Pattern Matching and Membership for Hierarchical Message Sequence Charts

Hierarhial Message Sequene Charts

BlaiseGenest andAnaMusholl

LIAFA,UniversitéParis VII2,pl.Jussieu,ase7014 F-75251Paris edex05e-mail:

{genest,musholl}liafa.jussieu.fr

Abstrat. Several formalisms and tools for software development use

hierarhy for system design, for instane stateharts and diagrams in

UML.Messagesequeneharts(MSCs) areastandardizednotationfor

asynhronouslyommuniatingproesses.ThenormZ.120inludesalso

hierarhialHMSCsinformofHigh-levelMSCs(HMSC).Algorithmson

MSCsrarelytakeinto aountall possibilities overedby thenorm. In

partiular,hierarhyis not takeninto aount sine the modelusually

onsidered are MSC-graphs that orrespond to the unfolding of (hier-

arhial) HMSCs. However, omplexity an inrease exponentially by

unfolding.Theaimofthispaperistoshowthatbasialgorithmsanbe

designedsuhthattheyavoidtheostlyunfoldingofhierarhialMSCs

and HMSCs. We onsider the membership and the pattern mathing

problemtoillustratethewaytoproeed.Weshowthatthemembership

problem for hierarhial HMSCs is PSPACE-omplete.Seond,we de-

sribeapolynomial-timealgorithmforthepattern-mathingproblemon

hierarhialMSCs.

1 Introdution

It isommonto usemaros towrite aprogramortospeifythe behaviorofa

system.Marosorhierarhialspeiationsallowamodulardesignofomplex

systemsandhavetheadvantageofbeingmoresuintanduser-friendly.Several

formalismsandtoolsforsoftwaredevelopmentusehierarhyforsystemdesign.

Oneofthemostprominentexamplesistheformalismofstateharts[11℄,whihis

aomponentofseveralobjet-orientednotations,suhastheUniedModeling

Language (UML). Besides stateharts, UML widely uses several kinds of dia-

grams(ativity,interationdiagramset),allbasedontheITUstandardZ.120

of message sequene harts (MSCs). Whilestateharts extendnite state ma-

hinesbyhierarhyandommuniationmehanisms,MSCsareavisualnotation

forasynhronouslyommuniatingproesses.TheusualappliationofMSCsin

teleommuniationisforapturingrequirementsofommuniationprotoolsin

formofsenariosinearlydesignstages.MSCsusuallyrepresentinompletespe-

iations, obtainedfrom apreliminary viewofthe systemthat abstrats away

severaldetailssuhasvariablesormessageontents.High-levelMSCs(HMSCs)

ombinebasiMSCsusinghoieanditeration,thusdesribingpossiblyinnite

olletions of senarios. For abstrat speiations aswith HMSCs, hierarhy

whihanbeveryabstrat,adesigner should be ableto merge dierentspei-

ation asesyielding thesameabstrat senarioandto use thissenarioasa

maro.Byusing marosdesignersmayidentify subsenarioswhihhaveto be

renedatalaterstage.

Algorithms on MSCs rarely take into aount the whole spetrum of the

HMSC standard denition. In partiular, hierarhy is not taken into aount

sinethemodelsusuallyonsideredareMSC-graphs,thatorrespondtotheun-

foldingof(hierarhial)HMSCs.However,omplexityaninreaseexponentially

by unfolding. Theaim of this paper is to show that this exponential blow-up

is avoidablein many ases,by avoiding theexpensive unfoldingand using the

hierarhyforomputingthedesiredresultsinamodularway.Weusetehniques

stemmingfromombinatorisonompressedtexts,sinehierarhialMSCde-

nitionsanbeseenasakindofompressionbymeansofStraight-LinePrograms

(SLP).

In this paper we onsider two fundamental problems for hierarhial HM-

SCs,that arealledherenestedhigh-level MSCs(nHMSCsforshort): member-

shipproblemandpatternmathing.However,wethinkthetehniquesdesribed

herean beused to solveotheralgorithmiproblems onnHMSCs aswell.The

membershipproblemisabasiquestion,askingforinstanewhetheranegative

senarioours inasystemspeiation,oraskingwhether apositive senario

isredundant,sinealreadyoveredbythespeiation.Withouthierarhy,the

membership problem for HMSCshas been shown to be NP-omplete,[1℄. The

reasonfor this omplexity blow-up (ompared to nite-state mahines) is that

MSCsarepartial-ordermodels.We showthathierarhyyieldsasmallinrease

in omplexity, preiselyweshowthat the membership problem fornHMSCs is

PSPACE-omplete. Surprisingly, hierarhyalone is the soure of theomplex-

ity.Weshownamelythat themembershipproblemforhierarhialautomatais

alreadyPSPACE-omplete.Thisresultshowsadierenebetweenmembership

and reahability,sinereahabilityforommuniatinghierarhialautomata is

alreadyEXPSPACE-omplete[12℄.

TheseondproblemonsideredinthispaperispatternmathingfornMSCs.

GiventwonMSCs

M, N

^{,wewanttoknowwhether}

M

^{oursasapatternof}

N

^.A

polynomialtimesolutionforthisproblemisnotimmediate.Weapplysomenie

ombinatorialtehniquesstemmingfrompatternmathingonompressedtexts

andweobtainanalgorithmoftime

O(|C _M | ² · |M | ² · |N | ² )

^,where

|M |, |N|

^denote

thesizes of thedesriptionof

M

^and

N

^{, and}

|C M |

^{is the numberof onneted}

omponentsintheommuniationgraphof

M

^{.Thisquestionsubsumesthetest}

ofequalityoftwonMSC,andshowsthatequalityisdeidableinPTIMEaswell.

Related work. Regarding the omplexity of extended nite state mahines,

[12℄ onsidersthereahabilityandtraeequivaleneproblemsfor ommuniat-

ing FSMs(Finite States Mahines). Model-hekinghierarhialFSMsagainst

LTL and CTL properties is the topi of [4℄. The paper [3℄ ombines hierar-

hyandonurreny,analyzingtheomplexityofseveralproblems(reahability,

equivaleneet.)forommuniating,hierarhialFSMs.

reently,e.g. deteting raes[2,18℄, model-heking[5℄, pattern mathing with

gaps[19℄,inferene[1℄andrealizability[17,9,14℄,model-hekingagainstpartial-

order logis[16,21℄.HierarhialMSCshavebeenalso onsideredin [5℄forthe

model-hekingproblem.WenotehoweverthatourdenitionofnestedHMSCs

apturesalargerlassofMSCspeiationsthan[5℄.

