A snap-stabilizing point-to-point communication protocol in message-switched networks

Size: px
Start display at page:

Download "A snap-stabilizing point-to-point communication protocol in message-switched networks"

Transcription

1 A snp-stilizing point-to-point ommunition protool in messge-swithed networks Alin Cournier MIS Lortory, Université de Pirdie Jules Verne 33 rue Sint Leu, Amiens Cedex 1 (Frne) lin.ournier@u-pirdie.fr Swn Duois LIP6 - UMR 7606/INRIA Roquenourt, Projet-tem REGAL Université Pierre et Mrie Curie - Pris Avenue du Président Kennedy, Pris (Frne) swn.duois@lip6.fr Vinent Villin MIS Lortory, Université de Pirdie Jules Verne 33 rue Sint Leu, Amiens Cedex 1 (Frne) vinent.villin@u-pirdie.fr Astrt A snp-stilizing protool, strting from ny onfigurtion, lwys ehves ording to its speifition. In this pper, we present snp-stilizing protool to solve the messge forwrding prolem in messgeswithed network. In this prolem, we must mnge resoures of the system to deliver messges to ny proessor of the network. In this purpose, we use informtions given y routing lgorithm. By the ontext of stiliztion (in prtiulr, the system strts in ny onfigurtion), these informtions n e orrupted. So, the existene of snp-stilizing protool for the messge forwrding prolem implies tht we n sk the system to egin forwrding messges even if routing informtions re initilly orrupted. In this pper, we propose snp-stilizing lgorithm (in the stte model) for the following speifition of the prolem: Any messge n e generted in finite time. Any emitted messge will e delivered to its destintion one nd only one in finite time. This implies tht our protool n deliver ny emitted messge regrdless of the stte of routing tles in the initil onfigurtion. Keywords: Distriuted protools, snp-stiliztion, fult-tolerne, messge forwrding, point-to-point ommunition, dedlok-free routing, messgeswithed networks. 1 Introdution The qulity of distriuted system depends on its fult-tolerne. Mny fult-tolernt shemes hve een proposed. For instne, self-stiliztion [8] llows to design system tolerting ritrry trnsient fults. A self-stilizing system, regrdless of the initil stte of the system, is gurnteed to onverge into the intended ehvior in finite time. An other prdigm lled snpstiliztion hs een introdued in [3, 2]. A snpstilizing protool gurntees tht, strting from ny onfigurtion, it lwys ehves ording to its speifition. In other words, snp-stilizing protool is self-stilizing protool whih stilizes in 0 time unit. In distriuted system, it is ommonly ssumed tht eh proessor n exhnge messges only with its neighors (i.e. proessors with whih it shres ommunition link) ut proessors my need to exhnge messges with ny proessor of the network. To perform this gol, proessors hve to solve two prolems: the determintion of the pth whih messges hve to follow in the network to reh their destintions (it is the routing prolem) nd the mngement of network resoures

2 in order to forwrd messges (it is the messge forwrding prolem). These two prolems reeived gret ttention in literture. The routing prolem is studied for exmple in [1, 4, 13, 14, 15, 29, 30, 20, 23, 25] nd self-stilizing pproh n e found (diretly or not) in [16, 18, 9, 17]. The forwrding prolem hs lso een well studied, see [12, 21, 22, 26, 27, 28] for exmple. As fr we know, the messge forwrding prolem ws never diretly studied with snp-stilizing pproh (note tht the protool proposed y [17] n e used to perform self-stilizing forwrding protool for dynmi networks sine it is gurnteed tht the routing tles remin loop-free even if topologil hnges re llowed). This is the sope of this pper. Informlly, the gol is to give protool whih llows ll proessors of the network to send messges to ny destintion of the network knowing tht routing lgorithm lultes the pth tht messges hve to follow to reh their destintions. Prolems ome of the following ft: messges trveling through messgeswithed network ([24]) must e stored in eh proessor of their pth efore eing forwrded to the next proessor on this pth. This temporry storge of messges is performed with reserved memory spes lled uffers. Oviously, eh proessor of the network reserves only finite numer of uffers for the messge forwrding. So, it is prolem of ounded resoures mngement whih exposes the network to dedloks nd liveloks if no ontrol is performed. In this pper, we fous on messge forwrding protool whih dels the prolem with snp-stilizing pproh. The gol is to llow the system to forwrd messges regrdless of the stte of the routing tles. Oviously, we need tht theses routing tles repir themselves within finite time. So, we ssume the existene of self-stilizing protool to ompute routing tles (see [16, 18, 9]). In the following, we sy tht vlid messge is messge whih hs een generted y proessor. As onsequene, n invlid messge is messge whih is present in the initil onfigurtion. We n now speify the prolem. We propose speifition of the prolem where messge duplitions (i.e. the sme messge rehes its destintion mny time while it hs een generted only one) re foridden: Speifition 1 (SP) Speifition of the messge forwrding prolem. Any messge n e generted in finite time. Any vlid messge will e deliver to its destintion one nd only one in finite time. The reminder of this pper is orgnized s follows: we present first our model (setion 2), then we give, prove, nd nlyze our solution in the stte model (setion 3). Finlly, we onlude y some remrks nd open prolems (setion 4). 2 Preliminries We onsider network s n undireted onneted grph G = (V, E) where V is set of proessors nd E is the set of idiretionl synhronous ommunition links. In the network, ommunition link (p, q) exists if nd only if p nd q re neighors. Every proessor p n distinguish ll its links. To simplify the presenttion, we refer to link (p, q) of proessor p y the lel q. We ssume tht the lels of p re stored in the set N p. We lso use the following nottions: respetively, n is the numer of proessors, the mximl degree, nd D the dimeter of the network. If p nd q re two proessors of the network, we denote y dist(p, q) the length of the shortest pth etween p nd q. In the following, we ssume tht the network is identified, i.e. eh proessor hve n identity whih is unique on the network. Moreover, we ssume tht ll proessors know the set I of ll identities of the network. 2.1 Stte model We onsider lol shred memory model of omputtion (see [24]) in whih ommunitions etween neighors re modeled y diret reding of vriles insted of exhnge of messges. In this model, the progrm of every proessor onsists in set of shred vriles (heneforth, referred to s vriles) nd finite set of tions. A proessor n write to its own vriles only, nd red its own vriles nd those of its neighors. Eh tion is onstituted s follows: < lel >::< gurd > < sttement >. The gurd of n tion in the progrm of p is oolen expression involving vriles of p nd its neighors. The sttement of n tion of p updtes one or more vriles of p. An tion n e exeuted only if its gurd is stisfied. The stte of proessor is defined y the vlue of its vriles. The stte of system is the produt of the sttes of ll proessors. We will refer to the stte of proessor nd the system s (lol) stte nd (glol) onfigurtion, respetively. We note C the set of ll onfigurtions of the system. Let γ C nd A n tion of p (p V ). A is lled enled t p in γ if nd only if the gurd of A is stisfied y p in γ. Proessor p is sid to e enled in γ if nd only if t lest one tion is enled t p in γ. Let distriuted protool P e olletion of tions denoted y, on C. An exeution of protool P is mximl sequene of onfigurtions Γ = (γ 0, γ 1,..., γ i, γ i+1,...)

