3 F G L 1 = [,,,,,,,,,, ] L 2 = L 1 [,,,,,,, f,, H F, f, ] L 3 = [,,,,,,,,, 4, f,, f, ] G 5 () Logs H I Fig. 1: Exmplry systms n logs pturing login pross. I tht fin n infinit numr of wors, th numrtor or 3 PRELIMINARIES nomintor of quotint my om infinity. Lving This stion prsnts forml notions us to support th si th ovious finitionl issus, ny finition of isussions in th susqunt stions. vlu for suh quotint woul not only ritrry, ut woul lso rsult in singl vlu for ll infinit lngugs, rgrlss of thir hrtristis. 3.1 Multists, Squns, n Lngugs Consiring our xmpls, w my omput th lngug A multist, or g, is gnrliztion of st, i.., olltion xtnsion using rinlity s Ex ution msur Ex1: ution 1: for th logs L 1 (,,, (,,,,,,,, ),,, ) A n AB tht B C n ontin multipl instns of th sm lmnt. By L 2, pturing tht L(L 2 ) ontins Ex ution twi 1: s mny wors s CD DE EF FG GH HI I (,,,,,,, ) A B(A), B w not C th st of ll finit multists ovr som st A. L(L 1 ). Howvr, lngug xtnsion Ex ution Ex2: ution s 2: on rinlity is D E For som multist B B(A), B() nots th multipliity not mningful for ny pir (,, of, (,, Ex lngugs,, ution,, ),, 2:, ) of systms S 1, J JK KL LM M of lmnt S (,,,,,, ) F in B. For xmpl, B 1 = [], B 2 = [,, ], 2, n S 3, sin L(S 1 ) n L(S 2 ) r infinit. G H Log l Log l In th sm n B 3 Systm = [ Systm 2 sl of, ] log sl r lof log multists l ovr th st {, }. Multist vin, omputing th lngug ovrg oflog spifition l n log, to ssss th fitnss of th log or th prision of th spifition, is not mningful for th systms S 1 n S 2, n ny of th logs. Byon th hllng to op with infinit lngug, w not tht quotints shll stisfy prtiulr proprtis. As mntion, th lngugs of th thr utomt r in sust rltion, L(S 3 ) L(S 2 ) L(S 1 ). Ths susumption rltions, for instn, shll rflt in th rsptiv quotints of lngug xtnsion: A quotint fin ovr th smllst lngug L(S 3 ) n th lrgst lngug L(S 1 ) shoul yil smllr vlu thn quotint ovr L(S 3 ) n th son lrgst lngug L(S 2 ). Sin lngug L(S 1 ) ontins L(S 2 ) n is stritly lrgr, th itionl hviour shll lowr th vlu of th rsptiv rtio. Dsir proprtis of quotints suh s thos isuss ov trnslt into rquirmnts on th msurmnt funtions tht ptur prtiulr spt of lngugs. As w will isuss in th rminr, monotoniity of th msurmnt funtion n th xistn of suprmum tht ouns th msurmnt sp r of prtiulr rlvn in this ontxt. Th formr mns tht ing hviour to systm stritly inrss (or stritly rss) th msur, whrs th lttr implis tht spifi vlu is fin to th mpty hviour. Mny msurs for hviourl omprison propos in th litrtur nglt suh proprtis, rising ts on how to intrprt th otin rsults. In th omin of pross mining,.g., it ws rntly shown tht non of th xisting msurs to ssss th prision of spifition ginst log stisfis st of wll-motivt proprtis [28, 31]. Aginst this kgroun, th funmntl hllng of using quotints for hviourl omprison is to om up with frmwork for thir mningful finition. Tht is, th frmwork shll provi gurnts on th quotints to stisfy olltion of sirl proprtis. B 1 is mpty, i.., it ontins no lmnts, whrs B 2 () = 2 = B 3 (), B 2 () = 1 = B 3 (), n, hn, it hols tht B 2 = B 3. Th stnr st oprtions hv n xtn to l with multists s follows. If lmnt is mmr of multist B, this is not y B; othrwis, on writs / B. Th union of two multists C n D, not y C D, is th multist tht ontins ll lmnts of C n D suh tht th multipliity of n lmnt in th rsulting multist is qul to th sum of multipliitis of this lmnt in C n D. For xmpl, [] B 2 = [ 2, 2 ]. Also not tht L 2 in Fig. 1 is th union of L 1 n th multist of thr squns with two instns of squn, f, ; mor info on squns is provi low. Th iffrn of two multists C n D, not y C D, is th multist tht for h lmnt x C ontins mx (0, C(x) D(x)) ourrns of x. For xmpl, it hols tht B 3 B 2 = B 1, n B 3 [] = [, ]. Givn multist B B(A) ovr st A, y St(B) w rfr to th st tht ontins ll n only lmnts in B, i.., St(B) = { A B}. A squn is n orr list of lmnts. By σ = 1, 2,..., n A, w not squn ovr som st A of lngth n N 0, i A, i [1.. n], whr [j.. k] = {x N 0 j x k}, j, k N 0. 1 By σ = n, w not th lngth of th squn. By σ [i], i [1.. n], w rfr to th i-th lmnt of σ, i.., σ [i] = i. Givn squn σ n st K, y σ K, w not squn otin from σ y lting ll lmnts of σ tht r not mmrs of K without hnging th orr of th rmining lmnts. For xmpl, it hols tht,,,, {,} =,. Givn two squns σ n σ, y σ σ, w not th ontntion of σ n σ, i.., th squn otin y ppning σ to th n of σ. For xmpl,,,, =,,,,, whr is th mpty squn. For two sts of squns X 1 n X 2 ovr 1 By N n N 0, w not th st of ll nturl numrs xluing n inluing zro, rsptivly.

4 : Ent r u sr n m I II III A B D : Entr psswor : Auth ntit : Rst login sssion IV E PRE-PRINT SUBMITTED TO ARXIV.ORG. f: Aort uthntition DECEMBER A, X 1 X 2 = {σ A σ 1 X 1 σ 2 X 2 σ = σ 1 σ 2 }. By suffix (σ, i), i N, w not th suffix of σ strting from n inluing position i. For xmpl, L 1 in Fig. 1 ontins squns σ 1 =,,, n σ 2 =,,,,,,. It hols tht suffix (σ 1, 3) =, n suffix (σ 2, 6) =,. If σ = 1, 2,..., n A is squn ovr A n f is funtion ovr A, thn f(σ) = f( 1 ), f( 2 ),..., f( n ). Similrly, if A A, thn f(a ) = {f() A }. An lpht is ny nonmpty finit st. Th lmnts of n lpht r its lls, or symols. By Ξ, w not univrs of symols. For xmpl, Fig. 1 spifis lpht Σ = {,,,,, f}. A wor ovr n lpht is finit squn of symols from th lpht. Th wor of lngth zro is ll th mpty wor n is not y ɛ. A (forml) lngug ovr n lpht Σ is st of wors ovr Σ. 3.2 Finit Automt W l with ommon notion of finit utomton [35]. Dfinition 3.1 (Nontrministi finit utomton). A nontrministi finit utomton (NFA) is 5-tupl (Q, Λ, δ, q 0, A), whr Q is finit nonmpty st of stts, Λ Ξ is st of lls, suh tht Q n Ξ r isjoint, δ Q (Λ {τ}) (Q) is th trnsition funtion, whr τ Ξ is spil ll suh tht τ / Q Λ, q 0 Q is th strt stt, n A Q is th st of pt stts. 2 An NFA inus st of omputtions. Dfinition 3.2 (Computtion). A omputtion of n NFA (Q, Λ, δ, q 0, A) is ithr th mpty wor or wor s = s 1, s 2,..., s n, n N, whr vry s i is mmr of Λ {τ}, i [1..n], n thr xists squn of stts q = q 0, q 1,..., q n, whr vry q j is mmr of th st of stts Q, j [1..n], suh tht for vry k [1..n] it hols tht δ(q k 1, s k ) = q k. W sy tht s ls to q n. By onvntion, th mpty wor lwys ls to th strt stt. An NFA B = (Q, Λ, δ, q 0, A) pts wor s iff s is omputtion of B tht ls to n pt stt q of B, i.., q A. Dfinition 3.3 (Lngug of n NFA). Th lngug of n NFA B = (Q, Λ, δ, q 0, A), is not y L(B), n is th st of wors tht B pts, i.., L(B) = {s Λ r (Λ {τ}) (B pts r) (s = r Λ )}. W sy tht B rognizs L(B). In n NFA, th trnsition funtion tks stt n ll to prou th st of possil nxt stts, whil in DFA, th trnsition funtion tks stt n ll n prous th nxt stt. Dfinition 3.4 (Dtrministi finit utomton). A trministi finit utomton (DFA) is n NFA (Q, Λ, δ, q 0, A) with th proprty tht for vry stt q Q n for vry ll s Λ {τ} it hols tht δ(q, s) 1. An NFA (Q, Λ, δ, q 0, A) is rgoi if its unrlying grph is strongly irruil, i.., for ll (x, y) Q Q thr xists squn of stts q 1,..., q n Q, n N, for whih it hols tht for vry k [1.. n 1] thr xists λ Λ {τ} suh tht δ(q k, λ) = q k+1, q 1 = x, n q n = y. 2 Givn st A, y (A), w not th powrst of A. F G Fig. 2: DFA S 4 tht rognizs th lngug of S 2 in Fig. 1. A lngug L Ξ is rgulr iff it is th lngug of n NFA. A lngug L Ξ is irruil if, givn two wors w 1, w 2 L, thr xists wor w Ξ suh tht th ontntion w 1 w w 2 is in L. A rgulr lngug L is irruil iff it is th lngug of n rgoi NFA [36]. Ex ution 1: An NFA B = (Q, Λ, (,,,,,,, ) A δ, q 0, B A) is τ-fr iff for ll q Q it C D E F G H hols tht δ(q, τ) =. Givn n NFA B, on n lwys onstrut τ-fr DFA B tht rognizs th lngug of B [37]. Ex ution 2: (,,,,,, ) J K L M In wht follows, w only onsir τ-fr DFAs. Th utomton inlog Fig. l 1 is τ-fr NFA S 2 = (Q, Λ, Systm δ, q 0 sl, A), of log fin l y stts Q = {A, B, C, D, E}, lls Λ = {,,,, }, trnsition funtion δ = {((A, ), {B}), ((B, ), {C, D}), ((C, ), {B}), ((D, ), {E}), ((E, ), {A})}, strt stt q 0 = A, n pt stts A = {A}. Fig. 2 shows τ-fr DFA tht rognizs th sm lngug s th utomton in Fig A FRAMEWORK FOR LANGUAGE QUOTIENTS This stion introus frmwork for hviourl omprison of systms using lngug quotints. As til in Stion 4.1, lngug quotint is fin s on msurmnt funtion ovr lngugs of systms. In Stion 4.2, w monstrt tht th propos quotints stisfy sirl proprtis for hviourl omprison of systms. Finlly, in Stion 4.3, w propos two msurmnt funtions for instntition of lngug quotints, on s on th rinlity of lngug n on s on th topologil ntropy of n utomton. 