Tree automata and discrete distributed games

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Tree automata and discrete distributed games

Julien Bernet, David Janin

To cite this version:

Julien Bernet, David Janin. Tree automata and discrete distributed games. Fundamentals of Com-

putation Theory (FCT), Aug 2005, Hungary. pp.540–551. �hal-00306410�

JulienBernetandDavid Janin

⋆

LaBRI,UniversitédeBordeauxI

351,oursdelaLibération

33405TaleneedexFRANCE

{bernet|janin}labri.fr

Abstrat. Distributedgames,asdened in[6℄,is areentmultiplayer

extension of disrete two player innite games. The main motivation

for their introdution is that they provide an abstrat framework for

distributed synthesis problems, in whih most known deidable ases

anbeenodedandsolveduniformly.

In the present paper, we show that this unifying approah allows as

wella better understandingofthe role playedby lassial resultsfrom

treeautomatatheory(asopposedtoadhoautomataonstrutions) in

distributedsynthesisproblems.More preisely,we use alternatingtree

automataomposition, andsimulationof analternatingautomaton by

anon-deterministione,astwoentraltoolsforgivingasimpleproofof

knowndeidableases.

Introdution

Distributedgames,asdenedin[6℄,isareentmultiplayerextensionofdisrete

two player innite games. The main motivation for their introdution is that

theyprovideanabstratframeworkfordistributedsynthesisproblems,inwhih

mostknowndeidableases[1,35,10℄anbeenodedandsolveduniformly.

Inthe presentpaper,weshow that this unifying approah allowsaswella

betterunderstandingof theroleplayedbylassialresultsfromtree automata

theoryin distributedsynthesisproblems.

Morepreisely,intheabovementionedworks,manydeisionalgorithmsrely

(moreorlessimpliitly)onautomataonstrutionsthatarenotexpliitlyrelated

tolassialautomatatheory.

Forinstane, in[3℄,themain onstrutiongivenbytheauthorstosolvethe

pipelinesynthesisproblemsounds likethesequentialompositionoftwotree-

automata.Similarly,oneofthemainonstrution(glueoperation)denedin[6℄

sounds like Muller and Shupp simulation of an alternatingautomaton by a

nondeterministione[7℄.

Thepurposeofthispaperistovalidatethis intuition,byexpliitlydening

theenounteredautomata(ortheirinputs)whentheyaremissingintheseworks,

andtoapply knownonstrutionsin ordertoreprovethesesynthesisresults.

⋆

ThisworkispartiallysupportedbytheEuropeanCommissionResearhandTraining

NetworkGamesandAutomataforSynthesisandValidation (RTNGAMES)

groundinto whih methods andapproahesanbe enoded andomparedone

withtheother.

Thetehnial relevane of our reformulation work is illustratedby the en-

odingandsolvingofthepipelinease[3℄.

Other Related Works

Peterson and Reif ([8℄, extended in [9℄) initiated the researh on multiplayer

gamesofinompleteinformation,onsidering nitegames,andintroduingthe

notionofhierarhial games:thesegamessatisfythepropertythatoneanlin-

early order theset of playerssuh that

p 1 ≤ p 2

^{ifand onlyif}

p2

^knowsmore

than

p 1

^,orequivalently

p 2

^{knowsthestateof}

p 1

^{.Theyprovethatthesegames}

aresolvable,byiterativelyremovingtheinompleteinformationassoiatedwith

eahplayer.

Subsequentresultsondistributedsynthesis(suhas[10℄,[3℄)essentiallyused

thesameideasandtehniques,exeptinthefatthattheyonsiderinniteplays

and/orbranhingtimespeiations.

Theommontehniqueistoutoutthelastplayerfromthegame(i.e.theone

thatknowsthestateofalltheother),modifyingintheproessthespeiation

sothatit reetsallmovesthatanbetaken bythisplayer,thendothesame

withthelastbut one,et....until asimple 2-playergameislefttosolve.