3 suh tht, i 0, γ i γ i+1 (lled step) if γ i+1 exists, else γ i is terminl onfigurtion. Mximlity mens tht the sequene is either finite (nd no tion of P is enled in the terminl onfigurtion) or infinite. All exeutions onsidered here re ssumed to e mximl. E is the set of ll exeutions of P. As we lredy sid, eh exeution is deomposed into steps. Eh tomi step is omposed of three sequentil phses: (i) every proessor evlutes its gurds, (ii) demon hooses some enled proessors, (iii) eh hosen proessor exeutes one of its enled tions. When the three phses re done, the next step egins. A demon n e defined in terms of firness nd distriution. There exists severl kinds of firness ssumption. Here, we present the strong firness, wek firness, nd unfirness ssumptions. Under strongly fir demon, every proessor tht is enled infinitively often is hosen y the demon infinitively often to exeute n tion. When demon is wekly fir, every ontinuously enled proessor is eventully hosen y the demon. Finlly, the unfir demon is the wekest sheduling ssumption: it n forever prevent proessor to exeute n tion exept if it is the only enled proessor. Conerning the distriution, we ssume tht the demon is distriuted mening tht, t eh step, if one or severl proessors re enled, then the demon hooses t lest one of these proessors to exeute n tion. We onsider tht ny proessor p is neutrlized in the step γ i γ i+1 if p ws enled in γ i nd not enled in γ i+1, ut did not exeute ny tion in γ i γ i+1. To ompute the time omplexity, we use the definition of round (introdued in [10] nd modified y [3]). This definition ptures the exeution rte of the slowest proessor in ny exeution. The first round of Γ E, noted Γ, is the miniml prefix of Γ ontining the exeution of one tion or the neutrliztion of every enled proessor from the initil onfigurtion. Let Γ e the suffix of Γ suh tht Γ = Γ Γ. The seond round of Γ is the first round of Γ, nd so on. 2.2 Messge-swithed network Tody, most of omputer networks use vrint of the messge-swithing method (lso lled store-ndforwrd method). It s why we hve hosen to work with this swithing model. In this setion, we re going to present this method (see [24] for detiled presenttion). Eh proessor hs B uffers for temporrily storing messges. The model ssumes tht eh uffer n store whole messge nd tht eh messge needs only one uffer to e stored. The swithing method is modeled y three types of moves: 1. Genertion: when proessor sends new messge, it retes new messge in one of its empty uffers. We ssume tht the network my llow this move s soon s t lest one uffer of the proessor is empty. 2. Forwrding: messge m is forwrded (opied) from proessor p to n empty uffer in the next proessor q on its route (determined y the routing lgorithm). As result of the move the uffer previously oupied y m eomes empty. We ssume tht the network my llow this move s soon s t lest one uffer uffer of the proessor is empty. 3. Consumption: A messge m oupying uffer in its destintion proessor is removed from this uffer (nd delivered to the proessor). We ssume tht the network my lwys llow this move. 2.3 Stiliztion In this setion, we give forml definitions of self- nd snp-stiliztion using nottions introdued in 2.1. Definition 1 (Self-Stiliztion [8]) Let T e tsk, nd S T speifition of T. A protool P is selfstilizing for S T if nd only if Γ E, there exists finite prefix Γ = (γ 0, γ 1,..., γ l ) of Γ suh tht ll exeutions strting from γ l stisfies S T. Definition 2 (Snp-Stiliztion [2, 3]) Let T e tsk, nd S T speifition of T. A protool P is snpstilizing for S T if nd only if Γ E, Γ stisfies S T. This definition hs the two following onsequenes. We n see tht snp-stilizing protool for S T is self-stilizing protool for S T with stiliztion time of 0 time unit. A ommon method used to prove tht protool is snp-stilizing is to distinguish n tion s strting tion (i.e. n tion whih initites omputtion) nd to prove the following property for every exeution of the protool: if proessor requests it, the omputtion is initited y strting tion in finite time nd every omputtion initited y strting tion stisfies the speifition of the tsk. We will use these two remrks to prove snp-stiliztion of our protool in the following of this pper. 3 Our protool 3.1 Informl desription We hve seen in setion 2.2 tht, y defult, the network lwys llows messge moves etween uffers. But, if we do no ontrol on these moves, the network n

4 d e d e Figure 1. Exmple of destintionsed uffer grph (on the right) on the network on the left. reh uneptle situtions suh s dedloks, liveloks or messge losses. If suh situtions pper, speifitions of messge forwrding re not respeted. Now, we quikly present solutions rought y the literture in the se where routing tles re orret in the initil onfigurtion. In order to void dedloks, we must define n lgorithm whih permits or forids vrious moves in the network (funtions of the urrent ouption of uffers) in order to prevent the network to reh dedlok. Suh lgorithms re lled dedlok-free ontrollers (see [24] for muh detiled desription). [21] introdued generi method to design dedlok-free ontrollers. It onsists to restrit moves of messges long edges of n oriented grph BG (lled uffer grph) defined on the network uffers. Then, it is esy to see tht yles on BG n led to dedloks. So, uthors show tht, if BG is yli, they n define dedlok-free ontroller on this uffer grph. For exmple, we n present destintion-sed uffer grph. In this sheme, we ssume tht the routing lgorithm forwrds ll pkets of Destintion d vi direted tree T d rooted in d. Eh proessor p of the network hs uffer p (d) for eh possile Destintion d (lled the trget of p (d)). The uffer grph hs n onneted omponents, eh of them ontining ll the uffers whih shred their trget. The onneted omponent ssoited to the trget d is isomorphi to T d (the reder n find n exmple of suh grph in Figure 1). It is esy to see tht this oriented grph is yli. Liveloks n e voided y firness ssumptions on the ontroller for the genertion nd the forwrding of messges. Messge losses re voided y the using of identifier on messges. For exmple, one n use the ontention of the identity of the soure nd twovlue flg in order to distinguish two onseutive identil messges generted y the sme proessor for Destintion d (sine ll messges follow the sme pth in T d ). The min ide tht leds our reserh is to dpt this solution in order to tolerte the orruption of routing tles in the initil onfigurtion. To perform this gol, we ssume the existene of self-stilizing silent (i.e. no tions re enled fter onvergene) lgorithm A to ompute routing tles whih runs simultneously to our messge forwrding protool. Moreover, we ssume tht A hs priority over our protool (i.e. proessor whih hs enled tions for oth lgorithms lwys hooses the tion of A). This gurntees us tht routing tles will e orret nd onstnt within finite time. To simplify the presenttion, we ssume tht A indues only miniml pths in numer of edges. We ssume tht our protool n hve ess to the routing tle vi funtion, lled nexthop p (d). This funtion returns the identity of the neighor of p to whih p must forwrd messges of Destintion d. We use ontroller sed on uffer grph similr to tht we presented efore (one routing tles re omputed). The uffer grph is omposed of n onneted omponents, eh ssoited to destintion d nd sed on the oriented tree T d. So, we re going to present only one onneted omponent, ssoited to destintion noted d (others re similr). We use two uffers per proessor for Destintion d. The first one, noted ufr p (d) (for proessor p), is reserved to the reeption of messges wheres the seond one, noted ufe p (d), is used to emit messges (see Figure 2). This sheme llows us to ontrol the dvne of messges. Indeed, we llow messge to e forwrded from ufr p (d) to ufe p (d) if nd only if the messge is only present in ufr p (d) nd we erse it simultneously. In this wy, we n ontrol the effet of routing tles moves on messges (duplition or merge whih n involve messge losses). To void liveloks, we use fir sheme of seletion of proessors llowed to forwrd or to emit messge for eh reeption uffer. We n mnge this firness y queue of requesting proessors. Finlly, we use speifi flg to prevent messge losses. It is omposed of the identity of the lst proessor ross over y the messge nd olor whih is dynmilly given to the messge when it rehes n emission uffer. In order to distinguish suh inoming messge of these ontined in reeption uffers of neighors of the onsidered proessor, we give to this inoming messge olor whih is not rry y suh messge. It is why messge is onsidered s triplet (m, p, ) in our lgorithm where m is the useful informtion of the messge, p is the identity of the lst proessor rossed over y the messge, nd is olor ( nturl integer etween 0 nd ). We must mnge ommunition etween our lgorithm nd proessors in order to know when proessor hve messge to send. We hve hosen to rete oolen shred vrile request p (for ny proessor p). Proessor p n set it t true when it is t flse nd when p