4.1 Frmwork Dfinition Bhviourl omprison of systms is usully rri out s on spts of thir lngugs. An spt of lngug n ptur y msur m (Ξ ) R + 0, whih is (st) funtion from th st of ll lngugs ovr Ξ to nonngtiv rl numrs. 3 A msur n sujt to ths two onstrints: A msur n monotoni. A msur m is (stritly monotonilly) inrsing iff for ll U Ξ n V Ξ suh tht U V, it hols tht m(u) < m(v ). A msur n mp th infimum of its omin to th infimum of its oomin. In this lin, w fin tht msur m strts t zro iff m( ) = 0. W sy tht msur ovr lngugs is lngug msur iff it is inrsing n strts t zro. 4 A lngug quotint sts spts of lngugs into rltion s follows: 3 By R + 0, w not th st of ll non-ngtiv rl numrs. 4 A lngug msur stisfis th proprtis of non-ngtivity n null mpty st ut is not rquir to ountl or finit itiv [38], s ths proprtis r not xploit in th susqunt nlysis of this rtil. If lngug msur m is ountly itiv, (Ξ, (Ξ ), m) fins msur sp, s it is stui in mthmtil nlysis. I H I 5 5

5 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER L 1 L 2 L 3 L 1 L 2 L 1 L 3 < L 1 L 2 L 3 L 1 L 3 < L 2 L 3 numrtor nomintor Fig. 3: Shmti rprsnttion of: Lmm 4.2 (top row) n Lmm 4.3 (ottom row). Dfinition 4.1 (Lngug quotint). Givn two lngugs L 1 n L 2, n lngug msur m, th lngug quotint of L 1 ovr L 2 inu y m is th frtion of th msur of L 1 ovr th msur of L 2, i.., quotint m (L 1, L 2 ) = m(l 1) m(l 2 ). Nomn st omn, lngug quotint is fin ovr lngugs, not systms. Th rtionl hin this formlistion is tht th frmwork of lngug quotints, on instntit with spifi msur, my ppli for ivrs lgri oprtions; xmpls inlu quotints tht r fin ovr th intrstion, union, or iffrn of lngugs, rfr to th notion of lngug ovrg in Stion 1 for illustrtion. In Stion 5.1, w provi furthr xmpls of quotints ovr th intrstion of lngugs tht r usful in th ontxt of pross mining. 4.2 Proprtis of Lngug Quotints Lngug quotints njoy usful proprtis tht rst on th proprtis of lngug msur. On n ompr quotints with th sm numrtors s follows. Lmm 4.2 (Fix numrtor quotints). If L 1, L 2, L 3 Ξ r lngugs suh tht L 1 is nonmpty, L 1 L 2, n L 2 L 3, thn it hols tht quotint m (L 1, L 3 ) < quotint m (L 1, L 2 ), whr m is lngug msur. Th sttmnt of th lmm is shown shmtilly in Fig. 3 (top row). If L 2 n L 3 r lngugs of two systms tht xtn th hviour of thir systm tht rognizs lngug L 1, thn, using th quotints, on n onlu tht th systm tht rognizs L 3 xtns th hviour of th systm tht rognizs L 1 mor thn os th systm tht rognizs L 2. Th iffrn twn th xtnsion hviours is ptur y quotint m (L 1, L 3 ) quotint m (L 1, L 2 ). Th mning of th iffrn pns on th mning of lngug msur m us to instntit th quotints. If m msurs th rinlity of lngug, thn th iffrn stns for th frtion of th hviour with whih L 3 xtns L 1 mor thn os L 2. Morovr, lngug quotints with th sm nomintors n ompr s low. Lmm 4.3 (Fix nomintor quotints). If L 1, L 2, L 3 Ξ r lngugs suh tht L 1 L 2 n L 2 L 3, thn it hols tht quotint m (L 1, L 3 ) < quotint m (L 2, L 3 ), whr m is lngug msur. Th sttmnt of th lmm is visuliz shmtilly in Fig. 3 (ottom row). For xmpl, if L 3 is lngug of spifition of systm, n L 1 n L 2 r lngugs of its two implmnttions, thn, s on th quotints, on n onlu tht th implmnttion tht rognizs L 2 is mor omplt thn th implmnttion tht rognizs L 1. In othr wors, L 2 hs ttr ovrg of th spifition thn L 1. Th xtnt to whih th implmnttion tht rognizs L 2 is mor omplt n quntifi y quotint m (L 2, L 3 ) quotint m (L 1, L 3 ). Th mning of th iffrn, gin, pns on th mning of lngug msur m us to instntit th quotints. Proofs of Lmmt 4.2 n 4.3 r in Appnix A. If on fixs th numrtor, lik in th s of ompring th mounts to whih vrious systms xtn givn hviour, th quotints r oun low. Corollry 4.4 (Fix numrtor quotints). If L 1, L 2 Ξ r lngugs suh tht L 1 L 2, thn it hols tht quotint m (L 1, Ξ ) < quotint m (L 1, L 2 ), whr m is lngug msur. Corollry 4.4 follows immitly from Lmm 4.2, s it hols tht L 1 L 2 n L 2 Ξ. Rntly, in pross mining, svrl proprtis tht prision n rll msurs for ssssing th qulity of pross spifition isovr from log shoul fulfil wr propos [28, 31]. As prision n rll in pross mining r fin s hviourl quotints, rfr to Stion 5 n [31], thy njoy th proprtis stt in this stion. Consquntly, on n sily vrify tht prision n rll tht r fin s hviourl quotints stisfy th proprtis propos in [28, 31]. For xmpl, Propositions 5 n 8 in [31] follow immitly from Lmm 4.2 n th ft tht lngug msur is trministi, rfr to Stion 4.1, whil Propositions 3 n 9 in [31] follow immitly from Lmm 4.3 n th finition of lngug msur. 4.3 Frmwork Instntitions This stion proposs two lngug quotints, s instntitions of Dfinition 4.1 using spifi msurmnt funtions. Thus, ths quotints njoy ll th proprtis propos in Stion 4.2. Th first quotint is s on th rinlity of lngug, whrs th othr on is groun in th notion of topologil ntropy. Crinlity quotint. As lngug L is st of wors, its rinlity, not y L, is proprty tht n srv s th sis for hviourl omprison. Clrly, rinlity is lngug msur, i.., it is inrsing n strts t zro. By fining lngug quotint s on this msur, w otin th rinlity quotint: Dfinition 4.5 (Crinlity quotint). Th rinlity quotint of lngug L 1 ovr lngug L 2 is th frtion of th rinlity of L 1 ovr th rinlity of L 2, i.., quotint r (L 1, L 2 ) = L 1 L. 2 Th rinlity quotint pturs th rtio of th sizs of two lngugs. It is wll-fin only for L 2. Not tht this is finitionl issu tht my rss xpliitly (.g., fining quotint (L 1, L 2 ) = 0 if L 2 = ). A mor

6 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER svr prolm is th omputtion of th quotint for infinit lngugs. Givn n lpht, finit y finition, rgulr lngug my fin ountly infinit st of wors [39]. For xmpl, th rinlity of n irruil rgulr lngug is infinity. Agin, on my rss th rsulting finitionl issus xpliitly,.g., y opting tht onstnt ivi y infinity is qul to zro n tht infinity ivi y infinity is qul to on. Howvr, ny suh onvntion is not usful for hviourl omprison in th ontxt of rgulr lngugs. For instn, th lngug xtnsion n lngug ovrg, s Stion 2, woul qul to on for ny pir of rgoi utomt, suh s thos givn in Fig. 1 n Fig. 1. W thus onlu tht rinlity quotints provi suitl mns for hviourl omprison solly for finit lngugs. Eignvlu quotint. To otin lngug quotints tht r usful for ompring infinit lngugs, w instntit thm with msur s on th topologil ntropy. Intuitivly, th topologil ntropy of lngug pturs th inrs in vriility of th wors of th lngug s thir lngth gos to infinity. Givn lngug L, lt C n (L), n N 0, th st of ll th wors in L of lngth n, i.., C n (L) = {x L x = n}. Thn, th topologil ntropy of L is fin s follows, rfr to [36, 40] for tils: nt(l) = lim n sup log C n(l) n. Topologil ntropy hrtriss th omplxity of lngug n is losly rlt to th proprtis of th DFAs tht rogniz this lngug. In prtiulr, th topologil ntropy of n utomton is qul to th topologil ntropy of th lngug tht it rogniss [36]. Tht is, for DFA B = (Q, Λ, δ, q 0, A), with C n (B), n N 0, s th st of ll th wors in L(B) of lngth n, i.., C n (B) = {x L(B) x = n}, it hols tht: nt(l(b)) = nt(b) = lim n sup log C n(b) n. Th topologol ntropy of DFA, n thus of its lngug, is furthr rlt to th strutur of th utomton. Blow, w shll l with squr non-ngtiv mtris G = {g ij }, i, j [1.. n], n N, i.., g ij 0 for ll i, j [1.. n]. Th jny mtrix of DFA (Q, Λ, δ, q 0, A), whr Q = {q 0, q 1,..., q n }, n N 0, is squr mtrix G = {g ij }, i, j [1.. Q ], suh tht g ij = {((q i, λ), q j ) δ λ Λ}, for ll i, j [1.. Q ]. 