Outline Y. Thierry-Mieg Octobre 2008 DDD, SDD and Applications Decision Diagrams for Model-checking: Data Decision Diagrams : Saturation

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1 Dt Decision Digrms, Set Decision Digrms, And Applictions Ynn Thierry-Mieg 3 Oct 8 Presenttion in Nntes

2 Outline Y. Thierry-Mieg Octobre 8 Decision Digrms for Model-checking: Dt Decision Digrms : A dynmic nd flexible DD librry Sturtion A more effective fixpoint strtegy Hierrchicl Set Decision Digrms Introduce structured descriptions Instntible Trnsition Systems A frmework to exploit SDD Dt Decision Digrms Sturtion Set Decision Digrms Instntible Trnsition System

3 Dt Decision Digrms ICATPN' J.M. Couvreur, P.A. Wcrenier E. Encrenz, E. Pviot-Adet, D. Poitrenud LIP6, LBRI Dt Decision Digrms Sturtion Set Decision Digrms Instntible Trnsition System

4 DDD: Dt Decision Digrms Y. Thierry-Mieg Octobre 8 4 Decision digrms:[bcm'9] Initilly BDD [Brynt86] Compct Structure to represent sets Unicity tble & opertion cche Complexity linked to number of nodes Exploits implicit symmetries between elements of the set Intermedite pek size problem Very widespred success SMV, Smrt, Uppl, Prism,... Dt Decision Digrm [Couvreur+] integer domin vribles, no ordering, vrible length pths Set Opertions + inductive homomorphisms 3 b 4 c

5 Dt Decision Digrm Y. Thierry-Mieg Octobre 8 5 E = { vrible }, Dom(e) = domin of e Inductive definition,, T re DDDs or d = (e,α) with e vrible in E, α : Dom(e) DDD α < is DDD ccepted Properties No vrible order A vrible my pper twice in decision pth A domin my be finite or infinite A DDD is finite structure Cnonicl representtion : if α = then (e, α) => zero suppressed b 3 T undefined Avoided in prctice

6 Undefined nd Better Defined Y. Thierry-Mieg Octobre 8 6 d is better defined thn d iff b d = d or 3 d = T or b d = (e,α), d =(e,α ) with 3 α(x) is better defined thn α (x) for ny x in Dom(e) DDD opertors A mpping f : DDD n DDD on DDD is DDD opertor if f is comptible with the better defined reltion: i : d i d i f(d,,d n ) f(d,,d n ) is better defined thn b 3 T

7 Union nd the undefined terminl T Y. Thierry-Mieg Octobre 8 7 Union "+" + T (e,α ) T (e,α ) T T T T T T T (e,α ) (e,α ) T T e=e? (e, α +α ) : T b 3 b = + b 3 b 3 3

8 Union nd the undefined terminl T Y. Thierry-Mieg Octobre T (e,α ) Union "+" T (e,α ) T T T T T T T (e,α ) (e,α ) T T e=e? (e, α +α ) : T b 3 b = b + 3 b 3 + 3

9 Union nd the undefined terminl T Y. Thierry-Mieg Octobre T (e,α ) Union "+" T (e,α ) T T T T T T T (e,α ) (e,α ) T T e=e? (e, α +α ) : T b 3 b = b 3 b 3 + 3

10 Union nd the undefined terminl T Y. Thierry-Mieg Octobre 8 Union "+" + T (e,α ) T (e,α ) T T T T T T T (e,α ) (e,α ) T T e=e? (e, α +α ) : T b 3 b = b 3 Dnger here! We void T by crefully choosing opernds of union. T

11 Other Elementry Opertors on DDD Y. Thierry-Mieg Octobre 8 Intersection "*", Set difference "/" defined in usul mnner, with dded cses to hndle T (not presented here) Conctention "." : specific opertion to extend pths b 3. = b 3

12 Homomorphism Y. Thierry-Mieg Octobre 8 Definition Φ: DDD DDD d,d Φ() = Φ(d ) + Φ(d ) Φ(d +d ) d d Φ(d ) Φ(d ) Becomes equlity when d nd d re well-defined DDD Exmples, hrd coded in the librry Id, Id + d, Id * d, Id \ d, Id.d, d.id Proposition If Φ, Φ re homomorphisms then Φ + Φ nd Φ ο Φ re homomorphisms

13 Inductive Homomorphism Y. Thierry-Mieg Octobre 8 3 Proposition Let (π i,j ), (τ i ) be fmilies of homomorphisms. The mppings (Φ i ) inductively defined by d Φ i (d) Φ i () T T (e,α) x Φ i (e,x)(α(x)) re homomorphisms where Φ i (e,x) = j π i,j ο Φ j (e,x) + τ i User defined homomorphisms re defined by: Φ(), constnt DDD tht is the evlution over terminl Φ(e,x), for ny vrible e nd vlue x, homomorphism to pply on the successor node

