1 Pross Winows Anry Mokhov, Jori Cortll, Alssnro Gnnro Nwstl Univrsity, Unit Kingom Univrsitt Politèni Ctluny, Brlon, Spin Astrt W sri mtho for formlly rprsnting th hviour of omplx prosss y pross winows. Eh winow ovrs prt of th systm hviour, i.. prt of th unrlying trnsition systm, n is sir to unrstn n nlys thn th omplt trnsition systm. Pross winows n ovrlp n hv shr stts n trnsitions so tht th omplt systm hviour is th union of winow hviours. W monstrt th vntg of suh rprsnttions whn ling with omplx systm hviours, n isuss potntil pplitions in iruit sign n pross mining. As motivtionl xmpl w onsir th prolm of ovring trnsition systms y mrk grphs, or mor gnrlly hoifr Ptri nts. Th otin winows orrspon to hoi-fr hviourl snrios of th systm, whrin on winow n tk ovr, or wk up, ftr nothr winow hs om intiv. Th orrsponing wk-up onitions n wk-up mrkings n riv utomtilly. () Trnsition systm. () Winow omposition. I. INTRODUCTION Unrstning n spifying th hviour of omplx onurrnt systm is iffiult tsk. Trnsition systms oftn suffr from th stt sp xplosion n vn Ptri nts struggl to rprsnt th hviour of mny rl systms in onis wy, us of multipl hviourl snrios ntngl in singl nt. In this ppr w show how trnsition systm n ompos into st of simplr hviourl mols, furthr rfrr to s winows. Th originl hviour n rovr s th union of th winows. A winow mols only on spt of th systm s hviour tht n hrtris y simpl vnt rltionships. By hoosing winows with simpl rprsnttions, suh s mrk grphs, hoi-fr nts, frhoi nts, t., on n mk sur tht h iniviul winow is simpl nough to unrstn, visulis n spify. A. Motivtionl xmpl Consir trnsition systm n its Ptri nt rprsnttion shown in Fig. 1(,). Arguly, th Ptri nt is mor iffiult to unrstn thn th trnsition systm, u to mix of onurrny n hoi. On n ompos th trnsition systm into two simplr ons, whih w ll winows, tht r hoi-fr, s shown in Fig. 1(). Th winows hv vry simpl Ptri nt rprsnttions W 1 n W 2 shown in Fig. 1(): th hoi spt of th systm hviour hs n strt wy from iniviul winows, n is impliitly rprsnt y th ltrntion of hviour twn winows. Th omposition n utomt iming to prou winows with spifi hviourl proprtis. In this xmpl, th rsulting winows r mrk grphs, lss of Ptri nts with prtiulrly wll-unrstoo struturl proprtis. Wk-up onition: p5 p8 Wk-up mrking: () Ptri nt. p0 = p7 p2 = p6 p3 = 1 Wk-up onition: p1 p3 Wk-up mrking: () Winows W 1 (lft) n W 2 (right). Fig. 1: Motivtionl xmpl. p6 = p2 p7 = p0 p8 = 1 An importnt proprty of th omposition is tht winows n hv ovrlpping hviours, whih provi rigs twn winows. As sn in Fig. 1() th winows ontin two shr stts: s0 (th initil stt) n s4. Th urrnt stt of th systm n thrfor osionlly sn in two winows simultnously, s shown in Fig. 2. By firing p or in s0 th systm lvs th shr stt n pros oring to on of th winows until it vntully oms to s0 or s4, wking up th intiv winow. Not: th systm n sty in th son winow infinitly y looping through stts {s6, s7, s8, s9}.