5 Th topologil ntropy of DFA B, i.., nt(b), is givn y th logrithm of th Prron-Fronius ignvlu of its jny mtrix, whih is uniqu lrgst rl ignvlu of th jny mtrix of B [36]. Not tht n jny mtrix of n rgoi DFA B hs n ignvlu r suh tht r is rl, r > 0, n r λ for ny ignvlu λ of th jny mtrix of B, rfr to Thorm 1.5 in [41]. Th rltion twn th ntropy of lngug n th ntropy of n rgoi DFA rognising this lngug, s outlin ov, is importnt for omputtionl rsons, s it provis us with strightforwr pproh to omput th ntropy of lngug, vi th Prron-Fronius thory. Importntly, topologil ntropy is monotoni msur ovr lngugs. Lt x, y Ξ two wors. If thr xist u, v Ξ suh tht x = u y v, thn y is su-wor of x, not y y x. Lt L Ξ lngug n lt K nonmpty st of wors (or su-wors) of L. By L K, w 5 Rll from Stion 3.2 tht w only onsir τ-fr DFAs Fig. 4: DFA S 5. not th st {x L y K (y x)}, i.., th lngug otin from L y foriing ll th wors in K. Thorm 4.6 (Monotoniity of ntropy, Thorm 1 in [36]). If L Ξ is n irruil rgulr lngug n K is nonmpty st of su-wors of som wors in L, thn it hols tht nt(l K ) < nt(l). Bus of Thorm 4.6, topologil ntropy ovr irruil rgulr lngugs is n inrsing msur; not tht L K L. Howvr, it os not strt t zro. In, nt( ) is unfin, us th ignvlu of zro mtrix is qul to zro; hr, w ssum tht th jny mtrix of n rgoi DFA tht inus th mpty lngug is th zro squr mtrix of orr on. By ig(l), whr L is n irruil rgulr lngug, w not th Prron-Fronius ignvlu of th jny mtrix of n rgoi DFA tht rognizs L, or th ignvlu msur of L. W lso sy tht ig(l) is th ignvlu of L. Corollry 4.7 (Eignvlu msur). Th ignvlu msur ovr irruil rgulr lngugs is lngug msur, i.., it is inrsing n strts t zro. Corollry 4.7 stms from Thorm 4.6 n th fts tht (i) th logrithm is stritly inrsing n (ii) th ignvlu of zro mtrix is qul to zro. Bus of Corollry 4.7, on n fin this lngug quotint: Dfinition 4.8 (Eignvlu quotint). Givn two irruil rgulr lngugs L 1 n L 2, th ignvlu quotint of L 1 ovr L 2 is th frtion of th ignvlu of L 1 ovr th ignvlu of L 2, i.., quotint ig (L 1, L 2 ) = ig(l 1 ) ig(l 2 ). As n xmpl, onsir utomt S 1, S 4, n S 5 in Fig. 1, Fig. 2, n Fig. 4. Ths thr utomt r rgoi n, thus, lngugs L(S 1 ), L(S 4 ), L(S 5 ) r irruil. Morovr, it hols tht L(S 5 ) L(S 4 ) n L(S 4 ) L(S 1 ). Not tht quotint ig (L(S 4 ), L(S 1 )) = 0.539, quotint ig (L(S 5 ), L(S 1 )) = 0.513, quotint ig (L(S 5 ), L(S 4 )) = 0.952, n quotint ig (L(S 5 ), L(X )) = 0.242, whr X is th st of symols {,,,, }. In, it hols tht: (i) quotint ig (L(S 5 ), L(S 1 )) < quotint ig (L(S 5 ), L(S 4 )) (rfr to Lmm 4.2), (ii) quotint ig (L(S 5 ), L(S 1 )) < quotint ig (L(S 4 ), L(S 1 )) (rfr to Lmm 4.3), n (iii) quotint ig (L(S 5 ), X ) < quotint ig (L(S 5 ), L(S 4 )) (rfr to Corollry 4.4). To show tht quotint ig (L(S 5 ), L(S 4 )) in quls to 0.952, Fig. 5 n Fig. 5 show jny mtris of S 4 n S 5, rsptivly. Not tht th Prron-Fronius ignvlu of th mtrix in Fig. 5 is , whil th Prron-Fronius ignvlu of th mtrix in Fig. 5 is Eignvlu quotints r fin ovr irruil rgulr lngugs. In th nxt stion, w show how on n us lngug msur ovr irruil rgulr lngugs to 5 6

7 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER F G H I F G H I () Fig. 5: Ajny mtris of DFAs in () Fig. 2 n () Fig. 4. inu lngug msur n, thus, lngug quotints ovr ritrry rgulr lngugs. 5 PRECISION AND RECALL Lngug quotints provi gnrl mns for hviourl omprison. To monstrt th us of th quotints, this stion proposs n isusss thir pplition in pross mining [26]. On of th prolms stui in pross mining is th prolm of pross isovry. Givn log of ror xutions of systm, isovry thniqu onstruts spifition of th systm tht rprsnts th hviour ptur in th log. As systm my xut sm squns of tions multipl tims, its log is multist of wors tht no th xutions. Dfinition 5.1 (Log). A log is finit multist ovr lngug. An lmnt of log is tr, whrs n lmnt of tr is n vnt of th tr. Givn log L, L(L) = St(L) is th lngug of L. For xmpl, logs L 1, L 2, n L 3, list in Fig. 1 ontin two, fiv, n thr trs, rsptivly. Not tht L 2 ontins tr, f, twi, whih nots tht this squn of tions ws ror in th log two tims. Th qulity of th gnrt pross spifition is typilly vlut using prision, fitnss ( spifi typ of rll), simpliity, n gnrliztion [26]. Nxt, w us th frmwork of hviourl quotints to fin prision n rll of spifitions w.r.t. logs (Stion 5.1) n instntit it s on th short-iruit lngug msur (Stion 5.2). Finlly, w monstrt tht our prision n rll quotints stisfy importnt rquirmnts for prision n rll msurs (Stion 5.3). 5.1 Dfinition of Prision n Rll This stion proposs two quotints for ompring hviours ptur in log n DFA, nmly prision n rll of DFA w.r.t. log. Ths quotints r inspir y th prision n rll msurs tht hv prov to usful in informtion rtrivl, inry lssifition, n pttrn rognition. Th prision n rll msurs propos hr n us to msur prision n fitnss, rsptivly, of spifitions isovr from logs. In informtion rtrivl, givn st of rlvnt oumnts n st of rtriv oumnts, prision is th frtion of rlvnt rtriv oumnts ovr th rtriv oumnts. Givn log n DFA, w propos to msur how prisly DFA (spifition) sris log s th frtion of xutions ror in th log n spifi in th DFA ovr () ll th xutions (of whih thr n infinitly mny) spifi in th DFA. Dfinition 5.2 (Prision of DFA w.r.t. log). Givn log L n DFA B, th prision of B w.r.t. L inu y lngug msur m is not y prision m (B, L) n is th lngug quotint inu y m of th intrstion of th lngugs of B n L ovr th lngug of B, i.., prision m (B, L) = quotint m (L(B) L(L), L(B)). Prision is th rtio of th msur of trs of th log tht r lso omputtions of th DFA (spifi n ror hviour) to th msur of ll th omputtions of th DFA (spifi hviour). For xmpl, th prision of utomton S 3 in Fig. 1 w.r.t. log L 2 in Fig. 1 inu y th rinlity of lngug is omput s follows: prision r (S 3, L 2 ) = L(S 3 ) L(L 2 ) L(S 3 = 1 ) 2 ; th lngugs of S 3 n L 2 shr on wor, σ =,,,, whil th lngug of S 3 hs two wors: σ n,,,,. In informtion rtrivl, givn st of rlvnt oumnts n st of rtriv oumnts, rll is th frtion of rlvnt rtriv oumnts ovr th rlvnt oumnts. Givn log n DFA, w propos to msur how wll th DFA pturs th hviour of th log s th frtion of xutions ror in th log n spifi in th DFA ovr ll th hviour ror in th log. Dfinition 5.3 (Rll of DFA w.r.t. log). Givn log L n DFA B, th rll of B w.r.t. L inu y lngug msur m is not y rll m (B, L) n is th lngug quotint inu y m of th intrstion of th lngugs of B n L ovr th lngug of L, i.., rll m (B, L) = quotint m (L(B) L(L), L(L)). Rll is thrfor th rtio of th msur of trs of th log tht r lso omputtions of th DFA (spifi n ror hviour) to th msur of th trs of th log (ror hviour). For xmpl, th rll of utomton S 3 in Fig. 1 w.r.t. log L 2 in Fig. 1 inu y th rinlity of lngug is omput s follows: rll r (S 3, L 2 ) = L(S 3) L(L 2 ) L(L 2 = 1 ) 4. This rsult is sy to vrify y hking tht th lngug of L 2 onsists of four wors. 5.2 Short-iruit Lngug Msur Th notions of fitnss n rll of DFA w.r.t. log, rfr to Stion 5.1 for tils, tk lngug msur s prmtr. Th lngug of log is finit. If th lngug of th DFA is lso finit, on n instntit th prision n rll with th rinlity of lngug, similr s propos in Dfinition 4.5 n xmplifi in Stion 5.1. To ount for irruil rgulr lngugs, on n instntit th quotints with th ignvlu msur, rfr to Stion 4.3 for tils. Unfortuntly, th lngug of log is not irruil. Morovr, th lngug of DFA is not gurnt to irruil. To ovrom ths limittions, in this stion, w propos short-iruit msur ovr lngugs. Lt L 1 n L 2 two lngugs. Th ontntion of L 1 with L 2 is not y L 1 L 2 n is fin y {w 1 w 2 Ξ w 1 L 1 w 2 L 2 }.