Ourpaper relyon the same priniple, making the automata onstrutions

expliit.

Organization ofthe Paper

Intherstsetion,afterreviewingsomeofthenotationsusedinthispaper,we

xthedenitions oftrees,treeautomataandinnitetwoplayergames.Muller

and Shuppnon determinization theorem is stated, and anotionof sequential

omposition oftreeautomataisalsodenedandanalyzed.

Distributedgamesanddistributedstrategiesarepresentedintheseondse-

tion.These gamesareplayedbyateamof proessplayersversusasingleenvi-

ronmentopponent.Eahproessplayeronlygetsinompleteinformationabout

thepositionoftheotherproesses.Theexisteneofawinningdistributedstrat-

egy inadistributed game isshownto be undeidable,evenforsimplewinning

onditionssuhas safetyand reahability.

Inthe third setion, werst show that using an(external) tree-automaton

inordertodenewinningstrategiesina(distributed)gameisessentiallyequiv-

alentto addinganadditionalproessplayer(internalizingtheautomaton)into

the game. Then, onversely, we show that when a proess player has enough

knowledgeto dedue the positions of theother proesses, then its loal arena

anbeexternalizedasatreeautomatonreadingthestrategiesoftheremaining

proesses, in whih non-deterministi hoies orrespond to the moves of the

proess; this automaton anbe omposed with any existing external winning

ondition.

orderto reduethenumberofproessplayersandthustosolvethedistributed

game.

The long-term goal of this approah is to gain benet from the high level

ofabstrationprovidedbygametheoryand, altogether,gainbenetfrom well-

knownonstrutionsof automatatheory(as itasbeendevelopedfrom Rabin's

seminal result[11℄), to help having abetter understandingof the fundamental

obstalesto thesynthesisofdistributedsystems.

1 Trees, Automata and Games

Forany alphabet

A

^{, let}

A ^∗

^and

A ^ω

^{be the set of all nite and innite words}

with letters from

A

^{. Let}

A ^∞ = A ^∗ ∪ A ^ω

^{, and}

A ^? = {ǫ} + A

^{. Standard nota-}

tionsonwordsandlanguagesofwordsareused.Inpartiular,givenalanguage

L ⊆ A ^∗

^{, weuse the notations}

L ⁺

^{and(when the emptyword}

ǫ 6∈ L

⁾

L ^ω

^that

stand,respetively,forthesetofwordsbuiltbyonatenatingnitelymanyand

innitelymanynitewordsof

L

^{.Foranyniteword}

w = a 1 . . . a n

^{, let}

|w| = n

bethelengthof

w

^{.Foranyinniteword}

w

^,let

inf(w) = {a ∈ A | w ∈ (A ^∗ .a) ^ω }

betheset oflettersthat ourinnitelyoftenin

w

^.

Foranytwosets

A

^and

X

^,foranyword

w ∈ A ^∗

^,dene

π X (w)

^{(theprojetion}

of

w

^over

X

^{)asthewordobtainedbydeleting anyletterthat isnotin}

X

^from

thewordrepresentationof

w

^.

Given

n

^numberedsets

A 1 , . . . , A n

^{, given}

A = A 1 × . . . × A n

^{, givenany set}

of indies

I = {i 1 , . . . , i k } ⊆ {1, . . . , n}

^with

i 1 < . . . < i k

^{, we write}

A[I]

^for

theset

A[I] = A _{i 1} × . . . A _i k

^{, forany}

x = (a 1 , . . . , a _n ) ∈ A

^{, wewrite}

x[I]

^forthe

elements

x[I] = (x i ₁ , . . . , x _i k ) ∈ A[I]

^{,and, forany}

P ⊆ A

^,wewrite

P [I]

^forthe

set

P[I] = {x[I] ∈ A[I] : x ∈ P }

^.

