MAD - Inriapeople.rennes.inria.fr/Eric.Fabre/Papiers/Mad 5.pptx.pdf · 4 Limitaons • the product of automata does not make concurrency visible (creates concurrency diamonds), and

MADModels&AlgorithmsforDistributedsystems

--5/5--

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1

Today…

•  Anewmodelfordistributedsystems:Petrinets

•  Mainfeatures–  concurrencynaturally(graphically)encoded–  runseasilyencodedasparFalordersofevents–  languagesencodedasbranchingprocessesandunfoldings

(FghtlyrelatedtotheformalnoFonofeventstructure)

2

Whatdowehavesofar?

3

Model•  networkofautomata•  language=setofruns,arun=asequenceofevents•  factorizaFon•  aMazurkiewicztrace:

–  onewaytorecoverconcurrency,arunbecomesaparFalorderofevents–  encodingoftracesastuplesoflocalwords

Algebra•  projecFon&producton(networksof)automataandlanguages•  richproperFesdistributed/modularcomputaFonsinthisalgebra•  workingwithfactorizedformsislikeworkingwithtraces•  applicaFon:distributeddiagnosis,distributedplanning

A = A1 ⇥ ...⇥AN

L(A) = L(A1)⇥ ...⇥ L(AN ) ✓ ⌃⇤

w 2 w1 ⇥ ...⇥ wN

)

4

Limita1ons•  theproductofautomatadoesnotmakeconcurrencyvisible

(createsconcurrencydiamonds),andleadstostateexplosion•  thenaturalsequenFalsemanFcs(runsassequencesofevents)doesnot

capturewellconcurrency•  tracesareanindirectwaytorecoveratrueconcurrencysemanFcsfrom

sequences,where“and”ismadeequivalentto“”;onemayneedtodisFnguishthesesituaFons:

a � b b � a b ? a

α1t2

bcac

ad bd

β

β

γ γt αβ

a

b

t3t4 α

c

d

γ

*(t , )2 c

*(t , )2 d

( ,t )4b*( ,t )4a*1 3(t ,t )

“I can go first” ^ “you can go first” 6) “we can go at the same time”

Petrinets

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changeofnota1on•  automaton

–  transiFonsset–  onetransiFon

•  newnotaFon(PetriNetinspired)–  arefinitesetsofstates(places),transiFons,labels–  flowconnectstransiFonsandstates–  presetandsym.forpostset–  labelingoftransiFons

T ✓ S ⇥ ⌃⇥ St = (s,↵, s0) = (•t,�(t), t•)

A = (S, T,⌃, s0, SF )

A = (S, T,!, s0,�,⇤)S, T,⇤

! ✓ (S ⇥ T ) [ (T ⇥ S)8x 2 S [ T,

•x = {y 2 S [ T : y ! x} x

•

� : T ! ⇤

b

a

α3 t2 t1

βγt

6

Productwhere

•  Places:–  disjointunion(nottheproduct!)–  iniFalplaces

•  Transi1ons:asinglecopyofeachprivatetransiFon–  synchrooncommonlabels–  privatetransiFonsin1stcomp.–  privatetransiFonsin2ndcomp.

•  Flow:–  isdefinedbyand–  whereand

Ai = (Si, Ti,!i, s0,i,�i,⇤i)N = A1 ⇥A2 = (P, T,!, P0,�,⇤)

P = S1 ] S2

P0 = {s0,1, s0,2}

T = {(t1, t2) : �1(t1) = �2(t2)}[ {(t1, ?) : �1(t1) 2 ⇤1 \ ⇤2}[ {(?, t2) : �2(t2) 2 ⇤2 \ ⇤1}

(t1, t2)• = t•1 ] t•2

•(t1, t2) =•t1 ] •t2!

?• = ; •? = ;

7

Remarks•  ingeneral,asfortheproductofautomata,theassociaFonoftransiFonsis

notonetoone•  thisdefiniFonofproductextendsto(safe)Petrinets…•  …andmakestheproductassociaFve

Example

cfe

d

b

a g

αt3 t2 t1 t4 t5 t6t’1

α β β

t’4

fc

g

b

a d

e

5 t6t3 t2 t1 t4 t

A1 A2 A3

8

DynamicsinaPetrinet•  Marking:

–  afuncFon–  assignsanumberoftokenstoeachplace–  notaFon:ifplacescontainatmostonetoken(safenet)