8 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER V VI VII V VI VII Fig. 6: Th jny mtrix of th utomton in Fig. 7. Dfinition 5.4 (Short-iruit msur). A short-iruit msur ovr lngugs ovr lpht Ψ Ξ inu y msur ovr lngugs m (Ξ ) R + 0 is th (st) funtion m (Ψ ) R + 0 fin y m (L) = m((l { χ }) L), whr L is lngug ovr Ψ, i.., L Ψ, n χ Ξ Ψ is shortiruit symol. Blow, w monstrt tht short-iruit msur ovr rgulr lngugs inu y lngug msur ovr irruil lngugs is lngug msur, rfr to Proposition 5.8. Hn, th quotints inu y suh short-iruit msurs njoy ll th proprtis propos in Stion 4.2. If short-iruit msur is inu y msur tht strts t zro, thn it lso strts t zro. Lmm 5.5 (Short-iruit msur strts t zro). If m is msur ovr lngugs ovr lpht Φ Ξ tht strts t zro, thn short-iruit msur m ovr lngugs ovr lpht Ψ Φ strts t zro. Morovr, if short-iruit msur is inu y n inrsing msur, thn it is lso inrsing. Lmm 5.6 (Short-iruit msur is inrsing). If m is n inrsing msur ovr lngugs ovr lpht Φ Ξ, thn short-iruit msur m ovr lngugs ovr lpht Ψ Φ is inrsing. Proofs of Lmmt 5.5 n 5.6 r in Appnix A. If m is lngug msur, thn m is s wll. Corollry 5.7 (Short-iruit msur). If m is lngug msur ovr lngugs ovr lpht Φ Ξ, thn short-iruit msur m ovr lngugs ovr lpht Ψ Φ is lngug msur. Th proof of Corollry 5.7 follows immitly from th finition of lngug msur, rfr to Stion 4.1, n Lmmt 5.5 n 5.6. Nxt, w show tht lngug msur ovr irruil rgulr lngugs n us to inu lngug msur ovr rgulr lngugs. Proposition 5.8 (Short-iruit msur). If m is lngug msur ovr irruil rgulr lngugs ovr lpht Φ Ξ, thn m is lngug msur ovr rgulr lngugs ovr lpht Ψ Φ. Th rr n fin proof of Proposition 5.8 in Appnix A. For xmpl, ig is lngug msur ovr rgulr lngugs, whr ig is th ignvlu msur; not Corollry 4.7 n Proposition 5.8. Thrfor, w rommn using ig msur to inu vrious lngug quotints ovr ritrry rgulr lngugs,.g., th prision n rll quotints propos in Stion 5.1. Consir utomton S 1 in Fig. 1 n log L 3 in Fig. 1. Fig. 7 n Fig. 7 show utomt with lngugs (L(S 1 ) { χ }) L(S 1 ) n (L(L 3 ) { χ }) L(L 3 ), rsptivly, whrs Fig. 7 shows n utomton with lngug ((L(S 1 ) L(L 3 )) { χ }) (L(S 1 ) L(L 3 )). It is sy to s tht givn τ-fr DFA B = (Q, Λ, δ, q 0, A), it hols tht L(B ) = (L(B) { χ }) L(B), whr B = (Q, Λ {χ}, δ (A {q 0 }), q 0, A). Not tht th utomton in Fig. 7 ws otin from th utomton in Fig. 1 using this simpl trnsformtion n susqunt minimiztion [42]. Suh minimiztion is possil us ny utomton with th lngug of intrst, in this s (L(S 1 ) { χ }) L(S 1 ), suffis. Fig. 6 shows th jny mtrix of th utomton in Fig. 7. Th Prron-Fronius ignvlu of this mtrix is Th Prron-Fronius ignvlus of th jny mtris of utomt in Fig. 7 n Fig. 7 r n 1.128, rsptivly. Thus, it hols tht prision ig (S 1, L 3 ) = n rll ig (S 1, L 3 ) = TABLE 1: Prision n rll vlus. Automton Log Prision Rll S 1 L S 1 L S 1 L S 2 L S 2 L S 2 L S 3 L S 3 L S 3 L All th prision n rll vlus inu y ig for h of th thr DFAs in Figs. 1 1 w.r.t. vry log in Fig. 1 r list in Tl 1. Not tht ths vlus oy ll th proprtis isuss in Stion Proprtis of Prision n Rll Prision n rll, s fin in Stion 5.1, r lngug quotints n, thus, possss ll th proprtis isuss in Stion 4.2. This stion proposs furthr proprtis spifi for th prision n rll of DFA w.r.t. log. Firstly, prision n rll tk vlus from th intrvl tht ontins zro n on. Proposition 5.9 (Prision intrvl). Givn log L, DFA B, suh tht L(B), n lngug msur m ovr rgulr lngugs, it hols tht 0 prision m (B, L) 1. Proposition 5.9 follows from Dfinition 5.2 n th ft tht m is lngug msur ovr rgulr lngugs. In, it hols tht L(B) L(L) L(B) n, thus, m(l(b) L(L)) m(l(b)); not tht m is inrsing. Proposition 5.10 (Rll intrvl). Givn log L, suh tht L(L), DFA B, n lngug msur m ovr rgulr lngugs, it hols tht 0 rll m (B, L) 1. Proposition 5.10 hols us of Dfinition 5.3, n th fts tht L(B) L(L) L(L) n m is inrsing. Sonly, prision n rll qul to on whn th lngugs of th DFA n log r in ontinmnt rltions. Proposition 5.11 (Mximl prision). Givn log L, DFA B, suh tht L(B), n lngug msur m ovr rgulr lngugs, L(B) L(L) iff prision m (B, L) = 1. If L(B) L(L), thn it hols tht prision m (B, L) = m(l(b))/m(l(b)) = 1. Convrsly, if prision m (B, L) =

9 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER χ, V VI,f VII () 1 2 1, thn m(l(b) L(L)) = m(l(b)). Thn, it hols tht L(B) L(L). Proposition 5.12 (Mximl rll). Givn log L, suh tht L(L), DFA B, n lngug msur m ovr rgulr lngugs, L(L) L(B) iff rll m (B, L) = 1. Th proof of Proposition 5.12 follows th strutur of th proof of Proposition 5.11 ut swps th rols of th lngugs of B n L. Thirly, prision n rll oth qul to on iff th lngugs of th DFA n log r intil. Corollry 5.13 (Mximl prision n rll). Givn log L, L(L), DFA B, L(B), n lngug msur m ovr rgulr lngugs, L(B) = L(L) iff prision m (B, L) = 1 n rll m (B, L)=1. Corollry 5.13 follows immitly from Proposition 5.11 n Proposition Finlly, prision n rll r r qul to zro whn th lngugs of th DFA n log o not ovrlp. Proposition 5.14 (Miniml prision). Givn log L, DFA B, suh tht L(B), n lngug msur m ovr rgulr lngugs, L(B) L(L) = iff prision m (B, L) = 0. If L(B) L(L) =, thn prision m = m( )/m(b) = 0, s m strts t zro. Convrsly, if prision m (B, L) = 0, thn m(l(b) L(L)) = 0. Thn, L(B) L(L) = us m strts t zro n is inrsing. Proposition 5.15 (Miniml rll). Givn log L, suh tht L(L), DFA B, n lngug msur m ovr rgulr lngugs, L(B) L(L) = iff rll m (B, L) = 0. Th proof follows th strutur of th proof of Proposition 5.12 ut swps th rols of th lngugs of B n L. 6 EXPERIMENTAL EVALUATION This stion strts with prsnttion of our implmnttion of th prision n rll msurs inu y ig. Thn, w mpirilly show th thortil vntgs of th propos ignvlu-s msurs y ompring thm with xisting pprohs for msuring prision n rll in pross mining; Tls 2 n 3 list th pprohs onsir for our omprtiv vlution, til ltr on in Stion 7. Finlly, th stion loss with th rmrks on th slility of th propos msur. 6.1 Implmnttion By t(b), w not th trministi vrsion of B, i.., th DFA with th lngug lng(b). Givn B, t(b) lwys 3 4 f χ () f 7 Fig. 7: Thr DFAs. 8 A B C D E TABLE 2: Prision msurs us in this vlution. Short ll f χ () Full nm n rfrn vbhappropritnss Avn hviourl ppropritnss [43] lignmntprision Alignmnt-s prision [44] ntialignprision Anti-lignmnts prision [45] stalignprision Bst optiml-lignmnts prision [46] ngtivevntprision AGNEs spifiity [47] onalignprision On optiml-lignmnt prision [46] prisioneig This ppr prisionetc ETC prision [48] projtprision PCC prision [49] simplbhappropritnss Simpl hviourl ppropritnss [43] xists, rfr to [37], n n onstrut using th Rin- Sott powrst onstrution mtho [35], whih hs th worst-s tim omplxity of O(2 n ), whr n is th numr of stts in th NFA [50]. For h rgulr lngug, thr xists uniqu (up to isomorphism) DFA with minimum numr of stts [37] tht rognizs th lngug. Lt B DFA. By min(b ), w not th miniml vrsion of B, i.., th DFA with minimum numr of stts tht rognizs lng(b ). Thr xist svrl lgorithms tht givn DFA B onstrut min(b ). For xmpl, th worst-s tim omplxity of th Hoproft lgorithm [42] is O(nm log (m)), whr n is th numr of stts n m is th siz of th lpht. In nutshll, w pt s input ny nontrministi utomton B n omput its trministi vrsion B, whih w susquntly minimiz to gt min(b ). 6 Thn, w omput th lrgst ignvlu of min(b ). To nsur tht th ignvlu is omputl, w trnsform th DFA min(b ) y short-iruiting it, s sri in Stion 5.2. Short-iruiting is simpl O(n) oprtion, whr n is th numr of th (sink-)nos in min(b ). Aftr short-iruiting, w rt th jny mtrix of th rsulting utomton. Th jny mtrix srvs s input to xisting numril Fortrn-s mthos for trmining th lrgst ignvlu of mtrix [51]. Not tht typilly, w woul xpt tht th mtrix of lngug is rthr sprs. Thus, w r l to hnl vry lrg grphs on prsonl omputrs n omput thir ignvlus with th hlp of mmory-frinly sprs t struturs for mtris. Comprss olumn storg is typil formt for sprs mtris. W us th Jv lirry Mtrix Toolkit Jv 6 Th minimiztion stp n skipp, s th omputtion of th topologil ntropy os not rquir DFA to miniml, rfr to Stion 4.3. Whil prforming th slility vlution rport in Stion 6, w noti tht it is fstr to minimiz DFA n thn omput th Prron-Fronius ignvlu of its jny mtrix thn to omput th ignvlu of th originl DFA. A til stuy of this phnomnon, spit importnt, is out of sop of this ppr. Th omputtion tims rport in TABLE 5 inlu th minimiztion tims. F G H

