Bayesian Networks of Dynamic Systems (Version du 28 Mars 2007)

Transcription

1 Hilittion à Diriger les Reherhes mention Informtique Université de Rennes 1 Byesin Networks of Dynmi Systems (Version du 28 Mrs 2007) Eri Fre Soutenue le 14 juin 2007, devnt le jury omposé de Président Mihel Rynl, Prof. d Informtique, Univ. Rennes 1 Rpporteurs Exminteurs Stéphne Lfortune, Prof. of EECS, Univ. of Mihign Aln Willsky, Prof. of EE, MIT, Cmridge (US) Glynn Winskel, Prof. of CS, Univ. of Cmridge (UK) Alert Benveniste, DR INRIA, IRISA, Rennes Christophe Dousson, Resp. d Eq. de Reh., Frne Teleom R&D Alessndro Giu, Prof. of EE, Univ. de Cgliri IRISA/INRIA Cmpus de Beulieu Rennes edex Eri.Fre@iris.fr

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3 Contents 1 Introdution Motivtion A distriuted pproh to monitoring prolems Overview of our ontriution Orgniztion of the doument Historil perspetive Grphil models of intertions Systems nd their grphs Systems Exmples Grphs of ompound system Distriuted redution lgorithms The redution Prolem Messge pssing lgorithm Turo lgorithms Involutive systems Summry Networks of dynmi systems Dynmi systems nd their ompositions A tegory of multi-lok utomt Composition y produt Composition y pullk Grphs ssoited to multi-lok system Diret grph Dul grph Dignosis prolem Semntis Ojetives The dignoser pproh Distriuted dignosis : the lnguge pproh Dignosis in terms of lnguges Dignosis in terms of trjetories Extensions nd drwks

4 3.5 Trjetory sets in the sequentil semntis Trellis utomton Time-unfolding of n utomton Vritions round the height funtion Ctegoril properties Distriuted dignosis : the trellis pproh Centrlized dignosis, single sensor Centrlized dignosis, severl sensors Distriuted dignosis Towrds true onurreny semntis Summry True onurreny semntis Networks of utomt s synhronous systems Multi-lok nets nd their omposition A tegory of multi-lok nets Composition y produt nd pullk Grphs ssoited to multi-lok system Trjetory sets in the true onurreny semntis Unfolding of net Ftoriztion property Distriuted dignosis : the unfolding pproh Projetion Centrlized dignosis Distriuted dignosis Exmple Involutivity Augmented rnhing proesses Definition Key property Opertions on ABP : produt, pullk, projetion Seprtion theorem Wek involutivity Summry Trellis unfolding for onurrent systems Trellis nets Definition Trellis proess nd time unfolding of net Ftoriztion properties Reltions to unfoldings Vritions round the height funtion Nested o-refletions Distriuted dignosis : n exmple Summry

5 6 Applitions, ontrts, tehnology trnsfer MAGDA MAGDA VDT Conlusion Summry of results Diretions for future work Tehnil extensions Reserh diretions Aknowledgemen41 5

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7 Chpter 1 Introdution (...) diviser hune des diffiultés que j exmineris, en utnt de prelles qu il se pourrit, et qu il serit requis pour les mieux résoudre. (...) onduire pr ordre mes pensées, en ommençnt pr les ojets les plus simples et les plus isés à onnître, pour monter peu à peu, omme pr degrés, jusques à l onnissne des plus omposés ; et supposnt même de l ordre entre eux qui ne se préèdent point nturellement les uns les utres. René Desrtes, in Disours de l méthode 1 (1637) 1.1 Motivtion The deomposition priniple splitting omplex tsk into simpler sutsks, hs founded dedes of tehnologil hievements. Most systems we ommonly use every dy involve omplex hins of retions etween omponents of different sles, tht omine to form the servie we expet from them. Assemling omponents to form more powerful system is so effiient nd widespred tht it lmost forms ommonple to mention it. However, onsidering with more ttention the omplexity levels of urrent systems, nd the wy they re designed tody, suggests tht this prdigm hs rehed some limittions. 1. Size. The most ovious limittion : The explosive numer of elements involved in some pplitions now rehes omplexity levels tht go fr eyond wht single person n mster. 2. Historil onstrints. Very often, redesign from srth hs eome intrtle or simply too expensive, or is impossile for mtters of downwrd 1 Among the four priniples of the method, these re the 2nd nd 3rd ones. In sustne, 1/ tke nothing for grnted, 2/ divide omplex prolem into simpler su-prolems, 3/ understnd eh elementry su-prolem, nd reomine, 4/ don t forget nything! Summrized s divide nd onquer y fst reders. 7

8 omptiility. So one is ound to ontinuously upgrde prts of n existing system or softwre, whih leds to disrepny etween reent nd old tehnologies, reent nd old design prdigms. 3. Heterogeneity. In the sme wy, there is lso trend to rpidly ssemle off-the-shelf omponents, of different genertions nd mnufturers, in order to follow the mrket demnd or new offers proposed y ompetitors. This produes systems tht sometimes hve unexpeted ehviors, whene the neessity of hevy test proedures. As mtter of ft, most softwres or servies tody re delivered in ontinuous flow of versions, following ontinuous flow of ug reports. 4. Open systems. In mny ses, ompound systems re no longer losed systems, like hip, omputer or plne, for whih the mnufturer ould theoretilly mster ll omponents. They re rther open systems, only prtilly known to people in hrge of their monitoring. In prtiulr in the field of teleommunition networks, or of distriuted softwres. 5. Unstrutured systems. After dedes of hierrhil deomposition into omponents, new design prdigms pper under the form of peer rhitetures, where omponents re oth lients nd servers of one nother. Muh less intuitive ojets in terms of mngement. 6. Dynmi rhitetures. Moreover, the struture itself of some urrent softwre systems is no longer stle nd designed one for ll, ut my e uilt on demnd. This ws prtly the se in teleommunition networks, ut it eomes entrl feture in peer-to-peer networks or in we servies, two fst growing pplition prdigms. One my e hppy with this sitution, s fr s the min funtion of omplex system is glolly stisfied, nd ontinuously improved y test nd modify proess. But of ourse, this n t e suffiient for ritil pplitions. New tools nd onepts re ontinuously needed to ssess eforehnd whether system fulfills its ojetives or not, whether it is error-free or not. And one lrge system hs een deployed, similr diffiulties remin t run time to monitor it nd nlyze its ehvior, whih is the topi of this doument. As typil domin where these issues re eoming of ritil importne, let us mention teleommunition networks nd servies mngement. It is y now onsidered tht network elements (NE) re so omplex nd offer so mny fetures nd djustle prmeters, menwhile networks inrese in size nd heterogeneity of equipment nd funtions, tht the trditionl monitoring of network y diretly essing nd tuning NEs hs eome impossile. It is ommonly onsidered tht humn opertor needs one yer to mster new NE tehnology nd e le to prmeterize the network he supervises. Reserh groups like the NMRG (Network Mngement Reserh Group) t IRTF (Internet Reserh Tsk Fore), or the Europen reserh network EMANICS (MANgement of Internet tehnologies nd Complex Servies), re now orienting reserh to new mngement onepts. Under 8

9 the generi nme of utonomi ommunitions, ojetives like self-onfigurtion, self-heling, self-dpttion, self-xxx revel trend to strt the inherent omplexity of systems nd serh for high-level progrmming mehnisms for teleommunition networks. Idelly, one should e le to progrm network y ssigning it servie-level ojetives. Wht tehnologies will ridge the gp etween these high-level ojetives nd low-level mngement is still unler nd remins very tive reserh field. One trend is to push down these high-level requirements, under the form of poliy-sed mngement (essentilly for performne mngement). Another strong tendeny fvors proilisti methods, oth to model network ehviors (performne nd onfigurtion mngement), or to understnd its ehvior (lerning methods or sttistil tehniques for fult mngement nd event orreltion). The reserh work we summrize in this doument ddresses omplexity issues in the reverse diretion. We strt from trditionl model-sed pprohes to some monitoring prolems, tht were suessful so fr for smll size systems, nd propose methodology to extend them to possily lrge networks of omponents. The ide is quite simple : the omplexity of networked systems preisely omes from the existene of mny interonneted funtions nd elements. Why not turning this to our dvntge nd imgining monitoring rhiteture tht would itself e distriuted? Idelly, this would oth solve slility issues, nd llow nturl upgrde of monitoring rhitetures s the struture of the supervised system is updted. As mtter of ft, we ll see tht our pproh nturlly leds to the fshionle ide of peer to peer monitoring rhitetures, ut with sound lgeri sis. Most of this reserh ws motivted y trget pplition : filure dignosis in teleommunition networks. The results we otined hve een suessfully implemented nd tested on different network tehnologies, in oopertion with industril prtners 2. But eyond these diret pplitions, the theory seems rih nd promising enough to ddress some of the diffiulties mentioned ove. For exmple systems with vrying struture, like We Servies. Very likely lso, off-line prolems like model heking for lrge omponents n e ddressed with this pproh. In summry, etween smll systems tht n e studied s whole, nd lrge ones tht n only e uilt nd tested, or modeled with proilisti methods, there seems to e some essile lnd to explore. 1.2 A distriuted pproh to monitoring prolems We fous on disrete event dynmi systems (DEDS), nd in prtiulr on distriuted systems otined y ssemling lrge numer of omponents, tht we ould lso ll networks of dynmi systems. Suh systems very rpidly eome intrtle s their size ugments. This is due to omintoril explosions tht tke ple oth in their stte spe, nd in their trjetory spe. Beuse of these om- 2 Experiments hve een rried out for SDH optil network, MPLS networks, sumrine line terminl equipment, nd GSM rdio ess network. Altel R&I is urrently evluting the introdution of this tehnology into ALMAP, its orporte network mngement pltform. 9

10 intoril explosions, glol (or entrlized) pprohes developed for monitoring prolems re no longer pplile. By monitoring prolem, we enompss prolems like supervisory ontrol, optiml ontrol, optiml stte/trjetory estimtion, or dignosis prolems. Severl uthors hve proposed to ddress the hllenge of distriuted systems y mens of distriuted (or modulr) methods. The entrl ide is to solve the trget monitoring prolem y prts, t the sle of single omponent, in suh wy tht omining lol/prtil solutions gives the glol one. Speifilly, there exist two strtegies to do so 3 : In the deentrlized monitoring rhiteture, lol supervisor is tthed to eh omponent (or group of), nd hs only lol knowledge : it only knows the model of tht omponent, plus interfe informtion with the rest of the system, nd only hs ess to oservtions/mesurements oming from tht omponent. Lol supervisors perform some omputtions nd forwrd their results to oordintor in hrge of ssemling them. This oordintor is supposed to ignore everything out the supervised system, nd hs miniml omputtion pilities, whih mens tht most of the work is performed y lol supervisors. In the distriuted rhiteture, one is not so muh interested in omputing glol solution to the monitoring prolem, like glol dignosis, estimtes of glol sttes, et. On the ontrry, only lol views of these glol solutions re of interest, tht is their projetions on eh omponent. Therefore the oordintor eomes useless. The monitoring rhiteture simplifies into olletion of lol supervisors, one per omponent, hving lol knowledge nd oordinting their work with supervisors of neighoring omponents to provide set of oherent lol views. The methodology we propose elongs to the seond lss, whih n e onsidered s generliztion of the first one, where the neessity of oordintor is relxed. The dvntge is ovious in terms of slility : eh time new omponent is inorported or repled into the system, one only hs to onnet/reple its orresponding lol supervisor to upgrde the monitoring rhiteture (Fig. 1.1). The ft tht the onnetivity of the supervising rhiteture must e isomorphi to the intertion struture etween omponents in the system is not sul nd will e ommented in the next hpters. Notie lso tht lthough the knowledge out the glol solutions to the monitoring prolem is distriuted in this pproh, the informtion is nevertheless present nd ville for stndrd post-proessings (like result report, tion deision, et.). At this point, we mde no distintion etween modulr nd distriuted proessings. The modulrity refers to prolem tht n e solved y prts, where eh omputtion module is sed on limited knowledge. Typilly, omputtions 3 In this lssifition, we omit ontriutions tht do not tke into ount the modulrity of the supervised system. For exmple pprohes where severl sensors ollet different oservtions on unique omponent. Although these pprohes re interesting in terms of oopertion etween sensors, the modulr proessing is generlly s omplex s glol proessing sed on ll oservtions. Our gol here is to redue omplexity, in order to pture lrge systems. 10

11 Distriuted monitoring plne Supervised system plne lol supervisor intertion omponent new omponent ollortion new lol supervisor Figure 1.1: A network of dynmi systems, nd its distriuted monitoring rhiteture. performed y lol supervisor, knowing only one omponent nd oservtions tht it produed. The expression distriuted proessing goes further y ssuming tht the prtil omputtions re performed t different lotions, nd thus require ommunitions etween modules. This introdues sheduling issues in the prolem : one first hs to determine wht should e omputed lolly, wht to ommunite to neighors, ut lso when ommunition should tke ple, nd wht to do with delyed (or lost) messges. These protool onerns re very sensitive in some pprohes [90]. In the frmework presented here, we will onsider synhronous distriuted systems, nd the distriuted monitoring lgorithms will lso e ompletely synhronous. Therefore modulrity will e equivlent to distriution, nd we shll not distinguish them. 1.3 Overview of our ontriution As suggested y the title, this doument desries n ttempt t ssemling two disonneted sets of results, developed in different ommunities nd with pprently unrelted ojetives. Byesin networks. The term Byesin Network 4 refers to grphil models displying the orreltion struture of olletion of rndom vriles. Let V = {V k, 1 k K} e finite set of vriles, nd let V = V 1... V N e overing of V, with V n V, 1 n N. We denote y v funtion ssoiting to eh vrile in V vlue of its domin, nd y v i we denote the restrition of v to V n. We lso denote v k vlue of vrile V k. A joint distriution P V on vriles V n e speified in terms of so-lled potentil funtions φ n defined on the v n nd tking vlues in R. Speifilly P(v) = 1 N Z exp { φ n (v n )} (1.1) 4 Sometimes lso lled Mrkov rndom field, grphil model, or elief network, ording to the ommunity using it. n=1 11

12 where Z is normlizing ftor. Eh suset V n is lled lique. Intuitively φ n defines (soft) onstrints on the elements of lique V n, nd y suitly omining ll these lol onstrints, one speifies the glol orreltion struture in V. To prepre the nlogy with dynmi systems, we ll the pir S n =(V n,φ n )omponent. S 2 S 1 V 2 V 5 V 2 V 5 V 1 V 4 V 7 V 8 V 1 V 4 V 7 V 8 V 3 V 6 S 4 V 3 V 6 S 3 d V 2 V 5 S 2 V 1 S 1 S 4 V 4 S 2 V 7 V 8 S 1 S 4 S 3 V 3 V 6 S 3 Figure 1.2: Four grphil representtions of dependenies etween vriles V k nd/or omponents S n. The intertions in V dmit severl grphil representtions. The most diret : s hypergrph H =(V, {V n } 1 n N ). Every vrile in V is node, nd eh suset V n defines hyper-edge (Fig. 1.2-). As grph G =(V,E), still with vriles s nodes. Two vriles re relted y n edge of E iff they pper in the sme V n, for some n. Or equivlently, G restrited to V n is omplete grph (Fig. 1.2-). As iprtite grph G =(V, S, E). One still hs vrile V s first set of nodes, nd the seond one S = {1,..., N} orresponds to the N systems defined y the V n. A vrile V V is relted y n edge to system n iff V V n (Fig. 1.2-). As dul grph G =(S, E) where nodes in S = {1,..., N} still represent the V n. There is n edge in E etween n nd m iff V n V m (Fig. 1.2-d). These grphil representtions hve two min dvntges. 1. First of ll, they n e diretly interpreted in terms of onditionl independene sttements. Consider Fig for exmple. Vriles {V 4,V 7 } seprte {V 1,V 2,V 3,V 5,V 6 } from {V 8 } in the sense tht removing V 4 nd V 7 from 12

13 the grph disonnets these two sets. This property immeditely entils tht (V 1,V 2,V 3,V 5,V 6 ) nd V 8 re onditionlly independent given (V 4,V 7 ) for P : P(v) = P(v 1,v 2,v 3,v 5,v 6 v 4,v 7 ) P(v 8 v 4,v 7 ) P(v 4,v 7 ) (1.2) s it n e heked diretly from (1.1). The grph is thus summry of set of onditionl independene reltions. Whene the nme Byesin network : (1.2) is Byes formul. 2. Seondly, preisely euse of these onditionl independene reltions, estimtion prolems n e resolved y prts. These prolems typilly tke the following form : one oserves the vlue of some vriles in V nd wishes to determine the most likely vlue of ll the others. In the simplest ses, i.e. when the intertion grph of fig. 1.2-d is tree, the resolution tkes the form of messge pssing lgorithms (MPA), where some omputtions re performed t the sle of single lique V n, nd where neighoring liques exhnge messges. The most fmous exmples of suh lgorithms pper for Mrkov hins, i.e. Byesin networks where the liques re orgnized in single string : the Viteri lgorithm, the soft output Viteri lgorithm (SOVA), the Klmn filter, the Rough-Tung-Strieel lgorithm, the Bhl- Coke-Jelinek-Rviv (BCJR) lgorithm, the forwrd-kwrd lgorithm, the [min,mx]-[sum,produt] lgorithm, the sum-produt lgorithm, dynmi progrmming, the elief propgtion re ll exmples of MPA 5. For more omplex grphs, MPA n still e pplied. They re theoretilly suoptiml, ut yield exellent results in prtie, s it ws reveled y the itertive lgorithms for deoding turo-odes. The reder will hve notied tht messge pssing lgorithms re form of distriuted proessings. We re preisely going to elorte on this remrk. Networks of dynmi systems. The simplest model of disrete event dynmi system (DEDS) tkes the form of n utomton A =(Q, T, v 0 ). A is omposed of single stte vrile V tking vlues in the finite set Q of possile sttes, nd initilized t v 0 Q. T Q Q is finite set of trnsitions : trnsition t = (v, v ) T n fire when V tkes vlue v. After the firing, A is in stte V = v. A run of A is thus sequene σ = v 0 [ v 1 [t 2 v 2... v l 1 [t l v l... suh tht t l =(v l 1,v l ) T. With very simple ide, this lss of DEDS n e extended to enompss muh more omplex systems : insted of single vrile V, we n define utomt operting on severl stte vriles. Speifilly, we define tile system S s triple S =(V, T, v 0 ), where V = {V k, 1 k K} is set of vriles with finite domins, T is finite set of tiles, nd v 0 is the initil stte of the system. Apprently, we only introdued vetor-vlued stte vrile. The originlity omes from the ft tht trnsitions of T, tht we ll tiles, do not operte on ll vriles t time. A tile t =(V t, vt, v+ t ) T modifies only vriles in V t V. Speifilly, t n 5 Often the sme lgorithm ppers with different nmes, ording to the ommunity tht (re)disovered it! 13

14 fire when the stte v of the system restrited to V t tkes vlue vt. After the firing, these vriles re hnged to vlue v t +, nd vriles in V \ V t remin unhnged (fig. 1.3). This formlism is very onvenient to define lrge systems, with numerous stte vriles, y lol dynmis. S 1 S 2 V 1 V 2 V 3 V 4 v 0 1 v 0 2 v 0 3 v 0 4 o 1 v v v v o 2 v 1 v 2 v 0 3 v 4 o 3 v v" v v 4 o 4 v 1 v" 2 v" 3 v" 4 Figure 1.3: A run of tile system mde of two omponents S 1 nd S 2, tht shre two vriles, V 2 nd V 3. The first tile firing in this run (top left) hnges only the vlue of V 1 nd V 2, nd produes the lel o 1. The min dvntge of this frmework is tht tile systems n e omposed very nturlly. There exists severl wys to define the omposition, tht we shll rell in this doument. The simplest one, t this point, is the following : let the S n =(V n, T n, vn) 0 e tile systems, 1 n N, their omposition S =(V, T, v 0 )= S 1 S 2... S N is otined y tking the union of vriles V = V 1... V N nd of tiles T = T 1... T N, nd y ssemling the initil sttes vi 0 (provided they oinide on shred vriles). The intertions ome from the ft tht two omponents S n, S m n oth red nd hnge the vlues of the stte vriles in the intersetion V n V m. The ltter thus ehve s ommunition ports. As for Byesin networks, one n ssoite grphil representtions to ompound tile system S, nd they turn out to e extly those of Fig. 1.2! Insted of potentil funtion φ n defining onstrints on suset V n of vriles, on hs omponent S n defining lol dynmis on the V n. But we need one more step to mke this nlogy opertionl. The join. In its simplest form, monitoring prolem for S ould e expressed s follows. Assume some of the tiles in the S n n emit possily rndom signl when they fire (the prodution of n lrm for exmple) tht we ll lel. S performs hidden run κ, nd the lels emitted y tiles in this run re olleted under the form of oservtions O (Fig. 1.3). Sine κ is hidden, the gol is to reover ll runs 14

15 of S tht ould hve produed O. Further, if eh omponent S n is stohsti system, one would like to reover the most likely run, s n estimte of κ. This prolem looks very muh like stndrd Hidden Mrkov Model (HMM) prolem. But here, we omplexify it little. First we ssume tht oservtions O re not olleted into single sequene of lels. Rther, lels re olleted on eh omponent S n y lol sensor, tht produes the lol oservtion set On. Our oservtion is thus the tuple O =(O1, O 2,..., O N ), nd the interleving of events in the O i is ssumed to e lost. Seondly, s mentioned in the previous setion, we im t possily lrge systems. Therefore the usul proedures for HMMs, tht operte on the stte spe of S, re just unffordle : the stte spe size explodes with the numer of omponents. We rther look for modulr methodology to solve the monitoring prolem, where omputtions would e performed t the sle of omponents S n. Finlly, we would like to perform this monitoring on-line, i.e. we wish to updte our estimtes of the hidden run κ on the fly, s new oservtions re olleted on the different sensors. This is lose enough to the MPA we skethed for Byesin networks. To estlish the onnetion, one must relize tht the ojets we re interested in re not so muh the omponents S n themselves, ut rther their sets of runs. So we must introdue time in our formlism. As illustrted in Fig. (1.3), run κ of S n e onsidered s tuple of trjetories (i.e. sequenes of events), one per vrile V k. So let us denote y V k vrile whose vlues re trjetories of V k, for V k V. We re going to use the ft tht vriles ( V k ) 1 k K form Mrkov field, in very speifi sense. Consider n opertor U tht would tke tile system S nd ompute in some form or nother the set U(S) of ll its runs κ. We will show in this doument tht U n e designed so s to e produt preserving funtor on tile systems. In other words, one hs U(S 1... S N ) = U(S 1 )... U(S N ) (1.3) where is n pproprite omposition opertor on trjetory sets. This ftoriztion property is the ounterprt of (1.1) for networks of dynmi systems, nd is t the ore of the methodology we present in this doument. Eh U(S n ) n e onsidered s potentil funtion or s the definition of lol onstrints on vriles V n = { V, V V n }. Beuse of (1.3), the intertion struture of the U(S n ) is identil to tht of the omponents S n themselves. Finlly, oservtions O n s well n e interpreted s some knowledge on the lol trjetories in U(S n ), i.e. on the vlues of vriles in V n. So we re lmost k to stti prolem tht n e solved y MPA. This forms the essentil messge of this doument : mny results otined for Byesin networks n e reyled into distriuted nd synhronous estimtion lgorithms for distriuted dynmi systems. 15

16 1.4 Orgniztion of the doument The next hpter desries n xiomti frmework designed to pture oth Byesin networks nd networks of dynmi systems. The ojetive is to desrie messge pssing lgorithms (MPA) in formlism tht enpsultes oth situtions. We define strt systems operting on vriles, tht we ompose y shred vriles. Only two opertors re useful on these systems : omposition nd projetion (or redution). We relte them y smll set of xioms, from whih mny lgeri properties n e derived. The intertion struture of ompound system n e desried y grph, on whih the stndrd seprtion riterion is equivlent to form of onditionl independene. This is suffiient to develop MPA, study their onvergene nd explore the properties of their sttionry points. Chpter 3 is first pplition of this frmework to dynmi systems, in the simplest possile setting. We define network of dynmi systems s the omposition of utomt y the usul prllel produt. This omposition is slightly modified to keep trk of omponents when they re ssemled. Suh systems re provided with the usul sequentil semntis : their runs re simply sequenes of events. The distriuted dignosis prolem is defined in this setting, nd solved in different wys. First of ll we reson on lnguges, whih highlights the rhiteture of the omputtions tht we pply ll long this doument. But lnguges re very ineffiient to enode lrge sets of runs of system, so they re inpproprite to on-line monitoring lgorithms. The notion of trellis proess is then introdued to desrie sets of runs in ompt mnner. We show tht these ojets enjoy nie ftoriztion property nd stisfy the xiomti frmework of hpter 2, whih llows us to ompute with them. The key point here,i.e. the ftoriztion property of trellis proesses, is derived y tegory theory rguments. We onlude this hpter y showing some drwks of the sequentil semntis, nd dvote true onurreny semntis to del with distriuted systems. Chpter 4 proposes first setting to hndle sets of runs in the true onurreny semntis. We first hnge the notion of omposition in order to preserve the stte vriles of omponents, rther tht merging them in ig produt stte vrile. As in the se of networks of utomt, omposition n e done either y produt or y shred omponents : oth situtions re equivlent. The notion of system this leds to turns out to e lmost equivlent to sfe Petri nets. In the true onurreny semntis, runs re prtil orders of events (or Mzurkiewiz tres), nd sets of runs n e enoded under the form of rnhing proesses. The ltter enjoy one gin nie ftoriztion property (still derived y tegory theory rguments), nd dmit nturl notion of projetion. However, the xioms of hpter 2 re stisfied only in very speifi ses. To perform omputtions in the generl se, one must introdue the notion of ugmented rnhing proess. Chpter 5 tries to go further nd explores the existene of trellis proesses for the true onurreny semntis, s n even more ompt wy of enoding sets of runs. Still in view of effiient distriuted nd on-line monitoring lgorithms. Trellis proesses n indeed e defined, nd enjoy very elegnt properties. In prtiulr, the ftoriztion property is preserved, nd nturl notion of projetion exists, whih 16

17 llows us one gin to pply the formlism of hpter 2 in some speifi ses. But the notion of ugmented trellis proess, tht would enompss the generl se, is still missing. Chpter 6 gives some snpshots t different ontrts tht guided this reserh. It desries the pplitions tht were onsidered nd some fetures of the proposed solutions. It underlines in prtiulr how the implementtions tht were experimented were either in dvne, or lte, or sometimes erroneous with respet to the development of the orresponding theory. As onlusion, hpter 7 summrizes the stte of this theory nd identifies some missing tiles in the puzzle. It lso lists numer of immedite or more futuristi extensions of this work. 1.5 Historil perspetive The ssemling of Byesin networks, distriuted dynmi systems nd tegory theory presented in this doument doesn t yet form ompletely smooth theory. But it lredy went through severl polishing phses. We riefly mention some of them to underline the enefits of rossing different sientifi ultures, ut lso to show how simple ides sometimes tke omplex wys to mterilize, wys in whih hne plys n importnt prt. The origins. The prolem tht triggered this reserh ws jointly rised in 1996 y Clude Jrd (kground in distriuted progrmming) nd Alert Benveniste (severl kgrounds, in prtiulr signl proessing nd rndom proesses). In teleommunition networks, filures generlly propgte in the net whih uses ursts of lrms t different lotions. These lrms re only prtilly ordered in time, due to the distriuted nture of the network. So the originl prolem ws to identify filures from ptterns of prtilly ordered lrms. The ide of interting omponents nd of stohsti systems were lso present t the very eginning, whih oriented us to Byesin networks. V 4 V V 1 V 2 V 4 3 t!1 V 4 V 1 V 2 V 3 t V 1 V 2 V 3 t+1 time Figure 1.4: Augmenting intertions in spe with intertions in time. Very soon however, we were fed with the diffiulty of introduing time, or dynmis, in Byesin networks. When there is single stte vrile, for exmple 17

18 in Mrkov hins, this is done y dupliting the vrile to represent its vlue t eh lok tik. Time is unfolded, in some sense. And the Mrkov hin dynmis is introdued y potentil funtions oupling vriles t time t nd t + 1. Applied to Byesin network, this priniple would mount to dd one more dimension to Fig. 1.2, perpendiulr to the pge, tht would represent the time xis (Fig. 1.4). But in distriuted system, time is not homogeneous for ll vriles : s illustrted in Fig. 1.3, some vriles my evolve while others remin onstnt. And the ples where trnsitions our depend the run! To pture this unusul feture, one would need Byesin network (or Mrkov rndom field) whose struture would depend on the vlue of some of its vriles. Unfortuntely, suh theory doesn t exist yet, nd seems diffiult to oneive. Petri nets. It ws thus hosen to ndon the ide of glol stohsti model pturing oth intertions in spe nd in time. The first frmework ws sed on prtilly stohsti sfe Petri nets. Petri nets re nturl model for onurrent systems : they desrie well the ide of lol trnsitions, nd the true onurreny semntis llows us to desrie their runs s prtil orders of events, lso lled onfigurtions. The term prtilly stohsti reltes to the ft tht trnsitions hve weight, relted to their likelihood, whih llows us to ompre runs. However no proper proilisti spe of runs ws derived. This setting ws suffiient to perform Viteri-like lgorithm, nd reover the most likely trjetory explining sequene (or tuple of sequenes) of lrms. Estimtion lgorithms were soon extended into distriuted proedures, for nets with severl omponents onneted y shred ples. The key oservtion ws tht onfigurtions of the glol net ould e split into lol onfigurtions of its omponents. The resulting lgorithm took the form of severl ooperting Viteri lgorithms : one per omponent, in hrge of reovering onfigurtions of this omponent, nd proposing to neighors its possile explntions for shred ples. After ll oservtions were proessed, elief propgtion phse ws initited to find the est omintion of lol explntions. Unfoldings. Hndling sets of onfigurtions n e quite hevy sine these ojets rpidly eome lrge. To minimize the memory spe, onfigurtions were enoded with k-pointer nottion, s in [35, 31]. In other words, the underlying dt struture we used ws the unfolding of the system, nd onfigurtion ws nothing more thn tuple of entry points in this dt struture. This reveled tht the dignosis ould proly e performed diretly in terms of unfoldings, whih ws done in [9]. Moreover, the distriuted version of the lgorithm suggested tht the unfolding of ompound system ftorized into lol unfoldings. This ws estlished diretly in [37], nd distriuted estimtion lgorithms were re-expressed in this setting : messges were now piees of unfoldings. Severl results were then otined in prllel. First of ll, the derivtion of n xiomti frmework to express generl messge pssing lgorithms (MPA) nd study their lgeri properties [38]. In prtiulr to express onditionl independene, nd study onvergene of MPA. Seondly, the introdution of ugmented rnhing proesses (ABP), s the orret frmework to express distriuted lgorithms 18

19 (ordinry rnhing proesses re not suffiient) [40]. Finlly, the expression of distriuted lgorithms in terms of event strutures, insted of rnhing proesses [41]. Ftoriztion properties nd tegory theory. By strike of fte, while I ws exploring different fmilies of event strutures to hek if ABP lredy existed, I me wre of Winskel s work on models for onurreny. I ws struk in prtiulr y result in [110], stting in ouple of lines the ftoriztion property (1.3) on unfoldings 6. The key rgument for this derivtion ws some osure result in tegory theory... whih I thus deided to investigte, motivted y the tedious proofs in [37]. And y Winskel s killing sentene in [110] : Proving these fts diretly from the unfolding onstrution is quite unwieldy - nd ompletely uninstrutive - so it is fortunte there is this strt hrteriztion of the ourrene net unfolding of [sfe] net. In sense, it ws there ll the time, euse (...), so it ws determined (...) y the tegoril set-up. The tegory theory pproh rought mny dvntges, y fousing developments on the essentil fetures, while sving us from tedious nd useless proofs. For exmple, intertions etween omponents n e expressed under the form of synhronous produts, or y shred vriles (pullks [43]), without hnging the theory. Severl other event strutures were lso quikly derived, like trellis proesses for the true onurreny semntis [45], or trellises for the usul interleving semntis [44, 42], nd their ftoriztion properties me lmost for free, thus mking them ville to distriuted lgorithms. Mrkov nets. These elements of history wouldn t e omplete without mentioning the ollortion with Stefn Hr, met t workshop on Petri nets (GDR ARP 7, June 2000) where I ws invited to give tlk out prtilly stohsti Petri nets. Just like us, Stefn ws trying t tht time to rndomize onurrent systems. The diffiulty ws to otin some equivlene etween onurreny nd stohsti independene. In other words, omponents tht do not intert should lso e independent in the stohsti sense, whih is not hieved in ny form of stohsti Petri nets. Comining our pprohes, Alert, Stefn nd I mnged to rndomize unfoldings of resonle lss of sfe nets, nd to define Mrkov property on unfoldings, sed on notion of stopping time. This resulted in Mrkov nets [39], tht ws lter refined y Smy Aes who introdued the simpler notion of rnhing ell (see Smy s thesis [1] nd [2]). Notie however tht the vilility of genuine stohsti frmework is importnt for identifition issues, or performne nlysis, ut it hs little influene on estimtion lgorithms, where one is only interested in ompring the reltive likelihoods of two trjetories ; so definition up to onstnt is suffiient. 6 This pper ws hnded to me y Smy Aes, for ompletely different purpose. 7 A CNRS funded reserh group, dedited to rhitetures, networks nd systems, nd prllelism. 19

20 20

21 Chpter 2 Grphil models of intertions The formlism presented in this hpter ims t doule ojetive. First of ll, we wnt to introdue the minimum mount of onepts llowing us to define grphil models of intertions nd messge pssing lgorithms. The ide is tht the simpler the formlism, the roder its sope. Seondly, with these simple tools, we wnt to go s fr s possile in terms of distriuted lgorithms, onvergene properties, et. We thus strt with simple definition of systems, operting on vriles, nd two opertions. The first one llows us to ompose systems, the seond one llows us to redue systems to prt of their vriles. With simple set of xioms on these two opertors, in prtiulr form of onditionl independene property, one n derive intertion grphs of systems nd messge pssing lgorithms. In some ses, onvergene properties n e estlished. This formlism is the sis on whih distriuted lgorithms will e uilt, when we move to networks of dynmi systems. 2.1 Systems nd their grphs Nottions. Speifying the nottions of the introdution, we onsider finite set V mx of vriles. A vrile V V mx tkes vlues v in domin D V. Vrile sets re denoted with sript letters V V mx, nd old-fe letters like v represent funtions over V, ssoiting vlue of D V to eh vrile V V. We ll v (lol) stte, nd, for onveniene, we sometimes represent it s tuple of vlues v =(v 1,v 2,..., v n ) ssuming there exists nturl ordering of vriles in V = {V 1,..., V n }, nd we denote y D V = D V1... D Vn the domin of vlues for v Systems We onsider n strt notion of system over these vriles, tht we generilly denote y S. To help intuition, systems n e understood s sets of tuples v mx D Vmx. Systems re provided with two opertions : omposition nd redution. The omposition S = S 1 S 2 is ssoitive nd ommuttive. The redution tkes the form of fmily of opertors Π V, indexed y sets of vriles V V mx, nd operting on single system. Intuitively, Π V (S) projets system S on vriles V. 21