14 Exmple : Increment the vlue of vrible v Y. Thierry-Mieg Octobre 8 4 Inc(v)(e,x) = Inc(v)() = v x+ > Id e x > Inc(v) if v=e otherwise 3 4 b Inc(b) c d

15 Exmple : Increment the vlue of vrible v Y. Thierry-Mieg Octobre 8 5 Inc(v)(e,x) = Inc(v)() = v x+ > Id e x > Inc(v) if v=e otherwise 3 4 b Inc(b) c d 3 4 b Inc(b) c d

16 Exmple : Increment the vlue of vrible v Y. Thierry-Mieg Octobre 8 6 Inc(v)(e,x) = Inc(v)() = v x+ > Id e x > Inc(v) if v=e otherwise 3 4 b Inc(b) c d 3 4 b Inc(b) c d b c d

17 User defined inductive homomorphisms Y. Thierry-Mieg Octobre 8 7 Cn be defined compositionnly : Simple opertions hrd coded bsic opertors Id, Id + d, Id * d, Id \ d, Id.d, d.id simple homomorphisms like Inc() Compose complex opertions using composition & union + intersection * No set difference in generl does not preserve linerity Use trnsitive closure h * newly introduced

18 Composition Exmple: Swp the vlues of two vribles v, w Y. Thierry-Mieg Octobre 8 8 Swp(v,w)(e,x) = Swp(v,w)() = Renme(v) ο Down(w,x) if v=e Renme(w) ο Down(v,x) if w=e e x > Swp(v,w) otherwise Renme(v)(e,x) = v x > Id Renme(v)() = T Up(v,z)(e,x) = e x > v z > Id Up(v,z)() = T Down(v,z)(e,x) = Down(v,z)() = T v x > v z > Id Up(e,x) ο Down(v,z) if v=e otherwise 3 4 b Swp(b,d) c d

19 Composition Exmple: Swp the vlues of two vribles v, w Y. Thierry-Mieg Octobre 8 9 Swp(v,w)(e,x) = Swp(v,w)() = Renme(v) ο Down(w,x) if v=e Renme(w) ο Down(v,x) if w=e e x > Swp(v,w) otherwise Renme(v)(e,x) = v x > Id Renme(v)() = T Up(v,z)(e,x) = e x > v z > Id Up(v,z)() = T Down(v,z)(e,x) = Down(v,z)() = T v x > v z > Id Up(e,x) ο Down(v,z) if v=e otherwise 3 4 b Swp(b,d) c d

20 Composition Swp the vlues of two vribles v, w Y. Thierry-Mieg Octobre 8 Swp(v,w)(e,x) = Swp(v,w)() = Renme(v) ο Down(w,x) if v=e Renme(w) ο Down(v,x) if w=e e x > Swp(v,w) otherwise Renme(v)(e,x) = v x > Id Renme(v)() = T Up(v,z)(e,x) = e x > v z > Id Up(v,z)() = T Down(v,z)(e,x) = Down(v,z)() = T v x > v z > Id Up(e,x) ο Down(v,z) if v=e otherwise 3 4 Renme(b) Down(d,) c d

21 Composition Swp the vlues of two vribles v, w Y. Thierry-Mieg Octobre 8 Swp(v,w)(e,x) = Swp(v,w)() = Renme(v) ο Down(w,x) if v=e Renme(w) ο Down(v,x) if w=e e x > Swp(v,w) otherwise Renme(v)(e,x) = v x > Id Renme(v)() = T Up(v,z)(e,x) = e x > v z > Id Up(v,z)() = T Down(v,z)(e,x) = Down(v,z)() = T v x > v z > Id Up(e,x) ο Down(v,z) if v=e otherwise 4 d Renme(b) Up(c,3) Down(d,)

22 Exmple: Swp the vlues of two vribles v, w Y. Thierry-Mieg Octobre 8 Swp(v,w)(e,x) = Swp(v,w)() = Renme(v) ο Down(w,x) if v=e Renme(w) ο Down(v,x) if w=e e x > Swp(v,w) otherwise Renme(v)(e,x) = v x > Id Renme(v)() = T Up(v,z)(e,x) = e x > v z > Id Up(v,z)() = T Down(v,z)(e,x) = Down(v,z)() = T v x > v z > Id Up(e,x) ο Down(v,z) if v=e otherwise 4 Renme(b) up(c,3) d d

23 Exmple: Swp the vlues of two vribles v, w Y. Thierry-Mieg Octobre 8 3 Swp(v,w)(e,x) = Swp(v,w)() = Renme(v) ο Down(w,x) if v=e Renme(w) ο Down(v,x) if w=e e x > Swp(v,w) otherwise Renme(v)(e,x) = v x > Id Renme(v)() = T Up(v,z)(e,x) = e x > v z > Id Up(v,z)() = T Down(v,z)(e,x) = Down(v,z)() = T v x > v z > Id Up(e,x) ο Down(v,z) if v=e otherwise 4 3 d Renme(b) c d