10 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER TABLE 3: Fitnss msurs us in this vlution. Short ll Full nm n rfrn lignmntfitnss Alignmnt-s fitnss [52] ngtivevntrll AGNEs rll [47] toknbsfitnss Tokn-s fitnss [43] prsingmsur Continu prsing msur [53] projtrll PCC rll [49] proprcompltion Propr ompltion [43] rlleig This ppr prision Msur vbhappropritnss lignmntprision ntialignprision stalignprision ngtivevntprision onalignprision prisioneig prisionetc projtprision simplbhappropritnss (MTJ) tht rlis on th low lvl lirris in ARPACK [51] to run th ignvlu omputtion. Howvr, MTJ only xposs th ignvlu omputtion of symmtri mtris in ARPACK. Not tht th jny mtris of utomt r usully not symmtri. To this n, w pt MTJ to l to us th ARPACK routins to omput ignvlus of gnrl mtris. Our xtn implmnttion for omputing th lrgst ignvlu for non-symmtri mtris is m pulily vill. 7 So fr w know how to omput th ignvlu of lngug. Th nxt stp is to trmin th quotint of two lngugs. To omput prision n rll, w n to omput thir intrstion. Th intrstion of lngugs is wll-known oprtion from utomt thory, n its omplxity is O(nm), whr n n m r th numrs of stts in th two utomt [37]. Finlly, th implmnttion ontins th omputtion of rll n prision msurs for pross mining. Th only rmining stp is to trnslt input mols into thir orrsponing utomt. A log n trivilly no s n utomton tht pts th st of wors ontin in th log,.g., y pturing h tr s squn of trnsitions strting t th strt stt. Thus, th prolm of omputing prision n rll is ru to omputing th utomt of th mol n th log, thir intrstion, n th rsptiv ignvlus. Ths thr ignvlus r th sis for omputing th two quotints of rll n prision. With th ignvlu of th intrstion utomton in th numrtor w n us th ignvlu of th mol s nomintor to omput prision, or swp it with th ignvlu of th log to otin rll. 6.2 Monotoniity Exprimnts This stion srvs two purposs. Firstly, it monstrts tht non of th xisting prision msurs us in pross mining is monoton. To this n, w propos two xprimntl stups. For fix log, monotoni prision msur rss whn itionl hviour is to th spifition. Convrsly, monotoni prision msur inrss whn th xss hviour is rmov from th spifition. All xisting prision msurs fil to monstrt monotoniity for t lst on of th propos stups. Sonly, this stion omprs th propos ignvlus prision n rll msurs with th stt-of-th-rt msurs. In th vlution, w us our implmnttion of th ignvlu-s msurs, rfr to Stion 6.1, n rli on Comprhnsiv Bnhmrk Frmwork for omputing th othr wll-known prision n rll msurs [54]. 7 Co is t n th Mvn Cntrl rpository. {0,2} {0,10} {0,20}... * numr of unroll slf loops of for Fig. 8: Inrsing numr of optionl s for. Strting with up to two s for, stpwis llow mor s up to th losur tht llows n ritrry numr of s for Monotoniity of Prision Msurs W ompr th propos prision n rll msurs inu y ig (f. Stion 5.3) with th prision msur nits propos in litrtur (f. Stion 7). In th following, w us rgulr xprssions to sri th spifition lngugs. In th first xprimnt, w hviour to th spifition strting with prftly fitting spifition to th log tht hs up to two s for. Mor prisly: L is th log with th lngug {0,2} ; 8 M x r th spifitions with lngug {0,x} ; M is th spifition with lngug. Fig. 8 shows th rsults of vrious prision msurs on th y-xis, plott for th vrious spifition lngugs strting from 0 2 possil rptitions of for up to 0 20 rptitions, rfr to th x-xis. Th lst msurmnt on th n of th x-xis nots th prision with rspt to th spifition tht llows n ritrry numr (i.., 0 ) of s for, tht is. Th msurs wr ror only if thy wr omput unr th thrshol of tn minuts. Th simpl hviourl ppropritnss msur [43] shows trn ontrry to th othr msurs, s th prision inrss with mor prmissiv spifition. Avn hviourl ppropritnss [43] fils to rogniz th growth of th spifition s lngug. Th nti-lignmnt msur [45] monstrts th orrt trn, ut hs fil to omput th prision for th spifition within th thrshol tim. Projt prision is stritly monoton in th rgion twn up to 2 n up to 20 s for, ut violts monotoniity t th losur. Th rmining msurs show similr hviour strting t 1.00 for th prftly fitting spifition n rsing ut stilizing quikly. Ths msurs, howvr, o not istinguish twn th spifitions {0,y}, whr y [3.. 20]. Our ignvlus prision msur shows sty stilizing lin th mor possil rptitions of r to th spifition n istinguishs ll th spifitions y thir prision vlus. Bsis itrtion, prlllism, ptur s possil intrlving of symols is nothr imnsion tht w invstigt. W ompr th vrying intrlving rorings of fix lpht of siz 5. This xprimnt orrspons to rwing 8 Nottion {<min>,<mx>} is short-hn for numrting th miniml n mximl numr of rptitions of symol. In this prtiulr s, w gt th log [<>,<,>,<,,>].

11 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER prision prlll numr of prmuttions of fiv symols Msur vbhappropritnss lignmntprision ntialignprision stalignprision ngtivevntprision onalignprision prisioneig prisionetc projtprision simplbhappropritnss Fig. 9: Th hviour of prision msurs for th log L with rspt to spifitions tht llow prmuttions of th sm fiv symols, inluing th prision of ll th xpliit 5! = 120 prmuttions, n th prision of th lngug quivlnt prlll spifition with 120 impliitly llow prmuttions. fiv out of fiv symols from n lpht without rplmnt, whr th orr mttrs. Hn, thr r 5! = 120 istint omintions of symols, i.., 120 istint wors. A pross spifition tht llows prlll xution of fiv tivitis lso prmits xtly 120 iffrnt xutions. W woul xpt spifition tht numrts ll 120 omintions to qully pris, s nothr spifition using th orrsponing prlll uiling lok tht sys tht th sm fiv tivitis n on in ny orr. Nxt, w us th following log n spifitions. L 5 is th log with th lngug {,,,, }. M x r th spifitions with th lngug of L 5 n furthr prmuttions, suh tht 5 x 120 n L(M x ) = x (i.., th numr of llow trs is x). M is th spifition with lngug of ll 120 omintions of th fiv symols,,,,. Most xisting prision msurs (f. list in Tl 2 n isuss in Stion 7) r monotonilly rsing for th fix log L 5 n n inrsingly prmissiv spifition, rfr to Fig. 9. Howvr, for th givn log, th spifition of 120 xpliit prmuttions n hv iffrnt prision thn th spifition with fiv tivitis in prlll, lthough ths two spifitions hv th sm lngug. Not tht only thr msurs rport th sm prision vlus for ths spifitions: vn hviourl ppropritnss, projt prision, n our ignvlu-s prision msur. Th monotoniity of th xprimnt is violt y th ETC prision [48], y th on-lign, n st-lign, n nti-lignmnt s prisions [45, 52], n lso th simpl hviourl ppropritnss [43]. Th nti-lignmnt s prision [45] fils to omput vlu for th fully prlll spifition in th givn tim. Simpl hviourl ppropritnss n only omput up to 110 prmuttions, n th vlu rops lmost in hlf whn looking t th prlll spifition. To stuy monotoniity in th nomintor, w invstigt th rl lif log of th BPI Chllng 2012 [55]. W min spifition M tht is l to rply th ntir log with th inutiv minr infrqunt n nois thrshol prmtr prision % 20% 30% 40% 50% 60% 70% 80% 90% 100% log siz in prnt Msur vbhappropritnss lignmntprision ntialignprision stalignprision ngtivevntprision onalignprision prisioneig prisionetc projtprision simplbhappropritnss Fig. 10: Expt inrs in prision with mor trs in th log of th min pross (A) in [55]. stting 0.0. Thn, w slt 5 prnt of th log L 5% n omput th prision of th spifition n th su-log. Bus w know tht th spifition is l to rply th ntir log, w know tht L(L 5% ) L(M), tht is th lngug of th log is ontin in th lngug of th spifition. W rpt this pross for inrsing su-logs in 5 prnt stps, suh tht L 5% L 10%... L 100%. Th rsulting prision vlus r rport in Fig. 10. Not tht vn if w inrs th numr of trs in th log in stp, nw hviour is not nssrily. At h stp, w n potntilly n up ing only trs tht th prvious log lry ontin. To mk th rsults sily ssil, w rt iffrn plot shown in Fig. 11, whih plots th lts to th prvious vlu (.g., if th vlu inrs y 0.1, whn inrsing th log y 5 prnt, w mrk t 0.1). Hn, for monotonilly inrsing msur on shoul osrv only non-ngtiv vlus. Only thr prision msurs r onsistnt in this stting, tht is thir grphs r monotonilly inrsing. Ths msurs r vn hviourl ppropritnss, lrgst ignvlu s prision, n projt prision. Som ngtiv vlus r u to th non-trministi ntur of som of th prision msurs, s isuss in [28]. Howvr, thr is lso systmti rror in th nti-lignmnt s prision msur [45] tht rports n unxpt ownwr trn in prision, spit th ft tht th spifition is fix n th numr of onsir trs (n with thm th hviour) inrss in this xprimnt. Msur vbhappropritnss lignmntprision ntialignprision stalignprision ngtivevntprision onalignprision prisioneig prisionetc projtprision simplbhappropritnss iffrns to prvious vlu Fig. 11: Eh ot rprsnts th rltiv inrs or rs in Fig. 10 t h susqunt msurmnt stp s th siz of th log inrss; r tringls no rltiv rs.