22 We provide this setting with the following xioms : V 1, V 2 V mx, Π V1 Π V2 = Π V1 V 2 (1) whih expresses tht redution opertors re tully projetions. S, V V mx : Π V (S) = S (2) System S is sid to operte on vriles of V. Using (1), one n derive the existene of smller vrile set on whih S opertes, denoted y V S. The entrl xiom onerns the reltion etween omposition nd redution. Let S 1, S 2 e two systems operting respetively on V 1, V 2, then V 3 V 1 V 2, Π V3 (S 1 S 2 ) = Π V3 (S 1 ) Π V3 (S 2 ) (3) (3) expresses tht the intertion etween systems S 1 nd S 2 is ompletely ptured y their shred vriles V 1 V 2, whih thus ehve s n interfe etween the two systems. Notie lso the striking similrity of (3) with the onditionl independene sttement P(V 1, V 2 V 3 )=P(V 1 V 3 )P(V 2 V 3 ), expressing tht V 3 ptures ll sttistil dependenies etween V 1 nd V 2 for distriution P. It is well known ft tht suh independene sttements form the sis of reursive estimtion lgorithms, nd we re indeed going to uild our lgorithms on this property. To illustrte the power of this xiom, let us reple S i y Π Vi (S i ) on the right hnd side of (3), nd pply (1). One gets V 3 V 1 V 2, Π V3 (S 1 S 2 ) = Π V3 V 1 (S 1 ) Π V3 V 2 (S 2 ) (2.1) Tking V 3 = V 1 V 2 in (2.1) yields Π V1 V 2 (S 1 S 2 ) = Π V1 (S 1 ) Π V2 (S 2 ) = S 1 S 2 (2.2) whih expresses tht S 1 S 2 opertes on vriles of V 1 V 2, nturl property one ould expet from omposition. The lst xiom we introdue is essentilly tehnil : it ssumes the existene of n identity elemeni for omposition : S, S 1I = S (4) It is lso nturl to require thi do not operte on ny vrile, i.e. V 1I =, or Π (1I) =1I 1. By (1), this indues Π V (1I) =1I for ll V V mx, nd so S Π V (1I) =S for ll S. 1 A more elegnt property would e S, Π (S) =1I, ut we tully don t need this stronger ssumption. 22

23 2.1.2 Exmples Constrint systems. Rell tht lol stte v is funtion v : V D V with V V mx. So v n e onsidered s set of glol sttes where the vlue is fixed on vriles of V nd free on V = V mx \V. More speifilly, we define the spn of v s the set of ll glol sttes v mx otined y extending v into totl funtions over V mx, in ll possile wys. We define onstrint system S s set of (lol) sttes, not neessrily fixing the vlue of the sme vriles, nd we sy tht v mx stisfies (or elongs to) S if it elongs to the spn of S : Spn(S) v S Spn(v). Systems with identil spns re onsidered s equivlent : we don t distinguish them. Let V V mx, the redution Π V (v) of stte v to V simply orresponds to the restrition v V : V V D V V, where only vlues over V V remin fixed. The redution of system follows : S =Π V (S) {Π V (v) :v S}. Oserve tht S =Π V (S ), whih underlines tht S speifies onstrints on vriles of V only. In tht se, the spn of S in D Vmx n e uniquely represented s the union of lol sttes of shpe v : V D V. We dopt nottion (S, V ) to express tht S opertes on V, nd tht elements of S re in the nonil form v. The omposition of S 1 nd S 2 is defined y the onjuntion of their onstrints, i.e. y the intersetion of their spns. In other words, ssuming (S i, V i ), the omposition of v 1 nd v 2 is non empty iff v 1 V2 = v 2 V1. And in tht se, the resulting stte v 1 v 2 is otined y merging the prtil funtions v 1 nd v 2 into prtil funtion over V 1 V 2. The definition of S 1 S 2 follows : S 1 S 2 = {v 1 v 2 : v i S i }. The unit system 1I is defined s the set D Vmx, i.e. 1I llows ll possile sttes. Oviously, V V mx,π V (1I) =1I, so 1I opertes on no vrile, nd its nonil form redues to the universl stte. It is strightforwrd to hek tht omposition nd redution of onstrint systems stisfy xioms (1) to (4). Proilisti systems. This lss extends the previous one. For simpliity, we only onsider onstrint systems (S, V) in nonil form, tht we extend into (S, V, C) where C : S Rssoites weight (or ost) to eh stte v. Let V V, the redution (S, V, C ) = Π V (S, V, C) remins the sme on sttes, nd the new weight funtion C is defined y : v S, C (v ) = min C(v) (2.3) v S : v V =v When V V, we simply define Π V (S, V, C) s Π V V(S, V, C). For the omposition, (S, V, C) = (S 1, V 1, C 1 ) (S 2, V 2, C 2 ) follows the sme priniple s ove to ompose sttes, with the extr rule C(v 1 v 2 ) = C 1 (v 1 )+C 2 (v 2 ) (2.4) when v 1 v 2 is non-empty. And nturlly V = V 1 V 2. We leve s n exerise the verifition of (3). The unit system is defined s 1I =({ },, 0) : it llows ll sttes, nd introdues no extr weight. 23

24 Systems with ost funtions re losely relted to Mrkov rndom fields, whene the nme of proilisti systems : Tking exp( C), nd renormlizing it y its sum over ll sttes v in S yields proility distriution on vriles V. For (S, V, C) =(S 1, V 1, C 1 )... (S N, V N, C N ), the ost funtion C i represents the solled potentil funtion of lique V i, while the glol ost funtion C is referred to s the energy funtion. We hve hosen the pir (min, +) to define redution nd omposition. In the proilisti interprettion ove, redution n thus e red s mximum likelihood opertion : the ost of stte orresponds to log of its proility, so, in (2.3), likelihood is mximized over disrded vriles. But other pirs thn (min, +) would work s well, for exmple (mx, +), (mx, ) or (+, ) (see [3]). For the ltter, the redution orresponds to mrginliztion : one integrtes the likelihood over the disrded vriles. Whtever the hoie one mkes, in prtie ost funtions re often hndled under renormlized form. This renormliztion n e inorported into the omposition nd redution opertors without ltering their properties. Lnguge systems. This lst exmple is orrowed to Rong Su s pproh to distriuted monitoring [102]. We define lnguge system S =(L, V) s regulr lnguge L over finite lphet V. So the letters in V define the vriles on whih this system opertes, nd we ssume the existene of mximl (finite) lphet V mx. The redution Π V (S) is given y the nturl projetion of words in L on the su-lphet V V. And the omposition S 1 S 2, with S i =(L i, V i ), is defined s S 1 S 2 =(L, V 1 V 2 ) with L = L 1 L L 1 Π 1 V 1 (L 1 ) Π 1 V 2 (L 2 ) (2.5) where Π 1 V i denotes the reverse projetion of (V 1 V 2 ) on V i. In other words, L is the usul prllel produt of lnguges L 1 nd L 2. It is simple exerise to hek xioms (1) to (4). Lnguge systems re tully very lose in nture to onstrint systems. We ll see lter tht they enjoy the sme properties. In the next hpters, we will enode runs of distriuted systems in wy similr to lnguge systems. But insted of regulr lnguges, we will hve more elorte dt strutures Grphs of ompound system Hypergrph. As mentioned in the introdution, severl grphil representtions n e ssoited to ompound system S = S 1... S N operting on set of vriles. In the hypergrph representtion, vriles in V mx give the verties, nd eh omponent (S i, V i ) defines the hyperedge V i (Fig. 1.2.). Without loss of generlity, one n ssume V i V j for i j (if inlusion hppens, we reple S i nd S j y the single omponent S i S j, operting on V j ). The interest of H =(V mx, {V 1,..., V N }) is to disply the interfes etween sets of omponents. Let X, Y, Z V mx e vertex sets, we sy tht Y seprtes X from Z (denoted X Y Z) when on the hypergrph H Vmx\Y, otined y removing verties Y, no onneted omponent ontins verties of oth X nd Z. For 24

25 exmple, {V 1,V 2,V 3 } {V 4,V 7 } {V 7,V 8 } in Fig This llows us to sy tht vriles {V 4,V 7 } pture ll the intertion etween omponents S 1 nd S 4. And sine {V 4,V 7 } V 2, omponent S 2 seprtes S 1 from S 4 s well, or is n interfe etween them. This property is ruil to distriuted lgorithms. But efore explining why, we introdue more onvenient grphil representtion to desrie the intertions etween omponents. This grph will form the support of our lgorithms. Connetivity grph, ommunition grph. The onnetivity grph G nx of S = S 1... S N hs {1,..., N} s verties, or equivlently omponents S i, nd (i, j) is n edge iff V i V j (Fig. 1.2.d). A ommunition grph G for S is otined y reursively removing redundnt edges in the onnetivity grph, until minimlity is rehed. An edge (i, j) is sid to e redundnt in grph iff there exists pth (i, k 1,k 2,,k L,j) suh tht V i V j V kl nd k l {i, j} for 1 l L. In other words, the diret intertion etween S i nd S j n e ptured y the lternte pth (S i, S k1,, S kl, S j ) of the grph. S 6 S S 4 1 S 3 S 5 S 2 S 6 S 4 S 5 S 3 S 1 S 2 S 6 S 4 S 3 S 1 S 6 S 4 S 3 S 1 S 5 S 2 S 5 S 2 Figure 2.1: The hypergrph H of system with 6 omponents (top left), nd the ssoited onnetivity grph G nx of omponents (top right). Below, two ommunition grphs for this system. In generl, system hs severl ommunition grphs, s illustrted in figure 2.1. But we ll see tht this is not relly othering. This is prtiulrly true for the sulss of tree shped systems, tht will ply n importnt role in the sequel : We sy tht S lives on tree iff one of its ommunition grphs is tree. This lss enjoys the following nie property : Proposition 1 If S lives on tree, then ll its ommunition grphs re trees. Seprtion property. Communition grphs of S re more helpful thn hypergrphs to identify interfes etween sets of omponents. This is tully their rison d être. Let us introdue some more nottions. For n index set I {1,..., N}, we define S I i I S i nd V I i I V i. Let I, J, K {1,..., N} e index sets, nd 25

26 onsider the ggregted omponents S I, S J, S K. We sy tht S J seprtes S I from S K in S iff V I V J V K on H. Proposition 2 If I J K on ommunition grph G of S, then S J seprtes S I from S K in S. The seprtion property red on G is tully fst wy of identifying (some of the) ses where xiom (3) pplies, nd forms the sis of messge pssing lgorithms, s we show in the next setion. 2.2 Distriuted redution lgorithms The redution Prolem Composition is nturl tool to uild lrge omplex systems from smll simple omponents. In mny pplitions, the lrge system S = S 1... S N eomes intrtle. Fortuntely, one is generlly not so muh interested in omputing the lrge system S, ut rther in understnding its influene on given omponent S i. Speifilly, one inserted into S, omponent S i hnges nd eomes S i Π V i (S). Computing these S i defines wht we ll the redution prolem. Nturlly, one would like to determine or pproximte these redued omponents without omputing S itself. Our ojetive is thus to otin the S i with lol omputtions, i.e. omputtions performed t the sle of omponent, nd to highlight some of their properties. Before, nd to illustrte the sope of this pproh, we give n pplition exmple of the redution prolem in oding theory, whih is muh different from the prolems we shll onsider in the next hpters. The exmple onerns the deoding of the so-lled low-density prity hek (LDPC) odes. These error orreting odes re onstruted in the following wy : odewords hve length of N its, represented s vriles B 1,..., B N tking vlues 0 or 1. The ode is otined y foridding some onfigurtions mong the 2 N possile ones. Speifilly, M (independent) liner onstrints S 1,..., S M re pplied to these vriles. Eh S i involves smll suset of its V i {B 1,..., B N } nd llows sttes stisfying B n V i B n = 0, where ddition is modulo 2. The numer of possile vlues for =( 1,..., N ) thus redues from 2 N to 2 K, with K = N M, whih orresponds to rte K N ode. Cse 1. Let us onsider first the deoding prolem when n LDPC ode is used over n ersure hnnel. This rndom hnnel erses trnsmitted it with proility p, nd trnsmits it perfetly with proility 1 p. The deoding prolem onsists in reovering the trnsmitted odeword =( 1,..., N ) from reeived vlues r =(r 1,..., r N ), where r n is either 0, 1 or x, stnding for ersed. This tkes the form of ig liner system, one eqution per onstrint, where B n is set to r n if 0 or 1 ws reeived, nd left s n unknown otherwise. In our setting, for eh oservtion r n let us uild system R n operting on vrile B n nd pinning its vlue to r n if 0 or 1 ws reeived, or llowing oth 26

27 vlues otherwise. The glol deoding mens omputing S R 1... R N, mde of odewords tht mth oservtions r =(r 1,..., r N ). There is no ovious wy to perform this glol omputtion effiiently. Moreover, it n result in huge set if r is not uniquely deodle. One would rther prefer to identify the vlue of its B n tht n e reovered, nd leve the others s unknown. Possily with omputtions involving only few its t time. The deoding of eh it B n is given y Π Bn (S R 1... R N ), whih is su-produt of the Π Vi (S R 1... R N ). If r =(r 1,..., r N ) is deodle, this projetion ssigns single vlue to eh B n. Otherwise, some undeodle its remin, tht n still tke oth vlues. Cse 2. Let us onsider now the trnsmission of n LDPC ode over, sy, Gussin hnnel. B n is modulted s +1 or 1 nd orrupted y the dditive Gussin noise Z n, whih yields oservtion R n = (2 B n 1) + Z n tking vlues in R. We now model systems S i s systems with weight funtion : they llow the sme lol sttes s ove, nd ssign null weight to eh of them. This stnds for the equiproility of ll odewords, weights eing homogeneous to log likelihood. We uild oservtion systems s follows : given the reeived vlue r n, system R n opertes on B n nd ssigns weights log P(r n B n = 0), log P(r n B n = 1) to vlues 0 nd 1 of B n. Then, in the (mx, +) setting, the redued system Π Bn (S R 1... R N ) llows vlues 0 nd 1 to B n with weights log mx P( 1,..., n 1, 0/1, n+1,..., N r 1,..., r N )+C (2.6) i, 1 i N, i n where C is onstnt. Therefore, these vlues llow mximum likelihood deoding of it B n. This sttement my e more onvining without the log, i.e. in (mx, ) setting. Let us ssign weigh to onfigurtions of S i, still for equiproility (ny onstnt vlue would work s well). Oservtion systems now ssign P(r n B n =0/1) to vlues 0 nd 1 of B n. Then Π Bn (S R 1... R N ) yields weights proportionl to mx P( 1,..., n 1, 0/1, n+1,..., N,r 1,..., r N ) (2.7) i, 1 i N, i n Messge pssing lgorithm The messge pssing lgorithm (MPA) solves the redution prolem relying on the seprtion riterion etween omponents. The ltter indues the following two omputtion rules. Consequenes of the seprtion riterion. Let I, J, K e pirwise distint index sets, nd ssume tht S J seprtes S I from S K in S, then : Π VJ (S I J K ) = Π VJ (S I ) S J Π VJ (S K ) (2.8) (2.8) is known s merge eqution. It expresses tht if system S J seprtes two (or more) omponents, the ltter hve independent influenes on S J. The proof 27

28 mostly uses (3) : Π VJ (S I J K ) = Π VJ (S I S J S K ) = Π VJ (S J ) Π VJ (S I S K ) = S J Π VJ (S I ) Π VJ (S K ) (2.9) The seond onsequene of seprtion expresses tht the influene of S K on S I n e propgted through the intermedite system S J, whih is known s the propgtion eqution : The proof uses oth (3) nd (1) : Π VI (S I J K ) = S I Π VI [S J Π VJ (S K )] (2.10) Π VI (S I J K ) = S I Π VI (S J S K ) = S I Π VI [Π VJ V K (S J S K )] = S I Π VI [Π VJ (S J S K )] = S I Π VI [S J Π VJ (S K )] (2.11) The key is tht Π VI Π VJ V K =Π VI Π VJ, due to the seprtion property. Of ourse, y (1) nd tking into ount tht S I opertes on V I, term like Π VJ (S I ) n e repled y Π VI V J (S I ). Distriuted redution lgorithm. Assume S = S 1... S N lives on tree, nd G is one of its ommunition grphs. Let N (i) denote the neighors of vertex i on G,1 i N. The redution lgorithm for S is sed on messges exhnged etween neighors : eh system S i mintins nd updtes messge M i,j for eh neighor S j, so there re two messges per edge (i, j) of G, one in eh diretion. The ide is tht M i,j ollets informtion out S j in systems loted on the side of S i with respet to edge (i, j), relying on the ft tht system S i seprtes S j from the rnhes ehind S i. The set of systems ontriuting to the messge grows until the whole rnh eyond i hs een overed. Algorithm A 1 1. Initiliztion M i,j = 1I, (i, j) G (2.12) 2. Until stility of messges, selet n edge (i, j) nd pply the updte rule M i,j := Π Vi V j [S i ( M k,i )] (2.13) k N (i)\j 3. Termintion S i = S i ( k N (i) M k,i ), 1 i N (2.14) 28

29 The termintion eqution (2.14) is oviously merge, while the updte eqution (2.13) mixes merge nd propgtion (Fig. 2.2). Oserve tht (2.13) merges inoming messges of ll edges round i exepted the edge (i, j) on whih new messge will e sent. S k1 S j S i S k2 S k3 Figure 2.2: Messges rriving t S i gther informtion of their su-tree, nd re omined to form messge to S j. Convergene. Theorem 1 Let S = S 1... S N live on tree, nd G e ommunition grph for S. Then A 1 onverges in finitely mny steps, nd t onvergene S i =Π V i (S), 1 i N. Steps refer to the numer of messge updtes, where only updtes hnging the vlue of messge re ounted. Notie tht the sheduling of the lgorithm, i.e. the hoie of the edge (i, j) t eh step, is left unspeified. This property will thus led to synhronous distriuted lgorithms in the sequel. Evolving systems. Surprisingly, this result n e esily extended to evolving systems, whih will e useful in the next hpters. Assume omponents S i re not fixed one for ll, ut my evolve in time, whih we denote y S i (t), t N. We mke no other ssumption on this evolution thn T i < : t >T i, S i (t) =S i (T i ) (2.15) i.e. the omponents stilize. Assume Algorithm 1 is strted t time t = 0, nd tht eh step of the reursion tkes one unit of time. So omponents hnge fter eh messge updte. Then theorem 1 ove still holds, with S repled y the stilized system S 1 (T 1 )... S N (T N ). Of ourse, in the trnsient prt of the lgorithm, the messges omined in (2.13) re desynhronized, i.e. they gther informtion olleted in different omponents t different times. This is the prie to py to get n synhronous lgorithm. One n proly refine this result with extr ssumptions. For exmple, with monotoni evolution of omponents nd speifi onstrints on the sheduling of the lgorithm 2, there ertinly exists some form of monotony on messges. We leve this for future reserh. 2 the ordering in whih messges re updted 29

30 2.2.3 Turo lgorithms In prtil pplitions, few systems hve tree struture. To solve the redution prolem in more generl ses, one my dopt severl strtegies. The simplest one onsists in ggregting some omponents into mro-omponents, in suh wy tht these mro-omponents intert ording to tree struture. There exists systemti wy to do tht, y first tringulting ommunition grph of S, nd then tking liques of nodes to form the mro-omponents. It n indeed e shown tht the liques of tringulted grph hve tree shped intertion struture. This is lso lled the juntion tree tehnique. The prie to py is tht omputtions of the MPA re then performed on lrger omponents, whih generlly mens loss in effiieny. And finding the est tringulted grph, or equivlently the tree with the smllest mro-omponents, hs een shown to e NP hrd. Moreover, even with resonly dense ommunition grph, the numer of ggregted omponents rpidly rings us k to n intrtle redution prolem. In the se of grid of omponents, for exmple, the typil ggregtion would gther ll omponents of row (or of olumn), nd orgnize them in hin. The less known onditioning method does similr thing. It freezes one (or severl) vrile(s) to prtiulr vlue, whih mounts to removing it (them) from S, nd my thus open yle. The redution prolem n e solved esily on the remining onditioned system, if it lives on tree. The omplexity is similr to the juntion tree method : ll omintions of vlues must e explored for the onditioning vriles. And tehnil diffiulty remins in the deonditioning step, whih omines ll redued omponents otined for ll vlues of the frozen vriles. In prtie, the omplexity of ext redution methods explodes with the numer of yles in the ommunition grph of the system. This doesn t men however tht we must stop here our quest for methods to hndle lrge distriuted systems. In the digitl ommunition ommunity, people hve disovered tht running the MPA on grphs with yles ould sometimes provide exellent results [10], even if this is theoretilly illegl! The MPA re then lled turo lgorithms, euse the messge sent on rnh my e propgted long yle of the grph nd eventully ome k to its trnsmitter fter some trnsformtions. Just like the power of exhust gses of n engine drives the ompression of fresh ir t the dmission side. We now exmine some of their properties. Conditions for onvergene. To study the onvergene of messge pssing lgorithms on grphs with yles, we introdue wek notion of topology on systems, with extr xioms defining its reltions with nd Π. Let us ssume the existene of prtil order on systems, where S S n e red s S ontins more informtion thn S. We require to stisfy the following properties : whih mens thi is the lest informtive system. S, S 1I (5) S 1, S 2, S 3, S 1 S 2 S 1 S 3 S 2 S 3 (6) 30

31 S, S, V V mx, S S Π V (S) Π V (S ) (7) Intuitively, (6) sttes tht omposition inorportes the sme mount of informtion to ll systems, nd (7) mens tht redution n t introdue informtion. As n exmple of this sitution, onsider onstrint systems : S S n e simply defined y Spn(S) Spn(S ), whih entils in prtiulr V S V S, i.e. S onstrins the vlue of more vriles. Under these onditions, one esily shows tht the messges M i,j in (2.13) hve deresing evolution for, regrdless of the struture of the ommunition grph. In other words, informtion ugments t eh node with the exhnge of messges. Moreover Theorem 2 Under xioms (5,6,7), lgorithm A 1 hs t most one essile sttionry point. Notie tht the updte eqution (2.13) my dmit severl sttionry sets of messges M i,j, ut t most one of them is essile from the initil vlue of A 1. The essiility refers to denumerle numer of steps 3. If the numer of possile sttes is ounded, (i.e. D Vmx < ), the sttionry point is rehed in finite numer of steps, whtever the ordering of updtes. For infinite stte spes, one my e le to refine this result nd show tht onvergene is tully grnted for wek topology. This is the se for lnguge systems for exmple (setion 2.1.2), ssuming the usul notion of wek onvergene for non ommuttive forml series. It would e nie if this simple pproh ould hold for rndom systems. Unfortuntely, this is not the se : simple ttempts t defining for systems with ost funtions fil, even if no renormliztion opertion is introdued on ost funtions. There is little hope of suess sine some uthors hve uilt exmples of systems for whih the MPA does not onverge to fix point [79]. Nevertheless, onvergene properties of MPAs on rndom systems re now quite well understood [104, 58, 92, 93, 94, 20], nd there exist tools to test it priori. Convergene deeply relies on the sprseness of the ommunition grph, or in other words, on the length of yles in the grph. Roughly speking, yles must e long enough to introdue deorreltion etween n outgoing messge nd its version oming k fter yli propgtion. This ensures tht the messges merged t node re lmost independent, s they re on tree. To summrize, one n onsider systems with ost funtions s hving doule nture. First, omponents in S = S 1... S N rry hrd onstrints, for whih onvergene of the MPA is gurnteed, whtever the grph of S is. But omponents lso rry soft onstrints, defined y the likelihood of the remining sttes, fter hrd onstrints hve operted their seletion. For these soft onstrints, onvergene must e studied with the stndrd tools dedited to turo lgorithms. Properties t onvergene. Let us now onsider properties of sttionry points of A 1, ssuming there exist some. 3 We only ount messge updtes tht hnge the ontent of messge. 31

32 S1 S1 S 4 d d S2 S 4 d d S2 S3 S3 Figure 2.3: Two exmples of onstrint systems defined on vriles {A, B, C, D}. Interprettion exmple for these grphis : omponent S 1 opertes on {A, B} nd ontins sttes (, ) nd (, ). Our im ws to otin the redued omponents Π Vi (S). Let us first remrk tht the S i otined y (2.14) t sttionry point of (2.13) generlly differ from the true Π Vi (S), when the glol system S doesn t live on tree, of ourse. Fig. 2.3 (left) gives ounter-exmple, in the se of onstrint systems : S = S 1... S 4 is empty, so the true redued omponents Π Vi (S) re lso empty. But the S i otined t onvergene of lgorithm A 1 re identil to the S i. In ft, the sitution is even worse : in some ses the true redued omponents n t e otined s sttionry point of A 1. Fig. 2.3 (right) illustrtes this se : Π V3 (S) ontins sttes (, d) nd (,d ), nd otherwise Π Vi (S) =S i for i 3. But there exists no set of sttionry messges M i,j tht would yield these redued omponents y (2.14). Nevertheless, nd despite these drwks, the S i omputed y A 1 re fr from eing meningless, s we show now nd in the next setion. The lol extendiility is proly the most striking property of the S i otined t sttionry point of (2.13). Theorem 3 Let G e ommunition grph of S = S 1... S N, let the M i,j e sttionry point of (2.13) on G nd let the S i e derived y (2.14). Selet J {1, 2,..., N} suh tht the sugrph G J is tree, nd define S i = S i ( k N (i)\j M k,i), 1 i N. Then j J, S j =Π V j ( S J ). The proof is intuitively simple : repling the S i y S i nd running A 1 yields the sme messges etween the remining nodes. Sine the ltter form tree, the result follows y theorem 1. To lrify the interest of theorem 3, Let us tke the exmple of onstrint systems. Consider lol stte v i in S i. Sine A 1 is n pproximtion, v i is not neessrily the projetion of glol stte v of S. Nevertheless, v i n e extended into lrger stte v J over ny tree round i (v J my not involve ll vriles of S, however). So only yle of G ould determine tht v i in some system S i doesn t elong to the true Π Vi (S) (see the exmple of fig. 2.3). In other words, A 1 is lind to yles. In the se of lnguge systems, the sme interprettion holds for words in the redued lnguge S i. The se of systems with ost funtions is exmined in more detils elow. The result my look wek sine J ould remin quite smll nd involve few omponents, nd thus few vriles of V mx (see Fig. 2.4, left). In relity, in mny 32

33 settings it is possile to duplite omponents nd relte them y n equlity onstrint, in order to introdue fke new omponents (see Fig. 2.4, right). A sttionry point of A 1 esily extends to sttionry point on this expnded system, for whih the J set now rehes ll omponents of the originl system. As result, v i n e extended to v J overing ll omponents of the originl system S, ut extremities don t mth, i.e. the vlues tken y v J in the vrious opies of omponent my differ. S 1 S2 S 1 S" 1 S 2 S 1 S 3 S 4 S 5 S 3 S 4 S 5 S 6 S" 8 S 10 S 7 S 8 S 9 S 10 S 7 S 8 S 8 S" 8 S 9 S 10 Figure 2.4: Left : Communition grph relting 10 omponents, nd nested tree defined y J = {2, 4, 5, 6, 7, 9} (solid edges). Right : y dupliting omponents S 1, S 8 nd S 10, nd relting opies y n equlity onstrint, one n tully define nested tree rehing ll omponents (possily severl times for the duplited ones). Moving to proilisti systems, the lol extendiility n e interpreted s lol-tree optimlity property, lredy mentioned in [108]. Let us onsider positive weight funtions in the (mx, ) setting, nd ssume systems re normlized : mx v S C(v) = 1. This requires tht the omposition ontin normlizing opertion (normliztion is lredy preserved y redutions). When S lives on tree, A 1 onverges nd yields the S i =Π Vi (S). It tully solves dynmi progrmming prolem. Let v (resp. vi ) denote stte of S (resp. S i ) hving weigh, i.e. most likely stte of S (resp. S i ) in the proilisti interprettion of weights. Clerly, the restrition of v to vriles V i neessrily yields vi, nd onversely v i is neessrily prt of t lest one v. So, if omponents S i ontin single v i t onvergene of A 1, these lol sttes n e omposed to form the unique optiml stte v of S : {v } = i {v i }. This is well known property in dynmi progrmming. In the se where S doesn t live on tree, ssume tht some lol optim vi n e ssemled into vlid glol stte v of S (in terms of hrd onstrints). There is priori no reson tht C(v ) = 1, so how good is this v in terms of ost funtion? Let J e n index set suh tht G J is tree, let I e its omplement in {1,..., N}, nd let us introdue vrile sets V I = V I \V J, V I,J = V I V J nd V J = V J \V I. Then v J, [(v J, v I,J, v I ) S C(v J, v I,J, v I ) C(v J, v I,J, v I ) ] (2.16) so hnging the vlue of vi on ny tree round node i will not improve the ost funtion to optimize. The MPA (lso lled elief propgtion in this se) thus onverges towrds lol optimum of the ost funtion of S, ut its sin of ttrtion is resonly lrge. 33

34 2.2.4 Involutive systems Some fmilies of systems enjoy useful property tht we ll involutivity. This property sttes tht systems do not hnge when omposed with prt of themselves : S, V, S Π V (S) = S (8) (We sy tht Π V (S) is sored y S.) Oserve tht Π V (1I) = 1I omes s n immedite onsequene of (8). Involutivity is strong property. It is lerly stisfied y onstrint systems, y lnguge systems, nd in generl orresponds to systems defined y some form of onstrint set. But systems with ost funtions re not involutive. Proposition 3 Assuming (8), let S i operte on V i, 1 i N, then S = S 1... S N S =Π V1 (S)... Π VN (S) (2.17) The proof essentilly uses (3). In words, if S ftorizes, the redued omponents S i Π V i (S) give nother ftoriztion of S, nmed the nonil ftoriztion. In the next hpters, we will lso use this property to uild miniml produt overing of system S. Involutivity hs mny other nie onsequenes. For exmple on the onvergene of A 1 : let us define y (i.e. S 1 sors S 2 ). Then S 1 S 2 S 1 S 2 = S 1 (2.18) Proposition 4 is prtil order reltion tht stisfies xioms (5,6,7). So theorem 2 holds. This mens tht messges ollet more nd more informtion in lgorithm A 1. Moreover, A 1 tully performs progressive redution of omponents S i : Theorem 4 Define the S i y (2.14) t ny step of A 1. In n involutive setting, one hs S = S 1... S N t ny time in A 1. Moreover, the S i omputed t eh step form deresing sequene for, nd they stisfy Π Vi (S) S i S i. As we hve lredy mentioned, this progressive redution my stop t (or onverge to) sttionry point loted ove the miniml term Π Vi (S). With theorem 2 in mind, this mens tht only yles of onstrints on G would llow to go further in the redution, s illustrted y the exmples in Fig Philosophilly, involutive systems re defined through onstrints. And sine dding redundnt onstrints is hrmless, one gets severl extr properties. For exmple, n lternte redution lgorithm n e derived. 34

35 Algorithm A 2 1. Initiliztion S i = 1I, 1 i N (2.19) 2. Until stility of redued systems, selet vertex i nd pply the updte rule S i := S i [ Π Vi V k ( S k )] (2.20) k N (i) This lgorithm derives diretly from A 1 : let us reple (2.13) y more symmetri expression tht would use ll inoming messges to ompute n outgoing messge : M i,j := Π Vi V j [S i ( M k,i )] (2.21) k N (i) In other words, one sends k to node S j its own messge Mj,i to S i. In n involutive setting, this will hve no impt on S j so the modified lgorithm will ehve like A 1. At this point, oserve tht messges eome useless : omputtions n e equivlently desried in terms of systems S i s in A 2. The ltter hs the sme shpe of Guss-Seidel proedure to solve liner systems of equtions. Not surprisingly, one hs Theorem 5 In n involutive setting, let G e ommunition grph for S = S 1... S N. Algorithms A 1 nd A 2 hve the sme essile sttionry point on G (in terms of omponents S i nd S i ), if this point exists for one of them. Rell tht there exists t most one suh sttionry point, y theorem 2. In prtiulr, if S lives on tree, lgorithm A 2 onverges in finite time to the ext redued omponents. One n go further in terms of using redundnt onstrints : insted of inorporting extr informtion in the updte eqution, let us now reinorporte redundnt edges in the ommunition grph of S : Theorem 6 In n involutive setting, let S = S 1... S N, nd G nx e the onnetivity grph of S. Let grph G G nx e otined y removing some redundnt edges in G nx. Whtever the hoie of G, A 1 (resp. A 2 ) yields the sme result in terms of omponents S i (resp. S i ). These results illustrte the power of (8). Mny other properties n e derived for involutive systems, in prtiulr in terms of lol extendiility. We refer the reder to [38] for detils. 35