5 3.2 Algorithm d e d e Figure 2. Exmple of our uffer grph (on the right) for Destintion on the network (on the left). hs messge to send. Otherwise, p must wit tht our lgorithm sets the shred vrile to flse (tht is done when messge is generted). The reder n find omplete exmple of the exeution of our lgorithm in Figure 3. Digrm (N) shows the network nd digrm (0) shows the initil onfigurtion for the onneted omponent ssoited to of the uffer grph. We oserve tht = 3, so we need 4 different vlues for the vrile olor, we hve hosen to represent them y nturl integer in {0, 1, 2, 3}. Remrk tht routing tles re inorret (in prtiulr there exists yle involving uffers of nd ) nd tht there exists n invlid messge m in the reeption uffer of (its olor is 0). Then, Proessor emits messge m (its olor is 0) in the reeption uffer of to otin onfigurtion (1). When the messge m is forwrded to the emission uffer of, we ssoite it the olor 1 (sine 0 is foridden, see onfigurtion (2)). During the next step, messge m is forwrded to the reeption uffer of (remrk tht it keeps its olor) nd emits (in its reeption uffer) new messge m whih hs the sme useful informtion s the invlid messge present on. So, we otin onfigurtion (3). Messge m n now e ersed from the emission uffer of nd m n e forwrded into this uffer (we ssoite it the olor 2). These two steps led to onfigurtion (4). Assume tht routing tles re repired during the next step. Simultneously, proessor is llowed to forwrd m into its emission uffer. We otin onfigurtion (5). Remrk tht the use of olor forids the merge etween the two messges whih hve m for useful informtion. Then, the system is le to deliver these three messges y the repetition of moves tht we hve desried: forwrding from reeption uffer to emission uffer of the sme proessor. forwrding from emission uffer to reeption uffer of two proessors. ersing from emission uffer or delivering. The sequene of onfigurtion (6) to (12) shows n exmple of the end of our exeution. We now present formlly our protool in Algorithm 1. We ll it SSMFP for Snp-Stilizing Messge Forwrding Protool. In order to simplify the presenttion, we write the lgorithm for Destintion d only. Oviously, eh destintion of the network needs similr lgorithm. Moreover, we ssume tht ll these lgorithms run simultneously (s they re mutully independent, this ssumption hs no effet on the provided proof). 3.3 Proof of the snp-stiliztion In order to simplify the proof, we introdue seond speifition of the prolem. This speifition llows messge duplitions. Speifition 2 (SP ) Speifition of messge forwrding prolem llowing duplition. Any messge n e generted in finite time. Any vlid messge will e deliver to its destintion in finite time. In this setion, we give ides to prove tht SSMFP is snp-stilizing messge forwrding protool for speifition SP. For tht, we re going to prove suessively tht: 1. SSMFP is snp-stilizing messge forwrding protool for speifition SP if routing tles re orret in the initil onfigurtion (Proposition 1). 2. SSMFP is self-stilizing messge forwrding protool for speifition SP even if routing tles re orrupted in the initil onfigurtion (Proposition 2). 3. SSMFP is snp-stilizing messge forwrding protool for speifition SP even if routing tles re orrupted in the initil onfigurtion (Proposition 3). In this proof, we onsider tht the notion of messge is different from the notion of useful informtion. This implies tht two messges with the sme useful informtion generted y the sme proessor re onsidered s two different messges. We must prove tht the lgorithm does not lose one of them thnks to the use of the flg. Let γ e onfigurtion of the network. We sy tht messge m is existing in γ if t lest one uffer ontins m in γ. We sy tht m is existing on p in γ if t lest one uffer of p ontins m in γ.

6 (N) (0) (12) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (m,,0) (m,,0) (m,,0) (m,,0) (m,,1) (m,,1) (m,,0) (m,,1) (m,,0) (m,,1) (m,,0) (m,,2) (m,,1) (m,,2) (m,,0) (m,,1) (m,,1) (m,,2) (m,,2) (m,,0) (m,,1) (m,,2) (m,,0) (m,,2) (m,,0) (m,,0) (m,,0) (m,,1) Figure 3. An exmple of exeution of our lgorithm.

7 Algorithm 1 (SSMFP): Messge forwrding protool for Proessor p with Destintion d. Dt: - n: nturl integer equls to the numer of proessors of the network. - I = {0,..., n 1}: set of proessor identities of the network. - N p : set of neighors of p. - : nturl integer equls to the mximl degree of the network. Messge: - (m, q, ) with m useful informtion of the messge, q N p {p} identity of the lst proessor rossed over y the messge, nd {0,..., } olor. The messge destintion is the uffer index. Vriles: - ufr p (d), ufe p (d): uffers whih n ontin messge. Input/Output: Mros: - request p : oolen. The higher lyer n set it to true when its vlue is flse nd when there is witing messge. We onsider tht this witing is loking. - nextmessge p : gives the messge witing in the higher lyer. - nextdestintion p : gives the destintion of nextmessge p if it exists, null otherwise. Proedures: Rules: - nexthop p (d): neighor of p given y the routing lgorithm for Destintion d. - hoie p (d): firly hooses one of the proessors whih n forwrd or generte messge in ufr p (d), i.e. hoie p (d) stisfies predite (hoie p (d) N p ufe hoiep(d)(d) = (m, q, ) nexthop hoiep(d)(d) = p) (hoie p (d) = p request p ). We n mnge this firness with queue of length + 1 of proessors whih stisfies the predite. - deliver p (m): delivers the messge m to the higher lyer of p. - olor p (d): gives nturl integer etween 0 nd suh s q N p, ufr q (d) does not ontin messge with s olor. /* Rule for the genertion of messge */ (R 1 ) :: request p (nextdestintion p = d) (ufr p (d) = empty) (hoie p (d) = p) ufr p (d) := (nextmessge p, p, 0); request p := flse /* Rule for the internl forwrding of messge */ (R 2 ) :: (ufe p (d) = empty) (ufr p (d) = (m, q, )) ((q = p) (ufe q (d) (m, q, ))) ufe p (d) := (m, p, olor p (d)); ufr p (d) := empty /* Rule for the forwrding of messge */ (R 3 ) :: (ufr p (d) = empty) (hoie p (d) = s) (s p) (ufe s (d) = (m, q, )) ufr p (d) := (m, s, ) 1 /* Rule for the ersing of messge fter its forwrding */ (R 4 ) :: (ufe p (d) = (m, q, )) (p d) (ufr nexthopp(d)(d) = (m, p, )) ( r N p \{nexthop p (d)}, ufr r (d) (m, p, )) ufe p (d) := empty /* Rule for the ersing of messge fter its duplition */ (R 5 ) :: (ufr p (d) = (m, q, )) (ufe q (d) = (m, q, )) (nexthop q (d) p) ufr p (d) := empty /* Rule for the onsumption of messge */ (R 6 ) :: (ufe p (p) = (m, q, )) deliver p (m); ufe p (p) := empty 1 The ft tht q my e different of s implies tht the messge ws in the system t the initil onfigurtion. We ould lolly delete this messge ut tht will not improve the performne of SSMFP.

8 Definition 3 (Cterpillr of messge m) Let m e messge of Destintion d existing on proessor p in onfigurtion γ. We define terpillr ssoited to m s the longest sequene of uffers tht stisfies one of the three definitions elow: 1. Cterpillr of type 1: (ufr p (d) = (m, q, )) ((ufe q (d) (m, q, )) (q = p)). 2. Cterpillr of type 2: (ufe p (d) = (m, q, )) (ufr nexthopp(d)(d) (m, p, )). 3. Cterpillr of type 3: (ufe p (d) = (m, q, )) q N p, (ufr q (d) = (m, p, )). The reder n find in Figure 4 n exmple for eh type of terpillr. Remrk tht n emission uffer n elong to severl terpillrs of type 3. Assume tht routing tles re orret in the initil onfigurtion. When we oserve the exeution of Protool SSMFP under wekly fer demon, we n see tht terpillr of type 1 ssoited to messge m (of Destintion d) on proessor p eomes terpillr C of type 2 ssoited to m on p within finite time (y rule (R 2 )). If p is the destintion of m, the messge is delivered (y rule (R 6 )) else, C eomes terpillr of type 3 on p (y rule (R 3 )) thnks to the firness of hoie nexthopp(d)(d). If severl neighors re implied in terpillrs of type 3 whih shre ufr p (d) (due to invlid messges in the initil onfigurtion), (R 5 ) is enled for eh neighor q of p suh tht nexthop p (d) q. It s why rule (R 4 ) is enled in finite time for nexthop p (d), its tivtion trnsform C into terpillr of type 1 ssoited to m on nexthop p (d). Then, we n give the following lemm: Lemm 1 Let γ e onfigurtion in whih routing tles re orret. Let m e messge existing on p in γ. Under wekly fir demon, the exeution of SSMFP will produt within finite time one of the following effets for ny terpillr of type 1 ssoited to m: m is delivered to its destintion. the terpillr disppered on p nd there exists terpillr of type 1 ssoited to the sme messge on nexthop p (d). Assume tht routing tles re orret in the initil onfigurtion. If uffer ufr p (d) ontins terpillr of type 3, we hve seen (in the proof of lemm 1) tht this terpillr eome terpillr of type 1 on p or dispper. Moreover, Lemm 1 implies tht uffer ufr p (d) whih ontins terpillr of type 1 is ersed within finite time. Then, the firness of hoie p (d) llows us to give the following result: Lemm 2 Under wekly fir demon when routing tles re orret, every proessor n generte first messge (i.e. it n exeute (R 1 )). Assume tht routing tles re orret in the initil onfigurtion nd tht proessor p hs generted messge m of Destintion d (with rule (R 1 )). This implies the retion of terpillr of type 1 ssoited to m in ufr p (d) when m hs een generted. The following result is otined y dist(p, d) + 1 pplitions of Lemm 1: Lemm 3 One messge is epted y SSMFP, it will e orretly forwrded to its destintion under wekly fir demon if routing tles re orret (when the messge ws epted). Assume tht routing tles re orret in the initil onfigurtion. To prove tht our lgorithm is snpstilizing messge forwrding protool for speifition SP, we must prove tht (R 1 ) (the strting tion) is exeuted within finite time if omputtion is requested. Lemm 2 proves this. After strting tion, the protool is exeuted in ordne to SP. If we onsider tht (R 1 ) hve een exeuted t lest one time, we n prove tht: the first property of SP is lwys verified (y Lemm 2 nd the ft tht the witing for the genertion of new messges is loking) nd the seond property of SP is lwys verified (y Lemm 3). By the remrk whih follows the definition 2, this implies the following result: Proposition 1 SSMFP is snp-stilizing messge forwrding protool for SP if routing tles re orret in the initil onfigurtion. We rell tht self-stilizing silent lgorithm A for omputing routing tles is running simultneously to SSMFP. Moreover, we ssume tht A hs priority over SSMFP (i.e. proessor whih hve enled tions for oth lgorithms lwys hooses the tion of A). This gurntees us tht routing tles will e orret nd onstnt within finite time regrdless of their initil sttes. As we re gurnteed tht SSMFP is snp-stilizing messge forwrding protool for speifition SP from suh onfigurtion y Proposition 1, we n onlude on the following property: Proposition 2 SSMFP is self-stilizing messge forwrding protool for SP (even if routing tles re orrupted in the initil onfigurtion) when A runs simultneously. By the onstrution of the lgorithm, it is ovious tht messge nnot e ersed from two distint uffers simultneously. Then, the onstrution of olor p (d) nd

9 m,r, m,p, m,q, m,r, q m,q, p m,r, m,p, p m,p, p p nexthop (d) p q q Figure 4. Exmples of terpillr ssoited to m on p (from left to right: two of type 1, one of type 2 nd one of type 3). of rule (R 4 ) gurntees us tht, when (R 4 ) is pplied y p, the messge in ufr nexthopp(d)(d) is opy of tht in ufe p (d). So, the messge hve een opied t lest one efore it is ersed, tht llows us to give the following lemm: Lemm 4 Under wekly fir demon, SSMFP does not delete vlid messge without deliver it to its destintion even if A runs simultneously. It is ovious tht the emission of messge y rule (R 1 ) retes only one terpillr of type 1. Then, the onstrution of rules (R 6 ) nd (R 4 ) implies the following property: if terpillr of type 1 ssoited to messge m is present on proessor p nd this messge is ersed from ll uffers of p, then only one neighor of p ontins terpillr of type 1 ssoited to m or m hve een delivered to its destintion (perhps m hve een opied severl times ut (R 5 ) ensure us tht there exists unique opy of m when (R 4 ) is enled). This llows us to give the following lemm: Lemm 5 Under wekly fir demon, SSMFP never duplites vlid messge even if A runs simultneously. Proposition 2 nd Lemm 4 llows us to onlude tht SSMFP is snp-stilizing messge forwrding protool with speifition SP even if routing tles re orrupted in the initil onfigurtion on ondition tht A run simultneously. Then, using this remrk nd Lemm 5, we n lim: Proposition 3 SSMFP is snp-stilizing messge forwrding protool for SP (even if routing tles re orrupted in the initil onfigurtion) when A run simultneously. 3.4 Time omplexities Let R A e the stiliztion time of A in terms of rounds. Proposition 4 In the worst se, 2n invlid messges will e delivered to Proessor d. Sketh of proof. In the initil onfigurtion, the system hs t most 2n distint invlid messges of Destintion d (sine the onneted omponent of the uffer grph ssoited to d hs 2n uffers). In the worst se, ll these messges will e delivered to their destintion, tht llows us to reh the nnouned ound. Proposition 5 In the worst se, messge m (of Destintion d) needs O(mx(R A, D )) rounds to e delivered to d one it hs een generted y its soure. Sketh of proof. In first time, we n show y indution the following result: if γ is onfigurtion in whih routing tles re orret nd C is terpillr of type 1 ssoited to messge m (of Destintion d) on proessor p suh s dist(p, d) = δ, then m is delivered to d or there exists terpillr of type 1 ssoited to m on nexthop p (d) in t most O( δ ) rounds. This result is due to the firness of hoie p (d) whih n llow t most messges to pss m. Then, onsider tht s is the soure of messge m of Destintion d. We hve dist(s, d) D y definition. We n onlude tht m 0 is delivered in t most O( δ ) O( D ) rounds if δ=d routing tles re orret when m is emitted. Finlly, we n dedue the result when m is emitted in onfigurtion in whih routing tles re not orret sine the messge is delivered in t most O( D ) rounds fter routing tles omputtion (whih tkes t most O(R A ) rounds if m is not delivered during the routing tles omputtion sine we hve ssumed the priority of A over SSMFP). Proposition 6 The dely (witing time efore the first emission) nd the witing time (etween two onseutive emissions) of SSMFP is O(mx(R A, D )) rounds in the worst se.