2 Fig. 2: Simulting winows from Fig. 1 (tiv winows r trnsprnt, intiv winows r opqu). Th min ontriutions of this ppr r s follows: W introu pross winows, oth informlly y xmpls n formlly, in Stions II n III. Pross winows n isovr utomtilly using th winow omposition lgorithm. Stion IV prsnts th lgorithm n provs its ompltnss. A mtho for utomt rivtion of wk-up onitions n mrkings is prsnt in Stion V. W xplor pplitions of pross winows in iruit sign n pross mining in Stion VI, n intgrt th prsnt lgorithms n sign mthoology into th opn-sour WORKCRAFT molling frmwork [1][2]. Stion VII isusss th potntil of pross winows for molling systms tht ontin rsour ritrtion n OR uslity. Th rlt work is rviw in Stion VIII. II. THE EXAMPLE IN MORE DETAIL In this stion w lrify th orrsponn twn trnsition systm n its rprsnttion using pross winows. Fig. 2 shows th trnsition systm from Fig. 1(), whr h stt is xpn into th orrsponing stt of th two pross winow nts in Fig. 1(). A winow is tiv whnvr it ovrs th urrnt stt, othrwis it is intiv; intiv winows r rwn opqu in Fig. 2. For xmpl, oth winows r tiv in stts s4 n s0, n th stt trnsition s4 s0 is rflt in oth winows y firing th trnsition. Dpning on whih trnsition ours in s0, ithr W 1 or W 2 oms intiv. In, W 1 os not ovr th trnsition, n W 2 os not ovr th trnsition p. Trnsitions {s7 y s4, s8 y s0} wk up winow W 1 n, similrly, {s2 s4, s3 x s4, s5 x s0} wk up winow W 2. Not tht thr is lwys t lst on tiv winow, us vry stt of th originl trnsition systm must ovr. Furthrmor, sin vry trnsition must ovr too, thr is lwys t lst on winow tht is tiv oth for n ftr trnsition ours. A. Wk-up mrkings Tl I provis n itionl illustrtion of th orrsponn twn th stts of th originl trnsition systm n th mrkings of th otin nts W 1 n W 2. As on n s, h stt is ovr y t lst on nt mrking. Whn TABLE I: Stts n nt mrkings in Fig. 1. Stt Mrking of W 1 Mrking of W 2 s0 {p0, p3} {p7, p8} s1 {p1, p4} - s2 {p1, p3} - s3 {p2, p4} - s4 {p2, p3} {p6, p8} s5 {p0, p4} - s6 - {p5, p8} s7 - {p6, p9} s8 - {p7, p9} s9 - {p5, p9}
3 winow oms intiv, th mrking of th orrsponing nt hs no mning n is forgottn. Whn th winow susquntly wks up, th nt must initilis with orrt mrking mthing th urrnt stt of th systm. Using Tl I on n otin th following wk-up mrkings: Whn W 1 wks up in stt s0, it must initilis with th mrking {p0, p3}. Whn W 1 wks up in stt s4, it must initilis with th mrking {p2, p3}. Whn W 2 wks up in stt s0, it must initilis with th mrking {p7, p8}. Whn W 2 wks up in stt s4, it must initilis with th mrking {p6, p8}. On n rmov ll rfrns to th originl trnsition systm n its stts s0 n s4, thry mking th pross winows s sription slf-ontin from th point of viw of th two nts, s follows: Whn W 1 wks up, it must hv tokn in p3, plus tokn in p0 (if p7 is mrk in W 2 ) or p2 (if p6 is mrk in W 2 ). All othr pls shoul unmrk. Whn W 2 wks up, it must hv tokn in p8, plus tokn in p7 (if p0 is mrk in W 1 ) or p6 (if p2 is mrk in W 2 ). All othr pls shoul unmrk. Th wk-up mrkings r shown in Fig. 1() low th nts. B. Wk-up onitions In ition to wk-up mrkings, w lso riv wk-up onitions: th wk-up onition of winow vluts to 1 in ll stts whr th winow n wk up n thrfor rquirs n initilistion using th wk-up mrking. In our running xmpl, oth winows n wk up in stts s4 n s0. From th point of viw of W 1, it ns to wk up whn W 2 hs mrking {p6, p8} or {p7, p8}. This n xprss y th Booln onition (p6 p7) p8, whih n simplifi to p5 p8 using Booln minimistion. Similrly, winow W 2 ns to wk up whn W 1 hs mrking {p0, p3} or {p2, p3}, s ptur y th onition (p0 p2) p3 or, quivlntly, p1 p3. Fig. 1() shows minimis wk-up onitions low h nt. A gnrl mtho for riving wk-up onitions n mrkings is prsnt in Stion V. III. LABELED TRANSITION SYSTEMS AND WINDOWS A. Lll Trnsition Systms A Lll Trnsition Systm (LTS) is tupl (S, Σ, T, s 0 ) whr S is finit st of stts, Σ is th lpht of lls, T S Σ S is th st of lll trnsitions n s 0 S is th initil stt. W us s s to not th lll trnsition (s,, s ) T. An vnt Σ is si to nl in s 1 S if thr xists s 1 s 2 for som s 2 S. Givn n vnt Σ, th Enling St of is th st of stts in whih is nl, i.., ES() = {s S s S : s s }. Similrly, w fin th Bkwr Enling St of : BES() = {s S s S : s s}. B. Winows Givn n LTS A = (S, Σ, T, s 0 ), winow of A is nothr LTS W = (S w { w }, Σ w, T w, s 0w ) suh tht S w S, Σ w Σ, n T w T. Morovr, Σ w stritly ontins th lls ssoit to T w n S w stritly ontins th stts ssoit to T w. w S rprsnts th intiv stt. If s 0 S w, thn s 0w = s 0, othrwis s 0w = w. Thus, ny winow is fully trmin y sust of trnsitions of th LTS. C. Winow Domposition Givn n LTS A = (S, Σ, T, s 0 ), Winow Domposition (WD) of A is st of LTS winows, {W 1,..., W n }, with W i = (S i { i }, Σ i, T i, s 0i ), suh tht S = S i, Σ = Σ i, T = T i i i i n ll i r iffrnt. Th finition implis tht vry stt n vry trnsition of A is ovr y t lst on winow. Aitionlly, th following onitions hol for vry W i : Th unrlying grph inu y T i is onnt. T i T j for ny j i. Intuitivly, th prvious onitions gurnt tht ll omponnts r miniml, i.., tht th stts r onnt (xpt i ) n no winow is runnt. D. Stt sp n trnsition stps A winow omposition W = {W 1,..., W n } is st of winows tht volv synhronously t vry trnsition stp oring to th trnsitions of th LTS thy rprsnt. W hs glol stt sp in whih h stt s is vtor of stt omponnts, s = (s 1,..., s n ), h on longing to iffrnt winow. Thr is on-to-on orrsponn twn th stts of W n th stts of th ssoit LTS. For vry stt s x S, th ssoit stt in W will s x = (s 1,..., s n ) suh tht for vry W i : s i = s x if s x S i, s i = i if s x S i i.., ll th tiv stts r intil. 1) Exmpl: If w onsir th LTS in Fig. 1() n th WD with two winows, W 1 (lft) n W 2 (right), in Fig. 1(), w osrv tht th initil stt s 0 is rprsnt y th vtor s 0 = (s 0, s 0 ). Whn firing trnsition s 0 s 6, th WD movs to stt s 6 = ( 1, s 6 ). W nxt sri possil tr of th WD: q (s 0, s 0 ) ( 1, s 6 ) ( 1, s 7 ) (s 4, s 4 ) p (s 0, s 0 ) (s 1, 2 ) As on n s from th ov tr, t vry stt h winow n ithr tiv or intiv. Som trnsitions p my tivt winow,.g., (s 0, s 0 ) (s 1, 2 ), whrs othr trnsitions my tivt (wk up) winow,.g., y ( 1, s 7 ) (s 4, s 4 ). Thr is lwys t lst on tiv winow tht kps trk of th urrnt stt so tht othrs n wk up n ppropritly initilis thmslvs whn thir tim oms. y
4 E. Struturl proprtis From th thory of Ptri nts, w know tht rtin sulsss, suh s Mrk Grphs or Fr-Choi Ptri nts, r wll suit for goo-looking visul struturs [3]. Aitionlly, ths struturs r lso sirl for prformn n rhility nlysis. An intrsting ontriution in this r is th work y Bst n Dvillrs hrtrising th stt sps of hviours tht n gnrt y Mrk Grphs [4] n hoi-fr Ptri nts [5]. Fr-hoi Ptri nts is nothr sulss with visully-frinly struturl proprtis [6]. Th synthsis of this sulss hs n propos in [7] y omining th thory of rgions n ll splitting to for ll hois to fr. In this work, w pply th rsults from [5] n [7] for th omposition of hviours into pross winows. On of our min gols is to rt frmwork in whih th nlysis of prosss n highly utomt y proviing lgorithms to xtrt winows. Th propos lgorithms r s on solving SAT formultion of th prolm tht nos lol 1 proprtis of th winows. W nxt prsnt lol proprtis prsnt in [5][7] to nfor struturl proprtis on th synthsis Ptri nts. Th first two proprtis (forwr n kwr prsistn) r nssry, ut not suffiint, for th synthsis of hoi-fr Ptri nts. Th thir proprty is nssry for th synthsis of Fr-Choi Ptri nts. 1) Forwr n kwr prsistn: Givn n LTS, two vnts n r si to forwr prsistnt if th following onition hols: s 1 ES() ES(), s.t. s 1 s 2 s 1 s 3 : s 2 ES() s 3 ES(). This onition gurnts tht os not isl, n vi vrs. Not tht forwr prsistn lwys hols whn ES() ES() =. A ul proprty is fin for kwr prsistn. Two vnts n r si to kwr prsistnt if th following onition hols: s 1 BES() BES(), s.t. s 2 s 1 s 3 s 1 : s 2 BES() s 3 BES(). An LTS is si to forwr (kwr) prsistnt if forwr (kwr) prsistn hols for ll pirs of vnts. 2) Fr hoinss: Whn two vnts r not forwr prsistnt, thn onflit (hoi) ours twn thm. In this s, w my sir th onflit to fr, i.., th hoi onitions to symmtri for oth of thm. Lt n two vnts tht r not forwr prsistnt. Thn, n r si to in forwr fr hoi if ES() = ES(). Similrly, if n r not kwr prsistnt, thy r si to in kwr fr hoi if BES() = BES(). 