36 2.3 Summry We hve introdued n strt notion of system, operting on redued set of vriles, nd provided with omposition opertion. This llows us the onstrution of lrge systems from omponents. The intertions of omponents n e desried s onnetivity grph, or more speifilly s ommunition grph. We hve lso introdued the onept of redution of lrge system to prt of its vriles. Sine lrge systems n t e studied glolly, this llows us to study them y prts. In prtiulr, for ompound system, the redution explins how omponent s ehvior is modified one the ltter is onneted to ll the others. Computing redued omponents of lrge system is powerful key to nswer mny prolems relted to distriuted systems. As soon s omposition nd projetion stisfy smll set of xioms, one n derive effiient lgorithms to ompute redued omponents. These messge pssing lgorithms re y nture distriuted (or distriutle) nd sed on synhronous omputtions. They onverge in finite time to the redued omponents for systems whose ommunition grphs re trees, whtever the ordering of omputtions. On generl grphs, they onverge when systems re defined in terms of onstrints, ut my provide only pproximte redued omponents. When systems lso inorporte notion of ost funtion, redution lgorithms generlly onverge if the ommunition grph is sprse enough, s reveled y the results on turo-lgorithms. Although the onvergene point is only n pproximtion of the desired redued omponents, severl properties of the ltter suggest resonle qulity of this pproximtion. The next hpters rely on this formlism to develop distriuted lgorithms for networks of dynmi systems. So the essentil hnges pper in the detils of system definitions : they eome dynmi systems, whih introdues the notion of time in this setting. Relted work The ide of messge pssing lgorithms hs ppered in severl ommunities, under different nmes nd sometimes independently. It is known s dynmi progrmming in omputer siene, s the Kruskl or Dijkstr lgorithm in grph theory, s the Viteri lgorithm in digitl ommunitions, et. So let us simply mention some referenes in sttistis [76, 77, 87] : This ommunity speifilly studied the topi, with fous on inferene prolems, of ourse, ut lso on different formlisms to desrie sttistil dependenies (vrious grphil models, semi-mtroids, et.). A lrge effort is dedited to lerning prolems, to disover the underlying grphs nd possily uslity reltions. Surprisingly, the generliztion of messge pssing lgorithms into turo lgorithms ourred elsewhere, nd for sul resons. The disovery ws triggered y the spetulr properties of turo-odes [10], nd it took (little) time to estlish the onnetion etween turo-lgorithms nd MPA [80, 69, 70]. Convergene properties of this pproximte proedure hve then een studied intensively [92, 94, 104], in prtiulr to design odes (i.e. grphs) tht would ehve in the est possile 36

37 wy when deoded with this lgorithm [20, 93]. The ide of pproximte inferene my look strnge in the lgeri ontext of this hpter, ut it hd n extrordinry impt in the field of digitl ommunitions, s well s in signl nd imge proessing. Let us simply mention the joint soure-hnnel deoding strtegies [57], or more generlly ll the itertive lgorithms to jointly perform synhroniztion, equliztion, detetion, user seprtion, deoding, et. The lol extendiility tht we mentioned in this hpter is n lgeri generliztion of lol optimlity riterion disovered y Weiss [108]. There exist severl extensions/generliztions of turo lgorithms. For exmple [113, 114], inspired from sttistil physis, the mixture of turo lgorithm with prtile methods [100, 101], or the omintion of spnning trees [99]. 37

38 38

39 Chpter 3 Networks of dynmi systems There exists n extremely undnt literture relted to Disrete Event Systems (DES), nd reserh is performed in different ommunities : DES of ourse, ut lso Computer Siene, Artifiil Intelligene, Automtis, et. A lrge set of ppers develop new models nd explore reltions etween model fmilies. Another lrge ody of results is relted to speifi pplitions : one ould mention supervisory ontrol (initited y the pioneering work of Rmdge nd Wonhm), optiml ontrol, fult dignosis, model heking (in prtiulr for iruit verifition), protool verifition, performne evlution, et. A powerful trend in most of these domins ims t ddressing lrge systems, whih mens the development of modulr, deentrlized or distriuted proedures to ope with the omintoril explosions inherent to lrge systems [8, 13, 21, 23, 24, 25, 54, 88, 103]. In ll ses, the first step onsists in modeling the system in modulr mnner. Different (generlly equivlent) formlisms hve een explored, for exmple ommuniting utomt (with ounded uffers), ounded or sfe Petri nets, networks of prtilly synhronized utomt, et. In this hpter, we hoose the lst one, for its simpliity nd fmilirity, in order to void the urden of nottions nd fous ttention on the onepts of distriuted omputtions. We lso limit ourselves to the distriuted dignosis prolem. As we will see, its entrl diffiulty mounts to 1/ finding ompt representtion of sets of trjetories for lrge system, nd 2/ finding n effiient wy to ompute with these sets. We elieve these ides re tully entrl to ll the prolems ove. 3.1 Dynmi systems nd their ompositions This setion exmines how ordinry utomt n e omined to form networks of dynmi systems. We first use stndrd omposition opertion, the prllel produt, tht we lter generlize into more omplex form of intertion, lled the pullk. One n ssoite severl grphil representtions to network of utomt, whih will mke the onnetion with the previous hpter. For resons tht will pper lter, we must keep trk of the elementry utomt in the omposition opertion, so we introdue the unusul multi-lok feture in the definition of n utomton. These multi-lok utomt re the formlized version of the tile systems we skethed in the introdution. 39

40 3.1.1 A tegory of multi-lok utomt Multi-lok utomton. An utomton is 4-tuple A =(S, T, s 0, ) omposed of stte set S, trnsition set T, n initil stte s 0 S nd flow reltion (S T ) (T S) stisfying t T, t = t = 1, where t nd t represent pre- nd post-sttes, s usul. Oserve tht S T S, so there my exist severl trnsitions etween two sttes 1. A leled utomton (LA) A =(S, T, s 0,, λ, Λ) is n utomton enrihed with leling funtion λ : T Λ. This is used elow for synhroniztion purposes. A multi-lok utomton (MCA) A =(S, T, s 0,, χ, I) is n utomton enrihed with n index set I nd n indexing funtion χ : T 2 I. These indexes will e used to keep trk of elementry utomt when they re ssemled to form lrger systems. So χ(t) represents utomt of A where trnsition t hs n influene, nd onversely χ 1 (i) ={t T : i χ(t)} identifies trnsitions of omponent i (see the produt susetion). Nturlly, leled multi-lok utomt (LMCA) enjoy the two extr funtions ove. When deling with severl utomt A 1, A 2, et., we shll use susripts to identify their elements, like S i,t i, et. Fig. 3.1 gives two exmples of MCA. The numers lose to trnsition nmes represent the vlues of χ, so the MCA on the left ontins two utomt, nd for exmple elongs to oth while t 2 only opertes in the first one. Morphism. Mny onstrutions nd results we use in the sequel n e more esily derived in the setting of tegory theory. The ltter n e regrded s lnguge to study ojets nd their reltions. In our se, the ojets will e MCA (or LMCA), nd reltions i.e. morphisms etween them re defined s run preserving mppings. Speifilly, let A 1, A 2 e two MCA, A i =(S i,t i,s 0 i, i,χ i,i i ), morphism φ : A 1 A 2 is triple (φ S,φ T,φ I ) where 1. φ S : S 1 S 2 is totl funtion on sttes, stisfying φ(s 0 1 )=s0 2, 2. φ T : T 1 T 2 is prtil funtion on trnsitions, where φ T ( )= denotes tht φ T is undefined t, 3. if φ T ( )=, then φ S ( )=φ S (t 1 ), i.e. one n erse trnsition only if pre- nd post-sttes re merged, 4. if φ T ( ), then φ S ( )= φ T ( ) nd φ S (t 1 )=φ T ( ), i.e. the flow reltion is preserved, 5. φ I : I 1 I 2 is reltion suh tht its opposite reltion φ op I : I 2 I 1 is prtil funtion nd T 1, φ I (χ 1 ( )) = χ 2 (φ T ( )), with the onvention χ i ( ) =. 1 The pre- nd post-stte nottions re orrowed to the Petri net formlism. In the sme wy, (S T ) (T S) prepres n extension of this formlism to networks of utomt, whih re tully lmost equivlent to sfe Petri nets, the formlism in whih this theory ws initilly developed. 40

41 Condition 5 is it omplex in order to llow the duplition of utomt indexes, property required y the produt we define lter. Condition 5 implies in prtiulr φ T ( )= φ I (χ 1 ( )) =. So when χ 1 ( ) Dom(φ I )=, one hs φ T ( )=, nd onversely. In summry, φ erses ll trnsitions tht hve no influene on the elementry utomt with indexes in Dom(φ I ), or in other words removes ll trnsitions lol to utomt I \ Dom(φ I ). This requirement will ensure tht φ is lok preserving, or doesn t modify the ounting of time in eh utomton, s it will eome ler in the sequel. A morphism φ is sid to e folding when φ T is totl funtion nd φ I = Id. s 0 t {2} 3 s 2 s 0 t {1} 5 {1,2} t {1} 2 t 2 {1} t 1 {1} s 3 s 1 t {2} 4 s 1 Figure 3.1: Two multi-lok utomt, relted y morphism (dshed rrows). Fig. 3.1 gives n exmple of morphism φ : A 1 A 2 relting two MCA. In A 1 (left), one hs χ 1 1 (1) = {,t 2,t 5 } nd χ 1 1 (2) = {,t 3,t 4 }, s indited y the (red) numers lose to trnsition nmes. φ ollpses s 0 nd s 2 into s 0, nd s 1,s 3 into s 1, nd t the sme time erses ll trnsitions outside χ 1 1 (1). For leled MCA, we introdue some extr requirements in the definition of morphisms : φ : A 1 A 2 is morphism of LMCA, with A i = (S i,t i,s 0 i, i, χ i,i i,λ i, Λ i ), iff it is morphism of MCA nd lso stisfies 6. Λ 2 Λ 1, i.e. φ redues the lel set, 7. Dom(φ T )=λ 1 1 (Λ 2), i.e. φ is defined on trnsitions rrying shred lel, nd only on them 2, 8. if φ T ( ) then λ 1 ( )=λ 2 (φ T ( )), i.e. φ preserves lels on its domin of definition Dom(φ T ). We denote y A the tegory hving the LMCA s ojets, nd the ove morphisms s rrows. By use of nottions, we will write φ insted of φ S,φ T or φ I Composition y produt The produt A = A 1 A 2 of two LMCA is defined s stndrd prllel produt : trnsitions rrying shred lel re synhronized, while trnsitions rrying privte lel remin privte (fig. 1.2) : 1. S = S 1 S 2 nd s 0 =(s 0 1,s0 2 ), 2 In onjuntion with point 5 in the definition of morphisms, this ondition mens tht the lels shred y A 1 nd A 2 re extly ssoited to the elementry utomt preserved y φ I. 41

42 2. T = T s T p where T s = {(,t 2 ) T 1 T 2 : λ 1 ( )=λ 2 (t 2 )} (3.1) T p = {(, s2 ): T 1,s 2 S 2,λ 1 ( ) Λ 1 \ Λ 2 } {( s1,t 2 ):s 1 S 1,t 2 T 2,λ 2 (t 2 ) Λ 2 \ Λ 1 } (3.2) The nottion 3 si stnds for (fke) loop t stte s i. 3. is defined y (s 1,s 2 ) (,t 2 ) (s 1,s 2 ) iff s i i t i i s i, i =1, 2, where one of the t i n e si nd ssuming s i i si i s i holds for every stte s i S i, 4. Λ = Λ 1 Λ 2 nd λ follows ordingly, 5. I = I 1 I 2 is the disjoint union of elementry utomt indexes, nd χ is defined y χ(,t 2 )=χ 1 ( ) χ 2 (t 2 ) with the onvention tht χ i ( si )= when t i = si. As mentioned in setion 3.1.1, the disjoint union of indexes ppering in point 5 llows to keep trk of the elementry utomt omposing n LMCA when produt is performed. This wkwrd feture inorported in the si definition of n utomton will e used to define vetor lok, neessry to the proessings we perform lter (whene the nme multi-lok utomton ). The lel-sed produt n e onsidered s speiliztion of more ordinry produt defined on MCA. The ltter is otined y removing the lel onditions in (3.1) nd (3.2) (nd lso removing point 4, of ourse). The two produts hve the sme lgeri properties : so lels, whih re extremely onvenient to uild lrge systems out of omponents, pper formlly s free feture. Consequently, most proofs n e derived for MCA, for the ske of simpliity, the extension to LMCA eing strightforwrd 4. (t, ) 2 * " t 2 "! t 4 # t 3! d (*,t 4) d #! # (t,t ) 1 3 " (t, ) 2 * d (*,t 4) d Figure 3.2: Two LMCA A 1, A 2 (left) nd their lel-sed produt (right). Trnsition lels re indited y Greek letters, nd we ssume Λ 1 Λ 2 = {α}. Let us denote y π i : A 1 A 2 A i the nonil projetions from the produt A 1 A 2 to their ftors A i. It is strightforwrd to hek tht they re morphisms, 3 This nottion voids introduing stuttering trnsitions t every stte, efore omputing the produt. 4 This ide is orrowed to [110], whih introdues the notion of synhroniztion lger insted of mthing lels. 42

43 with the onvention tht π 1 ( s1,t 2 )= nd symmetrilly. Moreover, one hs the following property Proposition 5 (universl property of the produt) Let A e n LMCA nd let the φ i : A A i, i =1, 2 e two morphisms. There exists unique morphism ψ : A A 1 A 2 suh tht φ i = π i ψ, i =1, 2. % 1 A & % 2 $ 1 A 1 A 1 A 2 $ x 2 A 2 Figure 3.3: Universl property of the produt in A. Proposition 5 mkes the tegoril produt in A, whih immeditely entils its ssoitivity. Oserve tht A 1 A 2 is defined up to unique isomorphism, s lwys in tegory theory, so we will write A 1 A 2 = A for ny other A stisfying the universl property (this redily proves the ommuttivity of ). Miniml Produt overing. Let us define the A i = π i(a 1 A 2 ), nd A = A 1 A 2. Then A is isomorphi to A 1 A 2, whih we lso write A 1 A 2 = A 1 A 2. More generlly, let us write A B when there exists n injetive morphism from A to B. Consider A A 1 A 2, nd A i = π i(a), then A A 1 A 2, nd tking ny A i A i insted of A i will rek this inlusion. Therefore we ll A 1 A 2 the miniml produt overing of A in A 1 A 2. Notie tht A = A 1 A 2 doesn t hold in generl, unless A lredy hs produt form in A 1 A 5 2. This notion of miniml produt overing is generl nd will e used with other tegoril produts Composition y pullk The produt of LMCA provides simple mnner to ssemle omponents y prtil synhroniztion of trnsitions. However, in some ses, it my e more nturl to define intertions y mens of shred resoures, for exmple in the se of omponents ommuniting y messges. Speifilly, nd in order to estlish onnetion with hpter 2, we would like to define the intertion of two LMCA y the ft tht they shre some elementry utomt, just like two omponents were shring vriles in hpter 2. We therefore need to define omposition sed on the notion of interfe, tht will enjoy dequte tegoril properties. The omposition y pullk is nturl ndidte. Consider three LMCA A 0, A 1, A 2 relted y morphisms f i : A i A 0, i {1, 2}. This expresses tht A 0 is n interfe etween A 1 nd A 2, or tht A 1, A 2 hve ommon prt represented in A 0 (for exmple ommunition uffer or shred 5 The literture relted to lnguges nd their produts would sy tht A is seprle lnguge. 43

44 resoures in generl). The pullk of this digrm is defined s terminl triple (A,g 1,g 2 ), with g i : A A i, suh tht f 1 g 1 = f 2 g 2. By terminl, we men tht the following universl property is stisfied (see fig. 3.4) : (A,h 1,h 2 ) with h i : A A i, f 1 h 1 = f 2 h 2!ψ : A A, h i = g i ψ (3.3) A h 1 & A g g 1 2 h 2 A 1 A 2 f 1 f A 2 0 Figure 3.4: Universl property of the pullk in A. Oserve tht Fig. 3.4 with the null LMCA 6 for A 0 oinides with Fig. 3.2, so the pullk generlizes the produt to the se where A 1, A 2 hve ommon prt. Its detiled definition is thus it more omplex thn the one of the produt. The pullk A = A 1 A 0 A 2 (or simply A = A 1 A 2 when there is no miguity) is given y : 1. S = {(s 1,s 2 ) S 1 S 2,f 1 (s 1 )=f 2 (s 2 )} nd s 0 =(s 0 1,s0 2 ), 2. T = T s T p where T s = {(,t 2 ):f 1 ( )=f 2 (t 2 )} (3.4) {(,t 2 ):λ 1 ( )=λ 2 (t 2 ),f 1 ( )= = f 2 (t 2 ),f 1 ( )=f 2 ( t 2 )} T p = {(, s2 ):λ 1 ( ) Λ 1 \ Λ 2,f 1 ( )=, f 1 ( )=f 2 (s 2 )} {( s1,t 2 ):λ 2 (t 2 ) Λ 2 \ Λ 1,f 2 (t 2 )=, f 1 (s 1 )=f 2 ( t 2 )} (3.5) (with the onvention tht t i rnges over T i nd s i over S i ), 3. is defined y (s 1,s 2 ) (,t 2 ) (s 1,s 2 ) iff s i i t i i s i, i =1, 2, where one of the t i n e si nd ssuming s i i si i s i holds for every stte s i S i, 4. Λ = Λ 1 Λ 2 nd λ follows ordingly, 5. On indexes, the onstrution is more tehnil to ount for the struture of the φ I, tht must llow index duplitions. We only mention it for ompleteness, ut the reder n sfely skip it. Consider I = I 1 I 2 { } nd define reltion R in I y i 1 Ri 2 i 0 I 0,f op 1 (i 0)=i 1,f op 2 (i 0)=i 2 (3.6) 6 The null LMCA hs single stte, no trnsition nd n empty index set I. 44

45 where i 1 or i 2 n e. Let e the equivlene reltion generted y R in I, then I is the quotient set I / minus the lss (the overline denotes lsses). χ is defined y χ(,t 2 )={i 1 : i 1 χ 1 ( )} { i 2 : i 2 χ 2 (t 2 )}, nd similrly for the other trnsitions. The morphisms g i : A A i, i =1, 2, re the nonil projetions on sttes nd trnsitions, nd on index sets g op 1 (i 1)=i 1, nd similrly for g op 2. #!! " # "!! d! " d dd #!!!!!!! d "! d!! " Figure 3.5: Exmple of omposition y pullk : the result A 1 A 2 is on top, the lowest LMCA is the interfe A 0, nd the two omponents A 1, A 2 pper etween them. Fig.3.5 illustrtes this omposition. For the lrity of drwings, only trnsition lels re mentioned. Morphisms f 1,f 2 re depited with dotted rrows (only stte mppings re represented, trnsition mppings n e dedued), nd morphism g 1,g 2 re suggested y stte nmes (not drwn either). A single lel, α, is shred y two LMCA A 1, A 2. Trnsitions outside the domins of morphisms f i synhronize s in n ordinry prllel produt. As for the produt, the definition vi universl property entils the ommuttivity of the pullk. The ssoitivity is less nturl : the nottion A 1 A 2 suggests inry opertor, ut tully hides 5-tuple, i.e. A 0, A 1, A 2 plus the two morphisms f 1,f 2. So there is no strightforwrd mening to (A 1 A 2 ) A 3 = A 1 (A 2 A 3 ). Nevertheless, the pullk is tegoril limit, nd the ltter n e omputed y prts. So if we n give mening to A 1 A 2 A 3, we know tht rkets n e introdued. We tke nother wy elow to get ssoitivity. Reltions etween produt nd pullk. Let us first simplify the setting to the se we use in the sequel, nd justify the utility of the pullk. 45

46 Proposition 6 Consider three LMCA A 0, A 1, A 2 nd the prtil produts A 1 = A 1 A 0, nd A 2 = A 0 A 2, with respetive nonil projetions f i : A i A 0. One hs A 1 A 0 A 2 = (A 1 A 0 ) A 0 (A 0 A 2 ) (3.7) (3.7) n e derived y strightforwrd pplition of the universl properties of the produt nd of the pullk (fig. 3.6). A 0 (A1 x A 0) (A 0 x A 2) g 1 g $ 0 A $ 0 0 A 1 x A 0 A 0 x A 2 10 $ 1 $ 10 $ 0 $ $ 2 A 1 $ 1 A 1 x A 0 x A 2 $ 2 A 2 Figure 3.6: Equivlene etween produt of three omponents nd pullk of prtil produts. Proposition 6 is importnt euse it llows us to rephrse intertions defined s produts in terms of intertions vi n interfe. Oserve however tht A 0 does not lwys pture ll intertions etween A 1 nd A 2 : the ltter n hve trnsitions rrying identil lels, tht will thus synhronize in the produt, s expressed y (3.4). Proposition 6 generlizes to the se of N LMCA. Consider A 1,..., A N, nd for I {1,..., N} let us define A I i I A i. Then, for I, J {1,..., N}, we define A I J A I A J A I A J By (3.7) we hve A I A J = A I J (3.8) nd onsequently I, J, K {1,..., N}, (A I A J ) A K = A I (A J A K ) (3.9) tht we n tke s definition for A I A J A K. So we re k to sitution where is ommuttive nd ssoitive inry opertor 7. A I A J A K n tully e defined diretly s tegoril limit, ut we don t detil this here. 7 As rule of thum, to ompute n expression with mny pullk signs, one must put rkets round pirs whih determines the interfes to use in eh pirwise pullk. The position of rkets doesn t hnge the result. 46

47 3.2 Grphs ssoited to multi-lok system Let us now ome k to n LMCA A defined s produt A = A 1... A N. Our ojetive is three-fold : 1. we wnt to represent grphilly ll the intertions etween the A i, 2. we need to express these intertions under the form of omponents interting y shred vriles, in order to use the formlism of hpter 2, nd 3. we must otin grph on whih the seprtion theorem holds. There exist different mnners to stisfy these onditions ; whih one is dequte depends on the hoie of omposition nd projetion opertors. Let us ll the A i sites, insted of omponents, in order to void onfusions. The intertions etween sites re of ourse due to shred lels. One n dopt severl viewpoints to red these intertions. For exmple, the λ s re resoures (vriles) tht sites (systems) n use to uild their trjetories. And sites re of ourse in ompetition for the λ s, in the sense tht they hve different ilities to omine them into sequenes. This governs the diret grph representtion. By ontrst, lel λ shred y two or more sites introdues synhroniztion onstrint etween them, in the sense tht ll these sites must fire synhronously when suh lel is onerned. So lels n e onsidered s lol onstrint systems defined on susets of sites, tht eome the resoures (or vriles). This governs the dul grph representtion. To illustrte these onstrutions on running exmple, let us strt from iprtite grph representtion of intertions : The first fmily of verties re the sites A i, the seond fmily is formed y lels λ in Λ = Λ 1... Λ N, nd n edge is drwn etween A i nd λ iff λ Λ i (Fig. 3.7). A 1 A 2 ' 1 ' 2 A 3 A 4 ' 3 ' 4 A 5 Figure 3.7: A iprtite grph illustrting the reltion etween lels nd sites A i Diret grph Here, lels in Λ = Λ 1... Λ N ply the prt of vriles nd sites A i re the omponents tht use them : this orresponds to setting where system omposition is defined y shred lels, nd projetions re defined with respet to lel sets, s for lnguge systems (setion 2.1.2). As lredy detiled in hpter 2, intertions 47

48 in A n e represented s hypergrph H, where eh A i defines the hyperedge Λ i (Fig. 3.8.). To form the support of distriuted omputtions, H must e turned into onnetivity grph G nx, with sites A i s verties, nd where (A i, A j ) is n edge s soon s Λ i Λ j (Fig. 3.8.). The ltter in turn must e simplified into ommunition grph G, y reursively removing redundnt edges. An edge (A i, A j ) is redundnt in grph G iff there exists nother pth (A i, A k1, A k2,..., A kl, A j ) in G suh tht Λ i Λ j Λ kl for every k l,1 l L(Fig. 3.8.). Nturlly, some of the A i n e grouped into lrger omponents if tree-shped ommunition grph is needed : on the exmple, one n reple A 3 nd A 4 y unique node orresponding to A 3 A 4. Grouping A 4 with A 5 would work s well. A 1 A 2 ' 1 ' A 2 1 ' 1 A 2 A 1 A 2 ' 1 A 3 A 4 ' 1 ' 1 ' ' 2 2 ' 1 ' 2 ' 2 ' 2 A 3 A 4 A 3 A 4 ' 3 A 5 ' 4 ' 3 ' 4 ' 3 ' 4 A 5 A 5 Figure 3.8: Hypergrph defined y the A i on the lel set Λ (left), its ssoited onnetivity grph etween sites A i (enter), nd ommunition grph (right). Seprtion property. Antiipting little on future setions, this is how the ommunition grph G etween sites will e used. Let I, J, K {1,..., N} e site index sets. We denote y I J K the seprtion property on G : there is no pth from node of I to node of K tht doesn t ross J. I J K entils tht the MC utomt A I, A J, A K re suh tht Λ I Λ K Λ J, so the ommon vriles (i.e. lels) of A I nd A K re ptured y A J. Sine the ltter sees ll intertions etween A I nd A K, one will e le to prove reltion like Π ΛJ (E I T E K ) = Π ΛJ (E I ) T Π ΛJ (E K ) (3.10) where E I nd E K will e trjetory sets tthed to A I nd A K, T omposition opertor on them, nd Π ΛJ projetion opertor on lels. This eqution orresponds to xiom (3) nd to diret pplition of results presented in hpter Dul grph This representtion orresponds to the se where projetions on lels re impossile nd must e repled y projetions on the A i. In other words, the sites A i must e interpreted s the vriles. And we must s well define notion of omponent, nd omposition opertion sed on shred vriles, i.e. shred sites. This is where the omposition y pullk eomes useful. 48

49 ' 1 A 1 A 2 ' 2 ' 1 A 3 ' 3 ' 4 A 4 A 1 A 2 A 3 ' 3 ' 2 ' 4 A 4 A 5 A 5 d A 1 ' 1 A 2 A 1 S 1,2 A 2 ' 1 ' 1 ' 2 ' 2 ' 2 S 1,3 S 2,4 A 3 A 4 S 2,3 A 3 A 4 ' 3 ' 4 A 5 S 3,5 A 5 S 4,5 S 3,4 e f A 1 ' 1 A 2 A 1 S 1 A 2 ' 1 ' 2 '2 S 2 S 3 A 3 A 4 A 3 A 4 ' 3 ' 4 S 4 A 5 A 5 S 5 Figure 3.9: Different wys of defining omponents s lol produts of sites A i. The ojetive is to pture y suh omponents the onstrints tht remin in ommunition grph on the A i, s in Fig

50 Let us define omponents S k = i Ik A i where I k {1,..., N}. By (3.8), one hs A = k S k s soon s the S k over ll the A i. However, this leves lot of flexiility. Considering tht every λ Λ introdues onstrint on some of the vriles A i, one n for exmple tke s omponents the S λ given y λ Λ, I λ = {1 i N : λ Λ i } nd S λ = i Iλ A i (3.11) in order to over the full onstrint grph of A. This hoie is illustrted in figures 3.9. nd This first definition, however, results in quite lrge omponents, whih n e ineffiient for the modulr omputtions desried in hpter 2. Another possiility is thus to onsider only pirwise synhroniztions nd tke { Ai A i, j {1,..., N}, i < j, S i,j = j if Λ i Λ j (3.12) null otherwise A = 1 i<j N S i,j is stisfied s soon s every site shres lel with t lest one omponent. This hoie is illustrted in Figs nd 3.9.d. In this formultion, the hypergrph indued y omponents S i,j on nodes A i redues to n ordinry grph : the ltter is nothing more thn the onnetivity grph G nx etween sites A i (ompre Fig. 3.9.d to Fig. 3.8.). To get the onnetivity grph etween omponents S i,j, we thus hve to tke the dul of G nx. Apprently, this seond definition of omponents involves lrger numer of smller omponents, nd genertes more omplex grph. So, t this point, there is no ovious gin. Rell tht some edges in G nx re redundnt, i.e. desrie synhroniztion onstrints etween the A i tht re ptured y others pths. So we redue the set of S i,j to those tht over ommunition grph G etween the A i (Fig. 3.9.e nd 3.9.f). The remining omponents still over ll the A i, nd their onnetivity grph is still the dul of G. This hoie not only redues the numer of omponents, ut lso stisfies nie seprtion properties (see elow). In prtiulr, if G is tree, then every ommunition grph etween the seleted S i,j is tree. ' 3 A 2 A 3 ' 1 ' 2 S 1 S 2 A 1 Figure 3.10: Two omponents S 1 = A 1 A 2 nd S 2 = A 1 A 3 tht shre only vrile/site A 1, lthough their intertion is due oth to A 1 nd to the externl lel λ 3, not ptured y A 1. Seprtion Property. Antiipting gin on future setions, this is how n expression like A = k K S k will e used, ssuming tht the S k over ommunition grph G etween sites A i. 50

51 The first importnt thing to notie is tht the shred vriles (= sites) etween two S k do not pture ll intertions etween these omponents, so we re not extly in the setting of hpter 2. Consider the exmple in Fig : S 1 nd S 2 shre only vrile/site A 1, ut they lso intert y lel λ 3 whih is not seen y A 1. This externl intertion orresponds to the seond line of (3.4) in the definition of pullk, nd kills xiom (3). In other words, it is not possile to diretly trnslte intertions y shred lels into intertions y shred sites. Nevertheless, there exists resonle rnge of situtions where xiom (3) holds. For K n index set, let us denote S K = k K S k = A K where K = k K I k. Consider the ggregted omponents S I = A I nd S K = A K, their shred vriles A J (with J = I K ) pture ll intertions etween S I nd S K iff Λ I Λ K Λ J 8. Under this ssumption, one will e le to write Π AJ (E I T E K ) = Π AJ (E I ) T Π AJ (E K ) (3.13) for E I nd E K trjetory sets of S I nd S K, respetively, nd T omposition funtion on these trjetory sets. Lemm 1 Assume omponents S k over ommunition grph G of the A i. Let S J seprte S I from S K on their own ommunition grph, then vriles A J seprte A I from A K on G, nd shred vriles of S I S J nd S J S K pture ll intertions etween these ggregted omponents (Fig. 3.11). This limited vlidity rnge of xiom (3) is tully suffiient to rederive ll results of hpter 2, in prtiulr the messge pssing lgorithms. S 3 S 4 A 6 A 5 S 5 S 1 S 2 ' 1 ' 2 ' A A A 3 A 4 Figure 3.11: Component S 4 seprtes S 1 from S 2 : its vriles see lel λ 3, responsile of diret intertion etween S 1 nd S 2. Consequently, ommon vriles of S 1 S 4 nd S 4 S 2 pture ll intertions etween these ggregted omponents. 3.3 Dignosis prolem As mentioned in the introdution of this hpter, we onentrte on dignosis prolems, in wide sense. We first stte the prolem, then exmine two pprohes to solve its entrlized version, nd their ility to e extended into distriuted tehniques. 8 This sitution is referened s the struturl ssumption in [37, 40]. 51

52 3.3.1 Semntis So fr we hve foused on the stti struture of systems nd didn t use t ll the ft tht they re dynmi systems. To introdue the time dimension, we must give mening to the notion of trjetory of n LMCA. For mtter of simpliity, we dopt the usul sequentil semntis, lso lled interleving semntis, nd define runs s sequenes of trnsitions. The next hpters will e sed on true onurreny semntis, where runs re prtil orders of events. The ltter re more dpted to distriuted or modulr systems, with sprse intertions nd thus high level of onurreny. As we will see t the end of the hpter, sequentil semntis do not fully tke dvntge of the distriuted pproh to monitoring prolems. But their simpliity llows us to present the mthemtil skeleton of this frmework nd the mehnis of omputtions, without the tehnilities due to prtil orders. One the priniples re estlished, it will e suffiient to hek few key results to dpt this setting to prtil order semntis. Let A =(S, T, s 0,, χ, I, λ, Λ) e n LMCA. In the sequentil semntis (SS), run (or trjetory) of n LMCA A is modeled s sequene σ =(,t 2,..., t N ) of trnsitions suh tht = s 0 nd t n = s n = t n+1, 1 n N 1 (lso denoted s 0 [... s n 1 [t n s n [t n+1 s n+1...[t N s N, nd s 0 [σ s N for sequenes) Ojetives Centrlized dignosis, single sensor. Consider n LMCA A, nd ssume A produes hidden run σ. One gets informtion out this run y mens of sensor tht oserves prt of the trnsition lels produed y σ. Speifilly, we ssume the sensor only ollets lels of Λ Λ, nd yields sequene O of oserved lels. The trditionl dignosis prolem is stted s follows (see Smpth et l. [95]) : ssume suset T T of trnsitions represents fults tht n our in A. Given O, one would like to determine whih of the following sttements holds : trnsition of T ws fired for sure in σ, σ didn t use ny trnsition of T, for sure, σ my hve fired trnsition of T, O doesn t llow to deide. There exist more elorte forms of the prolem, tht hek whether omplex property is stisfied y the hidden run σ given O, for exmple the ourrene of followed y two ourrenes of t 2, et. (see [61]). They re identil in nture nd do not introdue extr forml diffiulties. A more generl version of the dignosis prolem onsists in reovering ll runs of A tht ould explin the oserved sequene O, mong whih will lie the true hidden run σ. Reovering ll runs mthing O, or estimting the finl stte of σ given O re lmost identil prolems. And the ltter, lled the onstrution of n oserver for A, is the min uilding lok in the derivtion of dignoser for A (see next su-setion). So we fous on this version here, tht we ll the single sensor entrlized dignosis prolem. 52

53 Centrlized dignosis, severl sensors. In the se of modulr system A = A 1... A N = S 1... S K representing lrge distriuted mhine, it is more resonle to ssume severl sensors insted of single one. These sensors n e tthed to the A i, or equivlently to omponents S k. We onsider the first se for simpliity, sensor is tthed to eh A i nd oserves lels of Λ i Λ i (Λ i = represents the sene of sensor). When A performs the hidden run σ, the lels produed y trnsitions operting in A i re olleted y sensor i under the form of sequene Oi. So the oservtion is now tuple (O 1,..., O N ) of sequenes, nd the ojetive remins the sme : ompute ll runs of A tht explin these oservtions. Notie tht we lose informtion with respet to the single sensor se : the ext interleving of the different Oi is lost. Remrk : we ssume here tht lels tthed to trnsitions in the A i ply doule role. They re first responsile for trnsition synhroniztions, nd they lso orrespond to oservtions olleted y the sensors. These roles n e seprted, for exmple y introduing n extr lel on trnsitions to define their visile signture. An lternte solution onsists in using more elorte synhroniztion lger etween omponents. In ny se, these extensions omplexify nottions ut hnge very little in the forml spets of the prolem. So we prefer to keep the simplest formultion. Distriuted dignosis. The previous tsk n e extremely omplex : one must onsider ll possile interlevings of the O i, nd for eh of them the entrlized dignosis prolem requires to hndle possily huge system A. The distriuted (or modulr) dignosis pproh tkes dvntge of the intertion struture of A to void these diffiulties, provided intertions re sprse enough. In its simplest form, the distriuted pproh is sed on the following ides : lol supervisor D i is tthed to eh A i, eh D i hs limited knowledge out A (typilly the model of A i plus some interfe informtion with the other A j ), eh D i only knows lol oservtions O i, the D i ompute their lol view of the glol dignosis, i.e. they ompute trjetories of their omponent A i tht explin lol oservtions Oi nd t the sme time re oherent with explntions provided y the other lol supervisors. The next setions of this hpter will formlize this pproh nd nlyze its properties. Let us just mention tht, idelly, distriuted/modulr pproh to the dignosis prolem must e equivlent to the entrlized one, in the sense tht the omintion of lol results should yield the solution of the (multi-sensor) entrlized dignosis prolem. Historilly. Considering the evolution of ides, severl intermedite steps were explored efore the distriuted dignosis prolem ws stted in the ove terms. 53