24 Exmple: Swp the vlues of two vribles v, w Y. Thierry-Mieg Octobre 8 4 Swp(v,w)(e,x) = Swp(v,w)() = Renme(v) ο Down(w,x) if v=e Renme(w) ο Down(v,x) if w=e e x > Swp(v,w) otherwise Renme(v)(e,x) = v x > Id Renme(v)() = T Up(v,z)(e,x) = e x > v z > Id Up(v,z)() = T Down(v,z)(e,x) = Down(v,z)() = T v x > v z > Id Up(e,x) ο Down(v,z) if v=e otherwise 4 3 b c d

25 DDD for Petri nets Y. Thierry-Mieg Octobre 8 5 Study : Compute the rechbility set Problem Encoding stte set s DDD Encoding trnsition reltion s homomorphism Ordinry Nets Inhibitor rcs Cpcity plces Reset rcs Self Modifying Nets DDD = constnt depth TR = locl opertions Queuing Nets DDD = constnt depth TR = non locl opertions DDD = dynmic depth TR = locl opertions

26 Ordinry Petri nets Y. Thierry-Mieg Octobre 8 6 Technique : As MDD Encoding stte set s DDD One vrible per plce Define totl order on plces Encoding trnsition reltion s homomorphism One inductive homomorphism TestSet(p,pre,post) TestSet(p,pre,post)(e,x) = TestSet(p,pre,post)() = T p x+post-pre > Id if p=e, x pre if p=e, x<pre e x > TestSet(p,pre,post) otherwise Reltion for one trnsition R(t) = TestSet opertors Trnsition for Petri net = Id + Σ R(t) or (Id +R(t))

27 Petri net extensions Y. Thierry-Mieg Octobre 8 7 Inhibitor rcs, Cpcity plces, Reset rcs s ordinry Petri nets Self Modifying Nets DDD s ordinry Petri nets 6 new inductive homomorphisms pplying m(p) = m(p) + m(q) - m(r) strongly depend on the order between p, q nd r The 6 new homomorphisms re designed s the swp opertor (See the pper ICATPN', Couvreur et l. for detils)

28 Queuing nets Y. Thierry-Mieg Octobre 8 8 Encoding stte set s DDD One vrible per plce One vrible per queue but one occurrence per messge Define totl order on plces nd queues f f p q b # f f p q b # # p q Initil mrking p f.b q b Rech = 3 sttes

29 9 Y. Thierry-Mieg Octobre 8 Queuing nets b b f q p f q p # # f q p # f b b f p q Rech = 3 sttes One vrible per plce One vrible per queue but one occurrence per messge Define totl order on plces nd queues Encoding stte set s DDD f

30 Queuing nets Y. Thierry-Mieg Octobre 8 3 Encoding stte set s DDD One vrible per plce One vrible per queue but one occurrence per messge Define totl order on plces nd queues f f p q b # f b f p q # # p q p f q b Rech = 3 sttes

31 Queuing nets Y. Thierry-Mieg Octobre 8 3 Encoding trnsition reltion s homomorphism Two inductive homomorphisms Send, Rec Send = replce f # > d by f > f # > Id Send(f,)(e,x) = Send(f,)() = T f > f # > Id e x > Send(f,) if e=f, x=# otherwise Receive = remove the occurrence of f > Rec(f,)(e,x) = Rec(f,)() = T Id if e=f, x= if e=f, x e x > Rec(f,) otherwise Lossy chnnels re treted using other homomorphisms

32 Experimenttion (Vlues from ICATPN') Y. Thierry-Mieg Octobre 8 3 Model N Reched DDD No shring time Philosopher Sfe net.86* * * * Fms 5.89* * 6.76 Ordinry net.5* * * 74.87* Alternte bit Inhibitor, reset, cpcity * * Preemptive writer 5x Self modifying net x.94* * x5 4.95* * 37.5 Alternte bit Lossy Queuing net * *

33 Concluding remrks Y. Thierry-Mieg Octobre 8 33 DDD re very similr to MDD except for vrible length pths : useful in some contexts (FIFO...) forbids (level,index) ccess Homomorphisms re the rel difference opertions defined independently of the vlues they work with : x x + insted of,,... compositionl (lgebric) frmework good expressivity highly extensible/flexible trnsitive closure opertor llows sturtion type lgorithms

34 The Sturtion Algorithm for Decision digrms Dt Decision Digrms Sturtion Set Decision Digrms Instntible Trnsition System

35 Trnsitive Closure : Fixpoint Y. Thierry-Mieg Octobre 8 35

36 Sturtion Y. Thierry-Mieg Octobre 8 36 Model-checking using decision digrms => (nested) trnsitive closures over the trnsition reltion Optimizing complexity of this opertion criticl to efficiency [BCM 9] bsed on BFS style itertions, n itertions required where n is depth of deepest stte s N(s) N (s) [Roig 95] Chining my converge fster, bsed on clusters of trnsitions, no longer strict BFS s t(s) t(t(s)) t(t(t(t (s)))..)) [Cirdo ] Sturtion is empiriclly to 3 orders of mgnitude better