An extendedabstrat of this paper waspresentedat LATIN'02 [8℄.As ad-

ditional resultweshowherehowto extend thepolynomialtime algorithm for

patternmathingnMSCsto theasewherethepatternisnotonneted.

2 Syntax and Semantis of Nested MSCs

Weadoptthedenition of(basi)messagesequeneharts (MSCforshort), as

desribedin theITU-standard[13℄.

Denition1. (Message Sequene Charts.) A messagesequene hart isa

tuple

M = hP, E, C, ℓ, m, <i

^where:

P

^{isaniteset ofproesses,}

E

^{isaniteset ofevents,}

C

^{isanitesetof names formessages andloal ations,}

ℓ : E → T = {i!j(a), i?j(a), i(a) | i 6= j ∈ P, a ∈ C}

^{labels eah event with}

its type: on proess

i ∈ P

^{,the type iseither a send}

i!j(a)

^{of message}

a

^to

proess

j

^{,or a reeive}

i?j(a)

^{of message}

a

^{from proess}

j

^{,or a loal event}

i(a)

^{. The labeling}

ℓ

^{partitions the set of events by type (send, reeive, or}

loal),

E = S S ^· R S ^·

L

^{,andbyproess,}

E = S ^·

i∈P E i

^{.Wedenote by}

P (e)

^the

proessof event

e

^(i.e.,

P (e) = i

ⁱ

e ∈ E i

^).

m : S → R

^{isabijetion mathingeahsendtotheorresponding reeive.If}

m(s) = r

^{, then}

ℓ(s) = i!j(a)

^and

ℓ(r) = j?i(a)

^{for some proesses}

i, j ∈ P

andsomemessagename

a ∈ C

^{.Wedenotetheevents}

s, r

^{asmathingevents}

andthepair

(s, r)

^asmessage.

< ⊆ E × E

^{isanaylirelationbetween eventsonsistingof:}

•

^{atotalorder on}

E i

^{,for everyproess}

i ∈ P

^,and

• s < r

^,whenever

m(s) = r

^.

Theupperleft partofFigure 1depitsanMSC Monthree proesseswith

twomessagesandfourevents.Eahvertialline orrespondstoaproess,with

timeinreasingfromtoptobottom.

Forthequestionsonsideredhere,messagenamesareirrelevant.Thus,send

eventswillbeoftype

i!j

^{andreeiveeventsoftype}

i?j

^{.Moreover,wheneverwe}

referto anMSC in this paper, we meanatually its isomorphismlass, where

anisomorphismonthesetofevents

E

^{isabijetionthatisompatible withthe}

typefuntion

ℓ

^{and themessagefuntion}

m

^.

For ommuniationprotoolsit isnaturalto assumethateahommunia-

tion hannel delivers messages rst-in-rst-out (FIFO). We assume the FIFO

onditionthroughoutthepaper.Thatis,forallmessages

(e k , f k )

^,

k = 1, 2

^,suh

that

ℓ(e 1 ) = ℓ(e 2 )

^and

ℓ(f 1 ) = ℓ(f 2 )

^{we requirethat}

e 1 < e 2

ⁱ

f 1 < f 2

^{. The}

reexive-transitivelosure

≤

^{of the aylirelation}

<

^{is apartial orderalled}

visual order.Everytotalorder on

E

^extending

≤

^isthenalled linearizationof

M

^{. A onguration (prex)}

C

^{of an MSC}

M

^{is a downward losed subset of}

events,thatis,if

e < f ∈ E

^with

f ∈ C

^,then

e ∈ C

^.

Notethat with the FIFOmessageorder,anytotalorder on aset of events

E

^{denes at mostoneMSC. We obtainthis MSCfrom theeventsequene by}

mathing the

n

^{-th sendfrom}

i

^to

j

^{with the}

n

^{-th reeiveon}

j

^from

i

^{, foreah}

pairofdistintproesses

i, j

^.

Aspeialaseofthepatternmathingproblemonsideredinthepaperisthe

equalitytestoftwo(nested)MSCs.Inorderto hektheequalityoftwoMSCs

M, N

^(i.e.,uptoisomorphism)oneanhooseanylinearizationof

M

^andhek

whether it is a linearization of

N

^{, too. An} alternative approah, that will be used inouralgorithms,is tohekequalityoneahproess.Thus,foranMSC

M = hP, E, C, ℓ, m, <i

^{and a proess}

i ∈ P

^{we let}

M | i

^{denote the projetion}

of

M

^{on the set}

E i

^{of events loated on}

i

^{. We have}

M = N

^{if and only if}

M

^and

N

^{have thesame set of proesses,that is}

P(M ) = P(N ) = P

^{, and if}

theirprojetiononanyproessisequal,thatis

M | i = N | i

^foreah

i ∈ P

^(upto

isomorphism).NotethatbothtestsrelyontheFIFOorderofmessages.Without

theFIFOorder,alinearization(ortheprojetionsonproesses)doesnotsue

forreoveringtheMSC.Forexample,thelinearization

s 1 s 2 r 1 r 2

^where

s 1 , s 2

^are

sendsand

r 1 r 2

^{arereeivesfrom proess1toproess2,anproduetwoMSCs,}

onewhere

m(s 1 ) = r 1 , m(s 2 ) = r 2

^andonewhere

m(s 1 ) = r 2 , m(s 2 ) = r 1

^.

Wefollowthe ITU norm and dene nestedMSCs (nMSC for short) by al-

lowing the reuse of an already dened MSC in adenition. Thedenition we

givebelowaimsatpreservingthevisualharaterofMSCs(seealsoFigure1).

Denition2. (Nested MSC, nMSC.) A nested MSC

M = (M q ) ⁿ _q=1

^{is a}

nite sequene ofmarosof the form

M q = hP q , E q , B q , ϕ q , C, ℓ q , m q , < q i

^.

Eahmaro

M q

^onsistsof:

Anite set

E q

^ofevents.

Anite set

P q

^{of proesses.}

Anite set

B q

^{of referenes (boxes)usedby}

M q

^.

A funtion

ϕ q

^{that assoiates eah referene}

b ∈ B q

^{with an index}

q <

ϕ q (b) ≤ n

^{. Thus, referene}

b

^{refers to the maro}

M ϕ q (b)

^{.We require that}

P ϕ q (b) ⊆ P q

^.

The type funtion

ℓ q : E q −→ T

^{, that assoiates eah event with a type}

i!j, i?j

^or

i(a)

^{, with}

i, j ∈ P q

^,

i 6= j

^and

a ∈ C

^{. The labeling}

ℓ

^partitions

the set of events by type (send, reeive, or loal),

E q = S q S ·

R q S ·

L q

^{, and}

byproess,

E q = S ^·

i∈P E q,i

^{.Wedenote by}

P (e)

^{the proess ofevent}

e

^(i.e.,

P(e) = i

ⁱ

e ∈ E q,i

^).