1 By lol w rfr to proprtis tht n formult in trms of stts or trnsitions n smll nighourhoo roun thm. s0 s1 s2 s3 s4 s5 y y s6 x x Fig. 3: LTS with iffrnt prsistn n hoi proprtis. 3) Exmpl: Fig. 3 pits n LTS with iffrnt proprtis twn pirs of vnts. For xmpl, th pir (, ) is forwr prsistnt, ut not kwr prsistnt, sin n r oth kwr nl in s6, ut is not kwr nl in s4 n is not kwr nl in s5. Th pirs (, ) n (, ) r oth forwr n kwr prsistnt. Th pirs (, x) n (, x) r lso forwr n kwr prsistnt. Th pir (, ) is not forwr prsistnt, ut it is in forwr fr hoi. Finlly, th pir (, z) is not in forwr fr hoi, sin thy r not forwr prsistnt n z is not nl in s6. Similrly for th pir (, z). s7 s8 s9 x z s10 s11 s12 IV. ALGORITHM FOR WINDOW DECOMPOSITION This stion sris n lgorithmi mtho for isovring winow omposition of n LTS. Th mtho is inspir y th on prsnt in [3] for th xtrtion of LTS slis uring pross mining. It is s on SAT formultion of th prolm, using T vrils, tht mols ll possil winows tht onform to rtin st of proprtis. Th prolm is solv vi Psuo-Booln Optimistion [8]. With n us of nottion, w fin Booln vril t for h trnsition t T. A winow is fully trmin y sust of trnsitions, tht n mol s Booln ssignmnt to th orrsponing vrils. All thos vrils ssrt in th mol orrspon to th slt trnsitions for th winow. Th formul W (T ) tht mols ll possil winows tht n xtrt from n LTS is fin s: WINDOW(T ) = P 1 (T ) P n (T ) (1) whr h P i (T ) rprsnts st of onstrints ssoit to proprty. Nxt, th onstrints ssoit to iffrnt proprtis r sri. A ommon n importnt ftur of ths proprtis is tht thy r lol, i.., thy n spifi s Booln rltionships mong vrils tht rprsnt trnsitions in smll rgion of th LTS. A. Forwr n kwr prsistn Forwr prsistn nsurs tht no vnt isls nothr vnt in th xtrt winow (s St. III-E). Th Booln formultion of this proprty is s follows: Lt s, s 1, s 2 S n, Σ, with, suh tht t 1 = (s,, s 1 ) n t 2 = (s,, s 2 ). Lt T s2, = {t i = (s 2,, s i ) s i S} th st of trnsitions nl in s 2 with vnt. Thn th
5 following onstrint is to th formul to gurnt th prsistn of : (t 1 t 2 ) = (t i1 t ik ) whr t i1,..., t ik r th lmnts of T s2,. Noti tht, y symmtry, this onstrint will lso ppli for s prsistn. It lso works for nontrministi LTSs in whih T s2, > 1. In s T s2, =, th onstrint is ru to: t 1 t 2. Bkwr prsistn hs ul formultion tht intuitivly orrspons to th on for forwr prsistn whn th irtion of th trnsitions is rvrs. Th onstrints for forwr n kwr prsistn must formult for pirs of vnts nl in ny stt of th LTS. B. Dtrminism In som ss, it is sirl tht th xtrt winows r trministi. This implis tht ny non-trministi hoi nfors th systm to mov to iffrnt winow. Dtrminism n sily nfor y ing t-most-on onstrints ovr ll th trnsitions nl with th sm vnt in non-trministi stts. C. Conntnss This is proprty tht nnot gurnt with only lol onstrints. Inst, hyri pproh omining Booln onstrints n lgorithmi post-prossing is us. Th Booln onstrints gurnt tht no nw sour/lok stts r gnrt in winow. Formlly, for ny stt s S, w fin T in (s) = {t in1,..., t inm } n T out (s) = {t out1,..., t outn } s th st of inoming n outgoing trnsitions of s, rsptivly. For ny stt in whih T in (s) n T out (s), th following onstrint is : (t in1 t inm ) (t out1 t outn ). This onstrint gurnts tht ny stt with inoming n outgoing trnsitions in th originl LTS will lso hv inoming n outgoing