Inase

I = {i, i + 1, . . . , j}

^(where

1 ≤ i ≤ j ≤ n

^{), thesenotationssimplify}

to

A[i . . . j]

^,

x[i . . . j]

^and

P [i . . . j]

respetively(and even simplify to

A[i]

^,

x[i]

and

P[i]

^when

i = j

^{). Thesenotationsalso extendto wordsasfollows:for any}

word

w = a 1 .a 2 . . . . ∈ A ^∞

^{, for any}

I ⊆ {1, . . . , n}

^,

w[I] = a 1 [I].a 2 [I].a 3 [I] . . .

^,

andtorelations:foranyrelation

R ⊆ A × A

^,wewrite

R[I]

^{therelationon}

A[I]

dened by

R[I] = {(x[I], y[I]) ∈ A[I] × A[I] : (x, y) ∈ R}

^.

Giventwonitealphabets

D

^and

Σ

^,a

Σ

^-labeled

D

^{-tree(alsoalled}

D

^,

Σ

^-tree)

isapartial funtion

D ^∗ → Σ

^{whosedomainislosedunderprexoperation.In}

thesequel,elementsof

Σ

^{arealledlabelsandelementsof}

D

^{arealleddiretions.}

Foranytree

t : D ^∗ → Σ

^,thefuntion

f lat t : Dom(t) → Σ.(D.Σ) ^∗

^isdened

by:

f lat t (ǫ) = t(ǫ)

^{and, for any}

w ∈ D ^∗

^and

d ∈ D

^{suh that}

w.d ∈ Dom(t)

^,

f lat t (w.d) = f lat t (w).d.t(w.d)

^{. Observe that}

Dom(t)

^and

f lat t (Dom(t))

^or-

deredbytheprexorderingareisomorphi,and,asaonsequene,

f lat t (Dom(t))

uniquelydeterminestree

t

^.

Thefollowing denition isa variationon Mullerand Shupp's originaldef-

inition of alternatingautomaton[7℄. Ourgoal isto have atree-transduerlike

automatondenition,evenforalternatingautomaton.

Denition1 (Alternating tree automaton). A nite

(D, Σ)

-alternating treeautomatonisatuple:

A = hQ = Q ^∀ ⊎ Q ^∃ , D, Σ, q 0 , δ = δ ^∀ ∪ δ ^∃ , Acc ⊆ Q ^ω i

where

Q

^{isanitesetofstates,}

q 0 ∈ Q ^∃

^{istheinitialstate,}

δ ^∀ : Q ^∀ ×D → P (Q ^∃ )

and

δ ^∃ : Q ^∃ × Σ → P (Q ^∀ )

^{are the transition funtions, and the}

ω

^-rational

language

Acc

^{isthe innitary aeptane riterion.}

Automaton

A

^isanondeterministiautomaton (alsoallednonalternating) when

|δ ^∀ (q, d)| ≤ 1

^(forany

q ∈ Q ^∀

^,

d ∈ D

^).

Denition2 (Runs). A run of an automaton

A = hQ, D, Σ, i, δ, Acci

^overa

Σ

^-labeled

D

^-tree

t : D ^∗ → Σ

^isa

Q ^∀

^-labeled

(D ×Q ^∃ )

^tree

ρ : (D ×Q ^∃ ) ^∗ → Q ^∀

suhthat:

ρ(ǫ) ∈ δ ^∃ (q 0 , t(ǫ))

^,

for all

w ∈ Dom(ρ)

^{, if}

ρ(w) = q

^{, then for any diretion}

d ∈ D

^{suh that}

a = t(w[1].d)

^{is dened, and for any} existential state

q 1 ∈ δ ^∀ (q, d)

^{, there}

existsa universalstate

q 2 ∈ δ ^∃ (q 1 , a)

^{suh that}

ρ(w.(d, q 1 )) = q 2

^.