The messagefuntion

m q : S q −→ R q

^{that maps eah (send) event of type}

i!j

^{with a(reeive)event oftype}

j?i

^,forall

i 6= j

^.

Theaylirelation

< q

^{overthesetofeventsandreferenes}

E q ∪ B q

^,dened

by:

•

^Foreahproess

k ∈ P q

^,therelation

< q

^{isatotalorderovertheset}

E q,k

of eventsloated on

k

^{andthe setof referenes}

b ∈ B q

^with

k ∈ P ϕ q (b)

^.

• e < q f

^whenever

m q (e) = f

ⁱⁿ

M q

^.

The nesting depth of

M

^{isthe maximal}

d

^{suh that thereexists somesequene}

q 1 < · · · < q d+1

^with

ϕ q j (b) = q j+1

^{for some}

b ∈ B q j

^{,for all}

1 ≤ j ≤ d

^.

We dene the indies suhthat the lowest levels of hierarhy, whih stands

forlevelswhihdonotusereferenestootherlevel,orrespondstosmallindies.

M S

S

S M S

S M

S

P

M M h

h g

g g

b ₅

b ₂

b ₃ b ₁

b ₄ 1 2 3

1 2 3 1 2 3

1 2 3

k

k

k

e

f

Fig.1.AnnMSC

P

^{usingtworeferenes,}

S

^and

M

^.

Example 1. Consider thenMSC

P

^{in Figure 1.It uses three referenes,}

B P = {b 1 , b 2 , b 3 }

^{that orrespond to}

ϕ P (b 1 ) = ϕ P (b 3 ) = S

^and

ϕ P (b 2 ) = M

^{. The}

nesting depth of

P

^{is 2. The visual order}

< P

^of

P

^{requires on proess 1 the}

order

b 1 < P e < P b 2 < P b 3

^{.Notiethat thedenition ofanMSC forbids}

(f, e)

to beamessage,with

f

^thesendand

e

^{thereeive,sinethiswouldontradit}

theayliityof

< P

^{, evenintheasewhere}

M

^{wouldbeempty.}

Thesemantisof annMSC is theMSC dened byreplaing eah referene

of

M

^bytheorrespondingMSC.Indutivelyitsuesto denethesemantis of nMSCs of nesting depth one. Let

M = (M q ) ⁿ _q=1

^{be an nMSC of nesting}

depthone,with

M q = hP q , E q , B q , ϕ q , C, ℓ q , m q , < q i

^.Forsimplifyingthenotation below,wewriteinsteadof

ϕ 1 (b)

^just

b

^.

The MSC

hP, E, C, ℓ, m, <i

^{dened by}

M = (M q ) ⁿ _q=1

^{is given by}

P = P 1

^,

E = S ^·

b∈B ¹ E b S ·

E 1

^,

ℓ = ∪ ⁿ _q=1 ℓ q

^and

m = ∪ ⁿ _q=1 m q

^{.Thevisualorder}

<

^isdened

by

e < f

^{ifandonlyifeither}

m(e) = f

^,or

P (e) = P (f )

^{andoneofthefollowing}

onditionsholds:

e, f ∈ E 1

^and

e < 1 f

^,

e, f ∈ E b

^and

e < b f

^,

e ∈ E 1 , f ∈ E b

^and

e < 1 b

^,

e ∈ E b , f ∈ E 1

^and

b < 1 f

^,

e ∈ E b , f ∈ E b ^′

^and

b < 1 b ^′

^,

where

b, b ^′ ∈ B 1

^{.Forsimpliity, wedenote the MSCdened by}

M = (M q ) ⁿ _q=1

as

M

^,too.

Example 2. For the nMSC

P

^{in Figure 1, the lower right part of the piture}

showstheMSCdened by

S

^{.Note that event}

g ∈ E M

^ourstwiein

S

^for

simpliity,wedenotebothourrenesas

g

^.

Notealso that thesemantis requires that

b 1 < 1 e

^{, but this doesnotmean}

that all events of

S = ϕ P (b 1 )

^{must happen before}

e ∈ E P

^{. Forinstane, the}

rst ourrene of

g

ⁱⁿ

S

^{preedes event}

e

^of

P

^{, but the seond ourrene is}

onurrentwith

e

^.

Obviously,asyntatiallyorretnMSC

M

^{mightnotyieldanMSCbeause}

oftheFIFOorder.Forexample,themessage

(e, f )

^of

P

^{wouldviolatetheFIFO}

onditionif

M

^{wouldontainamessagefromproess1toproess3.F}ortunately, itanbeveriedeasily(polynomialtime)whether annMSCsatisestheFIFO

ondition. To test for the FIFO ondition, it sues to test that there is no

e < g < h < f

^{and no}

e < b < f

^with

b

^{ontainingasend from}

i

^to

j

^{, where}

(e, f ), (g, h)

^{aretwomessagesfrom}

i

^to

j

^.

Size of nMSC. For omplexity estimations we will denote by

℘

^{the overall}

numberof proesses.The size ofan nMSC

M

^{is denoted as}

|M |

^{. It represents}

the size of thesyntatial desription of

M

^{, where anevent is ofsize oneand}

thesizeofarefereneisthenumberofitsproesses.

3 Nested High-Level MSC

AnMSCanonlydesribeanitesenario.Forspeifyingmoreomplexbehav-

iors,in partiularinnitesetsof senarios,the ITUnorm proposesto ompose

MSCsinform ofMSC-graphs,byusing hoieanditeration.

Denition3. (MSC-graph)AnMSC-graphisgivenasatuple

G = hV, E, s, f, ϕi

^,

where:

(V, E)

^{isadiretedgraph withstartingvertex}

s ∈ V

^andnalvertex

f ∈ V

^.

Eahvertex

v

^{is labeledbythe MSC}

ϕ(v)

^.

eralizeMSC-graphstohierarhial HMSCs(ornestedhigh-levelMSCs,nHMSC

forshort).

Denition4. (Nested high-level MSC.) An nHMSC is a nite sequene

G = (G q ) ⁿ _q=1

^{, where eah}

G q

^{is either a labeled graph or an nMSC. A labeled}

graph

G q

^isatuple

hV _q , E q , ϕ q , s q , f q i

^onsistingof:

Adiretedgraph

(V q , E q )

^{withstarting vertex}

s q

^andnalvertex

f q

^.