trnsitions in ny winow. D. Algorithm W nxt prsnt n lgorithm for th isovry of winow omposition of n LTS (s Algorithm 1). Th lgorithm is s on th squntil xtrtion of pross winows tht fulfil st of proprtis. Th xtrtion is shphr y SAT formul tht nos th onstrints of th hosn proprtis (s lin 2 n qution (1)). Th vril T ontins th st of trnsitions tht hv not yt n ovr y th prviously xtrt winows. For h winow, th mximum numr of unovr trnsitions is slt (sis othr trnsitions). This sltion is str y ost funtion no s Psuo-Booln onstrint (lin 6). Th vril T i is th st of slt trnsitions. It is thortilly possil tht T i hs mor thn on onnt omponnt. In tht s, th lrgst omponnt is Algorithm 1 Gnrtion of Winow Domposition 1: funtion WINDOWDECOMPOSITION((S, Σ, T, s 0 )) 2: F SAT formul (1) for proprty noing 3: T = T T ontins th unovr trnsitions 4: i 1 Winow inx 5: whil T o whil unovr trnsitions 6: Cost t T t mx unovr trnsitions 7: T i PsuoBoolnOptimiztion(F, Cost) 8: T i LrgstConntComponnt(T i ) 9: Σ i Th lpht ssoit to T i 10: S i Stts from S jnt to T i 11: s ini s 0 S i? s 0 : i 12: W i = LT S(S i, Σ i, T i, s ini ) 13: T T \ T i 14: i i + 1 15: rturn {W 1,..., W n } Th WD slt to rt nw pross winow W i. Th initil stt n ithr s 0 or i pning on whthr s 0 longs to on of th slt trnsitions. Th min loop of th lgorithm itrts until ll th trnsitions of th LTS hv n ovr y som winow. E. Implmnttion tils Th urrnt lgorithm hs n implmnt to isovr winow ompositions with forwr n kwr prsistn tht n ltr l to hoi-fr Ptri nts. To voi n xssiv frgmnttion of th slt trnsitions into multipl onnt omponnts, th onntnss onition isuss in St. IV-C hs n to formul F. Th PBLi lirry [9] hs n us for Psuo-Booln optimistion n Minist [10] s SAT solvr. Th mximistion of th ost funtion hs n implmnt y inrmntlly strngthning th onstrints tht no th ost funtion until th prolm oms unstisfil [9]. F. Trmintion n Compltnss Lt us ll simpl trs thos tht orrspon to simpl pths in th LTS, i.., thos tht o not visit th sm vrtx mor thn on. Trmintion is lwys gurnt y onsiring th following osrvtion: ny simpl tr is winow with forwr n kwr prsistn. Thrfor, if T, it is lwys possil to fin winow onsisting of simpl tr tht ovrs t lst on of th trnsitions in T n fulfils prsistn. Hn, T is ru t h itrtion. Not tht th ov rgumnt lso provs th ompltnss of th lgorithm: it isovrs vli winow omposition for ny input LTS. Th prvious osrvtion lso provis n uppr oun on th numr of itrtions rquir to isovr WD: th numr of simpl trs nnot lrgr thn th minimum numr of trs rquir to ovr ll trnsitions of th LTS. In prti, th numr of itrtions is sustntilly smllr sin vry winow ovrs multipl trs xhiiting onurrnt hviours.
6 V. DERIVING WAKE-UP CONDITIONS AND MARKINGS In this stion w prsnt mtho for riving Booln qutions for wk-up onitions n mrkings. Th mtho is intgrt with th winow omposition lgorithm prsnt in Stion IV n is vill s prt of th WORKCRAFT frmwork [1]. A. Driving wk-up onitions Lt W winow ovring th st of stts S w. Th truth tl of its wk-up onition (W, s) n spifi s follows: (W, s) = 0 for ll stts s tht r outsi th winow, tht is: s / S w : (W, s) = 0. (W, s) = 1 for ll stts s whr w n ntr th winow, tht is: s S w : ( s / S w : s s) (W, s) = 1. Othrwis, for ll stts s tht r ovr y th winow ut nnot ntr from outsi, th vlu (W, s) is unonstrin, i.. it is on t r. In, thr is no hrm if th wk-up onition is tru whn th winow is lry tiv, n w n us this flxiility to otin simplr qution for th wk-up onition. For th xmpl isuss in Stion II-B, th ov finition yils th following onstrints for W 1 : { (W1, s6) = (W 1, s7) = (W 1, s8) = (W 1, s9) = 0, (W 1, s0) = (W 1, s4) = 1. Sin th winows r rprsnt y sf Ptri nts, it is nturl to rprsnt thir stts y sts of mrk pls. This ls to th following stnr Booln logi synthsis prolm tht n solv y th ESPRESSO tool [11], whr sts r no y Booln vtors: Mrking s Booln vtor Wk-up