10 Sketh of proof. Let p e proessor whih hs messge of Destintion d to emit. By the firness of hoie p (d), we n sy tht m will e generted fter t most ( 1) releses of ufr p (d). The result of Proposition 5 llows us to sy tht ufr p (d) is relesed in O(mx(R A, D )) rounds t worst. Indeed, we n dedue the result. The omplexity otined in Proposition 5 is due to the ft tht the system delivers huge quntity of messges during the forwrding of the onsidered messge. It s why we interest now in the mortized omplexity (in rounds) of our lgorithm. For n exeution Γ, this mesure is equl to the numer of rounds of Γ divided y the numer of delivered messges during Γ (see [5] for forml definition). Proposition 7 The mortized omplexity (to forwrd messge) of SSMFP is O(mx (R A, D)) rounds. Sketh of proof. In first time, we must prove the following property: if γ is onfigurtion in whih t lest one messge of Destintion d is present nd in whih routing tles re orret, then SSMFP delivers t lest one messge to d in the 3D rounds following γ. Assume now n initil onfigurtion in whih routing tles re orret. Let Γ e one exeution leds to the worst mortized omplexity. Let R Γ e the numer of rounds of Γ. By the lst remrk, we n sy tht SSMFP delivers t lest RΓ 3D messges during Γ. So, we hve n mortized omplexity of RΓ R Γ Θ(D). 3D Then, the nnouned result is ovious. 4 Conlusion In this pper, we provide the first lgorithm (t our knowledge) to solve the messge forwrding prolem in snp-stilizing wy (when self-stilizing lgorithm for omputing routing tles runs simultneously) for speifition whih forids messge losses nd duplition. This property implies the following ft: our protool n forwrd ny emitted messge to its destintion regrdless of the stte of routing tles in the initil onfigurtion. Suh n lgorithm llows the proessors of the network to send messges to other without witing for the routing tle omputtion. We use tool lled uffer grph whih hs een introdued in [21]. This pper proposed destintion-sed uffer grph tht we hve dpted in order to ontrol the effet of routing tle moves on messges. Our nlysis shows tht we ensure snp-stiliztion without signifint over ost in spe or in time with respet to the fult-free lgorithm. [21] lso proposed other uffer grphs. So, it is nturl to wonder if they ould e dpted to tolerte trnsient fults. In prtiulr, one of them (sed on the yli overing of the network, see lso [24]) is very interesting sine it needs less uffers per proessor in generl (3 for ring, 2 for tree...). But, uthors of [19] show tht it is NP-hrd to ompute the size of the yli overing of ny grph. So, this uffer grph nnot e esily pplied to ny network. An open prolem is the following: wht is the miniml numer of uffers per proessor to llow snp-stiliztion on the messge forwrding prolem? Another wy to improve our protool is to speed up the messge forwrding in the worst se (without inresing mortized omplexity). In this gol, we elieve tht we n keep our protool nd modify the fir sheme of seletion of messges hoie p (d). In ft, the omplexity of our lgorithm depends on the numer of messges whih n pss speifi messge t eh hop. Our protool hs the following drwk: when messge m is delivered to proessor p, p nnot determine if m is vlid or not. This n ring some prolems for pplitions whih use these messges. So, n interesting wy of future reserhes ould e to design protool whih solves this prolem. In [6] the uthors propose n effiient solution for the PIF prolem tht dels with similr prolem, unfortuntely their pproh does not seem suitle for our prolem. Finlly, it will e interesting to rry our protool in the messge pssing model ( more relisti model of distriuted system) in order to enle snp-stilizing messge forwrding in rel network. To our knowledge, in this model, only two snp-stilizing protools exist in the literture ([7, 11]). The prolem to rry utomtilly protool from the stte model to the messge pssing model is still open. Referenes [1] E. M. Bkker, J. vn Leeuwen, nd R. B. Tn. Prefix routing shemes in dynmi networks. Computer Networks nd ISDN Systems, 26(4): , [2] A. Bui, A. K. Dtt, F. Petit, nd V. Villin. Stteoptiml snp-stilizing pif in tree networks. In WSS, pges 78 85, [3] A. Bui, A. K. Dtt, F. Petit, nd V. Villin. Snpstiliztion nd pif in tree networks. Distriuted Computing, 20(1):3 19, [4] K. M. Chndy nd J. Misr. Distriuted omputtion on grphs: Shortest pth lgorithms. Commun. ACM, 25(11): , [5] T. Cormen, C. Leierson, R. Rivest, nd C. Stein. Introdution à l lgorithmique. Eyrolles, seonde edition, [6] A. Cournier, S. Devismes, nd V. Villin. Snpstilizing pif nd useless omputtions. In ICPADS (1), pges 39 48, 2006.

11 [7] S. Delët, S. Devismes, M. Nesterenko, nd S. Tixeuil. Snp-stiliztion in messge-pssing systems. CoRR, s/ , [8] E. W. Dijkstr. Self-stilizing systems in spite of distriuted ontrol. Commun. ACM, 17(11): , [9] S. Dolev. Self-stilizing routing nd relted protool. J. Prllel Distri. Comput., 42(2): , [10] S. Dolev, A. Isreli, nd S. Morn. Uniform dynmi self-stilizing leder eletion. IEEE Trns. Prllel Distri. Syst., 8(4): , [11] S. Dolev nd N. Tzhr. Empire of olonies: Selfstilizing nd self-orgnizing distriuted lgorithms. In OPODIS, pges , [12] J. Duto. A neessry nd suffiient ondition for dedlok-free routing in ut-through nd store-ndforwrd networks. IEEE Trns. Prllel Distri. Syst., 7(8): , [13] M. Flmmini nd G. Gmosi. On devising oolen routing shemes. Theor. Comput. Si., 186(1-2): , [14] P. Frigniud nd C. Gvoille. Intervl routing shemes. Algorithmi, 21(2): , [15] C. Gvoille. A survey on intervl routing. Theor. Comput. Si., 245(2): , [16] S.-T. Hung nd N.-S. Chen. A self-stilizing lgorithm for onstruting redth-first trees. Inf. Proess. Lett., 41(2): , [17] C. Johnen nd S. Tixeuil. Route preserving stiliztion. In Self-Stilizing Systems, pges , [18] A. Kosowski nd L. Kuszner. A self-stilizing lgorithm for finding spnning tree in polynomil numer of moves. In PPAM, pges 75 82, [19] R. Krlovi nd P. Ruzik. Rnks of grphs: The size of yli orienttion over for dedlok-free pket routing. Theor. Comput. Si., 374(1-3): , [20] P. Merlin nd A. Segll. A filsfe distriuted routing protool. IEEE Trns. Communitions, 27(9): , [21] P. M. Merlin nd P. J. Shweitzer. Dedlok voidne in store-nd-forwrd networks. In Jeruslem Conferene on Informtion Tehnology, pges , [22] L. Shwieert nd D. N. Jysimh. A universl proof tehnique for dedlok-free routing in interonnetion networks. In SPAA, pges , [23] W. D. Tjinpis. A orretness proof of topology informtion mintenne protool for distriuted omputer network. Commun. ACM, 20(7): , [24] G. Tel. Introdution to Distriuted Algorithms. Cmridge University Press, Cmridge, UK, 2nd edition, [25] S. Toueg. An ll-pirs shortest-pth distriuted lgorithm. RC , IBM T. J. Wtson Reserh Center, Yorktown Heights, NY, [26] S. Toueg. Dedlok- nd livelok-free pket swithing networks. In STOC, pges 94 99, [27] S. Toueg nd K. Steiglitz. Some omplexity results in the design of dedlok-free pket swithing networks. SIAM J. Comput., 10(4): , [28] S. Toueg nd J. D. Ullmn. Dedlok-free pket swithing networks. SIAM J. Comput., 10(3): , [29] J. vn Leeuwen nd R. B. Tn. Intervl routing. Comput. J., 30(4): , [30] J. vn Leeuwen nd R. B. Tn. Compt routing methods: A survey. In SIROCCO, pges , 1994.