•  EnablingofatransiFon–  transiFonisenabledatmarkingiff–  theresources/tokensneededbytarepresentinthecurrentmarking

•  FiringofatransiFon–  itchangesthecurrentmarkingmintom’with–  tconsumestokensinitspresent,andproducessomeinitspostset

N = A1 ⇥A2 = (P, T,!, P0,�,⇤)

m : P ! N

m ✓ P

t 2 T m ✓ P •t ✓ m

m0 = m� •t+ t•

e

da

b

g

cf

3 t2 t5 t6t4t1t

9

Trueconcurrencyseman1cs•  sequen1alseman1cs

–  arun=asequenceoftransiFonfirings,rootedatm0=P0

–  imposestheinterleavingofconcurrentevents–  differentinterleavings=differentruns

•  trueconcurrencyseman1cs–  arunisaparFalorderofevents–  encodedasanotherPetrinet,withoutcircuits,calledaconfigura1on

f

g

b

a

dc

a g e

b cd

c

g

b

a d

e

. . .

t3 t2 t1 t4 t5 t6

t4t3

t1

t5t1

Unfoldings

10

AsafePetrinet……andtwoofitsconfiguraFons(runs),asparFallyorderedevents

d

a g

dcb

b

a f

c

a g

dcb

ega

b

t1

t3

t1

t2

t6t4t3

t1

t5

fc

g

b

a d

e

5 t6t3 t2 t1 t4 t

Unfoldings

11

AsafePetrinet…mergingcommonprefixesyieldsanoccurrencenet

fc

g

b

a d

e

5 t6t3 t2 t1 t4 t

b

a g

dcb

b

a f

c

eg

d

t3

t1

t2

t6t4

t5t1

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Occurrencenet•  aspecialPetrinet•  placesarecalledcondiFons,transiFonsarecalledevents•  theflowisacyclic(parFalordering)•  andthisparFalorderiswellfounded

•  everycondiFonhasauniquecauseorisminimaland

•  noeventisinself-conflict

O = (C,E,!, C0,�,⇤)

!

8x 2 C [ E, |{y 2 C [ E : y !⇤x}| < 1

8c 2 C, | •c| 1 C0 = {c 2 C,• c = ;}

x#x

0 , 9e 6= e

0 2 E,

•e \ •

e

0 6= ;, e !⇤x, e

0 !⇤x

0

e e’

x’x

#

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concurrencyrepresentsnodesthatcanlieinthesameconfiguraFonco-set:suchthatrepresentsresources(tokens)thatareavailableatthesameFmeinsomerun/configuraFoncut:amaximalco-setforprefix:iffisacausallyclosedsub-netof,containingand

x ? y , ¬(x !⇤y) ^ ¬(y !⇤

x) ^ ¬(x#y)

X ✓ C 8c, c0 2 X, c ? c0

✓

O0 O C0 E0•O0 = (C 0, E0,!0, C0,�

0,⇤) v O

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configura1on:denoted,aconflict-freeprefixoflocalconfigura1on:[e]=smallestconfiguraFoncontainingevente,=causalpastofe

O

Lem:relaFngcutsandconfiguraFonsXisacutofsuchthatX=max(C’) ,O 9 = (C 0, E0, ...)

15

Branchingprocess•  abranchingprocessofnetisapair

whereisanoccurrencenet,andamorphismofnets(atotalfuncFon)

•  f“labels”condiFons/eventsofbyplaces/transiFonsofitturnsaconfiguraFonofintoarunof

•  parsimony:•  ifX =maximalcondiFonsinconfiguraFon(Xformsacut)

thenf(X) isthemarkingofproducedbyrun

(O, f)

f : O ! NO

N

NOO N

8e, e0 2 E, •e = •e0 ^ f(e) = f(e0) ) e = e0

N

f

c c

a g

bd

cb

b b bb d

egaa

t6t4t3 t3

t2 t1

t5t1t2t1t2

fc

g

b

a d

e

5 t6t3 t2 t1 t4 tf

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UnfoldingProof:mainideaistodefinetheunionofbranchingprocesses,ali8letechnical,butnotdifficult(seerefs.attheendofthelesson).Algorithm(unfolding)•  init

–  ,isomorphictothroughf– 

•  repeatunFlstability(extensionwithanewevent)–  foracosetandtransiFon t suchthat–  createevent(ifitdoesnotalreadyexist)suchthat–  createnewcondiFonsandextendfsothat