37 Sturtion vs BFS Y. Thierry-Mieg Octobre 8 37 Sturtion lgorithm: [Cirdo et l. TACAS ] Fire trnsitions from the leves (terminls) up to root Go to ncestor of node iff. The current node is sturted : ll events tht only ffect this vrible nd vribles below it hve been fired until fixpoint is reched Ech time node is ffected by n event, resturte it. Not BFS nymore, firing order of events follows dt structure Huge reduction of time nd spce complexity Good tckling of intermedite pek size effect However : Definition of sturtion lgorithm is complex Cnnot be implemented directly with public API of DD librries Our contribution : Automtic sturtion

38 Trnsitive Closure Y. Thierry-Mieg Octobre 8 38 The trnsitive closure or fixpoint noted * is unry opertor Evluted by h * (d): repet : d h(d) until : d = h(d) Evlution my not terminte depends on the homomorphism if it does, evlution described s finite composition : h * (d) = h h... h (d) Thus h * is homomorphism To cumulte sttes, use of common construction : (h + id) * Allows to implement lef to root sturtion strtegy

39 Fixpoint : n exmple Y. Thierry-Mieg Octobre 8 39 Seek(h,v)(e,x) = Seek(h,v)() = T h * e x > Id e x > Seek(h,v) if v=e otherwise Mx(z)(e,x) = Mx(z)() = T e x > Id if x<z otherwise h = Mx(3) Inc(d) // Increment d up to 3 b Seek(h+Id,d) c d

40 Fixpoint : n exmple Y. Thierry-Mieg Octobre 8 4 Seek(h,v)(e,x) = Seek(h,v)() = T h * e x > Id e x > Seek(h,v) if v=e otherwise Mx(z)(e,x) = Mx(z)() = T e x > Id if x<z otherwise h = Mx(3) Inc(d) // Increment d up to single trversl of these nodes b c 3 d Seek(h+Id,d)

41 Fixpoint : n exmple Y. Thierry-Mieg Octobre 8 4 Seek(h,v)(e,x) = Seek(h,v)() = T h * e x > Id e x > Seek(h,v) if v=e otherwise Mx(z)(e,x) = Mx(z)() = T e x > Id if x<z otherwise h = Mx(3) Inc(d) // Increment d up to 3 * b c d Id + Mx(3) Inc(d)

42 Fixpoint : n exmple Y. Thierry-Mieg Octobre 8 4 Seek(h,v)(e,x) = Seek(h,v)() = T h * e x > Id e x > Seek(h,v) if v=e otherwise Mx(z)(e,x) = Mx(z)() = T e x > Id if x<z otherwise h = Mx(3) Inc(d) // Increment d up to 3 * b c + Mx(3) Inc(d) d Id

43 Fixpoint : n exmple Y. Thierry-Mieg Octobre 8 43 Seek(h,v)(e,x) = Seek(h,v)() = T h * e x > Id e x > Seek(h,v) if v=e otherwise Mx(z)(e,x) = Mx(z)() = T e x > Id if x<z otherwise h = Mx(3) Inc(d) // Increment d up to Id 3 * b c + Mx(3) Inc(d) d

44 Fixpoint : n exmple Y. Thierry-Mieg Octobre 8 44 Seek(h,v)(e,x) = Seek(h,v)() = T h * e x > Id e x > Seek(h,v) if v=e otherwise Mx(z)(e,x) = Mx(z)() = T e x > Id if x<z otherwise h = Mx(3) Inc(d) // Increment d up to 3 b c d

45 Fixpoint conclusions Y. Thierry-Mieg Octobre 8 45 Trnsitive closure or fixpoint llows: single trversl of the top of the tree less intermedite nodes Itertion + h b c 3 d b c 3 d BFS style itertions Itertion + h 3 b c d 3 b c d 3 b c d

46 Fixpoint conclusions Y. Thierry-Mieg Octobre 8 46 Trnsitive closure or fixpoint llows: single trversl of the top of the tree => cost of + nd h less intermedite nodes Itertion + h b c 3 d b c 3 d BFS style itertions Itertion + h 3 b c d 3 b c d Useless intermedite nodes!! 3 b c d

47 Sturtion effect Y. Thierry-Mieg Octobre 8 47 Nested trnsitive closure or fixpoint = sturtion llows: single trversl of the top of the tree => cost of + nd h less intermedite nodes Itertion + h b c 3 d b c 3 d BFS style itertions + h b c 3 d [,] b c 3 d [,] Itertion Useless intermedite nodes!! [,,] 3 b c d

48 Performnce mesures : effect of sturtion Y. Thierry-Mieg Octobre 8 48 Trnsitive closure llows more efficiency Mnul Sturtion "à l Cirdo" (Tcs' nd '3) using * opertor Orgnize events by highest vrible ffected