54 Erly ontriutions onsidered first the se of severl sensors Oi on the sme mhine A, nd proposed lol supervisors mking deisions with the knowledge of A nd Oi. These gents would then forwrd their results to some entrl supervisor in hrge of ssemling them ording to vrious simple ggregtion rules or poliies [25, 115]. Despite the expensive use of A, there ws no reson why suh deentrlized pprohes should e equivlent to the entrlized one. Whene notions of deentrlized oservility, ontrollility, dignosility, et. Another pproh onsidered distriuted oservtions on modulr model, i.e. lol gents tking deisions with the knowledge of A i nd Oi, nd mking ssumptions on the intertions with neighoring omponents. One gin, lol gents would forwrd their results to entrl supervisor in hrge of ssemling them, now with the im of otining equivlene with the entrlized dignosis [8, 88]. The lst step ws then to get rid of the entrl supervisor, whih men/ distriuting its jo under the form of oopertion etween lol gents, nd more essentilly 2/ ndoning the ojetive of glol solutions ville somewhere, now repled y distriuted knowledge [103] The dignoser pproh Given n LMCA A nd suset of visile lels Λ Λ, n oserver for A is deterministi utomton D A, tking lels of Λ s input, nd whih sttes re susets of sttes of A. When A performs hidden run σ nd outputs the lel sequene O Λ, D A fed with O rehes unique stte tht lists ll possile sttes where A ould e, given O (Fig. 3.12).!! ( " (! d {,} {,,d}! " {}!! "! {,,d} {d} Figure 3.12: An utomton (left) nd its oserver (right) for visile lels Λ = {α, β}. D A n e otined esily in two steps [15] : 1. n epsilon-redution, tht reples eh stte of A y its invisile reh (sttes rehle through invisile trnsitions), followed y 2. determiniztion, tht ggregtes sttes rehle y the sme trnsition lel. To otin dignoser for A, let us onsider trivil two stte utomton B tht stys t stte green s long s trnsitions of T \ T re fired, nd otherwise moves from green to red nd stys there forever. The synhronous produt A B (synhroniztion on trnsition nmes) replites A with n ugmented stte, to memorize the firing of fult trnsition. The oserver of A B yields dignoser for A : when ll sttes of D A B re green, the nswer will e tht no fulty trnsition 54

55 ourred, et. Notie tht y strightforwrd extension, one n tully test ny regulr property on runs of A, i.e. properties tht n e expressed under the form of n utomton B [61]. There is simple se where this onstrution extends to modulr LMCA. Assume A = A 1 A 2, the oserver D A1 A 2 ftorizes s D A1 D A2 when shred lels re visile on oth sides, i.e. when Λ 1 Λ 2 =Λ 1 Λ 2. Indeed, under this ssumption the epsilon-redutions only remove lol trnsitions, nd the produt of deterministi utomt remins deterministi. Interestingly, the visiility of intertions ws often ssumed in pulitions relted to distriuted dignosis [25], ut it doesn t seem its importne for the modulrity of the dignoser hs een notied. The result is not mentioned in [22] for exmple, lthough it is fundmentl for the modulr dignosility tests developed in this work. And, similrly, the sme ssumption is hidden in [54, 55]. The extension of the modulrity property of the dignoser to the se of invisile synhroniztion events remins n open question. The result seems to hold with weker ssumptions, ut if modulr dignosers lwys exist, it is likely tht one ould derive from them joint oservility/dignosility test, in the se of distriuted oservtions, whih is shown in [105] to e undeidle. To summrize, dignosers offer the dvntge to e omputle off-line : the tul oserved sequene is not needed. Dignosers re reursive in nture, in the sense tht they n proess oservtions on the fly, nd t low ost. On the side of drwks lies the prolem of size, of ourse, sine they re designed to proess ny possile sequene of oservtions. And the modulrity of dignosers doesn t hold for most interesting ses, where omponent intertions re hidden. 3.4 Distriuted dignosis : the lnguge pproh Distriuted omputtions sed on lnguges were independently proposed y Rong Su in [102, 103]. This pproh proly forms the simplest frmework for the methodology we propose in this doument, so we re-express it here in our formlism. We then dpt it to perform omputtions on runs of omposite system Dignosis in terms of lnguges Centrlized dignosis. Consider leled utomton A = (S, T, s 0,, λ, Λ) produing hidden run σ tht is oserved through the lel set Λ Λ. This yields the oserved sequene O =Π Λ (σ) where Π Λ is the nturl projetion on lels of Λ. Let us denote y L(A) T the lnguge of A, i.e. the set of projetions Π Λ (σ) where σ rnges over ll runs of A. In this setion, we onsider simplified version of the dignosis prolem : we wish to reover ll words of L(A) tht yield O when projeted on Λ. We ll this the dignosis of O. It is given y D = L(A) Π 1 Λ (O ) = L(A) L O (3.14) where L is of ourse the prllel produt of lnguges. 55

56 Severl sensors. Let us now onsider modulr utomton A = A 1... A N, with A i =(S i,t i,s 0 i, i,λ i, Λ i ), nd distriuted oservtions Oi of the hidden run σ, Oi =Π Λ (σ) where Λ i i Λ i. As explined in the previous setion, distriuted oservtions men loss of informtion : one only knows tht there exists n interleving of the Oi tht orresponds to the hidden run σ. So let s tke O = O 1 L... L O N (3.15) s oservtion, whih omputes ll omptile interlevings of the Oi. The dignosis is gin given y D = L(A) L O, i.e. words of L(A) omptile with t lest one interleving of the Oi. Of ourse, oth O nd L(A) re generlly lrge, so in prtie there is no hope to solve the prolem in tht wy. Distriuted dignosis. Let us onsider insted the struture of A. From A = A 1... A N we derive Inserting (3.16) nd (3.15) into (3.14) yields L(A) = L(A 1 ) L... L L(A N ) (3.16) D = [L(A 1 ) L O 1 ] L... L [L(A N ) L O N ] (3.17) where L(A i ) L O i D i is lol dignosis, i.e. the set of words of omponent A i tht explin lol oservtions O i. We n now pply the results of hpter 2 : vriles re the lels in Λ, nd omponents re lnguges on theses lels (see the lnguge systems setion 2.1.2). D hs produt form, where ftors D i re omponents operting on lels Λ i. So D dmits miniml produt overing D = D 1 L... L D N with D i Π Λi (D) (3.18) The D i orrespond to the lol views of the glol dignosis : they ontin words of L(A i ) tht oth explin lol oservtions Oi nd tht re omptile with lol explntions in ll other omponents. As explined in hpter 2, the desired D i n e omputed in modulr mnner without omputing D itself, given the ft tht the prllel produt of lnguges L nd projetions Π Λ on susets of lels stisfy xioms (1)-(4). The support of modulr omputtions is ommunition grph G etween omponents A i, in the diret grph viewpoint (Fig. 3.8.), euse projetions re defined with respet to lels. If G is tree, onvergene to the D i is rehed in finite time. Otherwise, one n either group omponents to reover tree, or simply ignore yles nd pply turo proedure. Sine lnguge systems re involutive, turo lgorithms re known to onverge to unique point (t lest for the wek topology) nd provide resonle (upper) pproximtions D i of the D i : D i D i D i nd D = D 1 L... L D N Dignosis in terms of trjetories Let us ome k to our originl prolem : determining runs of A tht would explin the sequene O of oserved lels. 56

57 By post-proessing. Let L(A) denote the lnguge of A, now expressed in terms of sequenes of trnsitions insted of sequenes of lels. Oviously, the new dignosis we re looking for is the inverse projetion of D on L(A): Ḋ L(A) Π 1 Λ (O ) = L(A) Π 1 Λ (D) (3.19) where Π 1 Λ yields sequenes of trnsitions. In the sme wy, when A = A 1... A N the lol views of the glol dignosis re s well lerly given y reverse projetions of the D i on the L(A i ) Ḋ i = L(A i ) Π 1 Λ i (D i) (3.20) However, to prepre for the tehniques we develop in the sequel, it is worth determining diretly Ḋ nd the Ḋ i y omputtions on trjetory sets. By diret omputtion. We first refine the notion of produt L, to dpt it to trjetory sets. Consider two LMCA A 1, A 2 nd their produt A 1 A 2. There exist nturl projetions Π i : L(A 1 A 2 ) L(A i ) : they simply reple trnsitions (,t 2 ) y t i, nd erse the ltter when t i = si. Let T 1, T 2 e susets of L(A 1 ), L(A 2 ), respetively, we define their lel-sed produt y T 1 L T 2 = Π 1 1 (T 1) Π 1 2 (T 2) (3.21) Nturlly, when A = A 1... A N this produt preserves L(A) = L(A 1 ) L... L L(A N ) (3.22) For mtter of homogeneity, we ould enode O s leled sequene of trnsitions (s we will do in the sequel) in order to ompute its produt with L(A). For the ske of simpliity, let us rther extend L to sequenes of lels, whih llows us to write : Ḋ = L(A) L O (3.23) To express our prolem in the setting of hpter 2, we must now define systems, vriles nd projetions of systems on these vriles. We tke V mx = {1,..., N} Λ, so ompred to the previous setion, we dd site nmes to the vrile set. The systems we onsider re formed y words w of Λ, nd y leled sequenes of tuples (t i ) i I for some I {1,..., N}, tht form vlid runs of A I. We denote them y (σ, Λ I ) to keep trk of the lel set tthed to the run σ of A I. So we hve two types of sequenes, ut the systems we define elow will ontin sequenes of single type. The omposition opertion L pplies to oth types of sequenes. 57

58 Let us now ome to projetions. For J {1,..., N} nd Λ Λ we must define Π J Λ of the two ojets ove. On word w of Λ, we tke Π J Λ (w) Π Λ (w) (3.24) whih erses lels not in Λ. And on run (σ, Λ I ) of A I Π J Λ (σ, Λ I ) { (ΠI J (σ), Λ I Λ ) if I J Π Λ (σ) otherwise (3.25) So the run σ eomes pure lel sequene when no site of I is ommon to J. With this enoding, L nd Π stisfy xioms (1)-(4). Systems L(A I ) operte on vriles I Λ I nd re involutive. The dignosis Ḋ is given y Ḋ = [ L(A 1 ) L O 1] L... L [ L(A N ) L O N] nd dmits the miniml produt form Ḋ1 L... L Ḋ N (3.26) Ḋ = Ḋ 1 L... L Ḋ N with Ḋ i Π i,λi (Ḋ) (3.27) One gin, this redution prolem n e omputed y MPA, where messges re exhnged etween sites A i (or disjoint groups of sites A I ). The properties re the sme s for lnguge systems. Nevertheless, oserve very nie feture of this setting : eh vrile i is privte to site A i, i.e. sites only shre lels. So the messges exhnged etween sites re sequenes of lels, while sites tully ompute on sequenes of trnsitions Extensions nd drwks The two pprohes ove, y lnguge systems or y trjetory systems, n e extended in severl wys. Reursivity. First of ll, they n e mde reursive. Assume the Oi re not given one for ll, ut orrespond to growing sequenes of oservtions, tht we enode s Oi (t): O i (t) O i (t + 1), t 0. As explined t the end of setion 2.2.2, messge pssing lgorithms pplied to hnging lol dignoses Ḋi(t) = L(A i ) L Oi (t) remin vlid nd preserve their onvergene properties, provided oservtions sequenes stilize. And the Ḋi(t) n of ourse e omputed reursively. Distriuted optimiztion. Seondly, one ould ssoite ost funtions to words of L(A i ) or to lol runs of the L(A i ), nd dopt (min, +) setting to define the omposition. Messge pssing lgorithms would then ompute (or pproximte) the word (resp. the run) of miniml ost in L(A) (resp. in L(A)) explining ll oservtions. 58

59 Drwks. Both spets n of ourse e omined, so pprently we lredy hve the pproprite frmework to del with networks of dynmi systems nd we ould stop here. However, these pprohes sed on sets of sequenes suffer from two serious drwks. 1. The first drwk reltes to the size of the lol dignoses D i (t) or Ḋi(t): they grow rpidly with the length of the oservtion sequene Oi (t), euse of rnhings ourring in A i. So even if one performs modulr omputtions, the modules themselves eome enormous with time. We therefore need to find ompt mnner to enode sets of runs of omponent A i. This is preisely the ojetive of the next setion. 2. The seond drwk is onsequene of the sequentil semntis, inpproprite for onurrent systems. It multiplies the numer of runs y omputing useless interlevings. This effet ppers in prtiulr in produt opertions when few lels re ommon. Consider the extreme se of two omponents A 1, A 2 tht do not intert t ll (no shred lel), nd trjetory σ i of n events in eh of these omponents. There exist 2 n different interlevings of these two runsσ 1 nd σ 2, tht will e onsidered s different runs of the joint system A 1 A 2. Even with ompt wy of enoding lol dignoses D i, omintoril explosions remin due to the ft tht onurreny is not hndled properly (see setion 3.7). The next hpters will ddress this diffiulty y resting ll results in true onurreny semntis. 3.5 Trjetory sets in the sequentil semntis The mthemtil skeleton of the previous setion gives n urte view of the struture of omputtions tht we perform in the sequel. The min hnge tht we introdue now reltes to the ojets on whih these opertions pply : we reple lnguges y more omplex strutures tht enode sets of runs in ompt mnner Trellis utomton We hd lredy defined the tegory A, formed y leled multi-lok utomt ssoited to notion of morphism. A guiding property when one defines morphisms is tht they must preserve runs : Lemm 2 Let φ : A 1 A 2 e n LMCA morphism, nd let σ 1 e run of A 1, then σ 2 = φ(σ 1 ) is run of A 2. Nturlly, φ(σ 1 ) is defined y the reursion φ(σ 1 )=φ(σ 1 ) φ( ) if φ( ), nd φ(σ 1 )=φ(σ 1 ) otherwise. We now fous on the representtion of sets of trjetories for given LMCA. To do so, we introdue su-tegory of LMCA : A is sid to e trellis utomton (TA) when 1. the grph of A, determined y, is direted nd yli, or equivlently defines prtil order on nodes of A (i.e. sttes nd trnsitions), 59

60 2. this prtil order is well founded : x S T, {x S T : x x} <, 3. s 0 is the unique minimum of this prtil order : x S T, {x S T : x x, x x} =0 x = s 0. As onsequene, in ny run σ =(,..., t n ) of TA, ll trnsitions t i re different. In the sequel, sttes of TA re lled onditions (denoted y C insted of s S), nd trnsitions re lled events (denoted y e E insted of t T ) 9. A trellis utomton κ =(C,E,, 0,χ, I, λ, Λ) is lled onfigurtion when it is omposed of single sequene of events : C, 1 nd 1. We will sy tht κ is onfigurtion of T when this onfigurtion is su-ta of T. Chnging slightly nottions, we now identify runs of T with its onfigurtions. Multi-lok funtion. y the funtion To events of T we ssoite vetor-lok vlue defined e E, h e : I N i h e (i) =1I i χ(e) (3.28) So n event ounts 1 for omponent i iff it influenes this omponent. This definition leds to the notion of height of ondition in onfigurtion κ of T. Assume κ orresponds to the sequene of events (e 1,e 2,..., e n,...) nd goes through ondition immeditely fter e n, we define H κ () N h en (3.29) n=1 For every omponent index i I, H κ () thus ounts the numer of events ssoited to omponent i nd efore in run κ, whene the nme multi-lok funtion. It oviously reltes to vetor-loks of Mttern [84] nd Fidge [50]. A trellis utomton T is orretly folded iff C nd for ny pir κ, κ of onfigurtions of T ontining, one hs H κ () =H κ (). In other words, only sequenes of events with the sme multi-lok vlue n merge t the sme ondition (see Fig. 3.13). From now, we only onsider orretly folded trellis utomt ; they form the sutegory T of A. In (orretly folded) TA T, one n of ourse write H() insted of H κ () Time-unfolding of n utomton Let A A, T T, nd let f : T A e morphism. The pir (T,f) is trellis proess (TP) of A iff 1. f is folding of T onto A (i.e. totl funtion on sttes nd trnsitions), 9 These nmes ome from the definition of rnhing proesses, tht desrie runs of Petri nets in the true onurreny semntis. 60

61 t 2 t 2 t 3 t 4 t 5 t 6 d e f d t 7 t t 8 t g h i g t 3 t 4 t 5 e t 9 t 6 t 8 0 h i Figure 3.13: Two trellis utomt, with two omponents : thin rrows represent trnsitions ssoited to omponen, nd thik rrows those ssoited to omponent 2. The TA on the left is orretly folded, the other one isn t : the merge t h is illegl. f 2. T stisfies the prsimony riterion : 3. T is mximlly folded : e, e E, [ e = e, f(e) =f(e )] e = e (3.30), C, [ H() =H( ), f() =f( )] = (3.31) f n e viewed s leling of events of T y trnsitions of A. This definition thus ensures tht if κ is onfigurtion of T, then (κ, f κ ) enodes run of A (poin). Moreover, thnks to the prsimony riterion, Lemm 3 Let (T,f) nd (κ, f ) e trellis proesses of A, nd ssume κ is onfigurtion. There exists t most one injetive morphism φ : κ T suh tht f = f φ. This sttes tht run of A is represented t most one in trellis proess of A. As diret onsequene, Lemm 4 Two trellis proesses of A representing the sme trjetories of A re tully isomorphi. So trellis proesses provide the ompt struture we re looking for to desrie sets of runs of n LMCA. Nottion: for the ske of simpliity, we often omit mentioning the folding f in run (κ, f) or trellis proess (T,f) of A, nd simply tlk out κ or T. Let A =(S, T, s 0,, χ, I, λ, Λ) e n LMCA. One n uild trellis proess (T,f) of A y the following reursion : Proedure 1 1. Initiliztion : C = { 0 },f( 0 )=s 0,E= 2. Reursion : for C nd t T suh tht f() = t, if e E suh tht e = nd f(e) =t, then () rete e E, set e =, f(e) =t, χ (e) =χ(t) nd λ (e) =λ(t), 61

62 () rete C, set = e nd f( )=t, () if C with f( )=f( ) nd H( )=H( ), then merge nd. The three onditions of the definition re stisfied y onstrution, nd lerly ny trellis proess of A n e otined in tht wy, with suitle guiding of events to onnet. On the sis of the ove unfolding proedure, one n reursively define the union of two or more trellis proesses of A, whih produes nother TP of A. Notie right now tht the union T 1 T 2 my ontin onfigurtions tht were not present in ny of the T i, ut tht nevertheless remin vlid runs of A. We ome k on this point lter. Lemm 5 Let (T,f) e trellis proess of A, then T is isomorphi to the union of its onfigurtions (κ i,f i ). As onsequene, two trellis proesses of A tht hve isomorphi onfigurtions re isomorphi. Let us define the prefix reltion T 1 T 2 etween TP y the ft tht T 2 ontins more onfigurtions of A thn T 1, or equivlently tht there exists n injetive morphism φ : T 1 T 2. The ove lemms led to the following importnt result : Theorem 7 Given n LMCA A, there exists unique mximl trellis proess of A for the prefix reltion. We denote it ( UA st,f A) nd ll it the sequentil timeunfolding of A (ST-unfolding or STU for short). The proof is simple : Define ( U st A,f A) s the union of ll trjetories (κ, f) of A. Uniity is ovious from lemm 5, whih lso shows tht the STU ontins ll trellis proesses of A. t 2 t 5 t t 4 3 sttes t 2 t 2 t 2 t 2... t t t 5 3 t 5 3 t 3 t 4 t 4 time Figure 3.14: An ordinry utomton A(left) nd its sequentil time-unfolding U st A (right). The folding f A is represented y stte nd trnsition nmes lose to onditions nd events. Fig illustrtes the notions of trellis proess nd sequentil time-unfolding on n ordinry utomton, i.e. n LMCA with I = 1. The STU, ommonly lled the trellis of n utomton in digitl ommunitions, is the support of the Viteri lgorithm, for exmple in the mximum likelihood deoding of noisy its when trnsmissions re proteted y onvolutionl error orreting odes. Trellises lso form the impliit support of the Bellmn eqution, in optiml ontrol prolems. So only the multi-lok spets re new. 62

63 Oserve tht the size of n ordinry trellis grows linerly with its length. With multi-lok riterion to determine merge points, the numer of possile merges dereses nd one otins lrger struture, with growing width. But it remins smller thn the simple unfolding of A, tht unfolds oth time nd onflits (or hoies) nd yields deision tree. And the ltter is in turn smller thn the lnguge of A. Reltion etween trellis proesses nd su-lnguges. The time-unfolding of A enodes ll possile runs of A, or equivlently ll the lnguge of A. But rell tht union of onfigurtions of A generlly results in TP tht ontins more runs. So wht re the su-lnguges of A tht n e represented y trellis proess? Let (κ 1,f 1 ) nd (κ 2,f 2 ) e two runs of A, with 0 i [κ i i. They re sid to e H-equivlent, denoted y κ 1 H κ 2, iff f 1 ( 1 )=f 2 ( 2 ) nd H κ1 = H κ2, i.e. iff they hve sme length, nd strt nd finish t identil sttes of A. A su-lnguge L of A is sid to e suffix-losed iff κ = κ 1 κ 1 L, κ 2 L, κ 1 H κ 2 κ 2 κ 1 L (3.32) Lemm 6 Let L e su-lnguge of A, L n e enoded s trellis proess of A iff it is prefix- nd suffix-losed. The prefix-losure requirement is little emrrssing, ut n e esily disrded. Let us enrih the definition of trellis proess (T,f) with stop funtion St : C {0, 1} on onditions. We then ssoite to (T, f, St) the su-lnguge L of A formed y onfigurtions (κ, f) of T tht finish on stop point : 0 [κ nd St() = 1. Then Proposition 7 There is one to one orrespondene etween stopped trellis proesses of A nd su-lnguges L of A tht stisfy κ = κ 1 κ 1 L, κ 2 L, κ 1 H κ 2 κ 1 L κ 2 κ 1 L (3.33) (3.33) expresses oth suffix-losure nd weker form of prefix-losure. Notie however tht the ove limittions in the expressive power of trellis proesses will not prevent modulr omputtions nd dignosis sed on them Vritions round the height funtion The multi-lok funtion tht governs merge points of trjetories is somehow ritrry nd n e generlized, provided its vetor nture is preserved. So fr, eh event ffeting omponent i ws ounting for 1 in the lok of tht omponent. One n esily relx this ssumption. For exmple, let (E,,ɛ) e set provided with n internl omposition lw, nd ɛ s neutrl element. Let us tth to eh LMCA A funtion tht ssigns height vetor to ll trnsitions : t T, h t : I E suh tht i χ(t) h t (i) =ɛ, so the height of t is neutrl for omponent i when t doesn t ffet it. Nturlly, we provide these new LMCA with height preserving morphisms. 63

64 To define the height of ondition in onfigurtion κ, we simply omine these height vlues : for κ terminting t nd onsisting of events (e 1,..., e n ), we tke H κ () =h e1... h en (omponent-wise omposition of vetors). One n of ourse hoose (E,,ɛ) in order to hve monotoni loks, ut this is not neessry : iruits pprently don t other the theory 10. They n even e useful to void unfolding speifi prts of system. Assume for exmple tht one wishes to ount events tht produe lel in Λ Λ nd ignore ll other events. This n e done y h e = ɛ whenever λ(e) Λ. So silent loops re not unfolded in the time-unfolding of A. We refer the reder to [46, 47] for the interest of this property in dignosis pplitions. Notie tht the ide of not unfolding silent loops ws introdued y Lmperti nd Znell [72]. By ontrst, it is extremely importnt to preserve the vetor nture of height funtion. Andoning this struture mens reking the universl property stted in theorem 8 elow, nd onsequently loking the distriuted dignosis pproh sed on it. We ome k on the neessity of vetor loks t the end of the hpter (see lso [46, 47]). As remrk in pssing, let us mention tht one n esily design stritly inresing height funtion tht would llow no merge point t ll. For exmple height funtion tht stores ll the pst of ondition. In this se, the sequentil time-unfolding of A oinides with its ordinry unfolding, lso lled deision-tree. Consequently, ll results we stte on sequentil time unfoldings remin vlid for ordinry unfoldings Ctegoril properties The wkwrd funtion χ : T I introdued in the definition of LMCA ould hve ppered s useless deortion up to now. This feture is tully neessry to otin the following result. T A % A E!& U A st f A Figure 3.15: Universl property of the STU of A. Theorem 8 (Universl property) Let A A e n LMCA, let T T e trellis proess nd φ : T A morphism, there exists unique morphism ψ : T UA st suh tht φ = f A ψ. Theorem 8 essentilly expresses tht U st (A) desries ll runs of A, nd desries them only one (this explins the neessity of eing mximlly folded for trellis proesses). 10 This is only onjeture t this point, verified on set of exmples. For rigorous proof, one would hve to redefine trellis proesses to llow iruits, under the onstrint of height funtion. And then hek the preservtion of theorems 7 nd 8. 64

65 U Trellis utomt st U Automt Figure 3.16: Two funtors relting A nd T, tht form n djuntion. When designing struture to enode runs of A, s soon s its universl property is stisfied, some lssil tegory theory rguments re triggered nd led to very useful results. We sketh this resoning elow. T eing su-tegory of A, there exists n inlusion funtor F = : T A. Conversely, the ST-unfolding opertion on LMCA defines funtor G = U st : A T. By funtor, we men tht U st pplies lso to morphisms nd stisfies U st (f g) = U st (f) U st (g). The ft tht G = U st is funtor is diret onsequene of the universl property ssoited to ny UA st. The ltter lso entils tht (F, G) forms pir of djoint funtors (see [81] hpter 4, in prtiulr theorem 2.iv, p. 81). As the right djoint of the pir, G preserves tegoril limits, nd so preserves produts (see [81] hpter 5.5, in prtiulr theorem 1, p. 114). In other words, one hs A 1, A 2 A, U st (A 1 A 2 ) = U st (A 1 ) T U st (A 2 ) (3.34) where T denotes the tegoril produt in T. By ontrst with, the produt in T not only synhronizes events with shred lels, ut lso preserves the prtil ordering of the resulting struture. It must e understood s n opertion synhronizing trjetories insted of utomt. A 1 x A 2 A 1 A t 1 t 2 t 2 2 t t 1! t 2 " 1! t 2 " $ 1 $ 2 t 3 # t 3 t 3 * t t 3 # t 4 ) 4 f f 2 f 1 t 2 t 3 st U ( A 1 ) t t 2 t 2 1 $ 1 $ 2 t 3 t * t 4 3 st st U ( ) T U ( ) A 1 t 1 t 3 x st A A 2 2 U ( ) t 2 t 4 t 2 t 4 Figure 3.17: Top : two LMCA nd their produt. Bottom : the orresponding STU (on this exmple they oinide with lssil unfoldings). The 7 morphisms relting these LMCA form ommuttive digrm. Fig.3.17 illustrtes this property. The top line represents two LMCA A 1 nd A 2, with their produt (in the enter), nd the ssoited nonil projetions π i : A 1 A 2 A i. A 1 nd A 2 shre lels {α, β, γ} ut δ is privte to A 2. The 65

66 ottom line represents the orresponding sequentil time-unfoldings, together with their foldings (mterilized y stte nd trnsition nmes lose to onditions nd events). The STU UA st 1 A 2 is isomorphi to the produt U st (A 1 ) T U st (A 2 ), whih hs nonil projetions π i : U st (A 1 ) T U st (A 2 ) U st (A i ) to its ftors. The 7 morphisms mentioned on the piture form two ommuttive squres, thnks to theorem 8. Oserve tht UA st 1 A 2 hs two onditions pointing to the sme stte (, ) of A 1 A 2 : they orrespond to different multi-lok vlues, nd thus n t e merged. By doing so, one would get struture isomorphi to A 1 A 2, nd there wouldn t exist morphism π 1 nymore. In other words, tht would kill (3.34), whih shows the importne of seprte ounting of time in the vrious omponents of n LMCA. The djoint pir (F, G), where F orresponds to n inlusion, is lled orefletion. Sine dditionlly one hs T T, T U st (T ) (red isomorphi to ), one n tully define the produt T y T 1, T 2 T, T 1 T T 2 U st (T 1 ) T U st (T 2 ) = U st (T 1 T 2 ) (3.35) This lst reltion hs n importnt prtil onsequene : y omining Proedure 1, omputing ST-unfoldings, to the definition of the produt in A, one gets reursive lgorithm to ompute the produt in T. Finlly, just like produts, pullks re lso tegoril limits. So the resonings ove pply in the sme wy nd one hs A 0, A 1, A 2 A, U st (A 1 A 0 A 2 ) = U st (A 1 ) U st (A 0 ) T U st (A 2 ) (3.36) where the pullk T in the su-tegory T of trellis utomt is defined y T 0, T 1, T 2 T, T 1 T 0 T T 2 U st (T 1 ) U st (T 0 ) T U st (T 2 ) = U st (T 1 T 0 T 2 ) (3.37) And one gin, the omintion of Proedure 1 with the definition of gives reursive lgorithm to ompute pullks in T. 3.6 Distriuted dignosis : the trellis pproh The properties we hve otined on trellis proesses re now very lose to those of lnguge systems, so n we relly ompute with trellises? In this setion, we ome k to our dignosis pplition with these new tools. We follow the steps of setion 3.4 to highlight the forml similrities, ut lso the tehnil extensions tht re needed. As we will see, the mjor differene is tht we n t use of projetions on susets of lels. So projetions will now e defined with respet to sites A i, nd we will dopt the dul grph viewpoint of setion Centrlized dignosis, single sensor The setting doesn t hnge : A performs hidden run κ whose lels re oserved under the form of sequene O. The ltter n e enoded s trivil trellis utomton, formed of single sequene, i.e. O is onfigurtion. We re still looking 66

67 for runs of A tht would explin O. Sine we hve produt on trellis proesses, (3.14) eomes D = U st (A) T O (3.38) A few omments on this eqution. It is meningless to unfold A nd then ompute the produt. This forml reltion must e trnslted in prtie into reursive lgorithm : one omputes the unfolding of A guided y oservtions O, whih is loser to the expression D = U st (A O ) given y (3.35). Seondly, y ontrst with (3.14) or (3.23), D defined in (3.38) ontins runs of A tht explin only prefix of O, nd not the entire O. This effet n e orreted esily : it suffies to use stopped trellis proesses, s defined t the end of setion They form su-tegory of stopped utomt, ssuming morphisms tht preserve stop points. The theory is identil to wht hs een presented ove, so we omit this detil, for the ske of simpliity, nd ssume tht (3.38) does yield runs of A tht entirely explin O. We refer the reder to [46, 47] nd [48] for detils Centrlized dignosis, severl sensors When A = A 1... A N nd A i hs Λ i s lel set, ssume gin tht lels of the hidden run κ re olleted y independent sensors tthed to the sites A i. The sensor on A i ollets lels of Λ i Λ i nd produes the onfigurtion Oi. One n gin ssemle these oservtions y O = O 1 T... T O N (3.39) whih omputes ll possile interlevings of the sequenes Oi, nd then pply (3.38). The ltter will then mount to n unfolding of A guided y omplex trellis utomton O, insted of sequene. This is fesile, ut of ourse very ineffiient, euse of the omplexity of oth A nd O. Fig illustrtes this point : the two sequenes of oservtions shre single lel, β. The result of the produt is muh more omplex thn its ftors, euse it develops ll onurreny dimonds. So we re rther looking for dignosis methods tht operte on ftorized forms, nd void expnding lrge produts. This is in ft the hert of the pprohes we dvote in this doument Distriuted dignosis Before ddressing the distriuted dignosis issue, we must introdue the notion of projetion of trellis proess on sets of sites A i. Projetions. The site index set {1,..., N} eomes our vrile set V mx, nd we hve to define projetions Π I for I V mx. For simpliity, we will use nottion Π I insted of Π AI, when there is no miguity. 67

68 O! "! d 1 * ={!, "} 1 O # # " d # e 2 * 2 ={ ", #} O 1 x T O 2! # #! # dd #! # "! d #! de e Figure 3.18: The produt of two onfigurtions ompute ll possile interlevings of these sequenes, whih results in mny onurreny dimonds. Let K, L {1,..., N} e site index sets, nd ssume K L. From the produt form U st (A L )= U st (A K ) T U st (A L\K ) we know there exists nonil projetion Π L,K : U st (A L ) U st (A K ) (3.40) Every trellis proess of A L is prefix of U st (A L ), so Π L,K pplies to TP of A L nd yields TP of A K. When K L, we define Π L,K s Π L,K L, nd sine the strting point is generlly unmiguous, we simply write Π K insted of Π L,K. By ssoitivity of T, projetions stisfy Π K Π L =Π K L, whih orresponds to xiom (1). In prtie, projetions Π K re quite esy to implement. Consider for exmple Fig : it suffies to ollpse nodes relted y events tht do not elong to the sites we wnt to preserve. The result is lmost trellis proess of the seleted sites, exepted tht it violtes the prsimony riterion : isomorphi events my rnh out of given node. Therefore simple trimming opertion is lso neessry. Ojetive. The entrl result tht we invoke now is the ftoriztion property of the sequentil time-unfolding of A. Given A = A 1... A N, (3.34) eomes U st (A) = U st (A 1 ) T... T U st (A N ) (3.41) nd thus the dignosis dmits the ftorized form D = [U st (A 1 ) T O 1] T... T [ U st (A N ) T O N ] (3.42) In this eqution, eh D i = U st (A i ) T Oi is lol dignosis : it selets runs of site A i tht explin the lol oservtions Oi. The distriuted dignosis prolem simply mounts to omputing the miniml produt overing of D, i.e. to omputing the T i =Π Ai (D), whih yields D = [T 1 T O 1 ] T... T [T N T O N ] (3.43) Equivlently, one ould lso ompute the D i =Π A i Oi (D) =T i T Oi tht lso stisfy D = D 1 T... T D N. The ltter re lled the redued lol dignoses. 68