12 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER rll (fitnss) noisy trs in prnt Mtho lignmntfitnss ngtivevntrll toknbsfitnss prsingmsur projtrll proprcompltion rlleig TABLE 4: Prision n rll for spifition n two logs. L ( )? onsists of 3 trs, on tr, n on tr. L ()? onsists of 3 tims, n twi. Spifition Log Prision Rll S L ( )? S L ()? Fig. 12: Rll msurs for simplisti squntil spifition n inrsing mount of nois Monotoniity of Rll Msurs Rll of lngug spifition with rspt to log is fin s th frtion of th shr hviour y th hviour in th log. In this s, oth msurs ptur finit hviour, whih mks this prolm lss hllnging thn msuring prision. Fig. 12 shows th rsults of th following xprimnt. Givn squntil spifition of 10 tivitis n fitting log with no nois, w strt inrsing th mount of noisy trs in th log. Hr, nois is fin s rmoving, ing, or swpping vnts in th log, n th prntg shown on th x-xis rflts th rltiv numr of trs fft y nois. Th rll vlus r plott for iffrnt thniqus. Vry si msurs r th prsing msur n th notion of propr ompltion. Ths msurs simply ount th frtion of trs tht r ntirly fitting. In ontrst lignmnt s fitnss notions, togthr with th ngtiv vnt rll n th tokn s fitnss, look t th mislignmnts in finr grnulrity. Tht is, thy pnliz minor vitions only slightly. In ontrst, th prsing msur is s on inry ision for h tr. This mns tht smll vitions r ount s muh s ompltly unrlt trs. Th propos msur for rll pns on th siz of th lngugs of th log n spifition. Th ov xmpl shows tht th hviour in th log, y insrting som rnom nois, inrss rly on with th nois lvl. This ls to rpi rop in rll, s in th spifition fils to ptur th rnom hviour, ut only pturs its trministi squntil prt. Th fft of th inrs numr of noisy trs os not inrs th hviour of th log t highr nois lvls tht muh, s th proility tht nw noisy tr hs lry n sn inrss with th numr of noisy trs. In ontrst, th othr msurs show linr trn, s thy o not tk into ount th siz of th hviour, ut ount th numr of fitting trs w.r.t. th siz of th log. As onsqun, tritionl pprohs trt th two ss list in Tl 4 quivlntly, whil our msur jugs th rll of sitution ) lowr thn tht of ), s th vrin in th log is lowr in, vn though it hs th sm mount of viting ss. Whil th mount of noisy trs inrss linrly in this xprimnt, w r intrst in th hviour tht is in oth spifition n log vrsus th hviour in th log only. Th novl ignvlu-s msur pturs this nonlinrity in th hviour of th log. Thus, w onlu tht if on is intrst in th msur of how muh hviour of log, spifition is l to ptur, our msur is mor suitl, whil if w r intrst only in th fitting prt, n o not n to istinguish potntilly iffrnt rrors, th tritionl fitnss/rll msurs r prfrl. Lttr linrly ptur rsing numr of fitting trs w.r.t. givn spifition. 6.3 Slility Evlution Prtil lngug msurs n quotints must l to hnl lrg lngugs. Hn, w msur wll-lok tim of th ignvlu-s prision n rll for 12 lrg rllif logs n spifitions. Th logs r pulily vill 9 n r of iffrnt omplxitis. Th log with th lst vrition in trs (BPI Chllng 2013, opn ss) n trnslt into finit yli utomton with only 116 stts, whil th BPIC 17 log showing th most vrition in trs prous n utomton with stts. For ll ths logs, w min pross spifition with th inutiv minr [49] with th fult nois thrshol 0.2. Th spifitions r onsirly smllr in thir utomton rprsnttion, s th min spifitions o not llow uplit trnsitions. For ll ths log n spifition pirs, w ppli our mtho y first onstruting th rsptiv finit utomt n omputing th ignvlus of thir short-iruit rprsnttions. Th osrv wll-lok tims of th omputtions of th lrgst ignvlus for th log L, th spifition M, n thir intrstion utomton L M r shown in Tl 5. As n initor for th omplxity, th numr of stts of th rsptiv utomt r list in th tl. An jny mtrix of n utomton hs siz tht is qurti in th siz of th utomton, whih n pos prtil iffiultis whn storing it on omputr. Howvr, jny mtris r usully sprs, whih llow us to us thir mmory ffiint rprsnttions. 10 Th vrin in msur wll-lok tims is rmrkl. Th longst tim to omput th prision n rll ws tkn for th BPIC 17 log. In this prtiulr s, th numri trmintion of th lrgst ignvlu fil u to nononvrgn within th pr-onfigur itrtions. Th unrlying thniqu ll impliitly rstrt Arnoli itrtions [51] pprntly hs issus to hnl this spifi utomton mtrix. Not tht th numril mthos for omputing n ignvlu of gnrl mtrix provi no gurnts of onvrgn in fix numr of itrtions. Thus, th thrshol for th mximum numr of itrtions ws hosn for prtil rsons. In ft, th uthor of th softwr pkg stts tht: Th qustion of trmining 9 Logs r vill t: vnt logs rl 10 Th xprimnts n o r vill t nrs-solti/ign-msur.

13 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER TABLE 5: Msurmnts on lptop with 16GB of RAM n Intl i7-6600u prossor. Automton siz (# stts) Lrgst ignvlu Rll Prision Wlllok-tim (minuts) Log nm L M L M L M L M L M L M totl BPIC BPIC 13, los BPIC 13, inints BPIC 13, opn BPIC BPIC BPIC BPIC WABO WABO WABO WABO shift strtgy tht ls to provl rpi rt of onvrgn is iffiult prolm tht ontinus to rsrh [56]. In th rr ss of non-onvrgn of th omputtion, w us th stimt vlu of th ignvlu otin t th n of th omputtion. This insight n th xprimntl rsults tll us tht th siz of th input is not nough to trmin th runtim of th mtho. Rthr, th rt of onvrgn pns on othr proprtis of th jny mtris of th unrlying utomt, i.., on th iffrn in siz twn th lrgst ignvlu n th son lrgst ignvlu. If this is ngligil, th numri mthos n strt to osillt los to th lrgst ignvlus n fil to onvrg. Howvr, th propos mtho is not ti to th ARPACK implmnttion [56]. On novl solutions to lrgst ignvlu omputtion om vill, th improvmnts will irtly nfit this work. W onlu th slility xprimnts with th insight tht th mtho shows lrg vriility in slility pning on th onvrgn of th unrlying ignvlu omputtion, n tht nononvrgn issus my slomly ris. 7 RELATED WORK Th omprison of hviours hs ply mjor rol in th vrifition of softwr n hrwr rtfts ross svrl rs of omputr sin n softwr nginring, inluing th thory of onurrnt systms [57], rtiv systms [58], n gnt progrmming [59], to mntion ut fw. Stion 7.1 outlins notil notions of hviourl quivln n hviourl omprison, inluing inhritn n similrity. Thn, Stion 7.2 sris th volution of th prision n rll msurs for hviourl omprison in th fil of pross mining, long with highlights on ommonlitis n issimilritis to our pproh. Finlly, Stion 7.3 rports on prvious rsrh on hviourl omprison in softwr nginring, gin mphsising th similritis n iffrns with our thniqu. 7.1 Bhviourl Equivln In th ontxt of ynmi systms, thr r svrl notions of hviourl quivln, whih r roly lssifi into two tgoris: quivlns tht r s on th intrlving smntis n thos s on th tru onurrny smntis [60]. W rmrk tht th systms unr nlysis in this ppr fll unr th lss of finit-stt, ssum th prsn of finl/pting stts, n oprt with intrlving smntis. Proly th most importnt hviourl quivln twn two systms of omputtion in this ontxt is th on tht gurnts tht ny stp prform in on systm n mimik y th othr on, n vi vrs [61]. This i is th sis for th notion of isimultion [62]. On root lll trnsition systms ( supr-lss of th systms w nlys), isimultion imposs tht from th initil stt onwr, possil tions must oini twn th systms n inutivly l to stts tht r isimilr s wll. Wk isimultion [62] rlxs isimultion in tht it onsirs only osrvl tions, i.., it is prmitt tht systms gurnt isimultion on non-τ trnsitions only, s τ trnsitions n s prfixor suffix-movs to tht xtnt. Brnhing isimultion nfors wk isimultion y rquiring tht th sm st of hois is offr for n ftr h unosrvl tion [63]. Bisimultion xrts lss strit onitions thn grph isomorphism, whih is ijtion twn ll stts prsrving trnsitions. Howvr, it is lso mor spifi thn tr quivln, solly srtining tht osrvl tions mth, thus ing insnsitiv to non-trminism, intrnl tions, hois, n loks [57]. Complt tr quivln s th onition tht, if systms hv sink stts from whih no furthr tion is possil, thy must rhl in systms y rplying th sm trs. Our rsrh nfits from th multipl notions of hviourl quivln n invstigtions onut on th mttr so fr, yt it strts from th ision prolm on th mthing of hviours n rthr ims t ssssing how muh th hviour of first systm is xtn y son on. Kunz n Wsk [64] lr not only hviourl quivln, ut lso hviourl similrity n inhritn, s min hllngs prtining to hviourl omprison. In prtiulr, th uthors introu proprty for th lttr, nmly tr inhritn, whih is njoy only if th lngug of systm is inlu in th lngug of nothr systm t th sm lvl of strtion. In th light of tht finition, our rsrh thus fouss on hviourl inhritn [65], n spifilly tr inhritn, twn ynmi systms. Howvr, w im t proviing msur ssssing in how fr lngugs xtn on nothr, rthr thn hking whthr th proprty hols tru or not. This quntittiv spt typilly prtins mor to hviourl similrity. To msur it, pplying nïv pprohs s on st-similrity msurs suh s th Jr offiint [66] to th st of systms trs provs infsil: Loops l to tr sts of infinit rinlity. To ovrom tht prolm, pprohs