49 Hierrchicl Set Decision Digrms & Automtic Sturtion Alexndre Hmez, Ynn Thierry- Mieg, Fbrice Kordon June 8 - ICATPN 8 Alexndre Hmez, Ynn Thierry-Mieg, Fbrice Kordon June 8 - ICATPN 8 LIP6 - LRDE Xi n, Chin

50 Hierrchicl Set Decision Digrm (SDD) Y. Thierry-Mieg Octobre 8 5 Limits of DDD reched, need for structure Ide : hierrchy Lbel the rcs with SET = Set Decision Digrm Increses shring Memory gin Time gin cche trversls With hierrchy DDD SDD + DDD or SDD = referenced vlues

51 Set Decision Digrms Forte 5, Couvreur, Thierry-Mieg Y. Thierry-Mieg Octobre 8 5 More Formlly e E : set of vribles e in E, Dom(e) : its (possibly infinite) domin d is inductively defined s n SDD iff. d {,} or i d i d = <e,α> Given prtition π of Dom(e) α : π SDD, s.t. i,j, i j α( i ) α( j ) Arcs to nd rcs lbeled by not represented Fusing Arcs e j i e i j Splitting rcs e j i j \ i e j i i \ j d i d i d j d i d j d i d i d j

52 Set Decision Digrms : A compositionl Model Y. Thierry-Mieg Octobre 8 5 SDD rcs my be lbeled by DDD Hierrchicl Structure Adpted to composition cptures the similrity of repeted modules The similrity of behvior of the philosophers is cptured : 8*( Philosopher)

53 DDD shring & SDD shring Y. Thierry-Mieg Octobre 8 53 Stte spce, 4 philosophers (DDD) P WitLeft Mod_ Idle WitLeft mod With SDD, the stte of one philosopher is referenced. (P) HsLeft HsLeft HsLeft WitRight WitRight WitRight Mod_ Mod_3 Mod_ Mod_ HsRight HsRight HsRight HsRight P Fork Fork Idle Idle Fork Fork Idle Idle mod mod mod mod (P) WitLeft WitLeft WitLeft WitLeft WitLeft WitLeft HsLeft HsLeft WitRight HsLeft WitRight HsLeft HsLeft WitRight HsLeft WitRight Mod_ Mod_ Mod_3 Mod_ Mod_3 Mod_ Mod_ Mod_ Mod_3 Mod_ Mod_ Mod_ HsRight HsRight HsRight HsRight P3WitLeft Fork Idle WitLeft Fork Idle WitLeft Fork Fork Idle Idle WitLeft WitLeft WitLeft mod mod mod mod (P3) WitLeft Idle WitLeft Idle WitLeft Idle WitLeft Idle WitLeft WitLeft HsLeft HsLeft HsLeft WitRight WitRight HsLeft HsLeft HsLeft WitRight WitRight Mod_3 Mod_ Mod_ Mod_ Mod_3 Mod_ Mod_ Mod_ HsLeft HsLeft HsLeft HsLeft HsLeft HsLeft HsRight HsRight HsRight HsRight Fork P4 Idle3 WitLeft3 WitLeft3 Fork Idle3 WitLeft3 Fork Fork Idle3 Idle3 WitLeft3 WitLeft3 WitLeft3 mod 3 mod 3 mod 3 mod 3 (P4) WitRight HsRight WitRight HsRight WitRight HsRight WitRight HsLeft3 HsLeft3 HsLeft3 WitRight3 WitRight3 HsLeft3 HsLeft3 HsLeft3 WitRight3 WitRight3 Mod_3 Mod_ Mod_ Mod_ Fork Fork HsRight3 HsRight3 HsRight3 Fork3 Fork3 P // P // P3 // P4 one (P) // (P) // (P3) // (P4) Pi