69 In the sequel, we don t distinguish these two redution prolems, tht re identil in nture nd n e solved in the sme mnner, with the sme omplexity. Apprently, we re thus in well known lnd, s explored in hpter 2. In relity, there is mjor differene with setion 3.4.2, due to the ft tht we don t hve projetions on lel sets. New shpe of omputtions. Our ojetive is to reover the D j y mens of modulr omputtions. Consider two LMCA A 1, A 2 tht shre lels Λ 1,2 Λ 1 Λ 2, see (3.44). Let T = T 1 T T 2 e trellis proess of A 1 A 2, nd let us try to replite the modulr omputtions of setion in order to ompute the miniml produt overing of T. One would hve to projet T 1 on Λ 1,2, in order to form messge M 1,2 =Π Λ1,2 (T 1 ) tht would e some sort of lel struture. The ltter would then e omined y speil produt L to T 2 nd would produe T 2 = T 2 L M 1,2. And symmetrilly to otin T 1. A 1 Λ 1,2 A2 (3.44) As we hve seen, this pproh works on simple strutures like sets of sequenes, ut, unfortuntely, projetions of trellis proesses on lel sets re hrd to define : By simply ersing the events of trellis proess tht do not rry visile lel, one would kill the trellis struture. Even worse, this n rete iruits, nd produe struture ontining fke sequenes of visile events 11. So to trnsfer informtion from one side to the other, we ypss this undefined lel struture nd use trik, sed on more fmilir opertions : We first omine T 1 nd T 2, nd projet the result on A 2 to form the messge : M 1,2 =Π A2 (T 1 T T 2 ). One n onlude y T 2 = M 1,2 T T 2. This omputtion trik looks like tutology on suh trivil exmple. But it mkes sense to propgte informtion when one imgines third site A 3 eyond A 2. As we show elow, modulr omputtions re tully sed on this ide. Expression of A with pullks. Sine we n t projet on lels, we re ound to use projetions on sites or sets of sites A i. So the A i eome our vriles, nd to omply with the formlism of hpter 2, we must express our systems in terms of shred vriles, i.e. in terms of shred sites. We know tht A n e reformulted s A = S 1... S M, where omponents S j re defined y S j = A Ij = i Ij A i nd over ll the A i, i.e. 1 j M I j = {1,..., N}. By (3.36), one hs nd similrly U st (A) = U st (S 1 ) T... T U st (S M ) (3.45) D = [U st (S 1 ) T O I 1 ] T... T [ U st (S M ) T O I M ] D 1 T... T D M (3.46) 11 It is not exluded however tht more lever height funtions would solve this diffiulty. But this is still reserh topi. 69

70 where the lol dignoses D j = U st (S j ) T OI j re now omputed on omponents insted of sites. The expression ove ssumes tht every lol dignosis D j inorportes y OI j the oservtions on ll sites A Ij overed y omponent S j. In relity, the duplition of oservtions Oi is unneessry : it is suffiient to distriute them on omponents S j in suh wy tht they ll pper t lest one. However, tking ll OI j to ompute D j hs the dvntge to mximlly redue U st (S j ) efore messge exhnges strt. The distriuted dignosis prolem now mounts to omputing the T j =Π Sj (D), or equivlently the D j =Π S j OI (D) =T j T OI j, tht form the miniml pullk j overing of D : D = [T 1 T O I 1 ] T... T [T N M T O I M ] = D 1 T... T D M (3.47) Nturlly, the miniml produt overing n e otined y further projeting the D j on the individul sites they represent. Seprtion theorem. To relte things to hpter 2 in ler mnner, we now hve vrile set V mx = {1,..., N}, trellis proesses T I s systems, projetions Π I nd the omposition opertor T. So we re in the setting of the dul grph representtion (setion 3.2.2). But t this point, the entrl xiom (3) is missing... The reson is tht the intertions in A = A 1... A N re due to shred lels, nd we need to express them in terms of shred sites. When one omputes A I A K, the shred sites A I K do not neessrily pture ll intertions etween A I nd A K : shred lels Λ I Λ K my not ll lie within Λ I K (see Fig nd the disussions in setion 3.2.2). Nevertheless, one hs the following property Theorem 9 Let I, K {1,..., N}, nd tke J = I K. Let T I, T K e trellis proesses of A I, A K respetively, nd ssume tht shred sites A J pture ll intertion lels etween A I nd A K, i.e. Λ I Λ K Λ J. Then Π J (T I T T K ) = Π J (T I ) T Π J (T K ) (3.48) This weker form of xiom (3) is tully suffiient to derive modulr omputtions. Propgtion nd merge equtions now tke the following form : Proposition 8 Let I, J, K {1,..., N}, nd let T I, T J, T K e trellis proesses of A I, A J, A K respetively. If Λ I Λ K Λ J, denoted I J K, then Π J (T I T T J T T K ) = Π J (T I T T J ) T Π J (T J T T K ) (3.49) Π I (T I T T J T T K ) = Π I [T I T Π J (T J T T K )] (3.50) Compring these equtions to the true ones (2.8) nd (2.10), the reder will notie tht proposition 8 simply implements the trik desried t (3.44). 70

71 Support grph of omputtions. So fr, no hoie ws mde to deompose A into pullk of omponents nd otin (3.45). We now exploit this degree of freedom, tht ws lredy disussed in setion 3.2.2, dedited to dul grph representtions. Let us define omponents S k in suh wy tht they over ommunition grph G etween sites A i (Fig. 3.19). Then form ommunition grph G etween these omponents. The ltter forms the support of omputtions. Indeed, on grph G, the seprtion property holds, whih mens tht theorem 9 pplies, so propgtions nd merges of proposition 8 re legl nd the MPA leds us to the desired redued omponents S j. d A 1 A 2 A 1 ' 1 A 2 A 1 S 1 A 2 S 1 ' 1 ' 2 A 3 A 4 ' 1 '2 '2 A 3 A 4 A 3 S 2 A 4 A 2 S 2 ' 3 ' 4 ' 3 ' 4 S 3 A 3 A 5 A 5 A 5 S 3 Figure 3.19: From left to right : lel onstrints etween sites A i, ommunition grph G of the A i, definition of omponents S j overing G, nd onnetivity grph etween omponents S j. We didn t mention sites orresponding to mesurements, i.e. the O i. They tully ply no role in the struture of intertions, even if the O i oserve shred lels. Indeed, eh O i is seprted from ll other sites (inluding mesurement sites) y site A i. So introduing the O i in Fig simply mounts to putting n edge etween eh A i nd the ssoited O i ; ll edges etween the O i, if ny, re dotted (i.e. redundnt). Moreover, sine the O i orrespond to oservtions olleted on true run of A, the projetions Π Oi (D) neessrily yield O i. In summry, mesurements n just e ignored in the grphil representtions. Involutivity. It is not hrd to hek tht trellis proesses form involutive systems : for T I TP of A I, one hs T I T Π J (T I )=T I. So we re in the nie sitution where turo proedures onverge, if the support grph of omputtions is not tree. The progressive redution performed y MPA (theorem 4) nd the lol extendiility property of the limit (theorem 3) re thus grnted. 3.7 Towrds true onurreny semntis This setion omes k to the multi-lok spets, in order to (try to) onvine the reder of its neessity for modulr omputtions. Consider the two LMCA A 1, A 2 of Fig They synhronize on lels {α, β, γ}, 71

72 nd trnsitions t 5 nd t 4 re privte to A 1 nd A 2 respetively. Fig depits two trellis proesses T 1, T 2 of these LMCA, together with their produts for different hoies of height funtions. The first produt (3rd TP on the figure) orresponds to T 1 T T 2 in the multi-lok setting : time is ounted independently in eh omponent, whih prevents the merge of the two onditions leled (, ), sine they orrespond to different vlues of the vetor lok.! t 2 " t 5 d A 1 A 2 t 3 # t 1 t 3 t 2 " t 4! # Figure 3.20: Two leled multi-lok utomt, shring lels {α, β, γ}. Assume tht one would deide to ndon vetor loks nd hoose to use glol ounting of time in the definition of trellis proesses. The produt T 1 T T 2 would then result in the rightmost TP of Fig where the two onditions leled (, ) re now merged, sine they oth hve two events in their pst. Projeting k the result on A 1 (y ersing events of A 2 ) underlines the d properties of this onstrution : this projetion yields trellis proess isomorphi to A 1. So the result is not ny more orretly folded trellis proess of A 1. But worse thn tht, this projetion introdues the extr run [t 3 [t 5 d tht ws not present initilly in the ftor T 1. So there is little hope tht modulr omputtions sed on this notion of trellis proess would produe orret results.! t 2 " t 5 d T 1 T T 2 T 1 T T 2 T 1 T 2 multi!lok folding single lok folding t 3 # t 1 t 3 t 2 " t 4! # t 1 t 3 t 3 t 2 t 2 t 4 t 5 d t 1 t 3 t 3 t 2 t 2 t 5 d t 4 Figure 3.21: Two trellis proesses of the previous LMCA, their orret produt, nd produt sed on glol notion of time. To explin the d properties of this single-lok produt T = T 1 s T T 2 y the more forml spets of our onstrution, notie tht T doesn t orrespond to 72

73 tegoril produt. There doesn t exist morphisms from T to its two ftors T 1 nd T 2 (tully there is no morphism to T 1 ). In other words, onsidering T s vlid trellis proess would violte the universl property stted in theorem 8 on T 1. There exists morphism from T to A 1, tht doesn t deomposes into morphism to T 1 followed y the folding of T 1 into A 1. t 1 t 3 t 3 t 1 t 3 t 3 t 2 t 2 t 4 t 5 t 4 t 2 t 2 t 4 t 5 d d t 5 t 5 d d Figure 3.22: Unfolding nd time-unfolding of A 1 A 2 of Fig Sine one is ound to the use of vetor loks, this mens tht the relevnt notion of time for distriuted systems is itself distriuted [84, 50]. Rther thn universl liner time, one hs lol notions of time, plus synhroniztions on speil events. Pushing further this ide, it mkes sense to develop setting where trnsitions themselves would hve lol effets. This suggests to use distriuted notion of stte, rther thn glol sttes, nd dopt true onurreny semntis. Consider for exmple A 1 nd A 2 of Fig These new semntis will led to new notion of produt for A 1 A 2, s well s new notions of unfolding U(A 1 A 2 ) (Fig left) nd of time-unfolding U t (A 1 A 2 ) (Fig. 3.22, right). On this exmple, the produt A 1 A 2 would tully look like the ltter : simply merge the sunets elow the two onditions leled. These new representtions of sets of runs, U(A) nd U t (A), disply expliitly the onurreny of events : for exmple the onurrent firings of t 4 nd t 5. In this setting, the produt of two systems tht hve no intertion will mount to their juxtposition. Tht will void the useless omputtion of onurreny dimonds, nd result in even more ompt dt strutures to enode sets of runs. The next hpter exmines omputtions sed on unfoldings, in the true onurreny semntis, nd hpter 5 does the sme with trellises, or time-unfoldings. As n extr motivtion to move to true onurreny semntis, let us illustrte lst drwk of sequentil semntis. We hve seen tht vetor loks re neessry to modulr omputtions. Nevertheless, they hve very unplesnt side effets : in some ses, there my exist drift etween loks of two omponents. In prtiulr when these omponents hve onurrent ehviors (i.e. lol trnsitions). In terms of trellis proesses, this results in very lrge strutures, s illustrted in Fig These phenomen suggest not to reommend the pproh of this hpter, nd use insted true onurreny semntis, tht will erse suh weird ehviors. 73

74 t 2 t 3 t 2! t 4 t 1! t t t t 3 t 2 t 3! t 3 t 3 t 3! t 2! t 2 t 1! t 4 t 1! t 4 Figure 3.23: Top : two LMCA A 1, A 2 tht shre lel α, nd their produt A 1 A 2. Bottom : the STU U st (A 1 ), U st (A 2 ) nd their produt U st (A 1 A 2 ). 74

75 3.8 Summry We hve proposed methodology to del with distriuted systems y mens of distriuted or modulr lgorithms, with the dignosis prolem s guiding pplition. To do so, we first introdued the notion of network of utomt. Mthemtilly, one n esily derive modulr dignosis lgorithms sed on omponent lnguges : they tke the form of messge pssing lgorithms, originlly derived for Byesin networks. The keystone of the onstrution is ftoriztion property on sets of runs of the glol system, just like in Byesin networks the ftoriztion property of the glol likelihood funtion is the key to fst lgorithms. Nevertheless, omputing with omponent lnguges demnds importnt storge pilities, whih is not suited to prtil pplitions, in prtiulr if we im t on-line monitoring lgorithms. So we hve proposed more ompt wy of enoding sets of runs, y mens of trellis proesses. A well known notion in different reserh ommunities, tht hs een dpted here to networks of utomt. This dpttion required to introdue the onept of multi-lok, i.e. to keep trk of omponents when they re ssemled into networks. Without this onept, it seems diffiult to otin the ftoriztion property on trellis proesses of ompound system, whih is essentil to modulr omputtions. As soon s this property is derived, whih is most esily done with tegory theory rguments, the mthemtil frmework developed for lnguges pplies to trellis proesses, up to some tehnil modifitions. Introduing vetor loks in distriuted systems is first step to orret modeling of onurrent ehviors. This is wht we do in the next hpters, moving from sequentil semntis to true onurreny semntis. We will show tht the mthemtil skeleton of this hpter still pplies, up to some tehnil extensions, nd tht we n otin even more ompt desriptions of sets of runs. Relted work The dignoser pproh hs een first investigted in [95], with fous on dignosility issues. Dignosility mens tht ounded numer of oservtions fter the ourrene of fult is suffiient to detet the filure. This mounts to heking the sene of miguous iruits in the dignoser, i.e. (in the formlism of setion 3.3.3) iruits of the oserver of A B tht ontin stte estimtes of the two kinds, green nd red. Issues ppering with distriuted oservtions nd/or systems hve een soon onsidered. The protool-sed pproh of [25], for exmple, ssumes distriuted oservtions on seprte omponents, ut glol dignosers re used t eh sensor, nd the fous is on the informtion they should exhnge to merge their estimtes. By ontrst, [21, 22] introdue true distriuted pproh, where lol dignosers re sed on lol model. Similr results re lso derived in [54, 55], with slightly different model : omponents re defined s Petri nets insted of utomt. In oth ses, omponent intertions re ssumed to e visile, whih gurntees tht the dignoser/oserver of the glol model hs produt form, nd thus enles modulr solutions. Unfortuntely, the prolem is not so muh nlyzed from this ngle, whih hides the importnt phenomen tht mke things work. For exmple the ft tht, when intertions re oserved, no 75

76 mtter whih interleving of the oserved sequenes is hosen in Fig. 3.18, the sme estimted sttes re otined. The literture relted to distriuted dignosis is vst nd diverse. Severl pprohes mke use of messge pssing (or elief propgtion) lgorithms, in reltion to the topology of the supervised system, ut ssume no dynmis in the omponents. For exmple [89] deploys MPA on filure trees, nd [97, 98] use dependeny grphs etween the elementry funtions tht ompose network. All these models re stohsti ut stti. As lredy mentioned, the losest ontriutions to hpter 3 re due to Su nd Wonhm [102, 103]. They do use networks of dynmi systems, where eh omponent is speified s lnguge. The pproh is purely lgeri : no rndomiztion is ssumed, y ontrst with the previous ones. Let us lso mention the ontriutions of Penole [88], or Broni et l. [8]. They oth model distriuted system s network of ommuniting utomt, nd onsider the distriuted on-line onstrution of dignoser (si), i.e. system whose runs re extly the explntions to the distriuted oservtions. Although the ide is to progressively ssemle the lol dignosers, y produt, this is not properly messge pssing solution : there is no notion of projetion. But strtegies re deployed in [88] to void uilding useless prts in the reursive omposition of lol dignosers. Interestingly, the ide of not unfolding silent loops is present in these two ontriutions. Finlly, let us mention the very originl hronile pproh [28, 30, 49]. By ontrst with the previous ones, it doesn t look for n explntion to ll lrms or symptoms. It rther opertes s olletion of filters tht extrt relevnt ptterns of oservtions in sequene. These ptterns re speified s prtil orders of lels, relted y time onstrints. There exist oth fst hronile reognition lgorithms, nd lerning lgorithms. Another differene with model-sed pprohes is tht the expert knowledge must ome fterwrds, to evlute the relevne of lerned hroniles nd to ssoite them to filure dignosis. There exist tehniques, however, to merge hroniles with topology models [5, 56]. 76

77 Chpter 4 True onurreny semntis So fr, we hve defined distriuted system s network of omponents, with lolized intertions. We hve shown tht some monitoring prolems, like (mximum likelihood) trjetory estimtion, tht we lled the dignosis prolem, ould e solved in distriuted mnner. The modulr/distriuted resolution method exploits the ftoriztion property of sets of runs, under the vrious forms where these runs n e desried. In the sequentil semntis, run sets n e enoded either s sets of sequenes (of lels or trnsitions), s rnhing proesses or deision trees, or s trellis proesses, whih estlishes hierrhy in terms of omptness. Moving to true onurreny semntis, this hierrhy remins. It is lso remrkle tht the lgeri struture of distriuted omputtions is preserved. Extr gins re otined in terms of omptness of the ojets we hndle, t the prie of little more tehnility in the proof of the key results. This hpter presents the interest of rnhing proesses, nd the new fetures they introdue in the distriuted dignosis prolem. Chpter 5 will mke one more step nd study the ounterprt of trellis proesses. 4.1 Networks of utomt s synhronous systems The previous hpter introdued notion of omponent, somehow rtifiilly, into the ordinry definition of n utomton : in A =(S, T,,s 0, χ, I, λ, Λ), I is n index set tht gives nmes to omponents, nd the funtion χ : T 2 I defines on whih omponents every trnsition t T opertes. Although omponents re formlly dded y the produt of (leled) multi-lok utomt, the omposition nevertheless results in n utomton. Fig. 4.1 illustrtes this ide, in Petri-net-like representtion of utomt. This formlism suffers from severl drwks, some of whih hve een illustrted t the end of the previous hpter. Let us underline two of them. 77

78 (t,* ) 2 (t 1,t 3) t 2 " t! t t # 3! 4 (,t ) # (,t ) 1 * 4! " * 4 # d d (t,* ) 2 d " d Figure 4.1: Two LMCA (left) tht shre only lel α, nd their produt (right). Numers of sttes nd trnsitions re multiplied y the omposition, nd in prtiulr eh privte trnsition of one ftor is duplited n times, where n is then numer of sttes of the other ftor. So we expliitly represent the ft tht one LMCA is witing when the other performs privte trnsition (see the duplitions of t 2 nd t 4 in the exmple). Seondly, runs of produt LMCA A 1 A 2 re lmost ound to e sequenes of events : the ft tht the reset trnsitions t 2 nd t 4 re onurrent is not exploited. So one distinguishes (,t 3 )(t 2, d )(,t 4 ) from (,t 3 )(,t 4 )(t 2, ), wheres it would e preferle to write (,t 3 )t 2 t 4 nd (,t 3 )t 4 t 2, nd even onsider these two runs s equivlent, sine t 2 nd t 4 n e permuted without ltering the finl stte of A 1 A 2. The ojetive of this hpter is preisely to void the redundny introdued y the sequentil semntis, nd expliitly tke dvntge of the onurreny etween events. We ll tke s definition of run not sequene of trnsitions, ut n equivlene lss of sequenes, where equivlene mens identil up to the permuttion of two suessive onurrent events. In the exmple ove, { (,t 3 )t 2 t 4, (,t 3 )t 4 t 2 } will e onsidered s single run. In other words, sequenes will e repled y Mzurkiewiz tres [27]. t 2 " t! t t # 3! 4 1 (t,t ) t 2 " t # 4 1 3! d d Figure 4.2: Two leled utomt (left) omined into sfe Petri net (right). To do so, nturl step onsists in repling utomt y synhronous systems. This is simply otined y hnging the notion of omposition : in sustne, we tke the disjoint union of sttes (insted of the produt), nd glue trnsitions tht rry identil lels, in ll possile wys, to form joint trnsitions. Privte trnsitions remin unhnged. As illustrted in Fig. 4.2, this yields Petri net : we now hve severl tokens to identify glol stte of the system. Firing (,t 3 ) moves simultneously the two tokens to nd d, fter wht they n return to their originl 78

79 ples independently. Oserve tht the numer of tokens remins onstnt : this illustrtes the ft tht we now hve two omponents, or equivlently two (lol) stte vriles, insted of one. There exist severl formlisms to define nd hndle suh systems. Our originl ontriutions [37, 40] hose to emphsize the presene of severl stte vrile, whih led to the notion of tile system (see lso the introdution, setion 1.3). Alterntively, one ould simply onsider sfe Petri nets. But sine we need to preserve the notion of omponent, or of vrile, the ltter must e slightly enrihed. We therefore introdue the notion of multi-lok net (MCN). This hpter is orgnized s the previous one : we first define the tegory of (leled) multi-lok nets, nd their omposition. We then provide them with notion of trjetory, in the true onurreny semntis. As for multi-lok utomt, ruil prolem is the ompt representtion of sets of trjetories. We fous here on the notions of rnhing proess nd unfolding of MCN, the nturl ounterprts of the sequentil trellis proess nd the sequentil time-unfolding, exepted tht merges re not llowed. So these ojets re more on the side of deision trees. It is shown tht one n tully ompute with them, pretty muh like with sequentil trellis proesses. But speifi diffiulties of the true onurreny semntis lur the nie lgeri setting of the previous hpter. We show tht omputtions must tully e performed with more omplex ojets, tht we ll ugmented rnhing proesses. 4.2 Multi-lok nets nd their omposition A tegory of multi-lok nets Petri net. An ordinry Petri net is 4-tuple N =(P, T,,M 0 ) where P, T re respetively the ple nd trnsition sets 1, nd (P T ) (T P ) is the flow reltion relting ples nd trnsitions 2. M 0 : P N is finite multi-set on P representing the initil mrking of the net, i.e. the numer of tokens tht eh ple initilly holds. Given node x P T, x nd x stnd for pre- nd postsets, i.e. the immedite predeessors nd suessors of x in. By ontrst with the previous hpter, these sets re not ny more onstrined to e singletons. A trnsition t T is enled (or tivted ) in mrking M, denoted y M[t, iff p t, M(p) > 0. Firing t from M yields the new mrking M = M t + t, whih we denote y M[t M. M is rehle mrking iff there exists sequene σ =(,t 2,..., t n ) of trnsitions (or run) nd intermedite mrkings suh tht M 0 [ M 1 [t 2 M 2...[t N M N = M, revited into M 0 [σ M. We limit ourselves to sfe nets, i.e. nets for whih ples hold t most one token in ny rehle mrking. We thus identify mrkings to susets of P, nd M 0 to P 0 P. Multi-lok net. A multi-lok net (MCN), or net for short, is tuple N = (P, T,,P 0,ν) where 1 We do not require tht P nd T e finite. 2 We ssume eh ple is relted to one trnsition t lest, nd eh trnsition hs t lest one input ple nd one output ple. 79

80 1. (P, T,,P 0 ) is n ordinry sfe net, 2. ν : P P 0 defines prtition on ples, nd the restrition ν P 0 is the identity ; we denote y p the equivlene lss ν 1 (ν(p)) of ple p, 3. t T, ν is injetive on t nd on t, nd ν( t)=ν(t ). This definition deserves some omments. Oserve first tht every trnsition stisfies t = t. So the numer of tokens remins onstnt in MCN. Moreover, let M P e rehle mrking of N, one hs ν M is ijetive. In other words, let p P 0, t ny time there is extly one ple in ν 1 (p) holding token 3. Seondly, onsider N p, the restrition of N to ples of p, p P. N p is n utomton, i.e. Petri net where single ple holds token t ny time, s in Fig Therefore, multi-lok net n e regrded s synhronous produt of utomt, s illustrted in Fig Relting this formlism to the previous hpter, P 0 now reples the index set I to nme omponents, nd the χ funtion on trnsitions is repled y ν on ples. So trnsition t opertes on omponents ν( t)=ν(t ) insted of χ(t). By use of voulry, we will sometimes onsider p s the stte vrile of omponent N p (more rigorously, it orresponds to the set of possile vlues of this stte vrile). Nturlly, leled multi-lok net (LMCN) is MCN enrihed with leling funtion on trnsitions : N =(P, T,,P 0, ν, λ, Λ), with λ : T Λ. d d t 2 t 3 t 2 t 3 g g h t 4 e f g h t 4 t 4 i i i Figure 4.3: A typil morphism etween two multi-lok nets, indited y dshed rrows. In the left MCN, ν 1 defines the prtition {, e}, {, f}, {, g}, {d, h, i, i } on ples. Morphism. To turn the olletion of multi-lok nets into tegory, we need the extr notion of morphism etween nets. Let N 1, N 2 e two MCN, with N i = (P i,t i, i,pi 0,ν i), i =1, 2. A morphism φ from N 1 to N 2 is defined s pir (φ P,φ T ) where 3 Notie tht it is lwys possile to turn sfe net N into multi-lok net with essentilly the sme ehvior, simply y dding to eh ple of N omplementry ple. So multi-lok nets re lmost equivlent to sfe nets. 80

81 1. φ T is prtil funtion from T 1 to T 2, nd φ P reltion etween P 1 nd P 2, 2. P2 0 = φ P (P1 0 ) nd φop P : P 2 0 P 1 0 opposite reltion to φ P, is totl funtion, where φop P denotes the 3. if p 1 φ P p 2 then the restritions φ T : p 1 p 2 nd φ T : p 1 p 2 re totl funtions, 4. if t 2 = φ T ( ) then the restritions φ op P funtions, : t 2 nd φ op P : t 2 t 1 re totl 5. φ P preserves the prtitioning of ples : (p 1,p 2 ) P 1 P 2, p 1 φ P p 2 ν 1 (p 1 ) φ P ν 2 (p 2 ) Notie tht points 3 nd 4 entil tht the pir (φ P,φ T ) preserves the flow reltion (on its domin of definition), nd so morphisms will preserve runs, whih is the first property one expets from them. It my look surprising tht φ P is reltion nd not prtil funtion. This generliztion llows φ P to duplite ples, whih is neessry to the existene of tegoril produt (rell tht this duplition ility ws lso present in MCA morphisms). In onjuntion with the lst requirement, one n tully show tht Lemm 7 Let φ : N 1 N 2 e morphism of MCNs, suh tht every utomton N i p hs single onneted omponent, then φ erses or duplites stte vriles (or omponents) s whole, just like MCA morphisms (see Fig. 4.3). Speifilly, one hs. the inverse imge y φ of lss of ν 2 is inluded in lss of ν 1 ;. given lss of ν 1, φ is either defined on ll elements of this lss, or on none of them ;. when ple p 1 P 1 is duplited y φ, i.e. relted to (elements of) severl lsses of ν 2, eh ple in p 1 is duplited in the sme wy, i.e. relted to the sme lsses. Finlly, for leled MCN N i =(P i,t i, i,p 0 i,ν i,λ i, Λ i ),i =1, 2, the definition of morphism φ : N 1 N 2 must e reinfored y extr requirements (tht rephrse those of leled MCA) : 6. Λ 2 Λ 1, i.e. φ redues the lel set, 7. Dom(φ T )=λ 1 1 (Λ 2), i.e. φ is defined on trnsitions rrying shred lel, nd only on them, 8. if φ T ( )=t 2 then λ 1 ( )=λ 2 (t 2 ), i.e. φ preserves lels on its domin of definition Dom(φ T ). We denote y Nets the tegory hving the LMC nets s ojets, nd the ove morphisms s rrows. By use of nottions, we simply write φ insted of φ S or φ T, nd φ(x) to denote ples in reltion with t lest one ple in X. 81

82 4.2.2 Composition y produt nd pullk Produt. Let N 1, N 2 e two LMCN, their produt N 1 N 2 is defined s the triple (N,ψ 1,ψ 2 ) where N =(P, T,,P 0, ν, λ, Λ) is net nd ψ i : N N i morphism, suh tht 1. P = {(p 1, ) :p 1 P 1 } {(, p 2 ):p 2 P 2 } ; ψ 1 (p 1, ) =p 1 nd ψ 1 (, p 2 )= (i.e. undefined), nd symmetrilly for ψ 2, 2. P 0 = ψ 1 1 (P 0 1 ) ψ 1 2 (P 0 2 ), 3. T = T s T p where T s = {(,t 2 ) T 1 T 2 : λ 1 ( )=λ 2 (t 2 )} (4.1) T p = {(, ) : T 1,λ 1 ( ) Λ 1 \ Λ 2 } {(, t 2 ):t 2 T 2,λ 2 (t 2 ) Λ 2 \ Λ 1 } (4.2) nd ψ i (,t 2 )=t i if t i nd is undefined otherwise, 4. is defined y (,t 2 )=ψ1 1 ( ) ψ2 1 ( t 2 ), ssuming =, nd similrly for (,t 2 ), 5. Λ = Λ 1 Λ 2 nd λ follows ordingly, 6. ν is simply the union of prtitions ν 1 nd ν 2. In produt, eh omponent preserves its ples y the disjoint union in (1), nd omponents re dded (6). As for LMCA, trnsitions rrying shred lel synhronize (4.1), provided they find prtner, while those rrying privte lel remin unhnged (4.2). Notie tht privte trnsitions re not duplited, y ontrst with LMCA. See Fig. 4.4 for simple exmple. g d!! " " t 3 t 2 t t t 4 t 5 t Figure 4.4: Produt of three nets, eh one with single omponent. On this simple exmple, the produt mounts to gluing trnsitions with identil lels. e f Proposition 9 is the tegoril produt in Nets, i.e. the universl property stted in proposition 5 for LMC utomt holds lso for LMC nets. See [112] for proof in the unleled se, whih esily speilizes. 82

83 Pullk. The omposition y pullk, i.e. vi shred omponent, n lso e defined for LMC nets, nd it enjoys the sme properties s for LMC utomt. The generl definition is little omplex [43], so we fous on the more intuitive se where N 1, N 2 re relted to their interfe N 0 y prtil funtions f i : N i N 0 insted of generl morphisms. The onstrution resemles very muh the one of the produt : it tully oinides with it outside the domins of f 1 nd f 2. N nd the morphisms ψ i : N N i re given y 1. P = P P p where P = {(p 1,p 2 ):f 1 (p 1 )=f 2 (p 2 )=p 0 P 0 } (4.3) P p = {(p 1, ) :f 1 (p 1 )= } { (, p 2 ):f 2 (p 2 )= } (4.4) ψ i (p 1,p 2 )=p i if p i nd undefined otherwise, 2. P 0 = ψ 1 1 (P 0 1 ) ψ 1 2 (P 0 2 ), 3. T = T T s T p where T = {(,t 2 ):f 1 ( )=f 2 (t 2 )=t 0 T 0 } (4.5) T s = {(,t 2 ):f 1 ( )= = f 2 (t 2 ),λ 1 ( )=λ 2 (t 2 )} (4.6) T p = {(, ) :f 1 ( )=, λ 1 ( ) Λ 1 \ Λ 2 } {(, t 2 ):f 2 (t 2 )=, λ 2 (t 2 ) Λ 2 \ Λ 1 } (4.7) nd ψ i (,t 2 )=t i if t i nd is undefined otherwise, 4. is defined y (,t 2 )=ψ1 1 ( ) ψ2 1 ( t 2 ), ssuming =, nd similrly for (,t 2 ), 5. Λ = Λ 1 Λ 2 nd λ follows ordingly, 6. ν is the union of prtitions ν 1 nd ν 2 : they oinide on the ommon ples of ψ i (P ). The novelty ppers in the speil tretment of ommon ples (4.3) nd ommon trnsitions (4.5). Proposition 10 The omposition y orresponds to pullk in the tegory Nets. The proof of the universl property of the pullk, see (3.3) nd Fig. 3.4, n e found in [43]. So we re on solid ground, nd derivtions of the previous hpter sed on tegory theory rguments still hold : we only hnged ojets. In prtiulr, the reltions etween nd expressed in proposition 6 remin vlid for LMC nets, s illustrted in Fig

84 f 1 f 2 g g g d! "! "! " t 3 t 2 t t t 5 t t 4 t 4 N 1 N 0 N 2 Figure 4.5: Two nets, tht shre ommon omponent. The limit of this digrm (pullk) results in the net of Fig e f Grphs ssoited to multi-lok system The sitution is extly s for leled multi-lok utomt (setion 3.2). We strt from net N defined s produt N = N 1... N N, nd wnt to represent grphilly the internl intertions of N. The ojetive is of ourse to prepre the ground for distriuted/modulr lgorithms. There re essentilly two wys to do so : either y diret grph representtion (the ommon resoures, or vriles, re the shred lels) or y dul grph representtion (the ommon resoures re the sites N i ). Here, we will mostly use the ltter : the hoie is indeed governed y the type of projetions we use in the sequel, either projetions on lel sets, or projetions on sites. It is not exluded tht the rnhing proesses we introdue elow for modulr omputtions ould dmit projetions on lel sets ; we re tully working on the topi. But so fr this theory is not ville, nd we n only projet on (sets of) sites. This hoie thus imposes to se omputtions on pullks, s in setion In summry, thnks to proposition 6 nd property (3.8) expressed on nets, we n deompose N s N = j J S j with S j = i Ij N i (4.8) ssuming omponents S j over ommunition grph etween sites N i (Figs. 3.9-e nd 3.9-f). 4.3 Trjetory sets in the true onurreny semntis This setion defines the notion of trjetory of net, in the true onurreny semntis, nd proposes mens to represent sets of runs nd ompute with them. In the previous hpter, set of runs of n LMCA ould e understood nd hndled in four different mnners : s set of lel sequenes (su-lnguge), set of trnsition sequenes ( riher notion of su-lnguge), nd more omptly s rnhing proess, or s trellis proesses. The sme ides would pply here in the true onurreny semntis. We skip the first one, limited to lels, nd diretly move to runs expressed s tres on trnsitions, tht we will represent s onfigurtions. And 84