onition (W 1, s) {p5, p8} (1, 0, 0, 1, 0) 0 {p6, p9} (0, 1, 0, 0, 1) 0 {p7, p9} (0, 0, 1, 0, 1) 0 {p5, p9} (1, 0, 0, 0, 1) 0 {p7, p8} (0, 0, 1, 1, 0) 1 {p6, p8} (0, 1, 0, 1, 0) 1 By synthsising th ov into Booln qution, on otins vry suint wk-up onition: (W 1, s) = p5 p8. It is not iffiult to prov tht n qution with only positiv litrls n lwys otin for wk-up onitions. This oms from th ft tht th tivtion of winow lwys oinis with th rrivl of tokn in som othr winows. This monotoni hviour gurnts positiv rltionship twn st of vrils n th wk-up onitions. Th tils of th proof r out of th sop of this ppr. W implmnt th positiv mo in our tool to riv wk-up onitions without ngtiv litrls, whih prous (W 1, s) pos = (p6 p8) (p7 p8) in this s. On n s tht th otin qution my simplifi y ftoring out th ommon trm p8. Our tool n riv suh ftor qutions if rqust y th usr. In our xmpl th rsult is (W 1, s) opt pos = (p6 p7) p8, s xpt. B. Driving wk-up mrkings To riv wk-up mrkings w us similr pproh. Lt m(w, p, s) stn for th wk-up mrking of pl p in winow W n stt s. Its truth tl is s follows: m(w, p, s) = 1 if w n ntr W in stt s n p must mrk. m(w, p, s) = 0 if w n ntr W in stt s n p must unmrk. Othrwis, th vlu m(w, p, s) is unonstrin. In, w only r out th vlu in th stts whn w ntr winow W n wk it up. For th xmpl isuss in Stion II-A, th ov finition yils th following onstrints for W 1 n p0: { m(w1, p0, s0) = 1, m(w 1, p0, s4) = 0. This n quivlntly xprss y th following truth tl: Mrking s Booln vtor Wk-up mrking m(w 1, p0, s) {p7, p8} (0, 0, 1, 1, 0) 1 {p6, p8} (0, 1, 0, 1, 0) 0 It is not iffiult to s tht th truth tl n synthsis into vry simpl qution m(w 1, p0, s) = p7. VI. APPLICATIONS Pross winows r usful whnvr on ns to unrstn th hviour of omplx systm whr onurrny n hoi r intrtwin. Suh systms oftn mk ommonly us hviourl mols, suh s Ptri nts, iffiult to omprhn, s w hv lry monstrt in th motivtionl xmpl in Stion I. In this stion w isuss two pplition rs whr pross winows r prtiulrly usful: synhronous iruit sign n pross mining. A. Asynhronous iruit sign Asynhronous iruits oprt without glol lok signl, n iniviul gts n thrfor fir truly onurrntly. Dsign of suh iruits is vry hllnging tsk, n vn iruits with fw gts my rquir sustntil spifition n nlysis ffort from th signr. Signl Trnsition Grphs (STGs) r ommonly us s th spifition lngug in synhronous iruit sign [12]. STGs r Ptri nts whos trnsitions r lll with signl trnsitions, i.. vnts orrsponing to signls hnging thir vlu from 0 to 1 (th rising trnsition, not y + for signl ) n 1 to 0 (th flling trnsition, not y ). Th ky nfit of using STGs ompr to trnsition systms is tht STGs n rprsnt onurrny vry omptly using th tru onurrny smntis, whih is vry nturl for synhronous iruits. Trnsition systms, on th othr hn, us th intrlving smntis for rprsnting onurrny, whih ls to inqutly lrg mols. Dspit th ft tht STG mols of synhronous iruits r oftn ompt, thy my still hr to unrstn, prtiulrly y usrs who r not xprts in th onurrny thory, suh s inustril hrwr nginrs. Pross winows hlp
7 () z lt snrio. () z snt snrio. Fig. 4: STG spifition of si uk ontrollr [13]. mk STG mols sir to unrstn y strting wy th hois in th systm, n rprsnting h hviourl snrio sprtly. Consir n xmpl STG shown in Fig. 4. Th STG spifis th hviour of si synhronous uk ontrollr us in on-hip powr rgultors [13]. Th STG is rsult of rful nlysis of th ontrollr y th signr, who mnully xtrt thr hviourl snrios n rprsnt thm s sprt rnhs of th STG in orr to hiv lr rprsnttion of th ontrollr s hviour. Not tht th STG is not ntirly trivil: th two uppr rnhs strt with th sm signl trnsition uv +, whih inits tht th snrios prtilly ovrlp. Fig. 5: Buk STG synthsis from th trnsition systm. To illustrt tht riving th STG spifition in Fig. 4 mnully is not trivil, Fig. 5 shows th STG synthsis utomtilly from th trnsition systm of th uk ontrollr y