54 54 Y. Thierry-Mieg Octobre 8 C E E A A B B D D F F G G H C E E A A B B D D F F G G H C E E A A B D F F G G H C3 E3 A3 B3 D3 F3 G3 H3 C4 E4 A4 B4 D4 F4 G4 H4 B D A B B D D F G H F G H C3 E3 E3 A3 B3 D3 F3 G3 A3 B3 B3 D3 D3 F3 G3 H3 F3 G3 H3 C4 E4 E4 A4 B4 D4 F4 G4 A4 B4 B4 D4 F4 G4 H4 F F G G H C E E A B B D D F G F G A B B D D F G H F G H C3 E3 E3 A3 B3 B3 D3 D3 F3 G3 F3 G3 A3 B3 B3 D3 D3 F3 G3 H3 F3 G3 H3 C4 E4 A4 B4 B4 D4 F4 G4 D F F G G H B B D D D A B B D D F G H F G H C E E A B B D D F G F G A B B D D F G H F G H C3 E3 E3 A3 B3 B3 D3 D3 F3 G3 F3 G3 A3 B3 B3 D3 F3 G3 H3 F F G G H C E E A B B D D F G F G A B B D D F G H F G H C E E A B B D D F G F G A B B D D F G H F G H C3 E3 A3 B3 B3 D3 F3 G3 D F F G G H B B D D D A B B D D F G H F G H C E E A B B D D F G F G A B B D D F G H F G H C E E A B B D D F G F G A B B D F G H DDD to SDD Model "slotted ring", 5 prticipnts, sttes mod mod Mod_ mod Mod_ mod Mod_ mod Mod_ mod Mod_ mod Mod_ mod 3 Mod_ mod 3 Mod_ mod 3 Mod_ mod 4 Mod_ mod 4 Mod_ mod 4 Mod_ one Mod_ Mod_ Mod_ Mod_ Mod_ Mod_ Mod_ Mod_ Mod_ Mod_ mod 3 Mod_ Mod_ Mod_ Mod_ mod Mod_ mod Mod_ Mod_ Mod_ mod 3 Mod_ Mod_ Mod_ Mod_ Mod_ Mod_ Mod_ C E E A A B D F F G G H B D A B B D F G H C E A B B D F G C E Mod_ Mod_ 3 Mod_ pure DDD 35 nodes Pek 93k nodes 6 secondes SDD... 8 nodes...+ddd 3 nodes Pek t nodes.4 secondes (sturtion disbled)

55 Knbn exmple, low prmeter vlue (5) Y. Thierry-Mieg Octobre 8 55 SDD mod Mod_ Mod_ Mod_ Mod_3 Mod_ Mod_4 Mod_5 Mod_6 DDD Mod_ Mod_ mod Mod_3 Mod_4 Mod_5 Mod_6 P 5 Pout P 4 Pout 3 Pout P P P Pout Pout P P Pout mod mod mod mod mod mod Mod_ Mod_ Mod_3 Mod_4 Mod_5 Mod_6 Pbck Pbck Pbck Pbck Pbck Pbck mod Mod_ Pm Pm Pm Pm Pm Pm one Explosion of rcs per node s sttes per component increses Level 4 :(,56) Level 3 :(,56) Level 37:(9,56) 35:(46,44) 3:(,5) 8:(7,) 3:(,4) :(,) Level 7:(,56) MDD (Smrt) Level 7

56 Compositionl Approch & SDD Y. Thierry-Mieg Octobre 8 56 SDD well dpted to composition The modules give structurtion SDD put in reltion sets of sttes On the fly computtion f synchronized products Gin of n order of mgnitude over DDD

57 SDD Homomorphism Y. Thierry-Mieg Octobre 8 57 Definition Φ: SDD SDD d,d Φ() = Φ(d ) + Φ(d ) = Φ(d +d ) Exmples, hrd coded in the librry Id, Id + d, Id * d, Id \ d, Id.d, d.id Proposition If Φ, Φ re homomorphisms then Φ + Φ nd Φ ο Φ re homomorphisms

58 Inductive Homomorphism Y. Thierry-Mieg Octobre 8 58 Used to define user opertions Flexible, nd powerful Benefits from cche Φ is inductively defined by : Φ() SDD : constnt terminl cse Φ(e,x) Hom : e in E, x Dom(e): evlution for n rbitrry SDD rc returns homomorphism to pply on successor node

59 Skip predicte (NEW) Y. Thierry-Mieg Octobre 8 59 Skip(e) expresses locl invrince: Skip is true => the vrible is neither red nor written Extends to composition : f+g nd f g skip vrible e iff. both opernds skip(e) Miniml structurl informtion bout user opertions tht llows to enble sturtion utomticlly

60 Fixpoint opertor : * Y. Thierry-Mieg Octobre 8 6 Built-in opertor for trnsitive closure Φ * (d) = Φ n (d), where n is the smllest integer such tht Φ n+ (d) = Φ n (d) My not terminte (n infinite) Most often used s n ccumultor : (Φ + Id) * Trnsitive closure nturlly expressed s : (t + t + +tn + Id)* Allows librry to utomticlly enble sturtion

61 Homomorphisms for Petri Net Y. Thierry-Mieg Octobre 8 6 Pre rc homomorphism Post rc homomorphism h- returns terminl to prune pth if precondition not met Skip on e p : only one vrible (plce) is ffected by the rc For full trnsition, compose Post fter Pre, e.g.