Afuntion

ϕ q

^{thatassoiateseahvertex}

v

^{withareferene}

q < ϕ q (v) ≤ n

^,

representing

G ϕ q (v)

^.

Thus, a node in an nHMSC anbemapped either to some graphorto an

nMSC.Thisdenitionombineshierarhialautomataasdenedin[4℄withour

denition of nMSC. The speial ase where there is only one proess (i.e., no

onurreny)yields thehierarhialautomata usedin[4℄

1

.

We rst need to dene the omposition of two MSCs

N 1 N 2

^with

N k = hP _k , E k , C _k , ℓ k , m k , < k i

^.Intuitively,wejustgluetogetherthetwodiagramsproess- wise.Let

N 1 N 2 = hP, E, C, ℓ, m, <i

^with

E = E 1 S ·

E 2

^,

P = P 1 ∪P 2

^,

C = C 1 ∪ C 2

^,

ℓ = ℓ 1 ∪ ℓ 2

^,

m = m 1 ∪ m 2

^and

< = < 1 ∪ < 2 ∪ [

i∈P

E 1,i × E 2,i .

ThesemantisofannHMSC

G = (G q ) ⁿ _q=1

^{isa(possiblyinnite)setofMSCs}

L(G)

^denedreursively.If

G q

^{isannMSC,then}

L(G q )

^{isasingletononsisting}

of theMSCdened by

G q

^{. Letus onsider alabeledgraph}

G q

^{. Then}

L(G q )

^is

thesetofMSCsassoiatedwiththeaeptingpathsof

G q

^{,thatis,pathsstarting}

in

s q

^andendingin

f q

^{.With apath}

v 1 , . . . , v n

ⁱⁿ

G q

^{weassoiatetheset ofall}

MSCs

M 1 · · · M n

^,where

M i ∈ L(G ϕ q (v i ) )

^forall

1 ≤ i ≤ n

^{.Thesetofexeutions}

of

G

^isdenedas

L(G) = L(G 1 )

^.

Asin[1℄wealsoonsideraweakersemantisfornHMSCs,thatdoesnotuse

the omposition of MSCs (alled weak losure in [1℄). This semantisis based

on taking the produt of the sequential behaviors of single proesses. Several

algorithmiproblemsanbesolvedmoreeientlyfortheweaklosureofMSC-

graphs.Thismakesitinterestingtoompareitwiththeusualsemantisalsoin

thesettingofnHMSCs.

WeaklosureofnHMSC.Let

G

^{beannHMSC.Then}

L ^w (G)

^{denotestheset}

ofMSCs

M

^{suhthatforeahproess}

i

^{thereissomeMSC}

N ∈ L(G)

^suhthat

M | i

^isequalto

N | i

^.Notethat

L(G) ⊆ L ^w (G)

^{andthattheinlusionisstrit,in}

general(see[1℄).

1

Atually, [4℄ allows several nal nodes in eah automaton, whih ounts for the

omplexityoftheiralgorithms.

a b G i

G i

Fig.2.AnnHMSC

G i+1

^generating

(a + b) ^{2 i − 1}

^with

G 1 = ǫ

^.

4 Membership Problem

ChekingthemembershipofanMSC

M

^{inanMSC-graph}

G

^{isusedtypiallyfor}

hekingthatnobadsenarioanourinagivenspeiation.Anotherapplia-

tionishekingwhetheragoodsenarioisalreadyoveredbythespeiation.

Cheking membership is notan easy task already beause of the onurreny

impliedbytheMSComposition,allthemorein thepreseneofhierarhy.The

MSC membership problem

M ∈ ^? L(G)

^with

M

^{anMSC and}

G

^{an MSC-graph}

wasonsideredin[1℄,togetherwiththeweakmembershipproblem

M ∈ ^? L ^w (G)

^.

Theresultsof[1℄anbesummarizedasfollows:

TheMSCmembership problem isNP-omplete. Adeterministi algorithm

oftime

O(|G| · |M | ^℘ )

^{solvesit2,where}

℘

^{isthenumberofproesses.}

Theweak MSCmembershipproblemissolvablein time

O(|G| · |M |)

^.

SotheMSCmembershipproblemissolvableinpolynomialtimeifwexthe

numberofproesses.

4.1 Hierarhial MembershipProblem

Themembershipproblemseemsapriori morediultforannMSC

M

^against

an nHMSC

G

^{, sinethe naive approah of guessing apath of}

G

^{and heking}

equality with

M

^{is too expensive (both the path of}

G

^{and the MSC dened}

by

M

^anbeofexponentialsize).However,itis easyto showthat weantest membershipinpolynomialspae:

Theorem1(Hierarhial MSC MembershipProblem) Given an nMSC

M

^andannHMSC

G

^{,wean deide whether}

M ∈ ^? L(G)

^{inpolynomialspae.}

2

Thisisaslightlyimprovedruntimeomparedtotheresultstatedin[1℄.

L(G)

^{andwemathitagainstthenMSC}

M

^{,howeverexpandingneither}

M

^nor

G

^{.Reall thatfor testingequalityoftwoMSCs}

M, N

^{,itsuesto hooseone}

linearizationof

N

^{and hekwhether it isa}linearizationof

M

^{. Hene,wean}

hoose the linearization of the MSC in

G

^{(as long as we do not exlude any}

MSC in

L(G)

^{, that is as longas we do notexlude every} linearizationof one MSC).Weonsideronlythelinearizationsin

Lin ⁰ (G)

^,where

Lin ⁰ (G)

^isdened

reursively.If

G q

^{is annMSC, then}

Lin ⁰ (G q )

^{is theset of}linearization of

G q

^.

With apath

v 1 , . . . , v n

ⁱⁿ

G q

^{weassoiatethesetof all}linearizations

u 1 · · · u n

^,

where

u i ∈ Lin ⁰ (G ϕ q (v i ) )

^forall

1 ≤ i ≤ n

^{.Letus onsideralabeledgraph}

G q

^.

Then

Lin ⁰ (G q )

^isthesetoflinearizationsassoiatedwiththeaeptingpathsof

G q

^{,thatis,pathsstartingin}

s q

^andendingin

f q

^.Wedene

Lin ⁰ (G) = Lin ⁰ (G 1 )

^.

Intuitively,itmeansthatwedonotonsiderlinearizations

uavbw

^ofpath

v 1 · · · v n

where

a

^{belongstoanode}

v i

^and

b

^to

v j

^with

j < i

^{,thatiseverynodeneedsto}

befully exeutedbeforethenextnodeanbeonsidered.