PETRIFY [14]. As on n s, on th rful mnul lyout s on th insight into hviourl snrios of th systm is rmov, th STG oms hrr to unrstn. Th winow omposition mtho prsnt in this ppr n utomtilly isovr th thr snrios from th unrlying trnsition systm, without ny mnul intrvntion from th signr, thry sustntilly rsing th spifition n visulistion ffort. Th xtrt winows r shown in Fig. 6. Thy hv hv th sm wk-up mrking with th only tokn on th o uv + r, n symmtri wk-up onitions tht monitor ths pls. Th spt of ovrlpping snrios is vn mor vint in th winows rprsnttion: th mrk grph orrsponing to th z snt snrio is sugrph of th z lt snrio, whih mns tht whnvr th formr winow is tiv, th lttr is tiv too. Rprsnting iruit hviour y olltion of mrk grphs n lso nfiil for th following rsons: () z rly snrio. Fig. 6: Fully utomt winow omposition of th uk ontrollr (lyout gnrt y Grphviz [15]). Mrk grphs r sulss of Ptri nts with prtiulrly wll unrstoo struturl proprtis. For xmpl, on n sily hrtris mrk grphs in trms of prformn n nrgy. If th proportion of tim th iruit oprts in h snrio is known from th systm sription, it is possil to ggrgt iniviul hrtristis of th snrios, otining mtri for ovrll systm prformn n nrgy onsumption. Pross winows prmit inrmntl spifition of systm, whr h snrio is sign n vrifi sprtly s mrk grph. Suh inrmntl spifition llows to voi monolithi STG spifitions tht nnot sign in ntrlis mnnr y inpnnt tms of signrs. It is possil to synthsis synhronous iruits for h mrk grph sprtly n thn omin thm using orrt-y-onstrution pproh on th sis of wkup onitions n mrkings. This n potntilly prou iruits tht r sir to implmnt using gts vill in stnr thnology lirris. B. Pross mining Anothr potntil pplition r is pross mining [16], mor spifilly th isovry of pross mols from xution trs. Som of th prvious work in this r [3] inspir th onpt of pross winows prsnt in this ppr. Trs: Fig. 7: Log of trs n Lll Trnsition Systm. W illustrt th ppliility of pross winows to th r of pross mining with simpl xmpl in Fig. 7. On th lft, log of trs is shown with vnts longing to th lpht {,,,, }. On th right, th LTS otin from
8 () Trnsition systm. () Ptri nt mol utomtilly isovr y PETRIFY. () Winow omposition (shr stts highlight). () Pross winows {W 1, W 2, W 3}. :w 2 :w 3 :w 2 :w 1 () Winow ovrly. Fig. 8: Mining pross mol for th log in Fig 7. th trs is pit. This LTS hs n gnrt with th prfix multist onvrsion [17], in whih prfixs with th sm numr of ourrns of h vnt l to th sm stt (rgrlss of th ourrn orr). Fig. 8() pits th Ptri nt otin y PETRIFY [14] using th thory of rgions n ll splitting. Th strutur of th Ptri nt is vry intrit n givs no intuition on how th pross hvs. Th min rson is us th nt rprsnts multipl hviours of iffrnt ntur unr th sm strutur. By xtrting hoi-fr pross winows shown in Fig. 8(), th strutur of th hviour oms muh mor visil. In mny ss, th logs of systm ontin hviours of susystms tht intrt with iffrnt uslity/onurrny rltions. Pross winows ontriut to istill th hin su-prosss n show th vrity of hviours tht wr rtifiilly ln in th sm log. Furthrmor, it is possil to omin pross winows with othr thniqus for visulising snrios. For xmpl, Fig. 8() shows how th winows n ovrli y mthing thir ommon frgmnts n using Booln onitions to tivt iniviul vnts. By stting w 1 w 2 w 3 = 010, i.. Wk-up onition: p5 Wk-up mrking: p0 = p8 p1 = p9 p3 = 1 () Winow W 1. () Ptri nt. Fig. 9: Aritrtion. Wk-up onition: p0 Wk-up mrking: p5 = p3 p6 = p4 p8 = 1 () Winow W 2. hoosing winow W 2, on n rmov vnts n whos onitions vlut to 0, n otin th son snrio. Suh ompt ovrli rprsnttions n utomtilly prou y xisting pross mining thniqus [18]. VII. DISCUSSION In this stion w stuy two mor xmpls of winow omposition, tht highlight th spts of slility n flxiility of th propos pproh whn it is ppli to rllif systms tht oftn ontin suh sours of omplxity s rsour ritrtion n OR uslity.