Forany innitebranh

w

^ofarun

ρ

^of

A

^over

t

^,

states ρ (w)

^isthesequene

of (universal and existential) states enountered along

w

^{. A tree}

t

^{is aepted}

by

A

^{if and only if there exists a run}

ρ

^of

A

^over

t

^{suh that for any innite}

branh

w

ⁱⁿ

ρ

^:

states _ρ (w) ∈ Acc

^.Denoteby

L(A)

^{the language ofall trees that}

are aeptedby

A

^{.The sizeof an automaton}

A

^isdenotedby

|A|

^.

Observethatthese treeautomata(bothalternatingandnonalternating),if

slightlyunusual,havethesameexpressivepowerastheirstandardounterpart,

asin [7℄.Inpartiular:

Theorem1 (Simulation [7℄). Any alternating tree automaton

A

^{is equiva-}

lent to a non deterministi tree automaton

A ^′

^{, with}

|A ^′ | ≤ 2 ^{2 |A|}

^{(with Muller}

aeptaneondition).

Sinetherunsofanautomatonontreesarethemselvestrees, automataat

astreetransduersandanbesequentiallyombined.

Denition3 (Automata Composition). Given two tree automata

A 1 = hQ 1 , D 1 , Σ 1 , q _0,1 , δ 1 , Acc 1 i

^and

A 2 = hQ 2 , D 2 , Σ 2 , q _0,2 , δ 2 , Acc 2 i

^{, suh that au-}

tomaton

A 2

^isnon deterministi with

D 2 = D 1 × Q ^∃ ₁

^and

Σ 2 = Q ^∀ ₁

^{,we dene}

the ompositionof

A 1

^followedby

A 2

^{tobe theautomaton}

A 2 ◦ A 1 = h Q, D e 1 , Σ 1 , q e 0 , e δ, Acci g

denedasfollows:

Q f ^∃ = Q ^∃ ₁ × Q ^∃ ₂

^;

Q f ^∀ = Q ^∀ ₁ × Q ^∀ ₂

^;

q 0 = (q 0,1 , q 0,2 )

^,

(q ^′ ₁ , q ₂ ^′ ) ∈ δ e ^∀ ((q 1 , q 2 ), d) ⇔

q ^′ ₁ ∈ δ ^∀ (q 1 , d) {q ₂ ^′ } = δ ₂ ^∀ (q 2 , (d, q ^′ ₁ ))

(q ^′ 1 , q ₂ ^′ ) ∈ δ e ^∃ ((q 1 , q 2 ), a) ⇔

q ^′ ₁ ∈ δ ^∃ (q 1 , a) q ^′ ₂ ∈ δ ^∃ ₂ (q 2 , q ₁ ^′ )

Acc g = {w ∈ Q e ^ω | w[1] ∈ Acc 1 ∧ w[2] ∈ Acc 2 }

Theorem2. For any tree

t : D ^∗ 1 → Σ 1

^,

t ∈ L(A 2 ◦ A 1 )

^{if and only if there}

existsanaepting run

ρ : (D 1 × Q ^∃ ₁ ) ^∗ → Q ^∀ ₁

^of

A 1

^over

t

^suhthat

ρ ∈ L(A 2 )

^.

The proof, although tedious, is not ompliated, and is therefore omitted

here. Observe that it is ruial that

A 2

^is non-alternating ; nevertheless, by applyingTheorem1,oneanalwaysassumethatisisthease.

Denition4 (Simple(orTwoPlayer)Games).Asimplearenaisaquadru-

ple

G = hP, E, T P , T E i

^,where

P

^{isanitesetofProesspositions,}

E

^isanite

setofEnvironmentpositions,

T P ⊆ P ×E

^{isthesetofProessmoves,}

T E ⊆ E×P

is the set of Environment moves. A simple game

G = hP, E, T P , T E , e 0 , Wi

^is

built upon asimplearena

hP, E, T P , T E i

^{byequipping itwithan initial position}

e 0 ∈ E

^{andaregularwinningondition}

W ⊆ (P + E) ^ω

^.