85 sine onfigurtions will pper s speifi ses of rnhing proesses, we diretly move to the ltter nd show how to ompute with these ojets. Trellis proesses form the ody of the next hpter Unfolding of net Ourrene net. The LMC net O =(C, E,,C 0, ν, λ, Λ) is (leled multilok) ourrene net (LMCON, or ON for short) iff it stisfies : 1. C 0 = { C : = }, 2. the uslity reltion, irreflexive trnsitive losure of, is prtil order, nd x C E, [x] {x} {y C E : y x} is finite, 3. C, 1, 4. the onflit reltion # defined y the two properties elow is irreflexive : () e, e E, [e e, e e ] e#e, () x, x C E, [ e, e E, e#e,e x, e x ] x#x. The hnge in nottions ounts for the usul terminology of onditions, insted of ples, nd events, insted of trnsitions. In n LMCON, time is unfolded, s indited y (2). A ondition n e mrked y unique event (3). By ontrst, it my enle severl events, whih orresponds to onflit sitution. This retes rnhing in the net, nd the orresponding rnhes will never meet eh other gin (4). So onflits re lso unfolded. See Fig. 4.6, enter, for n exmple. Configurtion. LMCONs re generlly introdued to represent runs of net in the so-lled true onurreny semntis. To do so, we need extr elements of terminology out ONs. Two nodes x, x C E re sid to e onurrent, denoted y x x, iff neither x#x nor x x nor x x holds. A o-set is set of pirwise onurrent onditions, nd ut is mximl o-set for the inlusion. Finlly, onfigurtion is su-net κ of C E whih is onflit-free, uslly losed (i.e. leftlosed for ), nd suh tht e E, e κ e κ. By onvention we ssume onfigurtions lso ontin initil onditions C 0. See Fig. 4.6, right, for n exmple. One immeditely noties the one-to-one orrespondene etween onfigurtions of n ourrene net nd possile runs of tht net, i.e. possile irultions of tokens. Speifilly, onfigurtion must e red s prtil order of events. In Fig. 4.6 (right), the firing of preedes or is use to the firing of t 5, just like t 3, however, the ordering of nd t 3 is not speified. So they n fire in ny order. Every liner extension of the prtil order defined y onfigurtion yields vlid firing sequene on its events. In tht sense, onfigurtion represents n equivlene lss of sequenes, or tre [27]. This is preisely where the true onurreny semntis llows to sve with respet to the previous hpter : useless interlevings of onurrent events re not represented, nd single prtil order reples numerous sequenes. 85

86 t 2 t 3 t 4 t 2 t 3 t 4 t 2 t 3 t 4 d d d d d t 5 t 5 t 5 t 5 t 5 t 5 t 5 t 5 t 5 Figure 4.6: An LMC net N (left), rnhing proess O of tht net (enter) nd onfigurtion κ in O (right), orresponding to run of N. The folding of O into N is represented y trnsition nd ple nmes tthed to events nd onditions. LMC ourrene nets, equipped with morphisms of LMC nets, form the sutegory O of Nets. Oserve tht in this tegory, morphisms n only erse (not rete) uslity or onflit reltions etween two nodes. Conurreny reltions re preserved, nd onfigurtions re mpped to onfigurtions. Brnhing proess. The pir (O,f), where f : O N is morphism, is sid to e (leled multi-lok) rnhing proess (LMCBP or BP for short) of net N iff [31] 1. f is totl funtion on O, lso nmed folding of O, 2. e, e E, [ e = e,f(e) =f(e )] e = e. In other words, f lels ll events nd onditions of O y trnsitions nd ples of the net N (see Fig. 4.6). f eing morphism, this leling tully turns onfigurtion κ of O into run of the net N. The prsimony ondition 2. ensures tht ny run of N tht is represented in O ppers only one, or is represented y unique onfigurtion. There exist similrities etween the rnhing proesses of LMC nets defined ove for nets nd the sequentil rnhing proesses defined for LMC utomt. In prtiulr the tegoril rhitetures re identil, so the min results n e trnsported here. We riefly review them. First of ll, there exists simple nd intuitive lgorithm to uild rnhing proess (O,f) of net N, tht looks very muh like Proedure 1 (without the merge of onditions). We riefly mention it elow 4, sine it forms the sis of severl opertions we use in the sequel. Proedure 2 Initiliztion : 4 This reursive onstrution is expliit in [110] s well s in [35], where new events nd ples re nmed y kwrd pointer tehnique. It lso ppers in the definition of nonil rnhing proesses in [31]. 86

87 Crete P 0 onditions in C 0, nd define ijetion f : C 0 P 0. Set C = C 0, E = nd =. Reursion : Let X e o-set of C nd t T trnsition of N suh tht f(x) = t. If there doesn t exist n event e in E with e = X nd f(e) =t, rete new event e in E with e = X nd f(e) =t, then rete suset Y of t new onditions in C, set Y = e, nd extend f to hve the ijetion f : Y t. The prtition ν of onditions in O is of ourse inherited from the prtition of ples in N, s well s the leling λ of events. Lemms 3, 4 nd 5 remin vlid for rnhing proesses of net N, nd one esily modifies the reursion of Proedure 2 to ompute the union of rnhing proesses of A, whih leds to the notion of unfolding. Unfolding. A prefix O of n ourrene net O is defined s su-net of O whih is uslly losed, ontins the initil mrking (or equivlently ll miniml onditions), nd suh tht e E, e O implies e O. So onfigurtion of O is onflit-free prefix of O. The prefix reltion is denoted y O O. Theorem 10 Given net N, there exists unique mximl rnhing proess of N for the prefix reltion. It is lled the unfolding of N. We denote it y U(N ), or U N for short, nd its orresponding folding y f N : U N N. A proof of this theorem n e found in Winskel s works [110] or in Engelfriet s [31]. The unfolding is of ourse otined s the unique sttionry point of Proedure 2, nd its onfigurtions yield ll possile runs of N. Expressive power of rnhing proesses. We hd mentioned in the previous hpter tht not ll su-lnguges L of n LMCA A ould e represented s sequentil trellis (or rnhing) proess. In the sme wy : Lemm 8 Let L e set of onfigurtions of N. L n e represented s BP (O,f) of N iff κ L, κ 1,κ 2 L, κ κ κ L κ 1 κ 2 is onfigurtion κ 1 κ 2 L So ordinry BP represent lnguges tht re oth prefix-losed nd losed y onurrent suffix extension. In the previous hpter (setion 3.5.2), we introdued the notion of stop point for sequentil rnhing proess of A, in order to get one-toone orrespondene etween su-lnguges of A nd runs enoded in (stopped) rnhing proess of A. Similrly, to get one-to-one orrespondene etween onfigurtion sets of N nd rnhing proesses of N, one must enrih the definition of BP with notion of stop point. This tkes the form of funtion ssoiting 87

88 zero/one vlue to uts of O, i.e. onfigurtion extremities. We don t detil it here : this mehnism is very hevy nd lmost kills the dvntges of using rnhing proesses Ftoriztion property Theorem 11 (universl property) The unfolding ( U N,f N ) of net N stisfies O O, φ : O N,!ψ : O U N,φ = f N ψ (4.9) This property tully hrterizes U N mong ll ourrene nets O tht dmit morphism to N [110]. O A % N E!& f N U N Figure 4.7: Universl property of the unfolding of N. Following the sme tegory theory rguments s in the previous hpter (setion 3.5.4), this universl property estlishes o-refletion etween tegories O nd Nets. Speifilly, the unfolding opertion defines funtor G = U : Nets U O U Nets Figure 4.8: Two funtors relting Nets nd O, tht form n djuntion. O, nd there exists n inlusion funtor F = : O Nets. They oth form the djoint pir (F, G), nd sine G hs left djoint, we know it preserves tegoril limits, in prtiulr produts nd pullks. This entils one gin : N 1, N 2 Nets, U(N 1 N 2 ) = U(N 1 ) O U(N 2 ) (4.10) where O denotes the tegoril produt in O. The existene of the ltter is utomtilly grnted, nd it stisfies (or n e defined s) O 1, O 2 O, O 1 O O 2 U(O 1 O 2 ) (4.11) (4.11) is importnt in prtie : it expresses tht oupling Proedure 2 with the definition of the produt in Nets, one n tully ompute reursively the produt of two ourrene nets. In the sme mnner, onsidering pullks, one hs N 0, N 1, N 2 Nets, U(N 1 N 0 N 2 ) = U(N 1 ) U(N 0) O U(N 2 ) (4.12) 88

89 where the pullk O in O is defined y O 0, O 1, O 2 O, O 1 O 0 O O 2 U(O 1 O 0 O 2 ) (4.13) nd so O n gin e omputed reursively. As we hve seen in hpter 3, the ftoriztion property of struture enoding runs of ompound system is the first ingredient towrds distriuted dignosis lgorithms. So we re lredy in very good shpe. The next ingredient is n dequte notion of projetion, tht we exmine in the reminder of this hpter. t 2 t 3 t 4 t 5!! t 5 d Figure 4.9: The net in Fig. 4.6, expressed s produt of two omponents. But efore, let us illustrte the interest of reltion like (4.10). The net N in Fig. 4.6, left, n e deomposed s the produt of two smller nets N = N 1 N 2, see Fig Eq. (4.10) expresses tht U N is thus the produt, in the sense of ourrene nets, of the U Ni depited in Fig The ltter re simple trees, sine the N i re utomt. One first noties tht unfoldings grow in width s one progresses in length (the time dimension). This is preisely one of the drwks ddressed y the next hpter. Now, onsidering the numer of possiilities to fire t 5 (resp. t 5 ) in U N1 (resp. U N2 ), one finds two solutions for the first ourrene of t 5, then four for the seond ourrene, et. In the expnded produt U N, they result in 2 2=4 possiilities to fire the first ourrene of (t 5,t 5 ) (see Fig. 4.6, right), then 16 = 4 4 for the seond ourrene, et. In other words, the ftorized form of U N is more ompt thn the expnded one, just like ( ) p is more ompt thn its expnded form. Therefore it is likely tht proessings sed on the ftorized form of U N will e more effiient thn proessings on the expnded form, s it ws lredy the se in hpter 3 with sequentil semntis. 4.4 Distriuted dignosis : the unfolding pproh As soon s the produt is defined in O, nturl notion of projetion eomes ville. We first explore how fr one n go with this notion, in terms of modulr omputtions for the dignosis prolem. Nottions : notie tht we denote y O the oservtions, nd y O generl ourrene nets or rnhing proesses. 89

90 t 2 t 3 t 4 d d t 5 t 5 t 5 t 5 t 2 t 2 t 3 t 4 t 3 t 4 d d d d t 5 t 5 t 5 t 5 t 5 t 5 t 5 t Figure 4.10: Unfoldings of the two omponents of Fig Projetion Nturl projetion. Let us first onsider two nets N 1, N 2, nd rnhing proesses (O i,f i ) of these nets. The ourrene net O = O 1 O O 2 is ssoited to two morphisms ψ i : O O i. Comined to the foldings f i, they yield the f i ψ i : O N i, nd sine one hs morphisms from O to the N i, y the universl property of N = N 1 N 2 (prop. 9), there exists unique f : O N suh tht f i ψ i = f φ i, where φ i : N N i. It is esily heked tht f is folding, whih mkes O 1 O O 2 rnhing proess of N 1 N 2 (Fig. 4.11). O 1 & 1 & 2 O= O 1 x o O 2 O 2 f 1 f f 2 N 1 % 1 % 2 N= N 1 x N 2 N 2 Figure 4.11: The produt of two rnhing proesses yields rnhing proess of the produt. We tke s projetions of O the two ourrene nets O i = ψ i(o). They re prefixes of the originl ftors O i (O i O i) nd stisfy O = O 1 O O 2. Referring to notions introdued in the previous hpter (end of setion 3.1.2), the O i thus form the miniml produt overing of O, in the sense tht tking stritly smller prefix O i O i of one ftor kills the equlity. This is illustrted y the exmple in Fig. 4.12, tht depits the produt of two onfigurtions (κ 1, left nd κ 2, right). The produt lies in the middle, nd ll lels {α, β} re supposed shred. Oserve tht the event t 3 of κ 1, leled y β, finds no prtner in κ 2 nd thus vnishes. So ψ 1 (κ 1 O κ 2 ) redues to events nd t 2. This exmple lso illustrtes tht the produt of two onfigurtion is not neessrily 90

91 onfigurtion : there my exist different ssoitions of their events. Finlly, notie tht the two onurrent events,t 2 of κ 1 inherit uslity reltion of κ 2 in the produt, either in one diretion or the other. f f! t 2! t 4!! t 2 t 4! t 4 d g g d g t 3 " t 2 t 5!! t 5! t 5 e d h h h Figure 4.12: Produt (enter) of two onfigurtions (left nd right), tht shre ll event lels, i.e. α nd β. To define the projetion of ny rnhing proess of N = N 1 N 2, we onsider the sme sheme with U N insted of O. From U N = U N1 O U N2 stted in (4.10), nd its ssoited morphisms ψ i : U N U Ni, one n define the projetions O i = ψ i(o) for ny rnhing proess O U N of N. One hs O i U N i whih oviously mkes them rnhing proesses of the N i. They form the miniml produt overing of O in the sense tht O O 1 O O 2, nd repling one of the ftors O i y strit prefix O i O i kills this prefix inlusion of O. So the projetion keeps events of the U Ni tht re stritly useful to O, nd only them. Oserve tht O is inluded in its produt overing (whene the nme), nd is equl to it iff O ws lredy in produt form. Projetion opertor. We now move to the generl se N = N 1... N N nd onsider O = O 1 O... O O N where eh O i is BP of N i. For I {1,..., N}, we still use nottions N I = i I N i nd O I = O i I O i. By ssoitivity of O, we n write O = O I O O I nd define Π I (O) s ψ I (O). This projetion opertion generlizes to ny O J, for J {1,..., N}, y Π I (O J ) Π I J (O J ). And y ssoitivity of O, one esily heks tht I, J {1,..., N}, Π I Π J = Π I J (4.14) whih orresponds to xiom (1). In prtie, it is quite esy to ompute the projetion of BP : to get Π I (O J ), it suffies to remove onditions nd events not leled y N I, nd to trim the result, i.e. to reursively merge isomorphi events in order to stisfy the prsimony riterion (requirement 2 in the definition of BP). Miniml produt overing. This notion pplies s ove : O U N, O O 1 O... O O N where O i =Π i(o) U Ni (4.15) O i U Ni, j : O j O j O O 1 O... O O N (4.16) 91

92 In (4.15), one hs equlity iff O dmits produt form : O = O 1 O... O O N, nd in tht se O i O i. Moreover, one hs O I =Π I(O) = O i I O i with O i =Π i(o) Centrlized dignosis Single sensor. The setting is like in the previous hpter : net N produes hidden run κ, tht is only oserved through the lels of Λ Λ tht it produes. But wht does it men to oserve the prtil order of lels λ(κ) Λ? Things eome ler when one ssumes tht lels re logged in sequene y sensor. So insted of λ(κ) Λ one tully sees liner extension of this prtil order, under the ssumption tht the oservtion proess is not nti-usl : visile events tht re uslly relted n t led to lels oserved in the reversed order. However, two onurrent events ould produe lels oserved in ny order. Now, the oserved lels my not ll e ordered : indeed, when two visile lels re olleted on different sensors, the possile usl link relting these events is generlly lost, unless some speifi mehnism is deployed to preserve it. Therefore, s generl sitution we ssume tht our oservtion O is wekened version of n extension of λ(κ) Λ (see Fig for n exmple). d "! #!!! o 1 o 2 t 3! e t 2 " t 4 ) " ) " ) " ) o 3 ) f Figure 4.13: A hidden onfigurtion κ (left), nd n oserved prtil order of its lels, expressed s onfigurtion (right). γ is supposed to e invisile, i.e. Λ = {α, β, δ}. The seond plot is the prtil order on visile lels indued y κ. The ordering of α nd δ (third plot) my e reted y the sensor tht ollets them. By ontrst, the uslity reltion α β my e lost if these lels re olleted y different sensors (fourth plot). As prtil order of lels, O n e enoded under the form of onfigurtion ( LMCON), nd run κ of N is vlid explntion of O if it perfetly synhronizes with O, i.e. if ψ O (κ O O )=O. To ompute them ll t one, onsider D = U N O O (4.17) Not ll mximl onfigurtions of D explin entirely O, they my explin only prefix of O nd then e loked. But those tht do explin ll O form onfigurtion set losed y onurrent extension (see lemm 8). In other words, it suffies to ut off ded rnhes of D, then tke the remining mximl onfigurtions to get ll solutions to the dignosis prolem, one projeted k on U N. For simpliity, we ssume in the sequel tht the dignosis is given y (4.17) ove. 92

93 Multiple sensors. We now ssume tht N sensors re olleting lels produed y trnsitions of net N, nd possily oserve the sme lels. For exmple sensor i rets to lels of Λ i nd produes onfigurtion O i s oservtion. Speifilly, O i is wekened version of some extension of λ(κ), restrited to Λ i. It is importnt to notie tht the sme extension of λ(κ) lies ehind ll the Oi, otherwise the sensors ould oserve in different orders two onurrent events, whih would led to n inonsisteny. We esily get k to the previous se y defining s our joint oservtion. O = O 1 O... O O N (4.18)! "! O d 1 # # " # O d e 2 * 1 ={!, "} * 2 ={ ", #} O 1 x O O 2!! d " d e # # # Figure 4.14: Two oserved onfigurtions, on sensors shring lel β, nd the resulting equivlent joint oservtion. This deserves severl omments. First, oserve tht the Oi synhronize on the lels they jointly oserve. If the Λ i re disjoint, putting ll oservtions together simply mounts to juxtposing the Oi, whih is muh simpler thn the omplex omputtion of ll interlevings of Fig Trnslted in the true onurreny semntis, the ltter eomes Fig Seondly, tking s oservtion the produt of the Oi my yield generl ourrene net, not neessrily onfigurtion (see Fig. 4.12). This doesn t hnge muh the theory : it suffies to tke s solution to the dignosis prolem mximl onfigurtion of D tht projets into mximl onfigurtion of O, equivlently into mximl onfigurtions of the Oi (i.e. the O i themselves) Distriuted dignosis Algerilly, the sitution is pretty muh the sme s for sequentil semntis (setion 3.6.3). We now ssume tht the supervised system N is otined y ssemling smller nets N i, tht we ll sites : N = N 1... N N. Eh of the N i is tthed to speifi sensor olleting lels of Λ i Λ i nd produing oservtion O i, s ove. By injeting (4.10) nd (4.18) into (4.17), one gets D = [U N1 O O 1 ] O... O [ U NN O O N ] (4.19) 93

94 This reltion expresses tht the glol dignosis orresponds to the synhroniztion of lol dignoses D i = U Ni O Oi omputed for eh site. Our ojetive is to derive the miniml produt overing of D, tht is the rnhing proesses O i =Π Ni (D), or equivlently the D i =Π N i Oi (D) =O i O Oi. They orrespond to oherent lol views of D, in the sense tht they stisfy D = [O 1 O O 1 ] O... O [O N O O N ] = D 1 O... O D N (4.20) Of ourse, ruil point is to derive the O i or the D i without omputing D itself, sine the ltter n e huge ojet (rell tht produts generlly multiply trnsitions). So we im t deploying the modulr omputtion formlism of hpter 2. Trnsltion into pullks. As in the se of sequentil semntis (setion 3.6.3), we don t yet hve len theory tht would llow to projet rnhing proess on lel set, nd would llow s well to omine the resulting ojet (proly n event strutures rrying lels) with nother rnhing proess 5. By ontrst, there exists nturl notion of projetion on sites, whih suggests to use pullks to trnsmit informtion from omponent to omponent, see (3.44). We thus restte N s N = S 1... S M where omponents S j re defined y S j = N Ij = i Ij N i nd over ll the sites : j I j = {1,..., N}. One hs nd similrly U N = U S1 O... O U SM (4.21) D = [U S1 O O I 1 ] O... O [ U SM O O I M ] = D 1 O... O D M (4.22) where eh lol dignosis D j inorportes y OI j oservtions on ll sites N i of omponent S j. It is not neessry to duplite oservtions Oi when site N i ppers in severl omponents. It suffies to distriute the Oi on omponents, in suh wy tht eh Oi ppers t lest one in (4.22). But tking ll oservtions O I j on S j hs the dvntge of mximlly reduing U Sj, n interesting property for modulr omputtions. The distriuted dignosis prolem now mounts to omputing the O Ij =Π Ij (D), or equivlently the D j =Π S j O OI (D) =O Ij O OI j, tht form the miniml pullk j overing of D : D = [O I1 O O I 1 ] O... O [O IM O O I M ] = D 1 O... O D M (4.23) If neessry, the Π Ni (D) of the miniml produt overing n e dedued from the lrger D j. 5 This n proly e done, ut remins reserh topi. 94

95 Misleding projetions. This notion underlines prolem tht rises with nturl projetions defined on rnhing proesses. Computing projetion O I =Π I(O) mounts, in prt, to seleting some events nd onditions in O. But this opertion my forget uslities or onflits tht were linking them (see. Fig. 4.15). As result, two events or onditions my unduly pper s onurrent fter projetion, nd thus e llowed to pper in the sme onfigurtion, or with reversed uslity, wheres this would hve een impossile in the originl rnhing proess O. e e t 2 t 2 f g d d f t 2 g d Figure 4.15: Two rnhing proesses (left nd right) tht rete fke onurreny (enter) one the influene of the entrl vrile is disrded y the projetion. The BP on the left loses onflit reltion, the one on the right loses uslity reltion. We sy tht O I =Π I(O) is not misleding iff every onfigurtion κ in O I n e otined s the projetion of onfigurtion κ of O. One hs : Lemm 9 Let us sy tht N i is simple site if, s LMC net, it is omposed of single stte vrile. Projetions of rnhing proesses on simple sites re never misleding. This is due to the ft tht BP of simple site n t hve onurrent events. The next setion is preisely devoted to defining notion of projetion tht keeps trk of useful onflits nd uslities. Seprtion theorem. We re now well equipped to stte the seprtion theorem, tht forms the kone of modulr omputtions. Theorem 12 Let I, K {1,..., N}, nd tke J = I K. Let O I, O K e rnhing proesses of N I, N K respetively, nd ssume tht shred sites N J pture ll intertion lels etween N I nd N K, i.e. Λ I Λ K Λ J. If projetions Π J re not misleding, then Π J (O I O O K ) = Π J (O I ) O Π J (O K ) (4.24) This wek form of xiom (3) reprodues the forml setting of setion 3.6.3, so we n still derive propgtion nd merge primitives : 95

96 Proposition 11 Let I, J, K {1,..., N}, nd let O I, O J, O K e rnhing proesses of N I, N J, N K respetively. If Λ I Λ K Λ J, denoted I J K, nd if projetions Π J re not misleding, then Π J (O I O O J O O K ) = Π J (O I O O J ) O Π J (O J O O K ) (4.25) Π I (O I O O J O O K ) = Π I [O I O Π J (O J O O K )] (4.26) Compred to the hpter on sequentil semntis, the essentil hnge is thus the new ingredient of non misleding projetions. To show the importne of this ssumption, let us exmine the ounter-exmple of Fig The two BPs O 1 (left) nd O 2 (right) hve the sme projetion on vriles ā nd (enter), whih llows to fire nd t 2 onurrently. However, in relity O 1 imposes to fire efore t 2, while O 2 imposes the onverse, so their pullk O 1 O O 2 ontins no event t ll nd redues to the initil onditions {, e,, h}. e h f t 2 d t 2 i d t 2 g d j Figure 4.16: With misleding projetions, theorem 12 doesn t hold. Support grph for modulr omputtions. Identil to setion Exmple O 1 S 1 S 2 O 2 g g d! # ) # "!! # t 3 t 2 t 4 t 4 t 5 t 6 " e f ) # Figure 4.17: Two omponents S 1 = N 1 N 3 nd S 2 = N 3 N 2 formed of two sites eh, nd oservtions O1, O 2 on these omponents. 96

97 The enter of Fig represents two omponents S 1 = N 1 N 3, S 2 = N 3 N 2 tht shre site N 3, in hrge of flipping tokens from g to. Their pullk mounts to superimposing this ommon su-net. With slight use of nottions, we ssume the pullk is performed tking trnsition nmes s lels. Trnsitions of these nets produe lels (Greek letters) tht hve een olleted under the form of the two sequenes O1 nd O 2. On this exmple, we ssume tht the lels used for the pullk differ from those olleted y sensors, so nets re provided with two lel sets (or equivlently with synhroniztion lger [111]), whih doesn t hnge muh the theory ut introdues some flexiility.!! g g * g t 2 " " t 3 t 3 t 4 t 4 * t 4 # d # t 6 t 2 g!!!! t 2 g * g e ) f t 5... d # # t 4 t 6 g e ) f t 5... d Figure 4.18: Lol dignoses D i = U Si O Oi Π 3 (D 1 ) from S 1 to S 2 (enter). (left nd right), nd the messge The lol dignoses D 1, D 2 re depited in Fig They re otined y unfolding eh S i nd tking the produt with the orresponding Oi. Events nd onditions in gry re disrded y the produt. Oserve tht in D 2 the first firing of t 6 is disrded lthough it mthes the first oservtion γ. The reson is tht t 6 hs no future, nd so n t led to n explntion for the next oservtion δ. This orresponds to ded rnh in U S2 O O 2. The BP in the middle represents the projetion of D 1 on the interfe net N 3, whih is the messge from S 1 to S 2. It essentilly expresses tht oservtions O1 do not llow seond firing of. The strs next to onditions indite possile stop points 6, i.e. projetions of onfigurtions tht explin ll oservtions O1. One inserted into D 2, y pullk opertion, this messge removes some of the explntions : single onfigurtion remins in D 2 (Fig. 4.19). 6 As previously mentioned, the notion of stop point for BP is 0/1 funtion on uts of this BP. 97

98 !! g g * g t 2 " " t 3 t 3 t 4 t 4 * t 4 # d g g * g e ) t 5 # d # t 4 t 6 g e f Figure 4.19: Insertion of the messge Π 3 (D 1 ) into D 2 y D 2 =Π 3(D 1 ) O D 2. The messge in the reverse diretion (Fig. 4.20) is longer nd lso differs y the position of stop points : D 2 n t stop efore the first t 4 is fired. One inserted into D 1, this messge renders impossile the solution (t 2,t 3 ) euse this privte trjetory doesn t provide the sequene (,t 2 ) to S 2. So gin single onfigurtion remins possile in D 1. g g g "! t 2 t 3! " t 3 t 4 t 4 t 4 # d g g * g e ) t 5 * d # # t 4 t 4 t 6 g * g e f Figure 4.20: Messge from S 2 to S 1, nd its insertion into D 1 y D 1 =Π 3(D 2 ) O D 1. Notie tht putting together the redued lol dignoses yields unique solution to the prolem : D = D 1 O D 2 ontins single mximl onfigurtion (Fig. 4.21). The ltter revels tht α neessrily ppered efore the first γ in O2, while β is onurrent with O2. Moreover, the oserved sequenes do orrespond to true 98

99 uslities in the underlying system. So we hve redisovered the uslity reltions of our oservtions. g! " t 3 t 4 # d g e ) t 5 # d t 6 f Figure 4.21: Glol solution to the dignosis prolem. Of ourse, this exmple doesn t mnifest the interest of modulr omputtions : the ftorized form U S1 O U S2 hs the sme omplexity s the expnded form U S. The dvntge ppers when the two systems hve different wys of produing nd onsuming the resoures of their interfe N Involutivity MC rnhing proesses re involutive, i.e. stisfy O O Π I (O) =O. So we re in the setting where messge pssing lgorithms onverge on ny grph, nd produe n pproximtion of the redued lol dignoses. Insted of the D j =Π S j O O I j (D), one gets the upper pproximtions D j tht stisfy i, D j D j D j nd D = D 1 O... O D M (4.27) It n lso e shown tht MPA perform progressive redution (theorem 4), nd tht the limit D M stisfies the lol extendiility property (theorem 3). Misleding projetions. Wht hppens if the presene of misleding projetions is ignored? Theorem 12 is not ompletely lost : insted of (4.24), one gets Π J (O I O O K ) Π J (O I ) O Π J (O K ) (4.28) whih mens tht messge pssing lgorithms do propgte onstrints, ut not ll of them. This is due to the ft tht misleding projetions rete fke onurrenies, nd thus fke onfigurtions. MPA still onverge on trees in finite numer of (useful) steps. In reson of involutivity, onvergene is lso grnted on ny grph. And in oth ses the limit D j stisfies (4.27). 99

100 4.5 Augmented rnhing proesses [ The results desried in this setion re not yet pulished, t lest under this form. Prt of these ides ppered in [41] under the form of event struture sed omputtions. ] The BP omputtions presented in the previous setion remin vlid s long s projetions re not misleding, i.e. do not rete fke onurreny, nd onsequently fke onfigurtions. This phenomenon ppers when projetions erse omponents/sites tht were responsile for onflit (Fig left) or usl link (Fig right). In order to void the retion of fke runs fter projetion, one would like to preserve suh onflits or usl links etween events, when they re neessry. Augmented rnhing proesses re designed for this purpose : in ddition to the stndrd elements of BP, they lso rry extr usl links nd extr onflits, s illustrted in Fig e #. t 2 t 2 f g d d e t 2 f t 2 d g d Figure 4.22: Preservtion of onflits nd uslities in projetions Definition Augmented ourrene net. A (leled multi-lok) ugmented ourrene net (AON for short) Ȯ =(C, E,,C0, ν, λ, Λ,, # ) is otined y djoining uslity nd onflit reltion to the ourrene net O =(C, E,,C 0, ν, λ, Λ), in suh wy tht 1. is well founded prtil order reltion on E extending the prtil order + of O : e, e E, e e e e 2. # is symmetri nd nti-reflexive reltion on E, extending the nturl onflit # of O : e, e E, e#e e # e, 3. # is inherited vi uslity : e, e,e E, e # e nd e e e # e 100

101 The ugmenttion of O with nd # mounts to repling some onurreny links etween events either y uslity or y onflit. We ll # \ # (resp. \ ) the extr onflit (resp. uslity) reltions. Notie tht (E,, # ) is regulr event struture, therefore the ojets we hve introdued re mixture etween event strutures nd ourrene nets. The importne of preserving onditions will pper in the definition of projetions. Reltions nd # re defined on E ut nturlly propgte to ll nodes of Ȯ, y trnsitivity for nd y inheritne for #. As for stndrd ourrene nets, events e 1 nd e 2 re sid to e onurrent, denoted y e 1 e 2, iff (e 1 e 2 ), (e 2 e 1 ) nd (e 1 # e2 ). Co-sets nd uts re defined from in the usul wy. Prefix. The notion of prefix differs slightly from the one defined for stndrd ourrene nets. The AON Ȯ =(C,E,,C 0, ν, λ, Λ,, # ) is prefix of Ȯ, still denoted y Ȯ Ȯ, iff C nd E re left uslly losed in Ȯ for, nd e E,[e E e C ], # ontins the restrition of # to E. It is therefore suffiient to reinfore the onflit reltion # of Ȯ to otin strit prefix of Ȯ, provided this reinforement yields vlid ugmented ourrene net. Intuitively, tking prefix mens reduing the set of possile onfigurtions, whih this definition expresses (see lso elow). Configurtion. An ugmented onfigurtion κ of Ȯ (or onfigurtion for short) is onflit-free prefix of Ȯ, for #. The ugmented onfigurtion κ is thus lso onfigurtion of O enrihed with extr uslity reltions on its events. Oviously, the onverse doesn t hold : onfigurtion of O, my not e trnsformle into onfigurtion of Ȯ, even if uslity reltions re dded, euse # is stronger thn #. Morphism. A morphism φ : Ȯ 1 Ȯ2 of ugmented ourrene nets is MC net morphism φ : O 1 O 2 tht prtilly mps reltions 1 nd # 1 to 2 nd # 2 e 1,e 1 E 1, φ(e 1 ) 2 φ(e 1) e 1 1 e 1 nd φ(e 1 ) # 2 φ(e 1) e 1 #1 e 1 Notie tht uslity nd onflit reltions due to re preserved, only extr reltions of 1 \ 1 nd # 1 \ # 1 n e ersed nd trnsformed into onurreny. So morphisms still preserve runs, i.e. onfigurtions. Augmented rnhing proess. proess (ABP) of net N iff The pir (Ȯ,f) is n ugmented rnhing 1. f : O N is folding of O into N, 2. if κ nd κ re isomorphi (ugmented) onfigurtions of Ȯ, with identil folding into N, then κ = κ. 101

102 The seond requirement generlizes the prsimony riterion ppering in the definition of n ordinry BP, nd onstrins n ABP to represent t most one given ugmented run of N. Notie tht the prsimony riterion of ordinry BP doesn t hold here : given o-set X of Ȯ my e followed y two distint events e, e mpped to the sme trnsition of N y f. This hppens when e nd e re not relted to the other events of Ȯ in the sme mnner for, whih mens tht these events hve different psts. This is illustrted in Fig As onsequene, the pir (O,f) otined y removing the extr reltions, generlly violtes the prsimony riterion of ordinry BP. t 3 t 2 t 2 t 2 t 2 " " Figure 4.23: Projetion of BP tht removes omponent (nd thus t 3 ). This yields two runs : one where preedes t 2, nd one where nd t 2 fire onurrently. In the sequel, we will mke use of generlized ABP (GABP) of net N, whih mens tht point 2 of the definition my not e stisfied. Oviously, when this is not the se, there exists simple reursive trimming proedure tht mkes Ȯ true ABP of N. We will denote y T rim(ȯ) this opertion Key property Augmented rnhing proesses enjoy severl interesting properties. For exmple, removing the extr onflits in Ȯ, i.e. repling # y #, yields nother ABP of N, tht we ll the struture of Ȯ. One esily proves tht strutures hve finite width, t ny height, whih llows us to prove results on ABP y reursion on the height. t 2. U.. # # = t 2 t 2 t 3 t 4 " " Figure 4.24: Intersetion of two ABP (inherited onflits re not represented, for lrity). Severl importnt properties re relted to the ility of ABP to desrie sets of runs of N. If every onfigurtion of Ȯ 1 is isomorphi to onfigurtion of Ȯ 2, then 102

103 Ȯ 1 is isomorphi to prefix of Ȯ 2. In terms of opertions on ABP, one n define the intersetion Ȯ1 Ȯ2 tht extly ontins ugmented onfigurtions ppering in oth Ȯi. It is otined s the intersetion of the strutures of the Ȯi, followed y suitle definition of the extr onflits (see Fig.4.24). In the sme mnner, one n lso define the union Ȯ1 Ȯ2, tht extly ontins onfigurtions ppering t lest in one of the Ȯi (see Fig.4.25). This strongly ontrsts with ordinry rnhing proesses, where the union introdues redundnt runs. t 2.. # U = t 2. # t 2 t 3 t 4 t 3 t 4 " " " " Figure 4.25: Union of ABP (only miniml onflits re represented). In relity, these properties re onsequenes of Proposition 12 There is one to one orrespondene etween prefix-losed sets of (ugmented) onfigurtions of N nd ugmented rnhing proesses of N. So ABP re stronger tht ordinry BP in the sense tht one is not limited to onfigurtion sets losed y onurrent suffix extension (ompre to lemm 8). So unions nd intersetions of ABPs n e understood s operting on onfigurtion sets. Proposition 12 deserves one extr omment. We know tht distriuted omputtions re possile with onfigurtion sets, just like in the sequentil semntis one n ompute with su-lnguges. As ABP provide ompt enoding of ny onfigurtion set (up to prefix losure, whih is not severe limittion), it is likely tht one n ompute with them. The effort simply onsists in proving tht omputtions n e diretly performed on ABP, without trnsforming them into onfigurtion sets. Sine we hve the union of ABP, one ould define notion of ugmented unfolding, y tking the union of ll ugmented onfigurtions of N. This is of little interest, however. First of ll euse tht would yield huge ojet, ontining ll extensions of ll (stndrd) onfigurtions of N. Seondly, this ojet wouldn t hve the universl property, for the definition of morphisms tht we took. Consider for exmple the ABP Ȯ in Fig. 4.23, right. The ugmented onfigurtion κ of N formed y firing of followed y firing of t 2 n e sent into Ȯ with two morphisms : one tht preserves the uslity, nd one tht erses it Opertions on ABP : produt, pullk, projetion Produt. Let Ȯ1, Ȯ 2 e two AON, we define their produt Ȯ = Ȯ1 O Ȯ 2 in the tegory O of ugmented ourrene nets y extending the reursive proedure 103