Test Generation from Timed Input Output Automata

Test Generation from Timed Input Output Automata Chpter 8 Test Genertion from Timed Input Output Automt The purpose of this hpter is to introdue tehniques for the genertion of test dt from models of softwre sed on vrints of timed utomt. The tests generted

More information

System Validation (IN4387) November 2, 2012, 14:00-17:00

System Validation (IN4387) November 2, 2012, 14:00-17:00 System Vlidtion (IN4387) Novemer 2, 2012, 14:00-17:00 Importnt Notes. The exmintion omprises 5 question in 4 pges. Give omplete explntion nd do not onfine yourself to giving the finl nswer. Good luk! Exerise

More information

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

Behavior Composition in the Presence of Failure

Behavior Composition in the Presence of Failure Behvior Composition in the Presene of Filure Sestin Srdin RMIT University, Melourne, Austrli Fio Ptrizi & Giuseppe De Giomo Spienz Univ. Rom, Itly KR 08, Sept. 2008, Sydney Austrli Introdution There re

More information

TIME AND STATE IN DISTRIBUTED SYSTEMS

TIME AND STATE IN DISTRIBUTED SYSTEMS Distriuted Systems Fö 5-1 Distriuted Systems Fö 5-2 TIME ND STTE IN DISTRIUTED SYSTEMS 1. Time in Distriuted Systems Time in Distriuted Systems euse eh mhine in distriuted system hs its own lok there is

More information

Algorithm Design and Analysis

Algorithm Design and Analysis Algorithm Design nd Anlysis LECTURE 5 Supplement Greedy Algorithms Cont d Minimizing lteness Ching (NOT overed in leture) Adm Smith 9/8/10 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov,

More information

Arrow s Impossibility Theorem

Arrow s Impossibility Theorem Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

Algorithm Design and Analysis

Algorithm Design and Analysis Algorithm Design nd Anlysis LECTURE 8 Mx. lteness ont d Optiml Ching Adm Smith 9/12/2008 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov, K. Wyne Sheduling to Minimizing Lteness Minimizing

More information

NON-DETERMINISTIC FSA

NON-DETERMINISTIC FSA Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

More information

Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution

Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs

Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if

More information

Introduction to Olympiad Inequalities

Introduction to Olympiad Inequalities Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................

More information

Arrow s Impossibility Theorem

Arrow s Impossibility Theorem Rep Fun Gme Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties

More information

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6 CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized

More information

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version A Lower Bound for the Length of Prtil Trnsversl in Ltin Squre, Revised Version Pooy Htmi nd Peter W. Shor Deprtment of Mthemtil Sienes, Shrif University of Tehnology, P.O.Bo 11365-9415, Tehrn, Irn Deprtment

More information

Finite State Automata and Determinisation

Finite State Automata and Determinisation Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Non-deterministi Finite Stte Automt (nf) 4 Regulr Expressions

More information

The DOACROSS statement

The DOACROSS statement The DOACROSS sttement Is prllel loop similr to DOALL, ut it llows prouer-onsumer type of synhroniztion. Synhroniztion is llowe from lower to higher itertions sine it is ssume tht lower itertions re selete

More information

Automatic Synthesis of New Behaviors from a Library of Available Behaviors

Automatic Synthesis of New Behaviors from a Library of Available Behaviors Automti Synthesis of New Behviors from Lirry of Aville Behviors Giuseppe De Giomo Università di Rom L Spienz, Rom, Itly degiomo@dis.unirom1.it Sestin Srdin RMIT University, Melourne, Austrli ssrdin@s.rmit.edu.u

More information

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their

More information

Part 4. Integration (with Proofs)

Part 4. Integration (with Proofs) Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1

More information

CS 573 Automata Theory and Formal Languages

CS 573 Automata Theory and Formal Languages Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple

More information

Lecture Notes No. 10

Lecture Notes No. 10 2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite

More information

Chapter 4 State-Space Planning

Chapter 4 State-Space Planning Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different

More information

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of:

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of: 22: Union Fin CS 473u - Algorithms - Spring 2005 April 14, 2005 1 Union-Fin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x) - rete set tht ontins the single element x. 2. Fin(x)

More information

CS 491G Combinatorial Optimization Lecture Notes

CS 491G Combinatorial Optimization Lecture Notes CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,

More information

where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b

where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1}

More information

Lecture 6: Coding theory

Lecture 6: Coding theory Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those

More information

Graph States EPIT Mehdi Mhalla (Calgary, Canada) Simon Perdrix (Grenoble, France)

Graph States EPIT Mehdi Mhalla (Calgary, Canada) Simon Perdrix (Grenoble, France) Grph Sttes EPIT 2005 Mehdi Mhll (Clgry, Cnd) Simon Perdrix (Grenole, Frne) simon.perdrix@img.fr Grph Stte: Introdution A grph-sed representtion of the entnglement of some (lrge) quntum stte. Verties: quits

More information

Hyers-Ulam stability of Pielou logistic difference equation

Hyers-Ulam stability of Pielou logistic difference equation vilble online t wwwisr-publitionsom/jns J Nonliner Si ppl, 0 (207, 35 322 Reserh rtile Journl Homepge: wwwtjnsom - wwwisr-publitionsom/jns Hyers-Ulm stbility of Pielou logisti differene eqution Soon-Mo

More information

Chapter 3. Vector Spaces. 3.1 Images and Image Arithmetic

Chapter 3. Vector Spaces. 3.1 Images and Image Arithmetic Chpter 3 Vetor Spes In Chpter 2, we sw tht the set of imges possessed numer of onvenient properties. It turns out tht ny set tht possesses similr onvenient properties n e nlyzed in similr wy. In liner

More information

QUADRATIC EQUATION. Contents

QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

Spacetime and the Quantum World Questions Fall 2010

Spacetime and the Quantum World Questions Fall 2010 Spetime nd the Quntum World Questions Fll 2010 1. Cliker Questions from Clss: (1) In toss of two die, wht is the proility tht the sum of the outomes is 6? () P (x 1 + x 2 = 6) = 1 36 - out 3% () P (x 1

More information

8 THREE PHASE A.C. CIRCUITS

8 THREE PHASE A.C. CIRCUITS 8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the so-lled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),

More information

LIP. Laboratoire de l Informatique du Parallélisme. Ecole Normale Supérieure de Lyon

LIP. Laboratoire de l Informatique du Parallélisme. Ecole Normale Supérieure de Lyon LIP Lortoire de l Informtique du Prllélisme Eole Normle Supérieure de Lyon Institut IMAG Unité de reherhe ssoiée u CNRS n 1398 One-wy Cellulr Automt on Cyley Grphs Zsuzsnn Rok Mrs 1993 Reserh Report N

More information

Prefix-Free Regular-Expression Matching

Prefix-Free Regular-Expression Matching Prefix-Free Regulr-Expression Mthing Yo-Su Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST Prefix-Free Regulr-Expression Mthing p.1/15 Pttern Mthing Given pttern P nd text T, find ll sustrings

More information

Section 4.4. Green s Theorem

Section 4.4. Green s Theorem The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls

More information

CS241 Week 6 Tutorial Solutions

CS241 Week 6 Tutorial Solutions 241 Week 6 Tutoril olutions Lnguges: nning & ontext-free Grmmrs Winter 2018 1 nning Exerises 1. 0x0x0xd HEXINT 0x0 I x0xd 2. 0xend--- HEXINT 0xe I nd ER -- MINU - 3. 1234-120x INT 1234 INT -120 I x 4.