Weneedto storeaongurationof

M

^,orrespondingto theeventsalready mathedwiththeeventsfrom

G

^{.Sineaongurationisadownwardlosedset}

ofevents,itanbestoredasatupleof

℘

^{events(remindthat}

℘

^{isthenumberof}

proesses),representingthelasteventoftheongurationoneahproess.Suh

atupleisoflinearsizew.r.t.thesizeof

M

^.Eahevent

e

^of

M = (M q ) ⁿ _q=1

^willbe

representedbyasequene

b 1 , . . . , b m

^ofreferenesorrespondingtotheunfolding ofreferenesyielding

e

^{.Thatis,weindutivelyremind}

b m

^for

e ∈ E ϕ(b m )

^where

b m

^{isarefereneof}

ϕ(b m−1 )

^{,plustheposition of}

e

ⁱⁿ

M ϕ(b m )

^{.Thus,eahevent}

anbestoredusinglinear-sizememory.Inourgure1,therstourreneof

g

in

P

^orrespondsto

(b 1 , b 4 , g)

^{,theseondourreneto}

(b 1 , b 5 , g)

^,andsoon.

Similarly,weanstoretheurrentongurationofthelinearizationin

Lin ⁰ (G)

inspaepolynomialin

|G|

^(aneventof

G

^isrepresentedbyasequeneofnodes thenofreferenes).Sineanewnodeisstartedonlywhenthepreviousnodeis

fully exeuted, thelast eventfor everyproess belongsto thesamenode. The

non-deterministialgorithmonsistsin guessingasuessorongurationof

G

^,

obtainedbyextendingtheurrentongurationbyanevent

e

^{suhthatthenew}

onguration is still aprex of somelinearizationin

Lin ⁰ (G)

^{. Then we hek}

that

e

^{an extendthe urrent} linearizationof

M

^{as well. Thealgorithm stops}

whentheongurationthatorrespondstothepathbeingguessedin

G

^isequal

to

M

^andthepathof

G

^isaepting.

2

Theorem2belowshowsthatPSPACEisthelowestomplexityweanobtain

forthehierarhialmembershipproblem.Thelowerboundholdsevenifthereis

onlyoneproess(Theorem2),orifthegraph

G

^{isnothierarhial(Theorem3),}

but notboth(Theorem4).Thisshowsalsothat xingthenumberofproesses

doesnotlowertheomplexityoftheproblem,unlikeinthenonhierarhialase.

WeshowthePSPACElowerboundforthefollowingproblem:givenastraight-

line program

W

^{(see below) and a hierarhial automaton}

A

^{, test whether}

W ∈ L(A)

^{.This questionorrespondsto thehierarhialmembership problem}

withasingleproess.Notiealsothattheweakmembershipproblem

M ∈ ^? L ^w (G)

[1℄anbereduedto thisquestion.

Straight-line programs. A straight-line program (SLP for short) over the

alphabet

Σ

^{is a}ontext-freegrammar withvariables

V = {X 1 , . . . , X k }

^{, initial}

variable

X 1

^{and rules from}

V × (V ∪ Σ) ⁺

^{. The rules are suh that there is}

exatlyoneruleforeahleft-hand side variableand if

X i −→ α

^{, theneah}

X j

in

α

^satises

j > i

^.

Theonstraintsontherules makethat anyvariable

X i

^{generatesaunique}

word.For onveniene, we denote the word generated by the variable

X i

^also

as

X i

^{. Thelengthofavariable}

X i

^{representsthelengthofthewordgenerated}

by

X i

^{andisdenotedas}

||X i ||

^.Clearly,

||X i ||

^anbeatmostexponentialinthe number of rules. The size

|X i |

^{of an SLP is the sumof thesizes of the rules.}

Withoutlossofgenerality,weanassumethatrulesareofsize2,thatisofthe

form

X −→ Y Z

^with

Y, Z ∈ V ∪ Σ

^.

SineanyMSC

M

^{is determinedby itsprojetions}

(M | _i ) i∈P

^{, annMSC}

M

anbeidentiedwith

℘

^SLPs

L ¹ , . . . , L ^℘

^.TheSLP

L ⁱ

^{generatestheprojetion}

M | i

^of

M

^{onthe set of eventsof proess}

i ∈ P

^{. We denotethe variables used}

by

L ⁱ

^as

X | i

^{, where}

X ∈ {M q | q = 1 · · · n}

^{. The initial variable of eah}

L ⁱ

is

M n | i

^{. Atually, the SLPs are not in Chomsky normal form to preserve this}

representationofnMSCs.

Example 3. ForthenMSC

P

^{isFigure1wehavethefollowingSLPgenerating}

theprojetiononproess1:

P | 1 → S| 1 eM| 1 S| 1

^,

S| 1 → M | 1 hM | 1

^and

M | 1 → k

^.

Theorem2 ItisPSPACE-ompletetohekwhether

W ∈ L(A)

^forsomeSLP

W

^{andhierarhialautomaton}

A

^{.Ifthe alphabetisunary,thenthemembership}

problem isNP-omplete.

Remark1 TheNP-hardnessresultintheunaryasefollowsalsofrom [23℄.

Proof.Werstredue(1-in-3)SATtotheunarymembershipproblem,sine

weusethisredutioninthegeneralase,too.ThisproblemisNP-omplete,see

[24,6℄.

Let

ϕ = ∧ ^m _j=1 C(α j , β j , γ j )

^{be an instane of (1-in-3) SAT over}

n

^variables

(x i ) i=1,n

^.Here,alause

C(α j , β j , γ j )

^{istrueiexatlyoneoftheliterals}

α j , β j , γ j

istrue.Weusetheunaryalphabet

{a}

^{.Note thatanyword}

x ∈ a ^∗

^isuniquely

dened byitslength.

Weassoiatewitheahlause

C j = C(α j , β j , γ j )

^theword

w j ∈ a ^∗

^oflength

4 ^j

^{.ThiswordanbedenedbyanSLPofpolynomialsize.Let}

W = w 1 · · · w m ∈ a ^∗

^{be theword oflength}

P m

j=1 4 ^j

^{. Theautomaton}

A

^{onsists ofasequene of}

hoies with transitions labeledby

t i

^and

f i

^{, for}

i

^{varying from 1to}

n

^{, where}

t i = P

j∈R i 4 ^j

^and

R i = {j | x i ∈ {α j , β j , γ j }}

^{.Inthesameway,}

f i = P

j∈S i 4 ^j

and

S i = {j | (¬x i ) ∈ {α j , β j , γ j }}

^.

t

f

A ^t

f

1

1

n

n

Anypath

ρ

^of

A

^{orrespondstoavaluation}

σ

^{where eahvariable}

x i

^istrue

ifthepathhooses

t i

^{,andfalseifithooses}

f i

^.Let

n j

^{bethenumberofliterals}

of

C j

^{that areset trueby}

σ

^{. Reallthat}

σ

^{satisestheformula}

ϕ

ⁱ

n j = 1

^for

all

j

^{.Itiseasytoseethat}

ρ

^{islabeledbytheword}

L ∈ a ^∗

^oflength

P m j=1 n j 4 ^j

^.