104 omputing the produt in O. This tkes the following form, where the ψ i : Ȯ Ȯi s nonil mppings Proedure 3 Initiliztion : C 0 = C 0 1 C0 2, nd the ψ i : C 0 C 0 i follow ordingly, set C = C 0 nd E = (, nd # re empty s well). Reursion (until stility) : Let κ e onfigurtion of Ȯ nd X n extreml o-set of κ, let κ i = ψ i (κ) nd X i = ψ i (X), i =1, 2 if (e 1,e 2 ) E 1 E 2, λ 1 (e 1 )=λ 2 (e 2 ), e i = X i, nd e E, e = X, ψ i (e) =e i, rete suh e in E, rete set X = X 1 X 2 of e 1 + e 2 new onditions in C, with e = X nd ψ i (X i )=e i, e κ E, set e e if ψ i (e ) i ψ i (e) for some i, e E, set e # e if ψ i (e ) # i ψ i (e) for some i. if e 1 E 1, λ 1 (e 1 ) Λ 1 \ Λ 2, e 1 = X 1 nd e E, e = X, ψ 1 (e) =e 1 rete suh e in E, rete set X of e 1 new onditions in C, with e = X nd ψ i (X )=e 1, e κ E, set e e if ψ i (e ) i ψ i (e) for some i, e E, set e # e if ψ i (e ) # i ψ i (e) for some i. symmetrilly for privte events of Ȯ 2. Of ourse, Λ = Λ 1 Λ 2 nd the leling of events is inherited through the ψ i, s well s the prtitioning ν on onditions. Oserve tht the reursion does not rete ll extr uslities nd extr onflits, ut only the miniml ones. Therefore, in heking onfigurtions κ for possile extension, one must tke into ount the usl losure nd inheritne of onflits. This proedure resemles very muh wht one would otin for the produt in O, y oupling the unfolding proedure 2 to the definition of the produt in Nets. Aprt the tretment of nd #, the differene is tht events re onneted to pir o-set plus onfigurtion insted of o-set only. This trnsltes the differene of the prsimony riteri of BP ompred to ABP. Proposition 13 Proedure 3 omputes the tegoril produt in O. This entils the ssoitivity of O. Proposition 13 hs to e proved diretly sine O n t e derived from the produt in Nets vi suitle djuntion, this time. 104

105 Notie tht the produt of AON is very lose to produt of (leled) event strutures [17, 26, 106, 111], whih is onsistent with their nture. Considering ugmented rnhing proesses, the piture is not s nie s with ordinry BP, euse the prsimony riterion is lost y the produt. One hs Proposition 14 Let (Ȯ1,f 1 ), (Ȯ2,f 2 ) e ABP of N 1, N 2 respetively. Then Ȯ = Ȯ 1 O Ȯ 2 provided with the folding f =(f 1 ψ 1,f 2 ψ 2 ) is generlized ABP of N = N 1 N 2. A true ABP of N is otined y trimming O. This phenomenon is illustrted in Fig Fortuntely, however, omputing with generlized ABP is not so muh othering, exept for the size of the ojets involved : Lemm 10 Let Ȯ 1,..., Ȯ N e GABP of N 1,..., N N resp., then T rim(ȯ1 O... O Ȯ N ) = T rim[t rim(ȯ1) O... O T rim(ȯn )] (4.29) For distriuted omputtions on ABP, one n very well inlude trimming in the definition of the produt. This loses the nonil morphisms from the produt to its ftors, ut doesn t hnge the results in terms of onfigurtion sets of the underlying nets.! " " t 2 t 2 x o t 3 t 4! " = t 3! " " t 2 t 4 t 2 t 4 t 3! t 2 t 4 " " " " " Figure 4.26: From left to right, two ABP Ȯ1, Ȯ 2, their produt Ȯ 1 O Ȯ 2, nd T rim(ȯ1 O Ȯ 2 ). Pullk. Moving to pullks, it is not known whether they ll exist in O. But in the simple se where morphisms φ i : Ȯ i Ȯ0 re prtil funtions, the pullk Ȯ 2 n e otined with slight modifitions of Proedure 3 ove. And it enjoys the sme properties s the produt. This is the only importnt se for our omputtions. As remrk in pssing, notie tht the pullk ehves differently on ABP Ȯ 0 Ȯ 1 O thn on BP. For exmple, if O, O re two ordinry BP of the sme net, then O O O oinides with O O. This is not true nymore with ABP : one hs Ȯ Ȯ T rim(ȯ O Ȯ ). Miniml produt overing. This property doesn t hnge. Given Ȯ = Ȯ1 O... O Ȯ N, nd Ȯ i = ψi(ȯ), one still hs Ȯ i Ȯi, Ȯ = Ȯ 1 O... O Ȯ N nd the minimlity of the Ȯ i. Sme thing for the pullk. 105

106 Projetion. For produt ABP Ȯ s ove (or their prefixes), tking the imge ψ i (Ȯ) erses uslity nd onflit reltions inherited from the other Ȯj. This ws preisely the wekness of omputtions sed on ordinry BP. Therefore we rther define the projetion s follows : Π i (Ȯ) = T rim[ Ȯ Dom(ψ i ) ] (4.30) The restrition preserves ll inherited uslity/onflit reltions, while seleting only onditions nd events tht orrespond to omponent N i. The reltion Π Ȯ = 1 (Ȯ) O... O ΠN (Ȯ) is preserved. Moreover, for n ordinry BP O = O 1 O... O O N, one esily reovers miniml ftors O i =Π i(o) y removing ll extr reltions in Π i (O) nd trimming the result. Therefore, one n tke s intermediry ojetive the omputtion of the miniml produt overing in terms of ABP Seprtion theorem At this point, we hve projetions Π I defined on ABP, tht oviously stisfy xioms (1) nd (2), nd omintion y pullk tht stisfies xiom (4) with the empty ABP s neutrl element. Only the entrl xiom (3) is missing to enle distriuted omputtions. Theorem 13 Let I, K {1,..., N}, nd tke J = I K. Let Ȯ I, Ȯ K e rnhing proesses of N I, N K respetively, nd ssume tht shred sites N J pture ll intertion lels etween N I nd N K, i.e. Λ I Λ K Λ J. Then Π J (ȮI O Ȯ K ) = T rim[π J (ȮI) O Π J (ȮK) ] (4.31) This reltion resemles more xiom (3) if the trimming is introdued into O to form O. (4.31) eomes Π J (ȮI O Ȯ K )=Π J (ȮI) O Π J (ȮK). Notie tht when the seprtion riterion is violtes, one still hs Π J (ȮI O Ȯ K ) Π J (ȮI) O Π J (ȮK). With theorem 13, we re thus fully equipped to perform the distriuted omputtions of the previous setion, without the ssumption of non-misleding projetions Wek involutivity As mentioned setion (see lso setions nd 2.2.4), when system N = S 1... S M doesn t live on tree, messge pssing lgorithms (MPA) still provide good pproximtions of the redued lol dignoses, provided omputtions re performed on involutive ojets. This is the se of rnhing proesses : O I Π J (O I ) = O I (4.32) whih orresponds to xiom (8) setion 2.2.4, so ll MPA hve unique nd identil sttionry point. The ltter is formed of prtilly redued lol dignoses, tht form (non miniml) pullk overing of the glol dignosis. 106

107 Wht out omputtions sed on ugmented rnhing proesses? Unfortuntely, involutivity is lost, only wek form remins : For ȮI nd ABP of N I, one hs Ȯ I Π J (ȮI) ȮI (4.33) So omposing n ABP with prt of itself generlly introdues extr (reinfored) onfigurtions. This doesn t men however tht turo proedures re exluded for ABP. Let us define the reltion on GABP of net N y Ȯ 1 Ȯ2 Ȯ1 Ȯ1 Ȯ2 (4.34) Proposition 15 The pre-order on GABPs of N, stisfies xioms (5,6,7). So we re lmost in n involutive setting. Let us define the equivlene reltion on (G)ABP of N y : Ȯ 1 Ȯ2 Ȯ1 Ȯ2 Ȯ1 (4.35) Then the involutivity is reovered provided one reples = y, i.e. (4.33) eomes Ȯ I Π J (ȮI) ȮI. So turo lgorithms sed on ABP onverge to unique sttionry point, defined in terms of equivlene lsses of (see theorems 2 nd 4). These sttionry lsses re identil for ll lgorithms, ut the ltter my however reh different ABP in these lsses. t 2 t 2 t 2 t 2 Figure 4.27: Three equivlent ABP of the sme net. To help intuition, Fig depits equivlent ABP of the sme net. In sustne, two GABP Ȯ1, Ȯ2 re equivlent if onfigurtions of Ȯ 1 re otined y reinforing onfigurtions of Ȯ 2 with extr usl links, nd vie-vers. Or equivlently if eh GABP n e folded into the other. Proposition 16 Equivlene lsses of GABP re stle y, y trimming, y intersetion nd y union of GABP. So they dmit miniml element. In Fig. 4.27, the miniml net is the entrl one : the onfigurtions of the others re otined y reinforing onfigurtions of this miniml GABP. By xioms (5,6,7), we know tht the equivlene of GABP is preserved y pullk (whih inludes trimming), nd y projetion Π. But these opertions don t preserve the minimlity. Nevertheless, if ll omputtions re followed y minimiztion step, then involutivity in the strit sense is reovered. In onlusion, one n indeed run turo lgorithms on ABP, nd get the sme properties s efore, up to the minimlity of the result. Whih is not so muh othering sine in the end extr onflit nd uslity reltions re disrded to get the prtilly redued lol dignoses. 107

108 4.6 Summry This hpter is the equivlent of the previous one for the true onurreny semntis. We hve introdued seond wy of omposing utomt, in order to shpe the result s sfe Petri net nd revel the onurreny of trnsitions. Runs of these networks of utomt were defined s prtil orders of events, or onfigurtions. As for the sequentil semntis, sets of onfigurtions n e used s suh, or enoded into more ompt dt struture lled rnhing proess. One n go further in terms of omptness nd define the ounterprt of trellis proesses for the true onurreny semntis (this is exmined in the next hpter). Conerning the distriuted dignosis prolem, omputtions sed on rnhing proesses hve the sme lger s in hpter 3 : sine projetions on lels re impossile, one is gin onstrined to use projetions on sites to trnsmit informtion etween omponents, nd to use pullks to omine informtion. The min differene lies in the nture of projetions : using nive definition llows to reyle extly the pproh of hpter 3, ut this doesn t pture ll ses. In prtiulr, this strtegy fils when omponents shre n interfe tht enles onurrent events, whih my led to misleding projetions. To pture the generl se, we hve introdued the notion of ugmented rnhing proess, tht omines the definitions of BP nd of event strutures. Their properties give them the flexiility of onfigurtion sets, nd llow us to reover len definition of projetion. Up to this little modifition, the rest of the lger n e reovered. A slight differene ppers in the involutivity property, tht tkes weker form, ut still suffiiently strong to prove the onvergene of turo lgorithms on yli networks of utomt. The nture of ugmented rnhing proesses, tht resemle very muh event strutures, suggests tht one ould imgine working diretly with event strutures (see [41] for first solution). A ruil point would e to define projetions of these ojets on lel sets, whih would ring us k to the simple se of the diret grph setting (see the lnguge pproh in setion 3.4), nd would void the urden of using pullks. This is still reserh issue. Relted work The unfolding tehnique ws introdued in the erly 80 s [86, 110, 111] s onvenient wy to represent runs of onurrent systems, in the so-lled true onurreny semntis. The expression true onurreny refers to the ft tht runs re represented s prtil orders of events, y ontrst with other models of onurreny like Mzurkiewiz tres [27]. A tre is n equivlene lss of sequenes of events, where the equivlene is defined y the permuttion of onseutive independent events. Unfoldings were revisited in the 90 s s onvenient tool for verifition purposes [31, 82]. To this end, ruil step is the onstrution of finite omplete prefix of the unfolding, i.e. finite prefix tht is suffiient to hek given property on the system. The most frequent definition of ompleteness refers to prefixes tht ontin ll possile mrkings of the underlying net. M Milln proposed the first onstrution [83], sed on the key notion of dequte order (to ompre onfigu- 108

109 rtions). It ws lter refined y Esprz et l. [33, 34, 35] with the introdution of smrter dequte orders. It n then e proved tht finite omplete prefix is not lrger thn the mrking grph of the underlying net. These pprohes were lgorithmi : the prefix ws the output of proedure. Khomenko [64, 67] introdued the ide of nonil prefix, sed on the notion of utting ontext tht defines priori t whih events one should ut the unfolding. He lso proposed effiient unfolding lgorithms, mking use of SAT solvers to find o-sets nd possile extensions [60, 66]. Among vrious pplitions of unfoldings in model-heking, severl ontriutions onsider rehility nlysis [32, 36, 82, 83] nd dedlok detetion [65, 85]. The unfolding tehnique hs een pplied to vrious kinds of onurrent systems. Not only sfe nets, ut lso semi-weighted nets [31], produts of trnsition systems (s here) [35] or produts of symmetril nets [23]. More exoti onurrent models hve lso een explored. For exmple the importnt lss of Petri nets with red rs [7, 107], symoli (or high-level) nets [19], networks of timed utomt nd time Petri nets [16, 18], or grph grmmrs. Chtin, in prtiulr, proposed n importnt improvement with the ide of symoli unfolding [19], see setion We jointly proved tht symoli unfoldings enjoy the sme ftoriztion properties s stndrd unfoldings, whih mkes them nturl ndidtes for distriuted dignosis. Beside their intensive use in model-heking, prtil order methods nd unfoldings hve lso een explored in the disrete event systems ommunity, ut pprently with lower impt. Let us mention works in liveness enforement [59], nd severl interesting ontriutions in ontroller synthesis [51, 52, 53]. In distriuted dignosis pplitions, prtil order methods hve een used for stte reonstrution in the se of omponents interting y ples [14, 62, 63], with the interesting ide of kwrd unfolding. They hve lso een used, in prllel to our own work, for ommuniting utomt [88]. The lgeriztion of distriuted omputtions presented in this hpter is originl [9, 41]. The erly versions of this pproh hve muh enefited from the work of Winskel [109, 111, 112], nd in prtiulr [110], tht enled len nd onise re-expression of ll our results. Winskel introdued simple nd elegnt wy of deriving ftoriztion results on unfoldings, y mens of tegory theory, tht we use in different ples of this doument. So fr, it seems tht ftoriztion spets hve not een exploited for modulr model-heking pplitions. 109

110 110

111 Chpter 5 Trellis unfolding for onurrent systems The trellis of n utomton represents ll its possile runs in ompt struture, with omplexity tht is typilly liner in time (=length of runs). The ide is to merge not only the ommon psts, ut lso the ommon futures of two runs tht reh identil sttes t the sme time. Although suh strutures re fmilir in ommunities deling with Mrkov hins (ontrol, digitl ommunitions, et.), they re pprently new in the field of distriuted proessings, essentilly euse of the unnturl multi-lok ingredient tht we hd to introdue (hpter 4). This hpter presents the generliztion of this ide to true onurreny semntis, whih hs opened new nd promising reserh diretion. Trditionl unfoldings n e seen s operting doule expnsion of onurrent system. Time, of ourse, is unrolled (unfoldings re prtil orders of events), ut lso onflits re expnded : eh time there is hoie etween n possile events, n rnhes re reted, tht will never meet eh other gin, sine onflits re inherited in n unfolding. By ontrst, the notion of trellis is ment to unfold time, ut not onflits, nd thus results in more ompt struture. Cn suh simple ide e pplied to runs defined s prtil orders? Surprisingly, the nswer is positive. And s nie feture, ftoriztion properties n gin e proved, provided merge points re defined ording to vetor lok, s in hpter 3. In relity, the lgeri similrities re extremely lose, whih llows us to replite our pproh to distriuted dignosis, up to slight modifitions of the key theorems. The hpter is orgnized s follows. We first define trellis proesses, nd study their ftoriztion properties. We then exmine their reltions to unfoldings. Finlly, we show tht they dmit nturl notion of projetion, for whih the seprtion theorem n gin e estlished. This mkes trellis proesses suitle for distriuted dignosis pplitions, whih we illustrte on n exmple. 5.1 Trellis nets This setion defines the tegory of trellis nets, fmily of nets where time is unfolded, ut not neessrily onflits. This tegory is thus intermedite etween 111

112 O, ourrene nets, nd Nets, sfe Petri nets. We re still in the multi-lok setting, tht equips nets with prtition of ples, used to identify omponents Definition Pre-trellis net. The MC net T =(C, E,,C 0,ν) is pre-trellis net iff it stisfies : 1. C 0 = { C : = }, 2. for every C 0, the utomton T hs no iruit (i.e. its flow reltion defines prtil order). The definition of pre-trellis nets is muh less restritive thn the definition of ourrene nets. Speifilly, poin is preserved, point 2 is wekened sine is not ny more required to define prtil order, nd we hve ndoned points 3 nd 4 : onfliting rnhes re now llowed to merge on onditions. e 2 e 1 e 2 e 1 e" 1 Figure 5.1: A pre-trellis net ontining iruit (thik rrows). As n oriented grph, nd y ontrst with ourrene nets, pre-trellis net is not neessrily prtil order. Fig. 5.1 gives ounter-exmple of pre-trellis net ontining iruit. However, one hs the following property : Lemm 11 No run of pre-trellis net T n hve loop, i.e. n fill twie the sme ple. As onsequene, the restrition T σ of T to (nodes involved in) ny run σ defines prtil order of nodes. Therefore it mkes sense to express runs of T s onfigurtions κ rther thn sequenes σ of trnsitions. Configurtion, trellis net. In n ourrene net, every event elongs t lest to one onfigurtion, nd so is rehle. This is not gurnteed nymore in pre-trellis net (see T 1 in fig. 5.3), so we must refine our definition. We define onfigurtion κ of pre-trellis net T =(C, E,,C 0,ν) s su-net of T stisfying 1. C 0 κ, 2. e E κ, e κ nd e κ : eh event hs ll its uses nd onsequenes, 3. C κ, κ = 1 or C 0 : eh ondition is either miniml or hs one of its possile uses, 112

113 4. C κ, κ 1 : eh ondition triggers t most one event, 5. the restrition of T to nodes of κ is prtil order. e 1 e 2 e 2 e 1 e" 1 Figure 5.2: In the net of fig. 5.1, suset of nodes stisfying the first four requirements of onfigurtion, ut filing on the lst one. This definition is lose to the one introdued for ONs, prt from the ft tht 1 is not utomti nymore in pre-trellis net. So one must not only solve onflits forwrd (point 4) ut lso kwrds (point 3), to get vlid onflit-free ON. And the requirement tht onfigurtion is uslly losed is now spred on 2, 3 nd 4. The lst point is suggested y lemm 11, nd is indeed neessry sine points 1 to 4 lone do not gurntee this property (see ounter-exmple in fig. 5.2). With the ove definition, it is strightforwrd to hek tht sequene σ is run of T iff it orresponds to liner extension of some onfigurtion κ of T. And so n event of pre-trellis net is rehle iff it elongs to onfigurtion. We thus define trellis net (TN) s pre-trellis net where eh event elongs t lest to one finite onfigurtion (see fig. 5.3 for exmples). e 1 e 3 e 1 e 2 e 1 e 2 e 3 d f g d d f e 4 e 3 e 4 g h i f g h i T 1 T 2 T 3 Figure 5.3: T 1 is pre-trellis net ut not trellis net : event e 4 is unrehle. The other nets re trellis nets : ll events re rehle. In T 2, e 1 nd e 3 re not uslly relted... ut in onflit! T 3 displys non inry onflit : {d, f}, {f, g} nd {d, g} re ll pirs of onurrent onditions, ut the triple {d, f, g} ppers in no run. Removing e 2 in T 3 doesn t yield vlid prefix : we re k to T 1 whih is not trellis net. Conurreny nd onflit. From the definitions ove, one sees tht oth ONs nd TNs re grphil strutures enoding fmilies of onfigurtions in different 113

114 wys. TNs offer the dvntge of eing more ompt... t the expense of more omplex disply of onfigurtions. In prtiulr, the fmilir uslity, onflit nd onurreny reltions on events do not hve ny more simple grphil trnsltion (see T 2 in fig. 5.3). This is due to the ft tht, in TP, n event (or ondition) generlly ppers on top of severl histories. This phenomenon introdues strong ontrst with ONs, where node elongs to unique miniml onfigurtion, its usl losure. As onsequene, onurreny nd uslity re now ontext dependent : two events my e onurrent in one onfigurtion, nd pper s uslly relted in nother (fig. 5.4). e 1 e 2 e 1 e 2 e 1 e 2 d d d e 3 e 4 e 3 e 4 e 3 e 4 f g f g f g e 5 e 6 e 5 e 6 e 5 e 6 h i h i h i Figure 5.4: On this trellis net (left), events e 3 nd e 4 pper in severl onfigurtions. They n e onurrent in one of them (enter) nd uslly relted in nother (right). It is importnt to define o-sets in trellis net, i.e. to determine onditions tht n e used t the sme time to onnet one more event to the struture. To define this extended notion of onurreny, we thus hve to strt the ontext. Let x 1,x 2,..., x n e n nodes of T, they re onurrent in T, denoted y (x 1,x 2,..., x n ), iff there exists onfigurtion κ where they pper s onurrent nodes. In the exmple of fig. 5.4 (left), e 3 nd e 4 re thus delred onurrent for this extended notion. The notion of o-set (of onditions) derives from this definition. Oserve tht in TN, onurreny n no longer e derived from pirwise reltions, y ontrst with ONs (see T 3 in fig. 5.3). In the sme wy, n extended notion of onflit n e defined s follows : x 1,x 2,..., x n re in onflit, #(x 1,x 2,..., x n ), iff there is no onfigurtion ontining ll of them (for exmple #(e 5,e 6 ) in fig. 5.4). Agin, # nnot e derived from pirwise reltions, i.e. onflit is not inry in TNs. Prefix. Prefixes re less esy to define grphilly for TNs thn for ONs. Let T e TN, T is prefix of T (T T ) iff 1. T is su-net of T, 2. { ondition of T, = } = { ondition of T, = } 3. e event of T, e T [ e T nd e T ], 114

115 4. T is trellis net. The lst requirement imposes tht every event in the su-net T remins rehle. To illustrte its neessity, onsider T 3 in fig. 5.3 : if e 2 is removed, points 1-3 re stisfied, ut e 4 eomes unrehle. Of ourse, on TNs extends the reltion on ONs. Notie lso tht T T implies the existene of n injetive morphism φ : T T (whih mens here tht φ is totl funtion). Height funtion. The definition of trellis nets now llows us to merge onfliting onditions rehed y different runs. However, this leves lrge mount of flexiility. But if one wishes to get universl ojet to represent onfigurtion sets, some kind of guideline is neessry to indite where merges must e performed. Let us define string s onfigurtion σ =(C, E,,C 0,ν) in O where C 0 = 1. So σ hs single lss nd thus orresponds to sequene lternting onditions nd events. The height H σ () of ondition in σ is given y H σ () = { C, } (5.1) In generl onfigurtion κ, we define the height of ondition y fousing on the omponent κ tht ontins, so we set H κ () H σ () where σ = κ, or equivlently H κ () = { C, ν( )=ν(), } (5.2) A trellis net T is orretly folded for H iff for every ondition of T nd every pir of strings σ, σ ontining in T, one hs H σ () =H σ (). Equivlently, T is orretly folded iff C, κ, κ onfigurtions of T ontining, H κ () =H κ (). We denote this ommon vlue y H T () or simply H() when there is no miguity. Fig. 5.5 illustrtes this property. e 1 e 2 e 1 e 2 e 3 3 e e 4 d Figure 5.5: Two trellis nets ; the left one is not H-omplint, the other one is. Ctegory of trellis nets. In the sequel, we only onsider H-omplint leled trellis nets. The ltter, ssoited to the usul notion of morphism (of MCNs), form the tegory Tr. So we hve three nested tegories : O Tr Nets. 115

116 5.1.2 Trellis proess nd time unfolding of net Trellis proess. Reproduing the developments round ourrene nets, trellis nets n e used to represent runs of given LMC net N. Let T = (C, E,,C 0, ν, λ, Λ) e leled trellis net nd f : T N morphism, the pir (T,f) forms trellis proess (TP) of N iff 1. f is folding of T (i.e. totl funtion on T ), 2. T is prsimonious desription of runs of N : 3. T is mximlly folded : e, e E, [ e = e,f(e) =f(e )] e = e, C, [H() =H( ),f() =f( )] = The novelty with respet to rnhing proesses is thus the merge imposed y 3, tht opertes s n extr prsimony riterion to desrie runs of N. As f is folding of T into N, every onfigurtion κ of T represents run of N in the true onurreny semntis, nd hs ounterprt in U N. So trellis proess of N orresponds to olletion of runs of N. Conversely, run of N is represented y t most one onfigurtion in T : If κ 1 nd κ 2 re isomorphi nd folded into N in the sme wy, then they re identil. Indeed, one hs H κ1 = H κ2 whih shows tht onditions re identil (point 3), from whih events re lso identil (point 2). As efore, we will often omit mentioning the folding f when there s no miguity. Oserve the lose similrity etween the definition ove nd setion tht defines sequentil trellis proesses. Atully, most properties of sequentil TP n e trnsported to the prtil order semntis : TP is isomorphi to the union of its onfigurtions, two TP tht hve the sme onfigurtions re isomorphi, et. Time unfolding of net. One n esily uild ny trellis proess of net N =(P, T,,P 0,ν N,λ N, Λ) with simple reursion, yielding oth T =(C, E,,C 0, ν, λ, Λ) nd the folding f : T N. This reursion is simple refinement of proedure 2, designed for rnhing proesses. A single new feture ppers : the merge of newly reted onditions one new event hs een onneted, in order to stisfy the lst requirement of the definition. Proedure 4 Initiliztion : Crete P 0 onditions in C 0, nd define ijetion f : C 0 P 0. Set C = C 0, E = nd =. Reursion : Let X e o-set of C nd t T trnsition of N suh tht f(x) = t. 116

117 If there doesn t exist n event e in E with e = X nd f(e) =t, rete new event e in E with e = X nd f(e) =t, rete suset Y of t new onditions in C, with Y = e, extend f to hve f : Y t ijetive, then, for every Y, if C, f( )=f() nd H( )=H() then merge nd. The prtitioning of onditions ν : C C 0 nd the leling of events λ : E Λ re of ourse inherited from those of N through f. As for unfoldings, one n modify proedure 4 to ompute the union of n ritrry numer of TP of N. Tking the union of ll TP of N results in unique mximl TP, tht hs ll the others s prefixes. It is of ourse otined s the unique sttionry point of proedure 4. Theorem 14 Let N e multi-lok net, there exists unique mximl trellis proess of N for the prefix reltion. We ll it the trellis of N or the time unfolding of N, nd denote it y U t N, with orresponding folding f t N : U t N N. U(N ), the unfolding of N, nd U t (N ), the time unfolding of N, re different enodings for the sme onfigurtion set, formed y ll trjetories of N. Fig. 5.6 illustrtes the (eginning of the) time unfolding of our running exmple N. Despite the pprent loop k, tht my look surprising, there is no exeutle iruit in this net. Oserve lso tht t eh level one more possiility to fire ppers. This is due to the ypss trnsition t 2, tht llows us to use ondition g ritrrily fr in the pst. g g d t 2 d t 3 t 2 t 4 t 5 t 6 t 3 t 4 t 6 e f g e f t 2 t 5 d t 3 t 4 t 6 g e f... Figure 5.6: A net (left) nd the eginning of its time unfolding (right). 117

118 Expressive power of trellis proesses. Like ll other strutures we hve introdued, TP n t enode ny onfigurtion set. In ft, the more properties we sk to these strutures, the less flexiility they offer. Nevertheless, the su-lnguges of N they desrie re suffiient to perform distriuted omputtions. Let (κ 1,f 1 ) nd (κ 2,f 2 ) e two onfigurtions of N, leding respetively to uts C i, i.e. Ci 0[κ i C i. κ 1 nd κ 2 re sid to e H-equivlent, denoted y κ 1 H κ 2, iff f 1 (C 1 )=f 2 (C 2 ) nd H κ1 oinides with H κ2 on their terminl ut : ( 1, 2 ) C 1 C 2, f 1 ( 1 )=f 2 ( 2 ) H κ1 ( 1 )=H κ2 ( 2 ) (5.3) In other words, κ 1 nd κ 2 would finish t the sme ut in U t (N ). A su-lnguge L of N, s onfigurtion set, is suffix-losed iff κ = κ 1 κ 1 L, κ 2 L, κ 1 H κ 2 κ 2 κ 1 L (5.4) Lemm 12 A su-lnguge L of N n e enoded s trellis proess (T,f) of N iff it is prefix- nd suffix-losed. As efore, one n relx little the prefix-losure neessity y introduing stop funtion, tht ssigns zero-one vlue to uts of TP Ftoriztion properties Theorem 15 (Universl property of UN t ) Let N e n LMC net, for every trellis net T in Tr nd morphism φ : T N, there exists unique morphism ψ : T UN t suh tht φ = f N t ψ. By rguments we hve lredy detiled severl times, this is suffiient to estlish o-refletion of Tr into Nets. The djoint pir of funtors relting these two tegories re (F, G) where F = : Tr Nets is the inlusion funtor, nd in the reverse diretion G = U t : Nets Tr is simply the time-unfolding opertion. An importnt property one expets from trellis proesses onerns their ftoriztion, sine it forms the first pillr of modulr omputtions. As soon s the o-refletion ove is estlished, one mehnilly gets the preservtion of limits u U t, nd thus the preservtion of produts nd pullks. So one hs U t (N 1 N 2 ) = U t (N 1 ) T U t (N 2 ) (5.5) whih lso proves the existene of produt T in Tr for trellis nets. The ltter n tully e defined y T 1 T T 2 = U t (T 1 ) T U t (T 2 ) = U t (T 1 T 2 ) (5.6) One gin, this reltion is importnt in prtie : y inserting the definition of produt on nets into the time unfolding proedure, one gets n effetive lgorithm to ompute reursively produts like T 1 T T 2. Nturlly, the two reltions ove remin true if one reples produts y pullks, whih shows the interest of these strt lgeri pprohes. 118

119 g g d t 2 d t 2 t 4 e f t 6 t 3 t 4 t 6 t 3 t 4 t 5 g e f g d t 2 t 5 t 2 t 4 t 6 d e Figure 5.7: Left : Trellis of the net N depited in fig Right : Trellises of its omponents N ā, N ḡ, N d. The trellis on the LHS is the produt of the three other trellises, in the sense of λtr : U t N = U t N ā λt U t N ḡ λt U t N d. f Fig. 5.7 illustrtes the ftoriztion property. The net N in Fig. 5.6 ontins three elementry omponents N ā, N ḡ, N d, or sequentil mhines. Tking this deomposition for N shows tht its trellis UN t is the produt of the trellises of these omponents, whih re nothing more thn ordinry trellises, in the usul sense of tht word for utomt. The figure illustrtes lso the interest of ftorized representtions, tht we hve lredy underlined in the previous hpters : While UN t grows in width nd omplexity, euse of the extr tht ppers t eh level, its ftors hve liner omplexity in time. 5.2 Reltions to unfoldings Vritions round the height funtion As for the sequentil semntis, the definition of height funtion is quite flexile, provided one preserves its vetor nture. Consider the triple (E,,ɛ) formed y set E, n internl omposition lw in E nd neutrl element ɛ. Let us tth to eh LMC net N tuple of funtions h ssoiting vlues of E to the trnsitions of N. Speifilly, h is formed y the vetor (h p ) p P 0 nd h p : T E stisfies h p (t) =ɛ whenever p ν( t). In other words, when t doesn t influene the omponent N p, its vlue h p (t) is neutrl. The height vlues tthed to trnsition n depend on their lel, for exmple, nd we onsider net morphisms tht preserve these h vlues. To generlize trellis proesses, we simply omine the height vlues of events to define height funtion on onditions. To ondition of onfigurtion κ, we ssoite the height H κ () y onsidering the string σ = κ tht ontins. This string tkes the form 0 e 1 1 e 2... e n e n+1... nd we tke H κ () =h 0 (e 1 )... h 0 (e n ). One height funtion is defined, the theory remins unhnged with orretly nd mximlly folded trellis nets 1. 1 At this point, this sttement is only onjeture tht hs not een rigorously proved. 119

120 This degree of freedom llows us to uild trellises tht simply ignore some events in their ounting, for exmple events tht don t produe visile lels. So one n unfold time only when oservtions re produed, nd preserve silent yles unhnged. On the ontrry, one n define height funtion tht never llows merges in some omponents. For exmple, in onfigurtion (κ, f) of N onsider h (e) =f(e), the nme of the trnsition t = f(e) fired in event e, nd tke for the ontention of trnsition nmes. The trellis of eh omponent N p is then isomorphi to its unfolding, sine only isomorphi strings n led to merge point, nd there re none euse of the prsimony ondition. This is not suffiient however to mke the glol trellis U t (N ) isomorphi to the stndrd unfolding U(N ), s it ws erroneously sid in lemm 4 of [45]. A mehnism tht would ensure this nie property remins to e found Nested o-refletions Co-refletion of O into Nets. At this point, we hve three nested tegories O Tr Nets. By restriting Nets to Tr in the o-refletion of O into Nets, we n derive nother djuntion etween O nd Tr (nother o-refletion, in ft). Speifilly, we still hve the inlusion funtor F = : O Tr in one diretion, nd the unfolding funtor G = U : Tr O in the reverse diretion. Applying U to trellis net T performs n unfolding in the onflit dimension only, sine time is lredy unfolded, so we rther denote y G = U the restrition of U to Tr. In this djuntion, the universl property of onflit unfoldings still holds (y definition) : T T r, O O, φ : O T,! ψ : O U T, φ = f T ψ (5.7) where f T : U T T is the morphism tht refolds onflits of U (T ). Therefore limit preservtion theorems llow to stte U (T 1 T T 2 ) = U (T 1 ) O U (T 2 ) (5.8) nd give nother definition of the produt in O O 1 O O 2 = U (O 1 ) O U (O 2 ) = U (O 1 T O 2 ) (5.9) These expressions remin vlid of ourse will pullks insted of produts. F = U O F = 1 U G 1 = U Tr F = 2 U G 2 = U t Nets G = U Figure 5.8: Co-refletions relting tegories O, Tr nd Nets. 120