62 Homomorphisms for Lbeled Petri net Y. Thierry-Mieg Octobre 8 6 Built-in locl homomorphism llows to trget rc vlue(s) of given vrible e

63 DDD shring & SDD shring (reminder) Y. Thierry-Mieg Octobre 8 63 Stte spce, 4 philosophers (DDD) P WitLeft Mod_ Idle WitLeft mod With SDD, the stte of one philosopher is referenced. (P) HsLeft HsLeft HsLeft WitRight WitRight WitRight Mod_ Mod_3 Mod_ Mod_ HsRight HsRight HsRight HsRight P Fork Fork Idle Idle Fork Fork Idle Idle mod mod mod mod (P) WitLeft WitLeft WitLeft WitLeft WitLeft WitLeft HsLeft HsLeft WitRight HsLeft WitRight HsLeft HsLeft WitRight HsLeft WitRight Mod_ Mod_ Mod_3 Mod_ Mod_3 Mod_ Mod_ Mod_ Mod_3 Mod_ Mod_ Mod_ HsRight HsRight HsRight HsRight P3WitLeft Fork Idle WitLeft Fork Idle WitLeft Fork Fork Idle Idle WitLeft WitLeft WitLeft mod mod mod mod (P3) WitLeft Idle WitLeft Idle WitLeft Idle WitLeft Idle WitLeft WitLeft HsLeft HsLeft HsLeft WitRight WitRight HsLeft HsLeft HsLeft WitRight WitRight Mod_3 Mod_ Mod_ Mod_ Mod_3 Mod_ Mod_ Mod_ HsLeft HsLeft HsLeft HsLeft HsLeft HsLeft HsRight HsRight HsRight HsRight Fork P4 Idle3 WitLeft3 WitLeft3 Fork Idle3 WitLeft3 Fork Fork Idle3 Idle3 WitLeft3 WitLeft3 WitLeft3 mod 3 mod 3 mod 3 mod 3 (P4) WitRight HsRight WitRight HsRight WitRight HsRight WitRight HsLeft3 HsLeft3 HsLeft3 WitRight3 WitRight3 HsLeft3 HsLeft3 HsLeft3 WitRight3 WitRight3 Mod_3 Mod_ Mod_ Mod_ Fork Fork HsRight3 HsRight3 HsRight3 Fork3 Fork3 P // P // P3 // P4 one (P) // (P) // (P3) // (P4) Pi

64 Homomorphisms for Lbeled Petri net Y. Thierry-Mieg Octobre 8 64 Built-in locl homomorphism llows to trget rc vlue(s) of given vrible e Full trnsition reltion for synchroniztion is built s composition of locl opertions : e.g. philo P finishes eting:

65 Rewriting rules for homomorphisms Y. Thierry-Mieg Octobre 8 65 For Union : H = g_ +.. +g_n + f_ f_m such tht on current vrible g terms do not skip nd f terms skip. No priori vrible order => Prtition of opernds of union is cched

66 Effect of skip on (H + Id) * Y. Thierry-Mieg Octobre 8 66 Cse of interest : trnsitive closure of set of trnsitions + Id Additionl rules re defined for locl construction (see proceedings) Essentilly, (Locl(h,v) + id)* = Locl( (h+id)*, v )

67 Performnces Y. Thierry-Mieg Octobre 8 67

68 To finish : Compositionl nesting & hierrchy Y. Thierry-Mieg Octobre 8 68 SDD rcs my reference SDD Hierrchicl structure Arbitrry depth Exemple Philosophers : ^3 = 8 philosophes: 3 levels of depth + representtion of philosopher Shring t every level