Notie that sine eah lause hasthree literals,

n j ∈ {0, 1, 2, 3}

^{for all}

j

^{. The}

lengthof

L

^{inbase4isthus}

(n m n m−1 . . . n 1 0) 4

^.Wehave

W = L

ⁱ

(11 . . . 10) 4

⁼

(n m n m−1 . . . n 1 0) 4

^,thusi

n j = 1

^forall

j

^{.Thatis,thereisapathin}

A

^labeled

by

W

^{i there is avaluation satisfying}

ϕ

^{, whih implies that the membership}

problemforhierarhialautomatononaunaryalphabet isNP-hard.

WenowshowtherststatementofTheorem2.Wereduethe(1-in-3)QBF

(one-in-threequantiedboolean formula)to thehierarhialmembershipprob-

lem. Let

ϕ

^{be an instane of (1-in-3) QBF of the form}

ϕ = Q n x n · · · Q 1 x 1 ψ

^,

where

Q i ∈ {∃, ∀}

^{andtheformula}

ψ

^isoftheform

∧ ^m _j=1 C(α j , β j , γ j )

^.Asbefore,

alause

C(α j , β j , γ j )

^{istrueiexatlyoneliteralistrue.The}PSPACE-hardness ofthisproblemisshownin[24,6℄.

Theidea is to makethe valuations of thevariables orrespondto pathsin

the hierarhial automaton

(A i ) i=0,n

^{and to validate the valuations using the}

SLPs

(W i ) i=0,n

^{.Wedenetheautomata}

A i

^{and theSLPs}

W i

^{byindution on}

i = 0, . . . , n

^{.Here,weusethebinaryalphabet}

{a, b}

^.Theletter

a

^{will havethe}

samemeaning asin theNP-ase, and the letter

b

^{will be used asadelimiting}

symbol.

We dene the words

w j , t i , f i ∈ a ^∗

^{with respet to}

ψ

^{as before. That is,}

eah

w j

^{is assoiated with lause}

C j

^and

t i , f i

^{are assoiated withvariable}

x i

^.

Moreover, we assoiate with eah variable

x i

^{the word}

w i+m ∈ a ^∗

^{of length}

4 ^i+m

^{. Let}

W 0 = w 1 · · · w n+m

^bethewordof

a ^∗

^oflength

P n+m

j=1 4 ^j

^,andlet

A 0

beanautomatononsisting ofone

ǫ

-transition from itsinitial statetoits nal state.Letalso

S 0

^{beanautomatononsistingofonetransitionlabeledby}

b

^.The

SLP-ompressedwords

(W i ) i=1,n

^,aredenedby:

W i −→ W _i−1

^{, if}

Q i = ∃

^,

W i −→ W i−1 b W i−1

^,if

Q i = ∀

^.

Thereursive denition of the automata

(A i ) i=1,n

^and

(S i ) _i=0,n−1

^{is illus-}

tratedinthegurebelow.Transitionsareeitherlabeledby

ǫ

^,orbyxt

i = t i w i+m

orxf

i = f i w i+m

^{. The automatonon theleft denes}

A i

^when

Q i = ∀

^{, theau-}

tomatoninthemiddledenes

A i

^when

Q i = ∃

^{,andtheautomatonontheright}

denes

S i

^{.Notethatthesymbol}

b

^{isonlygeneratedby}

S 0

^{.Inthegurewereall}

itspositionbymarkinga

b

^asideeah

S i

^.

A _i A _i-1

xt i

xf i

A _i xt _i

A _i-1 xf i

Sn-i

n-i-1

S S n-i-1

xt _i+1

xf i+1

b b

A i-1

b S _n-i

Theoverall ideaisasfollows.Thevaluesof

x i+1 , . . . , x n

^{arealreadyhosen}

whenanautomatonalls

A i

^{(fromahigherhierarhylevel).Theautomaton}

A i

ontheleftsets

x i

^{true,thenuses}

S _n−i

^{toreoverthexedvaluesof}

x i+1 , . . . x n

^,

andnallyitsets

x i

^{false.Theautomaton}

A i

^{inthemiddleguesseswhether}

x i

^is

true(bytakingthetransitionlabeledbyxt

i

^{)orfalse(byhoosingthetransition}

labeledbyxf

i

^{).Ifithoosesboth}transitionslabeledbyxt

i ,

^xf

i

^{ornoneofthem,}

then thewordlabeling thispathwill notbeequalto

W n

^beause

W n

^ontains

exatlyoneourreneof

w i+m

^{betweenanytwoonseutive}

b

^{'s.Weillustrate}

how

A i

^{works on gure 3, that shows the unfolding of the automaton}

A 2

^for

ϕ = ∀x 2 ∀x 1 ψ

^{ontheleftandfor}

ϕ = ∃x 2 ∀x 1 ψ

^ontheright.

Toillustratehow

S n−i

^{reoversthevaluesof}

x i+1 , . . . , x n

^,weshow

S n−i

^for

n = 9, i = 7

^{inthegurebelow.}

S 2

xf ₉ b

b

xf ₈

xt ₉

xf ₉

xt ₈

b b

xt ₉

S 1 S 0

A _i

^and

S i

^{aredesignedsothatanypathof}

A _i

^{islabeledbyatmostonext}

_i

andatmostonexf

i

^{betweenanytwoonseutive}

b

^'s,foreah

i

^{(foronveniene,}

we suppose that eah automaton starts and ends with a tive

b

transition).