14 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER to hviour similrity wr introu tht rstrit th nlysis to lol rltions twn trs vnts [67]. Notil xmpls inlu th n-grm similrity [68], ompring systms y th shr llow n-long su-squns in systms rsptiv trs. Dspit th ffiiny of th solution, th trt to vliity is tht vn if n-grms oinis, not nssrily o th trs s wll. Nvrthlss, th st rsults r rportly hiv with th lst strit prmtr, nmly n = 2. Ltr on, hviourl profils similrity ws introu in [69]. Th i is to ompr footprints of systms, otin y mtris onnting pirs of vnt lls with mutully xlusiv rltions. Thos rltions r xlusivnss, strit orr, n intrlving orr, i.., th funmntl rltions of hviourl profils s of [70]. Dspit ing smntilly rih, Polyvynyy t l. [33] show tht th xprssiv powr of hviourl profils is stritly lss thn rgulr lngugs, thus ntiling tht thy nnot us to i tr quivln of finit stt utomt. Our pproh strts from th lol prsptiv on trs or rltions twn vnts in tht it rsorts on th topologil ntropy to ompr th vriility of lngugs. W rflt th omprison of ynmi systms into prision n rll. 7.2 Prision n Rll in Pross Mining Pross mining is th fil of sin tht ims t xtrting knowlg out prosss from th igitl t stor y orgnistions IT systms [4]. Pross mining is opt to isovr nw fts, inluing pross spifitions thmslvs tht wr not oumnt for, ompr th xpt pross hviour with rport rlity n tt vitions twn th formr n th lttr [26]. It shows thus th inhrnt im of fining n ssssing th mth twn th hviours of ynmi systm, in trms of to- pross spifitions vrsus s-is pross t. Thrfor, th intifition of quotints tht llow for omprtiv msurmnt of hviours nturlly suits th mttr. In prtiulr, Buijs t l. [71] intify (rply) fitnss, prision, gnrlistion, n simpliity s th four min qulity imnsions for ssssing th qulity of pross mining rsults [72]. A first prision msur ll hviourl ppropritnss is introu in th sminl work of Rozint n vn r Alst [43] s th gr of how muh hviour is prmitt y th spifition lthough not ror in th log. Th simpl hviourl ppropritnss uils on th osrvtion tht n inrs of ltrntivs or prlllism ntils highr numr of nl trnsitions uring log rply, whil th vn hviourl ppropritnss uss long-istn prn pnnis twn pirs of tivitis. In this wy, it is highr whn somtims-forwr n somtims-kwr rltion pirs shr twn spifition n log pproximt th totl mount of th spifition. Convrsly, it is lowr if th spifition llows for mor vriility. Th ssumption of totl fitnss of th log ntils tht th log nnot show mor vriility thn th spifition. Our pproh lso omprs th vilility of tions t givn stts, ut strts from th xt rply of trs y onsiring th ntropy of th lngugs. Th ETConformn pproh vois th omplt xplortion of th spifition hviour y trvrsl of th spifition to solly rflt th trs ror in th log [48]. To tht xtnt, finit (yli) root trministi lll trnsition systm nm prfix utomton is gnrt y foling trs s on prfix tr-quivln of th gnrt stts. Th ssumption of totl fitnss ntils tht th st of vill trnsitions ontins th ons prmitt y th prfix utomton. Th lolity of th pproh llows for highr ffiiny of th omputtion, with th ownsi tht only hviour los to th log is onsir. Similrly, our pproh sssss prision y quntifying th hviourl iffrns mong stts of finit-stt root lll trnsitions systm. Howvr, it strts from th ror runs of th involv spifitions. Rmrkly, Munoz-Gm n Crmon [48] lso introu vn ignosti msurs to ssss th svrity of imprisions n thir stility ftor with rspt to smll prturtions in th vnt log. An pproh omining th onpt of prfix utomton with th on of lignmnts [44] is propos y vn r Alst t l. [52] to l with non-ntirly fitting logs. Th propos lignmnt-s prision is th rithmti mn ovr ll vnts in th log of th rtio twn th tivitis tht wr llow y th spifition n th ons tht wr tully xut s pr th prfix utomton, givn th rply history. Arinsyh t l. [46] propos iffrnt prision msurs s on th ntur of th lignmnts to onsir. Th unrlying strutur rmins prfix utomton s in [48], hr ugmnt y ssoiting wights to stts. As in th pprohs of [52] n [46], th prision msur propos in this ppr os not tk into ount ivrging hviours. To tht xtnt, th log rpir givn y lignmnts oul nfiil to pr-prossing phs. Bus our solution rsorts on th ntropy of spifitions lngugs, it strts from th rply n ounting of vnts. Mor rntly, Lmns t l. [49] introu prision n rll msurs to ompr th hviour of spifitions or logs, rquiring finit stt utomton s th unrlying strutur for stt-to-stt omprison s in [46, 48]. To op with th high omputtionl ffort rquir y th intrstion oprtions, projtion of oth spifitions is pr-omput for vry sust of k tions in th joint lpht. Rsulting utomt ontin silnt trnsitions n prsnt non-trminism. Th rsulting Projt Conformn Chking (PCC) prision n orrsponing rll msur uil thn on k-susts projtions. As in [49], w nfit from minimistion of th unrlying strutur n provi ul finitions for prision n rll. Howvr, th omputtion of msurs s on ignvlus os not rquir th pproximtion vi k-projtions. Th nti-lignmnt s prision is fin y vn Dongn t l. [45] using th onpt of nti-lignmnt first propos in [73]. An nti-lignmnt is finit tr of givn lngth whih is pt y th pross spifition, yt not in th log n suffiintly istnt from ny tr thrin (whr th tr istn n omput y using it istn [74],.g.). To ssss prision, vry istint tr is rmov from th log n n nti-lignmnt of qul lngth is gnrt with mximum istn. Ths r vrg. Likwis, w rson on lngug proprtis of nlys spifitions, thus strting from th numr of ourrns of tr. Howvr, our pproh os not

15 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER rquir th itrtiv sn n omprison of spifitions xluing prts of th hviour, thus sving on omputtion tim. Th Artifiilly Gnrt Ngtiv Evnts thniqu (AGNEs) isovrs pross spifitions out of logs nrih with rtifiilly injt ngtiv vnts [47]. Th ssumption is tht th log inlus th omplt st of hviourl pttrns, whih mns tht vnts n only missing in log us thy r not prmitt y th pross. Th notion of rll n thn fin s th rt of tru positivs ovr ll vnts lssifi s positiv, n spifiity oringly. Bfor th omputtion, prliminry rution of mthing vnt squns to singl trs is onut suh tht trs o not up to th ovrll mount. Our finitions of prision n rll r lso ul n o not pn on th numr of ourrns of th sm tr. Howvr, no rtifiil injtion of nois is rquir in our pproh, thus ruing th is tht th ltrtion of th input hviour with ngtiv informtion my us. To vlut thir isovry lgorithm, nmly th Huristi Minr, Wijtrs t l. [53] introu th so-ll Prsing Msur (PM), whih is s on th frtion of orrtly prs trs ovr ll trs in th input log. As rivtiv, th Continu Prsing Msur (CPM) provis mor fin-grnulr nlysis, t th pri of ing oun to th spifition of th unrlying Huristi Minr. Our notion of rll for spifition is lso s on th msuring of th prt of lngug not ovring nothr hviour. Notily, PM n CPM wigh th mount of inorrtly prs trs, thus quntittivly ssssing to whih xtnt th ivrgns our in th vnt log. Owing to our lvl of strtion, w o not ount for this ssssmnt. Howvr, th msur w propos is lss pnnt on th ror trs n is not s on th ount of vnts. Th fitnss msur propos y Rozint n vn r Alst [43] ounts th numr of tokns onsum n prou uring th rply of trs ovr th Ptri nt spifition, n puts thm into rltion with missing tokns n tokns rmining ftr ompltion. It xtns simplr msur omput s th rtio of trs using missing or rmining tokns fin in th sm ppr n nm propr ompltion in [72]. Anothr tokn-s fitnss msur, us in gnti pross mining, ounts for tr frquny [75]. In ontrst to [43, 75], w im t fining msurs tht r not tilor to spifi hviour spifition lngug, thus w o not rly on Ptri nt smntis to fin rll. Th onpt of lignmnt-s fitnss introu y vn r Alst t l. [52] rlis on ost funtion to spifi y th usr, initing th pnlty for non-synhronous movs in th rply of trs on th spifition. Fitnss is thn omput for vry tr s th totl ost of th optiml lignmnt, ivi y worst-s lignmnt, init s th on onsisting of movs in th tr for vry vnt, follow y movs in th spifition from th strt to th n of shortst run. Log fitnss is thn lult y vrging th tr fitnss vlus ovr ll trs. Alignmnts r vlul mns to mk th pproh inpnnt on th spifition lngug, s in th rtionl of our invstigtion. Our thniqu os not llow th usr to init osts. Proviing this ftur in our pproh is n intriguing prolm tht oul rss in futur work. On th othr hn, our pproh os not rsort on th omputtionlly xpnsiv fining of optiml runs on th input spifitions. W rmrk tht spilly th pprohs sri in [43, 46, 49, 52, 52] not only propos prision n rll msurs n lgorithms for thir omputtion, ut provi lso thniqus to illustrt whr n in how fr vitions our twn th log n th spifition. Th intgrtion of thos powrful ignosti tools with our pproh lints intrsting plns for futur rsrh. To onlu, [28] rntly fin fiv rquirmnts (thr nm xioms) tht prision msur shoul gurnt, in striv for th gnrl finition of funmntl proprtis tht shoul njoy y pross mining qulity msurs. Th uthors show tht nithr of formntion simpl hviourl ppropritnss [43], vn hviourl ppropritnss [43], ETC prision [48], AGNEs spifiity [47], or PCC prision [49] omply with thir rquirmnts for prision. By sign, our pproh fulfils ll thos rquirmnts inst, s shown in Stion Bhviourl Comprison in Softwr Enginring In softwr nginring, notil oy of litrtur on utomton-s spifition mining hv propos highly rlvnt ontriutions towrs th hviourl