Thm:thereexistsauniquebranchingprocessofmaximalforprefixinclusion,itiscalledtheunfoldingof

(UN , fN ) NNv

•e = X, f(e) = t

E = ;, != ;C = C0 P0

X ✓ C f(X) = •t

e 2 E

X 0 = e• ⇢ C f(X 0) = t•

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Example

b c

ga a g

tt3 t2 t1 4

18

Example

c

ga

b c

a g

b

t3 t2 t1 t4 t1

19

Example

c

ga

b c

a g

b

t3 t2 t1 t4 t1

20

Example a a g

cbbc

g

b

1 4 t1t2t3 t2 t t

21

Example

bc

g

b

a a g

cb

4 t2 t1t3 t2 t1 t

22

Example

b

a

b c

ga

cb

ga

tt3 t2 t1 t4 t2

t3

1

23

Example

b c

ga

c

g

bb

a

a

tt3 t2 1t4 t2

t3

t1

24

Example

cb

b cc

g

b

a

b

ga

a

t3 t2 t1 t4 t2

t3

t1

t1

25

Example

b

bc

g

b

aa

a

cb

g

a

c

t3 t2 t1 t4 t2

t3 t3

t1

t1

26

Example

cb

c

g

b

a

g

b

ga

a

cb

a

t3 t2 t1 t4 t2

t3 t3 t4

t1

t1

27

Example

b

a

cc

g

b

b c

ga

a

b

a

b

c

g

t3 t2 t1 t4 t2

t3 t3 t4

t1 t1

t1

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Applica1onoftheunfoldingReachability/coverabilitytest•  onewishestoknowifthereexistsanaccessiblemarkingminnet

whereeachplaceofholdsatoken,i.e.•  bycreaFngintanewtransiFontwith

thisamountstocheckingiftisaccessible

Ques1ons1.  whatisthecomplexityofthistest?2.  howfarshouldonegointhecomputaFonoftheunfolding?

Q ✓ P Q ✓ m

Q = •t

N

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Proof:byreducFonofSATproblems(atleast3-SAT)example:encodingSATproblem•  Thecomplexityofunfoldingthisnet(beforetf )ispolynomial,

sothecomplexityoffindingaco-setwheretfisfirableisNP-hard.•  Asbuildinganunfoldingrequiresfindingco-sets,

onemustrelyonSATsolvers(whichmodernunfoldersdo).

Thm:thereachability/coverabilitytest(co-setconstrucFon)isNP-complete.

(x1 _ x2 _ x̄3) ^ (x̄1 _ x2)

falsetruefalsetrue true false

21x

OR

3

tf

OR clause 1

xx

clause 2

Finitecompleteprefix

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Idea:Aprefixissaidtobecompleteifallreachablemarkingsinarerepresentedas(theimageof)acutin.(OnewishestoavoiduselessrepeFFonsofsimilarpa8ernsin)Moreformally:iscompleteiff

–  reachablemarkingin,itappearsintheprefix

–  i.e.tfirablefromm,itappearsasaneventontopofmarkingm

O v UN

OO

O v UN

8m N9 2 O : m = Mark() = fN (max())

8t 2 T, m[tim0,

9,0 2 O : m = Mark(), 0 = � {e}, fN (e) = t

N

31

Howtobuildafinitecompleteprefix?Naiveidea:

–  applytheunfoldingalgorithm,andstopateventewhenthemarkingproducedby[e] isalreadypresentintheprefix:

–  thismakeseacut-offevent,ontopofwhichnomoreeventwillbeadded

Problem:itgenerallyyieldsanincompleteprefix…example:stopeventsinred,firingoft5notseen

9 2 O : Mark() = Mark([e])

a

b c

f

b c

eded

ff

ed

cb

a

t3

t1 t2

t4

t5

e1 e2

e3 e5 e6

e7 e8t5 t5

t3 t4t4t3

t2t1

e4

32

Solu1on:breakthesymmetry,byfavoringsomeconfiguraFonsforextensionAdequateorder:on(local)configuraFons[e]

–  wellfoundedparFalorder(finitenumberofpredecessors)–  refinesprefixinclusion:–  preservedbyisomorphicextensions:

Examples1.  takefortheprefixinclusion2.  definedbythenumberofevents(totalorder,proposedbyMcMillan)3.  takeforthelexicographicorder,whennetismadeofseveralcomponents,

byorderingcomponents,andcounFngeventsineachcomponent,asinMa8ern’svectorclocks(parFalorder,proposedbyEsparza)