Thatis,apathanbelabeledbyxt

i

^andxf

i

^{,butnotbytwoxf}

i

^ortwoxt

i

^.By

ontradition,assumethattherearetwoonseutive

b

^'sin

A _i

^{suhthatthereis}

apathfrom onetoanotherlabeledbytwoxt

j

^{(the asexf}

j

^{issymmetri).We}

taketheminimal

A i

^{whihensuresthis. Bytheminimalityof}

A i

^{,this anonly}

happeneitherbeauseoftherstxt

j

^transitionof

A i

^,orbetween

S _n−i

^andone

ofthetwo

A _i−1

^.Sinein

S _n−i

^allxt

k

^{ourafterthe(unique)}

b

^{,there isnoxt}

j

A 2

xt 2

xf ₂ b b

xt 2

xf 2

b xt 1

xf 1

A

1

xt 2

xf ₂ b b

xt 1

xf 1

A 1 xt

2

xf ₂ b b

xt 1

xf 1

A 1

A 2

xt 2

xf ₂ S 1

Fig.3.Unfoldingof

A 2

^for

Q 2 x 2 Q 1 x 1 = ∀ x 2 ∀ x 1

^{ontheleft,andontheright,unfolding}

of

A 2

^for

Q 2 x 2 Q 1 x 1 = ∃ x 2 ∀ x 1

in

A _i−1

^{before its rst}

b

^{(if any).It alreadyshowsa}ontraditionin thease where

Q i = ∃

^{. Considernow the ase}

Q i = ∀

^{. Forthe samereason asbefore,}

there an beat mostonext

j

^{betweenthelast}

b

^of

A _i−1

^{and the}

b

ⁱⁿ

S n−i

^,for

all

k < i

^{.Finally,betweenthe}

b

^of

S _n−i

^andtherst

b

^oftheseond

A _i−1

^there

anbeat mostonext

k

^with

k > i

^(from

S _n−i

^{)andatmostonext}

k

^with

k < i

(from

A _i−1

^{).Thus,in allasesweontradittheassumptionon}

A i

^.

Letusprovethat

W n ∈ L(A n )

^{ithereexistsasatisfyingvaluationtree}

V T

for

ϕ

^{. A valuation tree}

V T

^{is a binary tree of height}

n + 1

^{suh that its root}

(level

n

^)islabeledby

x n

^{andallnodesonlevel}

l

^arelabeledby

x l

^{.Theleavesare}

onlevel

0

^{,andareunlabeled.Anode}

v

^labeledby

x i

^{orrespondstoavaluation}

σ(v)

^{ofthevariables}

x i+1 , . . . , x n

^{.Forinstane,ifthevaluationforanodeis}

x n

istrue, thenitshildrenmustevaluate

x n

^{totrue,and evaluate}

x n−1

^{either to}

trueorfalse.Moreover,anodeonlevel

k

^{havetwohildrenif}

x k

^is universally quantied (one hild evaluate

x k

^{to trueand theother oneto false),and one}

hildif

x k

^isexistentiallyquantied.WesaythatavaluationtreesatisesaQBF formula

ϕ = Q n x n · · · Q 1 x 1 ψ

^{ifforeveryvaluationofeveryleaf,}

ψ

^istrue.

Usingthepropertywejustshowed,weannote that betweenanytwoon-

seutive

b

^{'sofanypathof}

A n

^{,thereareatmostthree}

w j

^andtwo

w i+m

^forany

1 ≤ j ≤ m, 1 ≤ i ≤ n

^{. Thus ouroding in base fourfor}determining whethera

lauseis true,isstill appliable.Hene,apath

ρ

^of

A _n

^islabeledby

W n

^ifor

all

1 ≤ k ≤ n + m

^{thereisexatlyone}

w k

^{betweenanytwoonseutive}

b

^'s.

Assumethat

V T

^{isavaluationtreeshowingthat}

ϕ

^{istrue.Avaluation}

σ(v)

denes twowords

T (v), F (v)

^{asfollows: theword}

T (v)

^isthe onatenationof allxt

j

^where

j > i

^and

x j

^istruein

σ(v)

^{. Theword}

F (v)

^istheonatenation of all xf

j

^where

j > i

^and

x j

^{is false in}

σ(v)

^{. Let}

v

^{be anode of}

V T

^labeled

by

x i

^{. Wedene the word}

ρ(v) = T ⁻¹ (v)W i F ⁻¹ (v)

^{. Wereall that}

T (v), F (v)

arewordsover

a ^∗

^,hene

T ⁻¹ (v)W i F ⁻¹ (v)

^{is thewordthatresultsfrom}

W i

^by

deleting

|T (v)|

^{many a's in the prex and by deleting}

|F (v)|

^{many a'sin the}

sux.

Letusshowbyindution on level

i

^that

ρ(v)

^{is in}

L(A i )

^foranynode

v

^of

V T

^onlevel

i

^.

If

v

^{is a leaf of}

V T

^{, then it denes an aepting valuation for}

ψ

^{, hene}

T (v)F (v) = W 0

^{using the sameargumentas in the} NP-hardness ase. Hene

ρ(v) = W 0 W ₀ ⁻¹ = ǫ ∈ L(A 0 )

^.

Consideran internal node

v

^{labeled by}

x i

^with

Q i = ∀

^{. Let}

v 1 , v 2

^{be the}

hildrenof

v

^,with

v 1

orrespondingto

x i

^true,and

v 2

^to

x i

^{false.Byindution}

letussupposethat

ρ(v 1 )

^,

ρ(v 2 )

^arein

L(A i−1 )

^.Then,

ρ(v) = T ⁻¹ (v)W i F ⁻¹ (v) = T ⁻¹ (v)W _i−1 bW _i−1 F ⁻¹ (v)

= T ⁻¹ (v)T (v 1 )ρ(v 1 )F (v 1 )bT (v 2 )ρ(v 2 )F(v 2 )F ⁻¹ (v)

=

^xt

i ρ(v 1 )F (v 1 )bT (v 2 )ρ(v 2 )

^xf

i

Weusedintheequationsabove

T ⁻¹ (v)T (v 1 ) =

^xt

i

^{forthepositivehild}

v 1

^of

v

^and

F ⁻¹ (v)F(v 2 ) =

^xf

i

^{forthenegativehild}

v 2

^of

v

^.Moreover,

F (v 1 )bT (v 2 ) =

F (v)bT (v) ∈ L(S n−i )

^{sine the indies of false variables in}

σ(v 1 )

^{and of true}

variablesin

σ(v 2 )

^{formapartitionof}

{i+1, . . . , n}

^{.Thisshowsthat}

ρ(v) ∈ L(A i )

^.

Consideraninternal node

v

^{that islabeledby}

x i

^with

Q i = ∃

^{. Assume by}

symmetry that

v 1

^{is the hild of}

v

ⁱⁿ

V T

^(thus,

x i

^{is true).By indution we}

assumethat

ρ(v 1 )

^isin

L(A _i−1 )

^{.Itiseasytoshownowthat}

ρ(v) ∈ L(A i )

^using:

ρ(v) = T ⁻¹ (v)W i F ⁻¹ (v) = T ⁻¹ (v)W i−1 F ⁻¹ (v)

= T ⁻¹ (v)T (v 1 )ρ(v 1 )F (v 1 )F ⁻¹ (v)

=

^xt

i ρ(v 1 )

Forthereversediretiontheargumentsaresimilar.Fromaword

W = W n

^of

A = A n

^{,weobtainsubwords}

ρ(v)

ⁱⁿ

L(A i )

^{asabove,labeledby}

T ⁻¹ (v)W i F ⁻¹ (v)

^.