121 Composition of djuntions. Gthering results otined so fr, we hve three djuntions relting tegories O, Tr nd Nets, s displyed y figure 5.8. It is well known ft tht djuntions n e omposed ([81], hp. IV-8, thm 1), so (F 2 F 1,G 1 G 2 ) defines nother pir of djoints etween O nd Nets. But sine F 2 F 1 = F, we hve tht G = G 1 G 2, up to nturl equivlene 2. This trnsltes into N Nets, U(N ) = U U t (N ) (5.10) nd nturlly the orresponding foldings n e omposed : f N = fn t f. Eqution (5.10) expresses tht the time-unfolding UN t of net n e reovered y re- UN t folding onflits on the full unfolding U N, whih we lredy used when we derived proedure 4 from proedure 2. Speifilly, f : U UN t N UN t merges onditions with the sme height nd representing the sme ple of N, then merges (or removes) redundnt events representing the sme trnsition onneted to given o-set. This n e heked in fig. 5.9 tht ompres the unfolding nd the trellis of our running exmple. g d t 3 t 2 t 4 t 5 t 6 e f g g t 2 d t 2 d t 3 t 3 t 4 t 6 t 3 t 4 t 6 g e f g e f t 2 t 2 t 5 t 2 t 5... d... d Figure 5.9: A net N (top), its unfolding U N (ottom left), nd its trellis U t N (ottom right). In terms of produt preservtion, the omposition of djoints yields, for ny pir 2 The djoint of funtor is unique up to nturl equivlene, see [81], hp. IV-1, or

122 N 1, N 2 in Nets nd similrly with pullks. U(N 1 N 2 ) = U(N 1 ) O U(N 2 ) (5.11) = U U t (N 1 N 2 ) (5.12) = U [ U t (N 1 ) T U t (N 2 ) ] (5.13) = U [ U t (N 1 )] O U [ U t (N 2 )] (5.14) In summry, the tegory of trellis nets ppers s n djustle intermedite etween ourrene nets nd sfe nets. Its position n e djusted y different hoies of height funtions, tht impose either numerous or sre merge possiilities. 5.3 Distriuted dignosis : n exmple As soon s one hs produt opertion on trellis nets, whih is now grnted, nturl notion of projetion omes for free. Moreover, thnks to the reltion etween the unfolding nd the trellis of net, one n reyle the theory developed for distriuted omputtions sed on rnhing proesses. Therefore, s long s one doesn t hve misleding projetions (of trellis proesses), the seprtion theorem holds nd llows us to perform distriuted omputtions. The dvntge is of ourse the omptness of these strutures. The prie to py lies proly in the diffiulty to identify o-sets in trellis net, key step in the reursion tht omputes produts. The lger tht supports distriuted omputtions hs een disussed severl times, so we rther illustrte the mehnism of trellis sed omputtions on n exmple. Notie tht we don t hve (yet) notion of ugmented trellis proess, tht would llow us to ope with misleding projetions, so the exmple ssumes interfes tht redue to single site. O 1 S 1 S 2 O 2 g g d! # ) # "!! # t 3 t 2 t 4 t 4 t 5 t 6 " e f )! Figure 5.10: A system S = S 1 S 2 = N 1 N 2 N 3 with two omponents, eh of them overing two sites : S 1 = N 1 N 3 nd S 2 = N 3 N 2. The sequenes O 1, O 2 represent oservtions on S 1, S 2 respetively. We onsider the sme setting s in setion 4.4.4, with different sequenes of oservtions (Fig. 5.10). The first step onsists in omputing lol dignoses D i = 122

123 US t i T Oi, s illustrted in Fig Oserve tht the onditions of the oserved sequenes Oi re not duplited y the produt, whih ontrsts with Fig The events in light gry orrespond to disrded prts of US t i. In the messge, the strs indite possile stop points for D 1. Integrting the messge into D 2 doesn t hnge this lol dignosis : D 2 = D 2 T Π N3 (D 1 )=D 2. g!! t 2 " t 3 t 4 g!!! t 2 " t 3 t 4... g g * * t 4 g * * t 4 g * g d # # t 4 t 6 g e f ) t 5 g # e d # t 4 t 6... f Figure 5.11: Lol dignoses D i = U t S i T O i, nd the messge Π N 3 (D 1 ) from S 1 to S 2. The irultion of messge Π N3 (D 2 ) in the reverse diretion essentilly prevents S 1 from firing twie trnsition t 4 (Fig. 5.12), whih results in the elimintion of single event of D 1 (in gry). A less visile phenomenon onerns possile stop points in D 1 (not represented in the figure). The messge imposes to produe t lest the susequene (,t 4 ), whih kills in D 1 the purely lol explntion (t 3,t 2,t 3 ). Tking the omintion D 1 T D 2 yields the glol dignosis D. It ontins three mximl onfigurtions, or explntions, depited in Fig Summry For the sequentil semntis, we hd proposed three strutures to enode sulnguges of system : sets of sequenes, rnhing proesses nd trellis proesses. With deresing degree of expressiveness, ompensted y n inresing degree of omptness. Algerilly, three properties trigger the possiility of distriuted omputtions : ftoriztion property on these strutures, the existene of projetions (onsequene of the preeding one), nd seprtion theorem. Surprisingly, this formlism n e lmost entirely shifted to the prtil order semntis, up to some speifiities of this frmework. Among the spets tht re preserved, the notion of height funtion, tht defines merge points in trellis proess, must still e lolized to eh elementry omponent (or site), otherwise the universl property of these ojets is lost, whih entils the loss of their lgeri 123

124 g g g t 2!! " t 3 t 4 t 4 t 4 # d! t 2! g! g * g e ) t 5 * d t 4 g Figure 5.12: Messge Π N3 (D 2 ) from D 2 nd its integrtion into D 1 y D 1 = D 1 T Π N3 (D 2 ). g g g! t 2! t 2! " t 3 t 4 d # g e!! ) d t 5! t 2! " t 3 t 4 d # g e!!! ) t 2 t 5 d! t 2! t 2! " t 3 t 4 d # g e!! ) t 5 d Figure 5.13: The 3 glol solutions to the dignosis prolem. 124

125 properties. By ontrst, the simple notion of projetion tht omes with the produt is not suffiient in the true onurreny semntis. In generl, it erses useful onflit or uslity informtion, nd my result in erroneous omputtions, exepted in the limited se where omponents intert y non onurrent interfes. This limittion motivted the introdution of ugmented rnhing proesses, in the previous hpter. Unfortuntely, we don t hve yet the equivlent notion for trellis proesses. Relted work Surprisingly, lmost t the sme time, Khomenko et l. hve proposed struture very similr to trellis proesses, under the nme of merged proesses [68]. These uthors develop unfolding-sed model heking tools, nd were motivted y memory onsumption issue, whene the ide to merge isomorphi futures of onfigurtions. The definition they proposed is slightly different from trellis proesses (see [45] for omprison). In prtiulr, it ounts heights seprtely for eh ple of N (insted of sites), nd my ontin exeutle iruits. The diffiulty to identify o-sets nd onfigurtions hs een identified, nd the uthors proposed solution sed on SAT-solvers. Overll, on trditionl enhmrks, experiments indite tht one doesn t sve muh in terms of omputtion time, ut does sve in terms of memory spe, whih is ruil in model heking pplitions. Merged proesses re not universl ojets, nd so do not enjoy ftoriztion properties. A hot topi is thus to investigte the interest of modulr proessings for trellis-sed model heking. They ould suggest more effiient strtegies to identify onfigurtions. Among relted works, let us lso mention plnning prolems, nd in prtiulr the Grphpln pproh [12]. Plning prolems onsist in orgnizing elementry tsks in order to reh n ojetive, so they orrespond to rehility prolems in model heking. A su-fmily of prolems, lled STRIPS-like domins, onsiders situtions where elementry tions onsume nd produe lol resoures, like Petri net. This ommunity hs soon disovered the superiority of prtil order methods on stte spe explortions. The Grphpln pproh goes further nd represents ll possile exeutions in domin s trellis net (see Fig. 2 in [12]). The prolem then oils down to optimlly exploring this grph, nd plning solution orresponds to onfigurtion ontining the gol. The relevne of prtil order semntis is ovious in this ontext, nd there is potentilly spe for distriuted/modulr plning lgorithms. 125

126 126

127 Chpter 6 Applitions, ontrts, tehnology trnsfer The reserh results presented in this doument originted nd were stimulted y reserh ontrts, gthering industril prtners s well s other demi tems. The origins dte k to CTI ontrts (Informl Themti Collortions) with Frne Teleom R&D, round 96, 97, tht grew up to tke the form of 3 onseutive RNRT 1 projets : MAGDA, MAGDA 2 nd SWAN. In the lst yers, diret ollortion ws initited with Altel to uild more relisti prototypes of our lrm orreltion lgorithms (VDT ontrt). The deision out their integrtion into Altel s mngement pltform is pending. This hpter gives n overview of these ontrts, for wht onerns their distriuted dignosis spets. 6.1 MAGDA MAGDA stnds for Modeling nd Lerning for Distriuted Mngement of Alrms (in Frenh). This projet (Nov. 98, Nov. 01) ws heded y Christophe Dousson (FTR&D). Its min ojetive ws to develop lrm orreltion methods, oth online nd off-line, to llow the nlysis of lrm logs. An importnt prt of the projet ws dedited to orreltion methods sed on hroniles [5, 28, 29, 78], nd to lerning methods for hroniles [11, 30, 49, 56]. Self-modeling. The distriuted dignosis prt foused on lrm orreltion in SDH/SONet networks, s the one depited in Fig Sine we use model-sed pproh, ruil step ws the derivtion of this model. This tsk turned out to e the most demnding in terms of reserh effort. The projet jointly me up with methodology to derive the model, tht we now ll self-modeling. The priniple is sed on the following ides : There my exist mny omponents in network, in relity there is smll numer of different types of omponents. Here, y omponent, we refer oth to 1 Ntionl Reserh Network in Teleommunitions, funded y the Frenh Ministries of Reserh nd of Industry. 127

128 Figure 6.1: A toy SDH ring, with different onnetions nd displying the vrious SDH trnsmission levels. physil equipment, trnsmission funtions in the different lyers, dpttion funtions, softwre, et. The monitoring model, t lest its topologil prt, n e otined y snning the network to disover omponents nd their onnetions, nd y instntiting the orresponding internl model tht will e used for monitoring. The mnul tsk of the modeler thus simply onsists in designing the uilding loks, i.e. generi omponents, nd speifying how their onnetion pilities. The network ehviors, orresponding to trnsitions in our monitoring model, n e defined t the sle of omponents. Elementry ehviors re desried prtly in the SDH stndrds, ut the most useful informtion ws otined under the form of filure senrios, given y n expert. The ltter were then deomposed into elementry trnsitions, desried s UML sequene digrms. Only filure propgtions nd there onsequenes were modeled, not the repir or reset tions. Apprently simple, disovering this methodology, nd tuning the model to otin the expeted ehviors ws n extremely demnding tsk. The experimentl system ontined round 100 elementry omponents (with 2 to 4 vriles per omponent), enoded into sfe Petri nets ommuniting y shred ples. Dignosis tehnology. This first prototype of distriuted dignosis lgorithm ssumed one lol supervisor per network element (so four susystems, s in Fig. 6.1). Eh supervisor ws hndling onfigurtion sets, nd not rnhing proesses. However, the dt strutures enoding these sets were in ft rnhing proesses. One BP is given, onfigurtions re simply identified y their ut (i.e. their mximl onditions). So we were tully hndling sets of uts. A drwk of this pproh is tht onurrent extensions of onfigurtions generlly rete trjetories 128

129 Figure 6.2: A zoom on the inner omponents of the network element Montrouge. The figure displys the pths of onnetions etween their injetion in the ADM, s high-rte or low-rte onnetions (HOP/LOP=High/Low Order Pth), down to the modultion (SPI=Synhronous Physil Interfe). They ross the Multiplex Setion (MS) tht ggregte onnetions, nd the Regenertion Setion (RS). All these omponents re self-mnged nd hve the ility to rise, trnsmit or ret to lrms. tht re lones one of nother (i.e. the underlying rnhing proess violtes the prsimony riterion). Therefore these lones hd to e deteted nd removed. By ontrst hndling onfigurtions hs the dvntge to filitte optimiztion strtegies. Eh ut ws provided with ost, ounting the numer of spontneous filures in the orresponding onfigurtion. It ws then esy to perform lol optimiztion (tully Viteri lgorithm) on runs exmined y eh lol supervisor : for two runs terminting t the sme mrking nd hving the sme ehvior on the interfes with neighoring susystems, only the est one ws kept. Finlly, speil tretment ws introdued for silent trnsitions, in order to proess piles of silent trnsitions s mro-tiles. In terms of ommunitions, eh extension of run, in lol supervisor, rised n updte messge when the newly onneted trnsition hd n influene on the interfe with neighoring susystem. Symmetrilly, messge reeived from neighoring supervisor ws used to updte lol runs, extly like lol extension. We did the exerise to fully distriute the lgorithm, i.e. not to ssume ny shred memory. Therefore the detetion of termintion ws itself distriuted, sed on Misr lgorithm [90], in order to hek tht eh lol supervisor hd finished extending its trjetories, nd tht t the sme time no more updte messge ws pending in ommunition hnnel etween lol supervisors. However, sine finite sequenes of oservtions were ssumed for eh supervisor, the issue of determining synhroniztion point where omptile results ould e olleted didn t pper. When termintion hd een deteted, seond phse of glol optimiztion ws strted. This phse extly implemented turo proedure, to determine the most likely glol run (rell tht the ost of trjetories ws optimized lolly, for eh 129

130 lol supervisor, not glolly). This lgorithm ws implemented in JAVA with RMI for ommunition etween lol supervisors. Figure 6.3: A usl grph of lrms. The root use in Gentilly (TF=Trnsmit Filure) orresponds to lser rekdown. Outomes. As nturl output, we proposed the initil filures in the most likely runs of the system, lssified y deresing likelihood. But the lgorithm provided more : the most likely runs tully reveled possile usl links etween lrms, whih ws of mjor interest to the users. We displyed them under the form of usl grphs of lrms, s the one in Fig The modeling effort ws quite rewrding : it llowed us to reognize long-rnge, topology-dependent orreltion ptterns, ut lso to disentngle ursts of lrms generted y multiple filures. 6.2 MAGDA 2 The ojetives of MAGDA 2 were more mitious : extending this pproh to heterogeneous networks, utomtilly instntiting the model nd deploying the distriuted supervisors, integrting this tehnology in stndrd mngement pltform. Self-modeling. In terms of self-modeling, the projet ws very suessful. It ws shown tht the omponents to supervise elonged to speifi lsses of mnged 130

131 ojets. The ltter n e found in the informtion model of the hosen network tehnology, nd these informtion models re quite strutured, Roughly speking, they inherit their elements from ommon high level definitions of mnged ojets, like Tril/Connetion Termintion Point (TTP/CTP), Adpttion Lyer, et. So gret prt of the model struture is known : the modeler just hs to speilize these lsses to the trget tehnology. This is true oth for the mnged omponents, for their onnetion pilities, nd for their ehviors : some ehviors re stndrd nd n e desried t high level, like the trnsmission of messges etween omponents, stte hnges in se of filure of neessry funtion, the notion of dependene of servie on nother, et. A tool (OSCAR) ws developed to llow UML drwing of the model. This tool llowed us to drw topology (deployment digrm), to red one in file, nd ws redy to sn network (whih wsn t experimented, however). It lso llowed us to refine the mnged ojets, their onnetion pilities, nd to design elementry ehviors (so-lled tiles), with sequene digrms. Finlly, one ould define with OSCAR the su-systems ssoited to the different lol supervisors. Our pplition se in this projet ws GMPLS/WDM network. Figure 6.4: Correlted lrms, s they ppered in the mngement pltform. Root uses were presented first, nd the field orrelted notifition flg ould e reursively liked to disover immedite suessors of n lrm. Dignosis tehnology. The omputtions were still sed on onfigurtion sets. The effort onentrted on the following fetures : The utomti deployment of the lol supervisors. The ltter were sent on different mhines, with knowledge of their su-system model nd of their 131

132 neighors, onneted together nd initilized. This ws progrmmed in Jv + CORBA. The distriuted dignosis lgorithm ws implemented ove rule engine, whih is the stndrd orreltion tehnology used in mngement pltforms. We me up with three sets of rules : lol extension rules, diretly derived from the su-system model, optimiztion rules (speifi to the mngement of trjetories), nd finlly ommunition rules etween lol supervisors. The diret onnetion of lol supervisors to true mngement pltform. Both to reeive lrms from the pltform, nd to visulize their orreltion t the end of omputtions. This ws done with CORBA. Outomes. Progrmming the distriuted dignosis lgorithm over rule engine ws rel hllenge, ut we proved its fesiility. In terms of visuliztion of the results, we only seleted the most likely explntion. We trnslted it into uslity grph etween the reeived lrms, whih ws used to fill the orrelted field of lrms s they re stored in the mngement pltform. This is illustrted in Fig VDT Figure 6.5: A sumrine network, with severl terminl equipment. Ojetives. VDT stnds for Viteri Dignoser Tehnology. The ojetive of this ontrt ws to propose n lrm orreltion method for terminl equipment of sumrine lines (SLTE, Fig. 6.5). These network elements ggregte in WDM trns132

133 missions high rte onnetions. They involve severl lyers of multiplexing nd mplifition, whih results in quite omplex systems, tht n esily fill severl rks (Fig. 6.6). The ojetive ws to orrelte filure informtion propgting etween these funtions. For exmple the ft tht low output signl of n mplifier will use extrtion errors on ll wvelengths multiplexed in this signl. The mjor diffiulty ws tht the propgtion of filures were esy to desrie t the physil level. Normlly, this is where the redundny of lrms should e voided, y dequte msking proedures involving simple ommunitions etween rds. Unfortuntely, this feture is often onsidered s seondry, nd delivery delys generlly prevent its development. So the orreltion must e performed t the mngement level. There is no one-to-one mpping etween the mnged ojets nd the underlying funtions tht support them. In generl, mnged ojet, like n Optil Pth or n Optil Chnnel Group, ollets informtion from severl rds, nd symmetrilly filure on given rd is refleted in severl mnged ojets. Moreover, lthough lrms re non miguous t the physil level, they re trnslted (when they re) into very generi filure inditions t the mngement level. Speifying the system model t the mngement level ws rel hllenge. Nevertheless, we mnged to reyle n importnt prt of the self-modeling methodology developed in Mgd nd Mgd 2. A few omponents, speifi to this tehnology, hd to e designed, s well s speifi onnetion pilities. Behviors were gin extrted from filure senrios given y n expert. And the ounterprt of the OSCAR tool ws redeveloped to esily drw the model. Figure 6.6: Two sumrine line terminl equipment (SLTE). Dignosis tehnology. This time we experimented true unfolding-sed pproh, in entrlized setting ( single supervisor). It ws fed on one side y the model produed y OSCAR, whih ws itself uilt from the desription of n equipment rhiteture. On the other side, the dignosis lgorithm ws fed y lrm logs, tking the form of prtil order (more preisely, tuple of sequenes). We experimented different unfolding lgorithms. The most effiient ws sed on reursive su-routine to disover o-sets where possile extension ould e performed. The unfolding ws guided oth y oservtions nd y ost funtion, in order to fvor 133

134 onfigurtions minimizing the numer of root filures. As output, the lgorithm provided oth the unfolding (Fig. 6.7) nd disply of the SLTE topology identifying the mnged ojets responsile for the root filures. Figure 6.7: Prt of the unfolding representing possile filure propgtions. 134

135 Chpter 7 Conlusion 7.1 Summry of results In sustne, this doument proposes methodology to represent the trjetory set of omplex system, nd n lger to perform omputtions on these trjetories. We strted y the definition of wht we ll omplex systems : they re otined y ssemling elementry omponents into network, nd thus n e desried s grph of omponents. The time dimension is introdued y unfolding the system, in different wys, whih is performed y funtor pplied to the system. Mny efforts hve een dedited to the design of suh funtors, in order to otin trnsformtions tht would preserve the intertion struture etween omponents. So in ll the proposed settings, runs of ompound system n e expressed s the omintion of runs of its omponents, whih is lled the ftoriztion property. Formlly, this property is equivlent to the ftoriztion of the proility distriution in Mrkov rndom fields, whih is itself the trnsltion of onditionl independene sttements etween omponents of the field (see the Hmmersley- Clifford theorem). The nlogy of these two domins hs een estlished y mens of ommon xiomti frmework. As onsequene, optiml estimtion lgorithms developed for Mrkov rndom fields (or Byesin networks) n e reyled into distriuted synhronous lgorithms for networks of dynmi systems. More speifilly, eyond ftoriztion spets, this forml nlogy requires the existene of notion of projetion tht llows us to express form of onditionl independene sttement on trjetory sets (xiom (3)). The methodology presented in this doument is very flexile nd dpts to different hoies of trjetory semntis, nd trjetory set representtions. The two rrys elow summrize results. In the se of sequentil semntis, one n represent trjetory sets s lnguge (i.e. olletion of sequenes), s rnhing proess (i.e. deision tree), or s trellis proess, whih extends the usul notion of trellis of n utomton. In ll ses, these strutures ftorize into simpler ftors, nd there exists nturl notion of projetion tht llows us to implement messge pssing lgorithms, for exmple to solve the dignosis prolem. When runs re provided with osts, it is sometimes possile to use the MPA to perform n optimiztion tsk, for exmple to determine 135

136 Sequentil semntis lnguge rnhing trellis systems proesses proesses ftoriztion property yes yes yes existene of projetions yes yes yes optimiztion tehniques yes yes? Figure 7.1: Aville results for sequentil semntis. the glol run of the system tht minimizes the ost. We didn t present it for trellis proesses, whene the question mrk, ut this result seems essile. It would tke the form of ooperting Viteri lgorithms, one per omponent, where severl est ost vlues (insted of one) would e rried y eh stte. They would orrespond to the different synhroniztion ptterns with neighoring omponents. However, sequentil semntis seem inpproprite for distriuted systems : they generlly indue lrge strutures, whtever the representtion one hooses. This is due to two phenomen : first of ll, the multi-lok feture, neessry to the ftoriztion property, imposes lrge trellises, nd seondly the nturl onurreny tht rises in modulr systems is not hndled ppropritely, whih uselessly multiplies the numer of runs to onsider. So the interest of sequentil semntis is essentilly theoretil : it provides simple setting where the struture of distriuted omputtions n e explined quite esily. True onurreny semntis Mzurkiewiz tres rnhing trellis (onfigurtions) proesses proesses ftoriztion property yes yes yes projetions on single vrile yes yes yes existene of generl projetions yes yes? optimiztion tehniques yes?? Figure 7.2: Aville results for true onurreny semntis. We rther reommend to use true onurreny semntis, tht onsiderly redue the numer of trjetories to onsider, sine the interlevings of onurrent events re not omputed. The three representtions ove remin possile. Insted of lnguges, one hs sets of onfigurtions, or equivlently sets of Mzurkiewiz tres. Brnhing proesses re now less trivil thn simple deision trees : they eome prefixes of the system unfolding, s it ws defined for Petri nets. And finlly, we hve lso defined the pproprite notion of trellis proess, n extension of the notion of trellis to onurrent systems. Projetions exist in lmost ll ses : they re esy to define when one projets on simple system, with single vrile (nd thus no possiility of internl onurreny). In the generl se, one must keep trk of some onflit nd uslity reltions, whih requires the more omplex notion of ugmented onfigurtion or ugmented rnhing proess. This generl projetion is still missing for trellis proesses. Optimiztion tehniques n lso e 136

137 implemented on sets of tres : they look pretty muh like those on lnguges. But t this point it is not known whether they extend to the other strutures. 7.2 Diretions for future work Tehnil extensions The frmework presented in this doument my look quite omplete nd stle, ut in relity mny tehnil extensions demnded y pplitions remin open questions. The most ovious ones re the empty entries in the previous rrys. In prtiulr, ler understnding of optimiztion issues is still missing. But severl other reserh diretions deserve ttention : Lel strutures. The forml frmework of hpter 2 is sed on systems interting y shred vriles. But the omputtions we hve presented, sed on ompt enodings of trjetory sets, do not fit extly this setting : intertions re etter desried y shred lels, nd projetions on lel sets re not (yet) defined. This motivted the use of pullks, to express omponent intertions y mens of shred vriles/sites, i.e. interfe utomt. These representtions remin it unnturl euse, in pullk, the shred sites do not neessrily pture ll intertions etween two omponents. Two wys to lrify this setting : either y re-expressing results of hpter 2 with new formlism sed on lel intertions (in the spirit of the seprtion riterion ppering in theorem 9), or y developing leled event strutures nd wy to ompose them with trjetory sets (s in setion 3.4.2). On-line omputtions. A nie lgeri trnsltion of reursive lgorithms in the tegory theory setting is missing : wht does it men to reple n oserved sequene y longer one. Understnding this hs importnt prtil onsequenes : When one runs reursive distriuted dignosis lgorithm, messges re exhnged nd new oservtions re introdued in hoti mnner. One would like to determine on-line whih prt of the lol dignoses omputed so fr re stle nd n e displyed. In the sme wy, where nd how n we set temporry stop point in the lgorithm to oserve its urrent result. This ertinly reltes to the sheduling of opertions in the lgorithm, tht ws n unused degree of freedom so fr. It lso reltes to pure spets of distriuted omputing, s studied in [90]. Symoli unfoldings. This ide ws introdued y Thoms Chtin nd Clude Jrd. Consider generi tile t tht would red the vlue of stte vrile V nd dd one to it. In the trditionl onstrution of unfoldings, ll vlues of V would pper s (onfliting) onditions, nd the tion of t would e onneted to eh of them. In symoli unfoldings, vrile V ppers only one, with n unknown vlue tht depends on its pst, nd the tion of t is represented s symoli event dding 1 to V. This results in muh more ompt strutures. The prie to py is more expensive test to deide whether tile n e onneted or not. In order to ompute with these ojets, Thoms nd I hve lredy proved 137

138 tht the ftoriztion property still holds. It remins to uild the dequte notion of projetion. Let us mention tht symoli unfoldings re nturl tool to unfold time systems. Multiresolution/hierrhil lgorithms. The dvntge of systems defined y onstrints is tht redundnt onstrints n only help redution lgorithms. The point is to understnd where! One n suspet tht extr long rnge onstrints ould help to reh etter onvergene of messge pssing lgorithms (MPA). One ould even imgine tht well designed redundnt omponents ould help seprte the intertion grph into smller prts. This is of ourse model trnsformtion issue, nd there re proly resoures out this question in the literture out dt-ses. Notie in prtiulr tht MPA re known in tht field under the nme of query/su-query lgorithms. Roustness issues. For our pplitions relted to teleom networks, we hve developed methodology to utomtilly uild model of the supervised network, y ssemling generi omponents. In prtie, omplete model is not lwys ville. One my only know prts of it, for exmple when single domin is monitored in multi-domin rhiteture, or when speifi intermedite network lyer is monitored. Moreover, modeling errors my lso introdue mismth etween wht model ould produe, nd wht the system tully outputs. Finlly, in the se of truly distriuted lgorithms, messges exhnged y lol supervisors ould e lost, for exmple when the network mngement informtion is trnsmitted in-nd. All these roustness issues hve not een explored so fr Reserh diretions Being le to ompute effiiently on the trjetory set of omplex system potentilly opens the wy to numerous pplitions. We hd distriuted dignosis issues s our trget, ut severl other domins seem essile, or re worth eing explored. Let us mention : Smooth systems. This nme suggests systems tht hve non rigid intertion struture. Networks n e reonfigured for exmple, rete or kill onnetions, whih mounts to reting or removing omponents in their model. In we servies, the tree (or grph) of servie requests, lso lled its horeogrphy, my depend on the prmeters of the initil request or on the nswer of some su-servies. The sme phenomen pper with tive XML douments (see the ASAX projet). The diffiulty here is not so muh to define notion of unfolding for suh systems, where some trnsitions my rete or disonnet omponents, ut rther to propose onvenient model to desrie the struturl evolutions. Some prtil nswers lredy exist, nd there is hope to otin ftoriztion properties. But we re still fr from pplitions. Ad-ho modeling. Relted to modeling issues, nother reserh diretion onerns the extension of this work to other formlisms of onurrent systems. Senrio 138

139 lnguges, or vrious forms of temporl logis ould e explored. Defining the dequte level of desription of system is lso n interesting topi : in generl, one is not so muh interested in omputing ll runs of system tht explin oservtions, ut rther in heking higher level property like did this phenomenon our in the runs tht produed this oservtion? In generl, system model ontins more detils thn neessry to hek given set of suh properties. There is no ovious wy so fr to utomtilly determine wht miniml grnulrity should e preserved in model in order to hek these properties with the sme ury. Modulr model heking. An importnt pplition re of unfolding tehniques onerns model heking prolems, for exmple in iruit design. Questions like dedlok freeness, liveness, essiility of stte or of trnsition re exmined. A entrl point to ddress these prolems is the onstrution of finite omplete prefix of the unfolding, in order to pture ll ehviors of interest for given system nd given property to hek. So fr, this notion doesn t exist for modulr systems. In oopertion with Vitor Khomenko (Newstle upon Tyne, UK) nd Agnes Mdlinski (ex Newstle student, now postdo in Rennes), we re urrently exploring the onstrution of prefixes in produt form, tht would of ourse e more ompt nd llow verifitions y prts. Distriuted optiml ontrol. This is more futuristi (optimisti?) ojetive, sed on the following intuition : The Bellmn eqution solves the optiml ontrol prolem of Mrkov hin, given pst oservtions on tht hin. A very similr reursion ppers in the Viteri lgorithm, used to reover the most likely trjetory of hin given oservtions. Both solutions re tully sed on the dynmi progrmming priniple. Pushing further this remrk, one ould imgine tht solution to the distriuted optiml trjetory reonstrution (one of the missing entries in the ove rrys) would suggest strtegies to design distriuted optiml ontrol lgorithms for modulr systems. Optiml plnning. The prolem onsists in orgnizing huge set of possile tsks in order to hieve given ojetive. Formlly, this is n essiility prolem, for trget stte (lled the gol), in system ontining lrge sets of vriles nd tiles. Conurreny is nturl in this setting. Trditionl pprohes, developed in the AI ommunity, use unfolding-like strutures to enode sets of runs. In prtiulr, in the Grphpln pproh, the proposed struture is very lose to wht we lled the time-unfolding. The possiility of modulr omputtions hs not een explored in this field. And modulr optimiztion would of ourse e rekthrough. A ollortion projet is in preprtion with Sylvie Thieut (Cnerr). Informtion theory for distriuted systems. Here we re eyond futuristi drems... In severl ples of this doument, we hve underlined the reltions etween Byesin networks nd distriuted dynmi systems. We would like to push further this nlogy, provided networks of dynmi systems n e properly rndomized, in order to introdue informtion theory in this setting. For exmple to 139

140 quntify the effet of omponent on the others, evlute the importne of messge in distriuted lgorithms, et. This is proly diffiult tsk, given the poor numer of results ville in network informtion theory, or in network oding. 140

141 Chpter 8 Aknowledgement 141

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143 The Shool of Athens in Distriuted Dignosis This piture ws tken on June 14th, 2007, on the wy out of IRISA s onferene room, where the defense ws tking ple. Yes, we do hve suh mgnifiient rooms t IRISA, ut we generlly dopt different dress ode. Exepted on speil osions. Tht dy ws one of these osions, s we were visited y mny people of wisdom. I m very indeted to Alert Benveniste, whom we see here, pointing his hnd down to pplitions. I feel unle to explin how muh he supported this work. As disoverer of fruitful industril prolems, s supporter of new pprohes to prolem, s sientifi referene, s lrifier of onfuse ides, et. My grtitute won t find dequte words. This is Glynn Winskel, pointing his finger up to the idel world of theoretil pprohes, nd rrying opy of his Leture Notes in Ctegory Theory. I lredy mentioned how muh his ontriutions hve een inspiring for my work. I feel sientifilly indeted to him, nd honored tht he epted to give his opinion on suh non-stndrd uses of forml methods. Aln Willsky, sying it s highly time to go to the resturnt. Aln hs the tlent to onjugte powerful sientifi inspirtion, with deep nd eduted tste for the good spets of life. Visiting him t MIT hs lwys een stimulting on oth topis! His sientifi enthousism is lwys refreshing plesure, nd I m gld he epted to evlute my deviting uses of elief propgtion lgorithms. Nothing is s stimulting s strong ompetitor. St ephne Lfortune hs lid orner stone in dignosis prolems nd dignosility issues, inluding distriuted spets. After severl yers of proximity in onferene sessions, where you wit for his lst novelties to nurture your own refletions, it s plesure nd hllenge to know his evlution of different point of view on the topi. 143

144 There re some people you re lwys plesed to meet in onferene. Alessndro Giu is one of them, oth for his plesnt ompny nd for his wide ulture in omputer siene, Petri nets, disrete event systems, ontrol... A hllenge to hve him s reviewer. Christophe Dousson hs een our light house on industril prolems for severl yers. He is himself ontriutor to dignosis solutions, nd shred with us his urte pereption out emerging tehnologies nd the prolems they rised. Industril prtnerships hve een ruil to my work. This is Mihel Rynl, tehing. His ooks out distriuted omputing helped me understnd how to mke the joint etween modulr proessings nd truly distriuted lgorithms. Among his (numerous) motto Corretness my e theoretil, ut inorretness hs prtil impt. I m honored he epted to venture it outside his lnd to evlute this doument. Your servnt, exhusted. All this work for prhment... There re mny other persons tht diretly or indiretly ontriuted to this work, s ollegues or s friends, with stimulting ides, friendly support or love. Let me just mention Armen Aghsryn, nd Clire. I m sure the others will reognize themselves on the piture. 144