More information

Nondeterministic Automata vs Deterministic Automata

Nondeterministic Automata vs Deterministic Automata Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n

More information

Alpha Algorithm: Limitations

Alpha Algorithm: Limitations Proess Mining: Dt Siene in Ation Alph Algorithm: Limittions prof.dr.ir. Wil vn der Alst www.proessmining.org Let L e n event log over T. α(l) is defined s follows. 1. T L = { t T σ L t σ}, 2. T I = { t

More information

Engr354: Digital Logic Circuits

Engr354: Digital Logic Circuits Engr354: Digitl Logi Ciruits Chpter 4: Logi Optimiztion Curtis Nelson Logi Optimiztion In hpter 4 you will lern out: Synthesis of logi funtions; Anlysis of logi iruits; Tehniques for deriving minimum-ost

More information

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri

More information

Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!

Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite! Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4 Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one

More information

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into

More information

CS 2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2014

CS 2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2014 S 224 DIGITAL LOGI & STATE MAHINE DESIGN SPRING 214 DUE : Mrh 27, 214 HOMEWORK III READ : Relte portions of hpters VII n VIII ASSIGNMENT : There re three questions. Solve ll homework n exm prolems s shown

More information

Behavior Composition in the Presence of Failure

Behavior Composition in the Presence of Failure Behior Composition in the Presene of Filure Sestin Srdin RMIT Uniersity, Melourne, Austrli Fio Ptrizi & Giuseppe De Giomo Spienz Uni. Rom, Itly KR 08, Sept. 2008, Sydney Austrli Introdution There re t

More information

Unit 4. Combinational Circuits

Unit 4. Combinational Circuits Unit 4. Comintionl Ciruits Digitl Eletroni Ciruits (Ciruitos Eletrónios Digitles) E.T.S.I. Informáti Universidd de Sevill 5/10/2012 Jorge Jun 2010, 2011, 2012 You re free to opy, distriute

More information

Figure 1. The left-handed and right-handed trefoils

Figure 1. The left-handed and right-handed trefoils The Knot Group A knot is n emedding of the irle into R 3 (or S 3 ), k : S 1 R 3. We shll ssume our knots re tme, mening the emedding n e extended to solid torus, K : S 1 D 2 R 3. The imge is lled tuulr

More information

s the set of onsequenes. Skeptil onsequenes re more roust in the sense tht they hold in ll possile relities desried y defult theory. All its desirle p

s the set of onsequenes. Skeptil onsequenes re more roust in the sense tht they hold in ll possile relities desried y defult theory. All its desirle p Skeptil Rtionl Extensions Artur Mikitiuk nd Miros lw Truszzynski University of Kentuky, Deprtment of Computer Siene, Lexington, KY 40506{0046, frtur mirekg@s.engr.uky.edu Astrt. In this pper we propose

More information

= state, a = reading and q j

= state, a = reading and q j 4 Finite Automt CHAPTER 2 Finite Automt (FA) (i) Derterministi Finite Automt (DFA) A DFA, M Q, q,, F, Where, Q = set of sttes (finite) q Q = the strt/initil stte = input lphet (finite) (use only those

More information

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

More information

A CLASS OF GENERAL SUPERTREE METHODS FOR NESTED TAXA

A CLASS OF GENERAL SUPERTREE METHODS FOR NESTED TAXA A CLASS OF GENERAL SUPERTREE METHODS FOR NESTED TAXA PHILIP DANIEL AND CHARLES SEMPLE Astrt. Amlgmting smller evolutionry trees into single prent tree is n importnt tsk in evolutionry iology. Trditionlly,

More information

Transition systems (motivation)

Transition systems (motivation) Trnsition systems (motivtion) Course Modelling of Conurrent Systems ( Modellierung neenläufiger Systeme ) Winter Semester 2009/0 University of Duisurg-Essen Brr König Tehing ssistnt: Christoph Blume In

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

University of Sioux Falls. MAT204/205 Calculus I/II

University of Sioux Falls. MAT204/205 Calculus I/II University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques

More information

Exercise sheet 6: Solutions

Exercise sheet 6: Solutions Eerise sheet 6: Solutions Cvet emptor: These re merel etended hints, rther thn omplete solutions. 1. If grph G hs hromti numer k > 1, prove tht its verte set n e prtitioned into two nonempt sets V 1 nd

More information

Linear choosability of graphs

Linear choosability of graphs Liner hoosility of grphs Louis Esperet, Mikel Montssier, André Rspud To ite this version: Louis Esperet, Mikel Montssier, André Rspud. Liner hoosility of grphs. Stefn Felsner. 2005 Europen Conferene on

More information

NEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE

NEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE NEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE V.S. Gordeev, G.A. Myskov Russin Federl Nuler Center All-Russi Sientifi Reserh Institute of Experimentl Physis (RFNC-VNIIEF)

More information

Global alignment. Genome Rearrangements Finding preserved genes. Lecture 18

Global alignment. Genome Rearrangements Finding preserved genes. Lecture 18 Computt onl Biology Leture 18 Genome Rerrngements Finding preserved genes We hve seen before how to rerrnge genome to obtin nother one bsed on: Reversls Knowledge of preserved bloks (or genes) Now we re

More information

Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths

Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t

More information

Metodologie di progetto HW Technology Mapping. Last update: 19/03/09

Metodologie di progetto HW Technology Mapping. Last update: 19/03/09 Metodologie di progetto HW Tehnology Mpping Lst updte: 19/03/09 Tehnology Mpping 2 Tehnology Mpping Exmple: t 1 = + b; t 2 = d + e; t 3 = b + d; t 4 = t 1 t 2 + fg; t 5 = t 4 h + t 2 t 3 ; F = t 5 ; t

More information

MAT 403 NOTES 4. f + f =

MAT 403 NOTES 4. f + f = MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn

More information

ANALYSIS AND MODELLING OF RAINFALL EVENTS

ANALYSIS AND MODELLING OF RAINFALL EVENTS Proeedings of the 14 th Interntionl Conferene on Environmentl Siene nd Tehnology Athens, Greee, 3-5 Septemer 215 ANALYSIS AND MODELLING OF RAINFALL EVENTS IOANNIDIS K., KARAGRIGORIOU A. nd LEKKAS D.F.

More information

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or

More information

5. Every rational number have either terminating or repeating (recurring) decimal representation.

5. Every rational number have either terminating or repeating (recurring) decimal representation. CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd

More information

SECTION A STUDENT MATERIAL. Part 1. What and Why.?

SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

More information

Learning Partially Observable Markov Models from First Passage Times

Learning Partially Observable Markov Models from First Passage Times Lerning Prtilly Oservle Mrkov s from First Pssge s Jérôme Cllut nd Pierre Dupont Europen Conferene on Mhine Lerning (ECML) 8 Septemer 7 Outline. FPT in models nd sequenes. Prtilly Oservle Mrkov s (POMMs).

More information

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points. Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You

More information

, g. Exercise 1. Generator polynomials of a convolutional code, given in binary form, are g. Solution 1.

, g. Exercise 1. Generator polynomials of a convolutional code, given in binary form, are g. Solution 1. Exerise Genertor polynomils of onvolutionl ode, given in binry form, re g, g j g. ) Sketh the enoding iruit. b) Sketh the stte digrm. ) Find the trnsfer funtion T. d) Wht is the minimum free distne of

More information

Algorithms & Data Structures Homework 8 HS 18 Exercise Class (Room & TA): Submitted by: Peer Feedback by: Points:

Algorithms & Data Structures Homework 8 HS 18 Exercise Class (Room & TA): Submitted by: Peer Feedback by: Points: Eidgenössishe Tehnishe Hohshule Zürih Eole polytehnique fédérle de Zurih Politenio federle di Zurigo Federl Institute of Tehnology t Zurih Deprtement of Computer Siene. Novemer 0 Mrkus Püshel, Dvid Steurer

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

More information

Implication Graphs and Logic Testing

Implication Graphs and Logic Testing Implition Grphs n Logi Testing Vishwni D. Agrwl Jmes J. Dnher Professor Dept. of ECE, Auurn University Auurn, AL 36849 vgrwl@eng.uurn.eu www.eng.uurn.eu/~vgrwl Joint reserh with: K. K. Dve, ATI Reserh,

More information

Expand the Shares Together: Envy-free Mechanisms with a Small Number of Cuts

Expand the Shares Together: Envy-free Mechanisms with a Small Number of Cuts Nonme mnusript No. (will e inserted y the editor) Expnd the Shres Together: Envy-free Mehnisms with Smll Numer of uts Msoud Seddighin Mjid Frhdi Mohmmd Ghodsi Rez Alijni Ahmd S. Tjik Reeived: dte / Aepted:

More information

AP CALCULUS Test #6: Unit #6 Basic Integration and Applications

AP CALCULUS Test #6: Unit #6 Basic Integration and Applications AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret

More information

Abstraction of Nondeterministic Automata Rong Su

Abstraction of Nondeterministic Automata Rong Su Astrtion of Nondeterministi Automt Rong Su My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 1 Outline Motivtion Automton Astrtion Relevnt Properties Conlusions My 6, 2010 TU/e Mehnil Engineering,

More information

CIT 596 Theory of Computation 1. Graphs and Digraphs

CIT 596 Theory of Computation 1. Graphs and Digraphs CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege

More information

Section 3.6. Definite Integrals

Section 3.6. Definite Integrals The Clulus of Funtions of Severl Vribles Setion.6 efinite Integrls We will first define the definite integrl for funtion f : R R nd lter indite how the definition my be extended to funtions of three or

More information

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P. Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time

More information

Computational Biology Lecture 18: Genome rearrangements, finding maximal matches Saad Mneimneh

Computational Biology Lecture 18: Genome rearrangements, finding maximal matches Saad Mneimneh Computtionl Biology Leture 8: Genome rerrngements, finding miml mthes Sd Mneimneh We hve seen how to rerrnge genome to otin nother one sed on reversls nd the knowledge of the preserved loks or genes. Now

More information

Propositional models. Historical models of computation. Application: binary addition. Boolean functions. Implementation using switches.

Propositional models. Historical models of computation. Application: binary addition. Boolean functions. Implementation using switches. Propositionl models Historil models of omputtion Steven Lindell Hverford College USA 1/22/2010 ISLA 2010 1 Strt with fixed numer of oolen vriles lled the voulry: e.g.,,. Eh oolen vrile represents proposition,

More information

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions. Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene

More information

Line Integrals and Entire Functions

Line Integrals and Entire Functions Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series

More information

Probability. b a b. a b 32.

Probability. b a b. a b 32. Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,

More information

Petri Nets. Rebecca Albrecht. Seminar: Automata Theory Chair of Software Engeneering

Petri Nets. Rebecca Albrecht. Seminar: Automata Theory Chair of Software Engeneering Petri Nets Ree Alreht Seminr: Automt Theory Chir of Softwre Engeneering Overview 1. Motivtion: Why not just using finite utomt for everything? Wht re Petri Nets nd when do we use them? 2. Introdution:

More information

Coalgebra, Lecture 15: Equations for Deterministic Automata

Coalgebra, Lecture 15: Equations for Deterministic Automata Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined

More information

Lecture 6. CMOS Static & Dynamic Logic Gates. Static CMOS Circuit. PMOS Transistors in Series/Parallel Connection

Lecture 6. CMOS Static & Dynamic Logic Gates. Static CMOS Circuit. PMOS Transistors in Series/Parallel Connection NMOS Trnsistors in Series/Prllel onnetion Leture 6 MOS Stti & ynmi Logi Gtes Trnsistors n e thought s swith ontrolled y its gte signl NMOS swith loses when swith ontrol input is high Peter heung eprtment

More information

2.4 Theoretical Foundations

2.4 Theoretical Foundations 2 Progrmming Lnguge Syntx 2.4 Theoretil Fountions As note in the min text, snners n prsers re se on the finite utomt n pushown utomt tht form the ottom two levels of the Chomsky lnguge hierrhy. At eh level

More information

INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable

INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd

More information

Single-Player and Two-Player Buttons & Scissors Games (Extended Abstract)

Single-Player and Two-Player Buttons & Scissors Games (Extended Abstract) Single-Plyer nd Two-Plyer Buttons & Sissors Gmes (Extended Astrt) Kyle Burke 1, Erik D. Demine 2, Hrrison Gregg 3, Roert A. Hern 4, Adm Hestererg 2, Mihel Hoffmnn 5, Hiro Ito 6, Irin Kostitsyn 7, Jody

More information

A Study on the Properties of Rational Triangles

A Study on the Properties of Rational Triangles Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn

More information

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL: PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles

More information

Bisimulation, Games & Hennessy Milner logic

Bisimulation, Games & Hennessy Milner logic Bisimultion, Gmes & Hennessy Milner logi Leture 1 of Modelli Mtemtii dei Proessi Conorrenti Pweł Soboiński Univeristy of Southmpton, UK Bisimultion, Gmes & Hennessy Milner logi p.1/32 Clssil lnguge theory

More information

Discrete Structures Lecture 11

Discrete Structures Lecture 11 Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.

More information

Foundation of Diagnosis and Predictability in Probabilistic Systems

Foundation of Diagnosis and Predictability in Probabilistic Systems Foundtion of Dignosis nd Preditility in Proilisti Systems Nthlie Bertrnd 1, Serge Hddd 2, Engel Lefuheux 1,2 1 Inri Rennes, Frne 2 LSV, ENS Chn & CNRS & Inri Sly, Frne De. 16th FSTTCS 14 Dignosis of disrete

More information

The Word Problem in Quandles

The Word Problem in Quandles The Word Prolem in Qundles Benjmin Fish Advisor: Ren Levitt April 5, 2013 1 1 Introdution A word over n lger A is finite sequene of elements of A, prentheses, nd opertions of A defined reursively: Given

More information

Linear Algebra Introduction

Linear Algebra Introduction Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +

More information

Symmetrical Components 1

Symmetrical Components 1 Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent

More information