Aspartiularasesofwinningondition,areahabilityonditionisawinning

ondition of the form

W = (P + E) ^∗ .X.(P + E) ^ω

^{for some set of positions}

X ⊆ P + E

^{tobereahedforProesstowin,andasafety onditionisawinning}

onditionoftheform

W = ((P +E) −X )) ^ω

^{forsomesetofpositions}

X ⊆ P +E

tobeavoidedforProesstowin.

Aplay

w ∈ (P + E) ^∗

^{in asimplegameis anynon-emptypathin thearena}

beginningon

e 0

^.Aplay

w

^{iswinningforProesswheneitheritisniteandends}

in anEnvironmentposition,oritis inniteandbelongsto

W

^{.Otherwise, itis}

winning forEnvironment.

A strategy forProess is apartial funtion

σ : (E.P) ⁺ → E

^{suh that for}

any

w.p ∈ Dom(σ)

^{, foranyposition}

e ∈ σ(w.p)

^{, then}

(p, e) ∈ T _P

^{,and for any}

suessor

p ^′

^of

e

^,

w.p.e.p ^′ ∈ Dom(σ)

^{. A play}

w = e 0 .x 1 . . . .

^{is onsistent with}

strategy

σ

^{when, forany}

i ∈ N

, if

σ(e 0 . . . . .x _i )

^and

x _i+1

^{arebothdened then}

theyareequal.Astrategy

σ

^{isawinningstrategyforProesswhenanymaximal}

play(w.r.t.theprexordering)onsistentwith

σ

^{iswinning forProess.}

Givenastrategy

σ

^insomegame

G

^{,thestrategytree}

t σ : P ^∗ → E

^of

σ

ⁱⁿ

G

isdened indutivelyby

t σ (ǫ) = e 0

^,and

t σ (u.x) = σ(f lat t σ (u).x)

^.

Theorem3 ([2℄). Onnite two-player games with regularwinning ondition,

either Proess or Environment has a winning strategy, whih an be omputed

eetively.

2 Distributed Games

Denition5 (Distributed Arena). A distributed arena isa free asynhro-

nous produt where the possible Environmentmoves may have been restrited.

More preisely, given two arenas

G 1 = hP 1 , E 1 , T P,1 , T E,1 i

^and

G 2 = hP 2 , E 2 , T P,2 , T E,2 i

^,a(two-proess)distributedarenabuiltuponthearenas

G 1

^and

G 2

^is

any simplearena

G = hP, E, T P , T E i

^{ofthe form}

Environmentpositions:

E = E 1 × E 2

^,

Proessespositions:

P = (E 1 ∪ P 1 ) × (E 2 ∪ P 2 ) − (E 1 × E 2 )

^,

Proesses moves:

T P

^{is the set of all pairs}

(p, e) ∈ (P × E)

^{suh that,for}

i = 1

^and

i = 2

^:

•

^either

p[i] ∈ P i

^and

(p[i], e[i]) ∈ T P,i

^(Proess

i

^isativein

p

^),

•

^or

p[i] ∈ E i

^and

p[i] = e[i]

^(Proess

i

^isinativein

p

^),

andEnvironmentmoves:

T E

^{issomesubsetof thesetofall pairs}

(e, p) ∈ (E × P)

^suhthat,for

i = 1

^and

i = 2

^:

•

^either

p[i] ∈ P i

^and

(e[i], p[i]) ∈ T P,i

(EnvironmentativatesProess

i

^),

•

^or

p[i] ∈ E i

^and

p[i] = e[i]

(EnvironmentkeepsProess

i

^inative).

When the set

T E

^of Environmentmoves ismaximal, we all suhan arenathe freeasynhronousprodut ofarenas

G 1

^and

G 2

^{anditisdenotedby}

G 1 ⊗ G 2

^.

Thesedenitionsextendto

n

^-proess distributedarena.