69 Idle WitLeft WitLeft HsLeft HsLeft HsLeft WitRight WitRight WitRight HsRight HsRight HsRight HsRight Fork Fork Fork Fork Idle Idle Idle Idle WitLeft WitLeft WitLeft WitLeft WitLeft WitLeft HsLeft HsLeft HsLeft HsLeft HsLeft HsLeft WitRight WitRight WitRight WitRight HsRight HsRight HsRight HsRight Fork Fork Fork Fork Idle Idle Idle Idle WitLeft WitLeft WitLeft WitLeft WitLeft WitLeft HsLeft HsLeft HsLeft HsLeft HsLeft HsLeft WitRight WitRight WitRight WitRight HsRight HsRight HsRight HsRight Fork Fork Fork Fork Idle3 Idle3 Idle3 Idle3 WitLeft3 WitLeft3 WitLeft3 WitLeft3 WitLeft3 WitLeft3 HsLeft3 HsLeft3 HsLeft3 HsLeft3 HsLeft3 HsLeft3 WitRight3 WitRight3 WitRight3 WitRight3 HsRight3 HsRight3 HsRight3 HsRight3 Fork3 Fork3 Fork3 Fork3 Idle4 Idle4 Idle4 Idle4 WitLeft4 WitLeft4 WitLeft4 WitLeft4 WitLeft4 WitLeft4 HsLeft4 HsLeft4 HsLeft4 HsLeft4 HsLeft4 HsLeft4 WitRight4 WitRight4 WitRight4 WitRight4 HsRight4 HsRight4 HsRight4 HsRight4 Fork4 Fork4 Fork4 Fork4 Idle5 Idle5 Idle5 Idle5 WitLeft5 WitLeft5 WitLeft5 WitLeft5 WitLeft5 WitLeft5 HsLeft5 HsLeft5 HsLeft5 HsLeft5 HsLeft5 HsLeft5 WitRight5 WitRight5 WitRight5 WitRight5 HsRight5 HsRight5 HsRight5 HsRight5 Fork5 Fork5 Fork5 Fork5 Idle6 Idle6 Idle6 Idle6 WitLeft6 WitLeft6 WitLeft6 WitLeft6 WitLeft6 WitLeft6 HsLeft6 HsLeft6 HsLeft6 HsLeft6 HsLeft6 HsLeft6 WitRight6 WitRight6 WitRight6 WitRight6 HsRight6 HsRight6 HsRight6 HsRight6 Fork6 Fork6 Fork6 Fork6 Idle7 Idle7 Idle7 Idle7 WitLeft7 WitLeft7 WitLeft7 WitLeft7 WitLeft7 WitLeft7 HsLeft7 HsLeft7 HsLeft7 HsLeft7 HsLeft7 HsLeft7 WitRight7 WitRight7 WitRight7 WitRight7 HsRight7 HsRight7 HsRight7 Fork7 Fork7 Philosophers & Hierrchy : Potentil of SDD Y. Thierry-Mieg Octobre 8 69 P - P - P3 - P4 - P5 - P6 - P7 - P8 (P) - (P) - (P3) - (P4) - (P5) - (P6) - (P7) - (P8) Mod_ mod Mod_3 Mod_ Mod_ mod mod mod mod Mod_ mod Mod_ Mod_3 Mod_ mod Mod_ mod Mod_ Mod_ Mod_3 mod Mod_3 Mod_ Mod_ Mod_ mod 3 mod 3 Mod_ Mod_3 Mod_Mod_ mod 4 mod 4 Mod_ mod 5 Mod_ Mod_3 Mod_ mod 5 Mod_ Mod_ Mod_3 Mod_ mod 3 mod 3 Mod_ Mod_3 Mod_ Mod_ mod 4 mod 4 Mod_ Mod_ Mod_3 Mod_ mod 5 mod 5 Mod_3 Mod_ Mod_ Mod_ Mod_ Mod_3 Mod_ Mod_ mod 6 mod 6 mod 6 mod 6 Mod_3 Mod_ Mod_ Mod_ Mod_ Mod_ Mod_ Mod_3 mod 7 mod 7 mod 7 mod 7 Mod_3 Mod_ Mod_ Mod_ ((P - P)-(P3 - P4)) - nd hlf ((P5 - P6)-(P7 - P8)) (P i - P i+ ) - (P i+ - P i+3 ) (P i ) - (P i+ ) P i st hlf nd hlf st hlf nd hlf PhiloSttes Mod_ nd hlf st hlf st hlf nd hlf st hlf nd hlf PhiloSttes Mod_ nd hlf st hlf nd hlf st hlf nd hlf PhiloSttes Mod_ nd hlf st hlf nd hlf st hlf nd hlf PhiloSttes Mod_4

70 Conclusion Y. Thierry-Mieg Octobre 8 7 Ltest evolution of decision digrms : SDD Suitble for very lrge systems Well suited to hierrchicl/compositionl specifictions Trnsprent nd efficient trnsitive closure User defined homomorphisms Automtic sturtion : generlize [Cirdo ] with definition independent of given formlism Recursive folding for logrithmic complexity on some exmples SDD nd DDD distributed s n open-source LGPL C++ librry :

71 Instntible Trnsition Systems to Exploit SDD DDD Sturtion SDD ITS

72 A Frmework to exploit SDD Y. Thierry-Mieg Octobre 8 7 Encourging results obtined with model instnce philosophers Need for generliztion A frmework to define structured models Notion of type nd instnce Composite types contin nested instnces Trnsition reltion defined s : Locl events to n instnce Synchroniztions of trnsition lbels of contined instnces

73 ITS Type Contrct Y. Thierry-Mieg Octobre 8 73 SDD Stte Encoding, Locls() nd Succ(t+ +tn) s homomorphisms

74 An exmple Y. Thierry-Mieg Octobre 8 74

75 A composite type Y. Thierry-Mieg Octobre 8 75

76 An elementry type (LTS) Y. Thierry-Mieg Octobre 8 76

77 XOR non deterministic Synchroniztions Y. Thierry-Mieg Octobre 8 77

78 Closing ring Y. Thierry-Mieg Octobre 8 78

79 A full system Y. Thierry-Mieg Octobre 8 79

80 Experimenttions (time) Y. Thierry-Mieg Octobre 8 8

81 Experimenttions (Memory) Y. Thierry-Mieg Octobre 8 8

82 Conclusions ITS Y. Thierry-Mieg Octobre 8 8 Still work in progress Provides (generl) wy of cpturing ptterns of similr behviors Experiments with hierrchy show tht most models of the benchmrk cn be encoded more efficiently with SDD An esy wy to profit from ITS : define your own (elementry) type(s) SDD stte encoding Homomorphism trnsition encoding Extensions of composite type lso possible Reset trnsitions, priorities