omprison of stt mhins. Th sminl work of Lo n Khoo [24] proposs frmwork ll QUARK (QUlity Assurn frmwork) for mpirilly ssssing th utomt gnrt y iffrnt minrs. Thir ssumption is, two mols hv to ompr: on rfrn n on rvrsnginr from API intrtions. This ontxt is similr to ours in tht w lso ompr rfrn pross spifition with nothr hviourl strtion, in our s stmm from st of xution trs of pross. In thir pproh, thy omput ury in trms of tr similrity. Thy first ollt two smpls of rnomly gnrt trs, on pr mol. Th prision is th proportion of smpls gnrt y th rvrs-nginr mol tht r pt y th rfrn utomton. Dully, th rll is th proportion of trs tht r gnrt y th rfrn utomton, n r pt y th rvrs-nginr on. Our pproh movs in th opposit irtion: w strt from trs n ompr systms, rthr thn ompring trs gnrt y th systms. Rmrkly, Lo n Khoo [24] lso propos msurs tht l with proilisti finit utomt, s upon th Hin Mrkov Mols omprison. Thir stuy suggsts th xtnsion of our pproh towr th nlysis of proilisti mols s n opportunity for futur rsrh. Th us of simult trs for systm omprison, first rport in [76] n ppli in QUARK [24], hs n ltr ritiis y Wlkinshw t l. [34]. A thrt to its vliity is, it is virtully impossil to ovr th whol hviour of systm y rnom wlks. This prolm is of high svrity spilly us som fulty xutions might rmin unxplor y rnom smpl, whih is of high rlvn in softwr tsting [22, 23]. To rss this is, Wlkinshw t l. [22] propos n pttion of th originl Vsilvski/Chow W-Mtho [77, 78]. Thir thniqu is im t gnrting tst sts tht ovr ll istinguishl

16 PRE-PRINT SUBMITTED TO ARXIV.ORG. DECEMBER runs of th mol. Furthrmor, thy rfin th notions of prision n rll to ount for not only th trs tht r mutully pt y th ompr mols, ut lso to inspt th pility of th two to rjt trs tht r not omplint with th trgt hviour. In our ontxt, to-rjt trs r not onsir s w ssum th log to stm from rgistr orrt systm runs. Howvr, w s in this spt n nvour for futur work: n xtnsion of our lngug-quotint s pproh tht ounts for th smnti isrimintion of runs tht n up in positiv outoms from thos tht o not, similrly to wht ws on y Pon Lón t l. [79] n Chsni t l. [80]. Wlkinshw n Bognov xtn thir sminl work [34] in two irtions [25]. Firstly, thy xpn th omprison msurs with lssil t mining ons suh s spifiity n ln lssifition rt. Sonly, thy introu th LTSDiff lgorithm, whih omprs mols unr struturl prsptiv, rthr thn hviourl on. In this ppr, w o not onsir th struturl similrity, thus ing mol-gnosti n not imposing rquirmnts on th trminism or minimlity of input systms. Howvr, our thniqu oul improv y intgrting th ognitiv-lik, itrtiv pproh of th LTSDiff lgorithm, s on n intrmit rsults xpnsion strting from lnmrks [81] (i.., mthing susts of th inputs). Qunt n Koshk [82] first onsir msur for mol omprison tking into ount th lngug of involv utomt without th nlysis of gnrt trs. Thy vis to tht xtnt n pproh similr to tht of it istn. A minimis union utomton is first rt twn th input ons. Thrupon, onurrnt synhronous run is xut on h of th mols n th union utomton. It trmins th numr of its, tht is, th trnsitions to rmov from th union (nvr trvrs) or to th input mol (unfol slf-loops). Th finl msur is omput y vrging th istns in trms of its of th mols from th union utomton. Our pproh rvolvs roun lngug omprison s on th nlysis of utomt s wll. Howvr, it isrimints twn prision n rll, thus giving mor pris pitur of th ury of th min mol with rspt to th rfrn of th log. Prl t l. [16] us vrint of th k-tils lgorithm [83] for ompring min n rfrn mols. To tht xtnt, thy first gnrt th union of th finit utomt givn s input mols. Thn, thy pt th k-tils lgorithm to pproximt th mthing of thos stts from whih ommon (su)squns of lngth k n gnrt. Suh stts r thn mrg. Prision is omput s on th numr of shr trnsitions twn th min mol n th intrstion of th rfrn mol with th utomton sujt to k-tils mrging. Rll is omput nlogously ut swithing min n rfrn mol. Th usg of k-tils to mrg stts llows for th prossing of mols min from noisy or inomplt trs. On th othr hn, th ft tht mths r not xt n sujt to propr hoi of k my l to n inury of rsults, s mphsis y Wlkinshw n Bognov [25]. As in [16], our pproh onsirs lngug strtion of systms for omprison purposs, without gnrting tr sts. In ontrst to it, w o not rsort to struturl pproximtions ovr th input spifitions. On th on hn, it fvours ury. On th othr, n pttion of our pproh to ount for nois, s in [16], is n intrsting irtion for futur work. 8 CONCLUSION This rtil propos hviourl quotints s mns to rlt th hviours of ynmi systms. A quotint tks lngug msur s prmtr, whih is rsponsil for mpping th systm s hviour onto th numril omin for furthr omprisons with othr hviours. Thr xmpl lngug msurs r put forwr in th rtil: on ovr finit, on ovr irruil rgulr, n on ovr rgulr lngugs. Th lngug msur ovr rgulr lngugs is s on th notion of topologil ntropy n is us to instntit hviourl quotints into prision n rll msurs for pross mining. Th xtnsiv vlution monstrts tht th propos prision n rll n omput in rsonl tim n qulittivly outprform ll th xisting msurs for prision n fitnss us in pross mining. Futur work on hviourl quotints shoul im t xtning n improving thm in svrl wys. First of ll, hviourl quotints n xtn to hviourl rprsnttions of ynmi systms othr thn thir lngugs,.g., hviourl profils [33, 70], lrtiv mols [84, 85], n hyri rprsnttions [86, 87]. Son, on n propos nw lngug msurs for instntiting hviourl quotints, n stuy intrprttions n omputtionl omplxitis of ths msurs. Thir, lngug quotints n improv to ount for multipliitis n similritis of wors. Not tht th quotints propos in this rtil strt from multipliitis of wors n onsir wors s ing istint, vn if thy iffr only in singl hrtr. To tkl this prolm, w n lrn from th is propos in [88]. A solution to this prolm shoul llow rssing phnomn lik rptitiv ourrns of th sm or similr trs in n vnt log. Finlly, on n sign nw qulity msurs tht rlt ritrry numrs of hviours,.g., to stlish sis for ompring rsults of vrious pross qurying mthos [89] n iffrnt hviourl rprsnttions [90]. Th rnt osrvtion tht ll th so fr propos prision msurs in pross mining fil to stisfy vn si proprtis [28], initit isussion on wht proprtis shoul th stnr qulity msurs in pross mining possss [31]. Th prision n rll fin s lngug quotints,.g., th ntropy-s msurs, rfr to Stion 5, stisfy ll th proprtis propos in [28, 31]. This rsult is u to th ft tht ths msurs r fin s rtios ovr lngug msurs, whih possss th proprtis of non-ngtivity, null mpty st, n strit monotoniity [38]. Consquntly, with this work, w propos to shift th fous of th isussion from th sir proprtis of th qulity msurs in pross mining to th sir proprtis of msurs ovr lngugs us to fin th qulity msurs n l to thir usful proprtis. For xmpl, on n xplor whthr n itionl rquirmnt of itivity or su-itivity rflts som usful proprtis of prision n rll in pross mining.

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Lomzov, Trnsition systms rution: Blning twn prision n simpliity, ToPNoC, vol. 12, pp , [89] A. Polyvynyy, C. Ouyng, A. Brros, n W. M. P. vn r Alst, Pross qurying: Enling usinss intllign through qurys pross nlytis, DSS, vol. 100, pp , [90] J. Prshr, C. Di Ciio, n J. Mnling, From lrtiv prosss to imprtiv mols, in SIMPDA, 2014, pp Artm Polyvynyy Dr. Artm Polyvynyy is snior lturr t th Shool of Computing n Informtion Systms, Mlourn Shool of Enginring, t th Univrsity of Mlourn, Austrli. H riv PhD gr in Computr Sin from th Univrsity of Potsm, Grmny. His rsrh intrsts inlu Distriut n Prlll Systms, Conurrny Thory, Ptri Nts, Forml Mthos, Informtion Systms, Softwr Enginring, Workflow Mngmnt, n Businss Pross Mngmnt. Anrs Solti Dr. Anrs Solti is post otorl rsrhr with th Institut for Informtion Businss t th Vinn Univrsity of Eonomis n Businss (Austri). His rsrh intrsts r in th r pross mining, pross nlytis, optimiztion, n snsor t nlytis. H is lso intrst in ptiv lrning thnologis tht n support with mhin lrning. Mtthis Wilih Prof. Mtthis Wilih is junior profssor t th Dprtmnt of Computr Sin t Humolt-Univrsität zu Brlin (HU). H ls th Pross-Drivn Arhitturs group, whih is fun y th Grmn Rsrh Fountion (DFG) through th Emmy-Nothr Progrmm. H hols PhD from th Hsso Plttnr Institut, Univrsity of Potsm. H is Junior- Fllow of th Grmn Informtis Soity (GI) n th ripint of th Brlin Young Rsrhr Awr Cluio Di Ciio Dr. Cluio Di Ciio is n Assistnt Profssor with th Institut for Informtion Businss t th Vinn Univrsity of Eonomis n Businss (Austri). His rsrh intrsts inlu pross mining, lrtiv pross molling, n srvi-orint omputing. H otin his Ph.D. t Spinz Univrsity of Rom, with thsis on th utomt isovry of flxil workflows from smi-strutur txt t sours. H is mmr of th IEEE Tsk For on Pross Mining. Jn Mnling Prof. Jn Mnling is Profssor in th Dprtmnt of Informtion Systms n Oprtions, Wirtshftsunivrsität Win, Austri. H riv iplom grs from th Univrsity of Trir, n PhD gr from Wirtshftsunivrsität Win. H hs pulish rsrh on usinss pross mngmnt n informtion systms in vrious journls n is o-uthor of th txtooks Funmntls of Businss Pross Mngmnt (Springr) n Wirtshftsinformtik ( Gruytr, in Grmn).