�

@ 0 ) � 0

� 0 ^ Mark() = Mark(0) ) � e � 0 � e0 where fN (e) = fN (e0)

@�

� N

�

33

Cut-offevent:eventeisacut-offeventintheBPofiffthereexistsanotherevente’insuchthatExample(conFnued)

–  assume,whichmakese5 acut-offevent–  thisentails,byisomorphicextension

–  ,whichwouldmakee4acut-offevent,isfalse,asthiswouldentailbyisomorphicextension,whichisfalse

(O, f)N(O, f)

Mark([e0]) = Mark([e]) ^ [e0] � [e]

[e3] � [e5][e3, e4] � [e5, e6]

[e6] � [e4][e6, e5] � [e4, e3]

aa

b

f

c b c

eded

f

ed

cb

f

4t3

t1 t2

t4

t5

e1 e2

e3 e5 e6

e8t5

t3 t4t4t3

t2t1

e

e7 t5

34

Proof:seereferencesattheendofthelesson;finitenessandcompletenessareprovedseparately,andheavilyrelyonproperFesofadequateorders.

Thmtheprefixobtainedbystoppingtheunfoldingalgorithmatcut-offeventsisfiniteandcomplete[McMillan,Esparza].

O v UN

35

Proof:seereferencesattheendofthelesson;finitenessandcompletenessareprovedseparately,andheavilyrelyonproperFesofadequateorders.

Thmtheprefixobtainedbystoppingtheunfoldingalgorithmatcut-offeventsisfiniteandcomplete[McMillan,Esparza].

O v UN

ThmiftheadequateorderusedtobuidtheFCPisatotalorder,thenthenumberofnon-cut-offeventsinisboundedbythenumberofreachablemarkingsin.

O v UN�O

N

Proof:fortwoeventssuchthateitherorholds,sooneoftheseeventsisacut-off

Mark([e]) = Mark([e0])e, e0 2 O[e] � [e0] [e0] � [e]

36

Applica1ontodeadlockcheckingDeadlock:amarkingofwherenomoretransiFoncanbefiredN

ThmLetbeafinitecompleteprefix.ThereisnodeadlockiniffeveryconfiguraioncanbeextendedintoaconfiguraFonthatcontainsacut-offevent.[McMillan]

O v UN v O

v 0 v ON

Proof:(sketchof)•  amaximalconfiguraFonwithnocut-offcan’tbeextended:

theterminalmarkingisadead-end•  conversely,atacut-off,onereachesamarkingthatispresentandextended

elsewhereintheprefix,whichmeansthataconFnuaFonispossible

Takehomemessages

(Safe)Petrinets•  areanaturalmodelforconcurrentsystems•  canbebuildbyproduct,asnetworksofautomata•  admitnatural(builtin)trueconcurrencysemanFcsfortheirruns•  setsofrunscanbehandledbybranchingprocesses(unfoldings)

insteadoflanguages

Extraresults•  byrestricFngbranchingprocessestoevents,onegets

(prime)eventstructuresacompletetheoryofeventstructuresexists

•  onecandefineaproductonunfoldings,andprojecFonsexistsaswell,whichenablesdistributedcomputaFonsbasedbranchingprocesses,oronotherstructures(e.g.eventstructures).

37

U(N1 ⇥ ...⇥NK) = U(N1)⇥ ...⇥ U(NK)

E = (E,!,#)

References

38

Aboutprefixconstruc1on1.  AnunfoldingalgorithmforsynchronousproductsoftransiFonsystems,Esparzaand

Romer,proceedingsofCONCUR’99,pp2-202.  AnimprovementofMcMillan’sunfoldingalgorithm,EsparzaandRomer,LNCS1055,

pp87-106,19963.  CanonicalprefixesofPetrinetunfoldings,Khomenko,KoutnyandVogler,Acta

InformaFca40,pp95-118,2003

Moreorientedtomodel-checkingapplica1ons

1.  Modelcheckingusingnetunfoldings,Esparza,ScienceofComputerProgramming23,pp151-195,1994

2.  Reachabilityanalysisunsingnetunfoldings,SchroterandEsparza3.  Deadlockcheckingusingnetunfoldings,MelzerandRomer,LNCS1254,pp352-363,

19974.  UsingnetunfoldingstoavoidthestateexplosionproblemintheverificaFonof

asynchronouscircuits,McMillan,LNCS663,pp164-174,1992