Foreah leaf

v

^{thismeansthat}

σ(v)

^{satisesexatlyone literalperlause.}

2

Theorem2showsimmediatelythat thehierarhialmembership problemis

PSPACE-hard evenwith oneproess,byenodingthe alphabet

{a, b}

^{by loal}

ationsonasingleproess.Similarargumentsanbeusedfortheasewhere

G

isanMSC-graphwithnohierarhy,asshowninthefollowingtheorem.

Theorem3 ThehierarhialMSCmembershipproblem

M ∈ ^? L(G)

^isPSPACE-

omplete. The lower boundholdseven if

G

^{isan MSC-graph, orifthere isonly}

oneproess.

Proof.Theproblemweredue fromis(1-in-3)QBF.Let

F

^{beaninstaneof}

(1-in-3)QBF of the form

F = (Q n x n ) . . . (Q 1 x 1 )ϕ

^{, where}

Q i ∈ {∃, ∀}

^{and the}

formula

ϕ

^isoftheform

∧ j=1...m R(α j,1 , α j,2 , α j,3 )

^,with

α j,k

^literals.

Theideaistoletvaluationsofthevariablestoorrespondtopathsof

G

^and

to validate thevaluationsusing the nMSC

M

^{. Wedene thegraph}

G

^andthe

nMSC

M

^byindutionon

F = F n

^.Let

F i = (Q i x i ) F _i−1

^,with

F 0 = ϕ

^.Eah

F i

willdetermine

G i , M i

^.

TheproessesusedintheonstrutionareSC

1 , . . . ,

^SC

m

^andRC

1 , . . . ,

^RC

m

^,

plusVY

1 , . . . ,

^VN

n

^andVN

1 , . . . ,

^VN

n

^.Here

V

^{meansavariableand}

C

^alause,

S

^{standsforsend,}

R

^forreeive,

Y

^foryesand

N

^forno.

For all

i

^{, let MY}

i

^{be the MSC onsisting of amessage from VY}

i

^{to VN}

i

^,

then bak from VN

i

^{to VY}

i

^{, and a message from SC}

j

^{to RC}

j

^{for all}

j

^suh

that

x i ∈ {α j,1 , α j,2 , α j,3 }

^. Symmetrially,letMN

i

^{be theMSConsisting ofa}

messagefromVN

i

^toVY

i

^{,thenbakfromVY}

i

^toVN

i

^{,andamessagefromSC}

j

toRC

j

^forall

j

^suhthat

¬x _i ∈ {α j,1 , α j,2 , α j,3 }

^.

M 0

^{is an MSC onsisting of one message from SC}

j

^{to RC}

j

^{, for all}

j

^{. The}

MSC-graph

G 0

^{onsists of}

4n

^{verties,labeledby MY}

i

^{, MN}

i

^,or

∅

^{. Thegraph}

hoosesbetweenMY

i

^andMN

i

^forall

i

^{,asdepitedbelow:}

MY

MN

G _{0 MY}

MN

1

1

n

n

Notethatallmessagesdened aboveommute,exeptfortheonesbetween

VY

i

^andVN

i

^.Let

a i

^{bethemessagefromVY}

i

^toVN

i

^,and

b i

^{themessagefrom}

VN

i

^toVY

i

^{. Wewillusetheorderbetween}

a i

^,

b i

^{asfollows:Thesequene}

a i b i

meansthat

x i

^istrue,while

b i a i

^meansthat

x i

^isfalse.

Assumenowthat

G _i−1 , M _i−1

^{arealreadydened,andthat thereare}

f

^uni-

versalquantiersin

F _i−1

^{.Forsimpliity,wedenote}

a = a i

^and

b = b i

^.Notethat

inavaluationtreefor

F

^showingthat

F

^{istrue,eahvalue0or1assignedtothe}

variable

x i

^isusedby

2 ^f

^{leaves.Avaluationtreeisdenedasusual,byassigning}

eah universally quantied variable two hildren labeled 0 and 1, respetively

eahexistentiallyquantiedvariableonehildlabeled0or1.

If

F i = ∀x i F i−1

^{, then let}

M i = (ab) ^{2 f} M i−1 S i (ba) ^{2 f} M i−1

^{(see Figure 4.1).}

The MSC

S i

^{is used for} synhronizing proesses ourring in

M i

^{. It ontains}

a message between eah (ordered) pair of proesses of

M i

^{(in some arbitrary}

order). Note that using thehierarhywe andesribe

(ab) ^{2 f}

^{, andthus}

M i

^{, by}

anexpressionofpolynomialsize.

Let

G i = (V i , E i )

^,where

V i = V _i−1 ∪{e 0 }

^and

E i = E _i−1 ∪{(

^Fin

, e 0 ), (e 0 ,

^In

)}

^.

The initial node In (the nal node Fin , respetively) of

G i

^{is the sameas for}

G _i−1

^.Thevertex

e 0

^{islabeledbythe}synhronizationMSC

S i

^.

G _i

G _i-1

M

VY VN

i

M _i-1

ba

S S

VY VN

M i-1

e ₀

i i i i

i i

ab

Thedenition of

M i , G i

^{anbe explained}intuitivelyasfollows.Let

ρ

^bea

pathof

G i

^labeledby

M i

^{.Note thattheMSC}

S i

^ourringin

M i

^{hasto math}

theMSC

S i

^of

e 0

^.Thus

ρ = ρ 1 e 0 ρ 2

^,with

ρ 1

^{anaeptingpathof}

G i−1

^labeledby

(ab) ^{2 f} M i−1

^and

ρ 2

^{anaeptingpathof}

G i−1

^labeledby

(ba) ^{2 f} M i−1

^.Eahtime

ρ j

^{goes through}

G 0

^{(whih happens}

2 ^f

^times),

ρ j

^{onsumes either}

ab

^{of MY}

i

or

ba

^{of MN}

i

^{, so}

ρ j

^{onsumesallourrenesof}

a, b

ⁱⁿ

(ab) ^{2 f}

^{. Inpartiular,all}

ourrenesonsumedby

ρ 1

^areoftheform

ab

^{,whihensuresthatthevaluation}

of

x i

^{assoiatedwith}

ρ 1

^isonsistent(

x i

^{is true).Thesameholdsfor thepath}

ρ 2

^{,wherethevalueof}

x i

^{isforedtobefalse.}