Sineadistributedarenaisbuiltupon

n

^{simplearenas,weneedadenition}

tospeakaboutitsloalomponents:

Denition6 (Projetion of distributedarena).Given adistributedarena

G = hP, E, T P , T E i

^,with

E = E 1 × . . . × E n

^and

P = ((P 1 ∪ E 1 ) × . . . × (P n ∪ E n )) − E

^{,given anonemptyset}

I ⊆ {1, . . . , n}

^{,denethe anonialprojetion}

G[I]

^of

G

^on

I

^{as the arena}

G[I] = hP ^′ , E ^′ , T _P ^′ , T _E ^′ i

^{given by:}

P ^′ = P[I] − E[I]

(possibly smaller than

P[I]

^!),

E ^′ = E[I]

^,

T _P ^′ = T P [I] ∩ (P[I] × E[I])

^{, and}

T _E ^′ = T _E [I] ∩ (E[I] × P [I])

^.

Remark. Observethat a

n

^-proess distributed arena

G

^{as aboveanalways}

beseenasadistributed arenabuiltupon thegames

G[1]

^,...,

G[n]

^.Moreover,

in the sameway Cartesianprodut of sets is (up to isomorphism) assoiative,

given anarbitrarynon empty set

I ⊂ {1, . . . , n}

^,given

I = {1, . . . , n} − I

^{, the}

n

^-proessdistributed arena

G

^{an,aswell,beseenasa}distributedarenabuilt uponthetwo(distributed)arenas

G[I]

^and

G[I]

^.

Example 1 (ThePipeline:Beginning).Adistributedarhiteture(asdened

in [10℄, [3℄)is a set of sites linked together by some ommuniation hannels.

Eahsiteanhostaprogram,whihisessentiallyasequentialfuntion 1

mapping

asequeneofinputstoasequeneofoutputs.Asatypialexample,inapipeline

arhiteture,thesites arelinearlyorderedfrom lefttoright,eah sitetakingits

inputfromthesiteonitsright,andwritingitsoutputtothesiteonitsleft.

Tobemorepreise,supposeeahommuniationhannel

x i

^anarryvalues

that rangeoversomeset

X i

^.Thesite

s i

^{reeivesitsinputfrom thehannel}

x i

^,

and writesits outputsto thehannel

x i−1

^{; thus, aprogram forthesite}

s i

^isa

1

reallthatasequentialfuntionisafuntion

f : A ^∗ → B ^∗

^{thatisrealizedbyaword}

transduerwithinputalphabet

A

^{andoutputalphabet}

B

^.

s 1 s n − 2 s n − 1 s n

x n

x n − 1

x n − 2

x 0

Fig.1.Apipelinearhiteture

sequentialfuntion

f _i : X _i ^∗ → X _i−1 ^∗

^.Theenvironmentwritesinputtothesystem onhannel

x n

^{,andthesystem'soutputisread onhannel}

x 0

^.

For any pipeline arhiteture

A

, we an build a distributed arena

G _A = hP, E, T P , T E i

^{whereeahproessplaystheroleofaprogram:onitsloalarena,}

theenvironment'smovesorrespondtothepossibleinputsforthissite,andthe

proessmovesorrespondtothepossibleoutputs:

P = X 1 × . . . × X n

^;

E = X 0 × . . . × X n−1

^.

((v 1 , . . . , v n ), (v ₁ ^′ , . . . , v _n ^′ )) ∈ T E

ⁱ

v _i ^′ = v i+1

^foreah

i ∈ {1, . . . , n − 1}

^and

v ^′ _n ∈ X n

^.

Observethat byrestritingtheEnvironmentmoves,weensurethat theen-

vironmentarriesorretlythevaluesalongthehannels.

Denition7 (Distributed Games). A

n

^-proess distributed game

G

^{is a}

tuple

G = hP, E, T P , T E , e 0 , Wi

where

hP, E, T P , T E i

^isa

n

^-proess distributedarena,

e 0 ∈ E

^{isthe initial (En-}

vironment)position,and

W ⊆ (E.P ) ^ω

^{isthe (regular)winning innitaryondi-}

tion.