9 ( n ) ( k, us thr r xtly n ) k iffrnt hoi-fr snrios. This shows th limittion of th propos pproh whn winows r rstrit to hoi-fr nts. A possil solution to this prolm is to ovrly winows, s sri in Stion VI-B, whih rsults in ompt rprsnttion vn for xponntilly mny winows. Altrntivly, on n pt th winow omposition lgorithm to llow simpl rsour ritrtion pttrns within winows. () Trnsition systm. () Ptri nt. () Winow omposition (shr stts highlight). B. OR uslity OR uslity [19] is known to iffiult to mol with Ptri nts, s it rquirs ithr 2-sf pls or vnt splitting. Fig. 10 shows n xmpl of systm with OR uslity: th vnt my us ithr y or y. Th Ptri nt in Fig. 10() splits vnt n to mol OR uslity, whih mks th nt iffiult to unrstn n introus n spt of hoi into th mol, vn though th originl trnsition systm is prsistnt. Fig. 10() shows how th trnsition systm n ompos into union of two kwr-prsistnt trnsition systms. Th rsulting winow omposition is shown in Fig. 10(); it ontins two winows us th originl trnsition systm is not kwr prsistnt. Our implmnttion of th winow omposition lgorithm llows th usr to hoos whthr to rspt kwr prsistn uring th omposition pross or not, us in som situtions it my nfiil to isovr winows tht ontin OR uslity, s thy my orrspon to nturl hviourl snrios of th systm. Wk-up onition: p8 Wk-up mrking: () Winow W 1. p0 = 1 p3 = 1 A. Rsour ritrtion Fig. 10: OR uslity. Wk-up onition: p3 Wk-up mrking: () Winow W 2. p5 = 1 p8 = 1 Fig. 9() shows trnsition systm of rqust-grnton ritr. Th ritr pts two onurrnt rqusts r1 n r2 n issus t most on grnt g1 or g2 for pning rqust. Th rqust is susquntly rmov y th orrsponing on vnt 1 or 2. Th orrsponing Ptri nt in Fig. 9() is wll-known n pturs th hviour of th systm in vry lr n onis wy. Furthrmor, it is sll rprsnttion in th following sns. If on ns to gnrlis th mol to sri k-of-n ritr for hnling n rqusts y issuing t most k grnts t tim, on n simply us n rqust-grnt-on loops ritrting vi shr rsour pl initilly ontining k tokns. Now onsir th winow omposition shown in Fig. 9() n th winows in Fig. 9(,). Th winows r hoi-fr, s sir, howvr, this rprsnttion is not sll. In, k-of-n ritr rquirs n xponntil numr of winows VIII. RELATED WORK Pross winows r rlt to n inspir y ro oy of work on snrio-s systm spifition n nlysis mthos [20]. In prtiulr, Mssg-Squn Chrts (MSCs) [21] n Liv Squn Chrts (LSCs) [22] r wily us for th spifition of protools of th ommunition twn systm omponnts y mns of mssgs. MSCs n LSCs r support y tools n n utomtilly trnsform to trnsition systms for furthr mol-hking n synthsis. Som pprohs us Ptri nt s mols to rprsnt iniviul snrios, for xmpl olts [23] us prtil runs to rprsnt snrios n nti-snrios (i.. snrios tht must not our). Untnglings [24] lso rprsnt systm hviour y olltion of yli prtil runs, ut for th purpos of ffiint stt rprsnttion n mol-hking, inst of spifition. Strutur Ourrn Nts [25] introu fmily of rihr rltions twn snrios, suh s hviourl strtion. In th r of synhronous iruit sign, on n lso highlight th work on Conitionl Prtil Orr Grphs [26] n Prmtris Grphs [27], whr prtil orrs r us for th spifition of multi-snrio hrwr systms, suh s prossors n on-hip ommunition ontrollrs. Th ky iffrntiting spt of this ppr is th utomt isovry of snrios from trnsition systms, irtly inspir y [3], whih llows to unrstn th hviour
10 of omplx xisting systms tht hv not n mnully ompos into snrios. Furthrmor, unlik mny of th ov-mntion pprohs, th propos mtho is not limit to yli snrios n n smlssly hnl oth yli n yli snrios, hoosing th most pproprit formlism for thir rprsnttion, s hs n monstrt in Stion VI. Th min iffrn from th pproh prsnt in [3] is th ovrility of hviours. In [3], th xtrtion of slis ws orint to isovring pross mols otin from logs. In tht s, h tr of th log ws ompltly ovr y t lst on sli. Th work prsnt in this ppr is mor gnrl in th sns tht winows ovr ll hviours of th LTS ut trs my trvl ross iffrnt winows. IX. CONCLUSIONS Complx prosss oftn hv intrit rltionships mong th prtiipting vnts. Hving monolithi mol to rprsnt suh hviours oftn rsults in struturs tht o not giv lr intuition on th su-prosss hin insi th omplxity of th glol pross. Pross winows is formlism tht hlps istilling n isovring su-prosss with struturlly simpl rltionships. Evry winow mols prtil hviour of th omplt systm n hs simpl proprtis tht ontriut to ttr unrstning of th intrtion mong vnts. An importnt spt of th pross winows mtho is tht it is not rstrit to ny prtiulr st of proprtis of isovr winows. In this ppr w hv xplor prsistnt winows to otin hoi-fr snrios. Howvr, on n nvisg othr winow proprtis to us to riv winows with othr usful fturs. Automtion is nothr importnt spt of pross winows. If th sir proprtis r simpl n lol, ffiint lgorithms for th isovry of winows n riv, thry mking th intrtion with th usr simpl, prtil n intrtiv. ACKNOWLEDGEMENTS W r grtful to Alx Ykovlv for inspiring isussions. This rsrh ws prtilly support y EPSRC grnt EP/L025507/1 (A4A), n y funs from th Spnish Ministry for Eonomy n Comptitivnss n th Europn Union (FEDER funs) unr grnt TIN2013-46181-C2-1-R n th Gnrlitt Ctluny (2014 SGR1034). REFERENCES [1] Th Workrft frmwork hompg. http://www.workrft.org/, 2009. [2] I. Polikov, D. Sokolov, n A. Mokhov. 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