Adistributedgameisapartiularaseofsimplegame.Itfollowsthatprevi-

ousnotionsof playsand strategiesarestill dened.However,in order to avoid

onfusionwithwhat mayhappenintheloal arenaadistributed gameisbuild

upon,weshallspeaknowofaglobal play andaglobal strategy.

Theloal view Proess

i

^{hasofaglobalplayina}distributedgame

G

^isgiven

bythemap

view i : (E.P) ^∗ .E ^? → (E i .P i ) ^∗ .E _i ^?

^{dened inthefollowingway:}

view i (ǫ) = ǫ

view i (x) = x[i]

view i (w.x.y) =

view i (w.x)

^if

x[i] = y[i]

view i (w.x).y[i]

^otherwise.

A play

w ∈ (E.P) ⁺

^{is said to be ative for Proess}

i

^when

w

^{ends in a}

position

p ∈ P

^suhthat

p[i] ∈ P[i]

^.

Denition8 (Loal and distributed Strategies). Given a

n

^{-tuple of loal}

strategies

(σ i : (E[i].P[i]) ⁺ → E[i]) i∈{1,...,n}

^{,the induedglobal strategy}

σ 1 ⊗ . . . ⊗ σ _n : (E.P ) ⁺ → E

is dened as follows: for any play of the form

w.p ∈ (E.P) ⁺

^{, given the set}

I ⊆ {1, . . . , n}

^{of ative proesses in the global Proesses position}

p

^(i.e.

I =

{i ∈ {1, . . . , n} : p[i] ∈ P i }

^),dene

σ(w.p) = e

^by:

e[i] = σ i (view i (w))

^for

i ∈ I

e[i] = p[i]

^for

i ∈ {1, . . . , n} − I

(providedeverything iswell-dened, otherwise

σ(w.p)

^{isleftundened).}

Aglobal strategy

σ : (E.P ) ⁺ → E

^isa distributed strategyif

σ

^{equals the}

omposition

σ 1 ⊗ . . . ⊗ σ _n

^{of some}

n

^loal strategies.

Note that global strategiesare not alwaysdistributed. Moreover,there are

distributedgamesinwhihtheProesseshaveawinningstrategy,butnowinning

distributed strategy.

From this, wean derive an important fat: the distributed game are not

determined,inthesensethatevenwhentheenvironmentdoesnothaveawinning

strategy,theproessesmaynothaveawinningdistributedstrategy.Furthermore,

usingthefatthattheproessesdonotsharethesameinformation,weareable

toprovidethefollowingundeidabilityresult:

Theorem4. The problem of nding a winning distributed strategy in a

3-proessdistributedgame withsafety orreahability winningonditionisunde-

idable.

Theproof is omitted heredue to spae restrition. Sue it to say that it

proeeds by redutionto the Post orrespondene problem, and relies heavily

on the fat that there are three proessesin the game. It is an open problem

whethersolvinga

2

^-proessdistributedgameisdeidableornot.

3 Tree Automata and Distributed Games

We rst mix games and automata,dening a winning ondition by means of

a tree-automatonthat reognizesthe set of trees of winning strategies. We il-

lustrate this new oneptby dening a pipeline gameoverthe pipeline arena.

Then,wepresentanalgorithm tosolvesuhagame,usingthenotionofleader

in adistributed game.

Denition9 (External Winning Condition). A game with external win-

ningonditionisatuple

G = hP, E, T P , T _E , e 0 , Ai

where

hP, E, T P , T _E i

^{isasimplearena,}

e 0 ∈ E

^{istheinitial position,and}

A

^isa

(P, E)

^{-tree automaton.Insuhagame, astrategyiswinning ifits strategytree}

belongsto

L(A)

^{.Thisdenition extendsto}distributedgames.

Inthesequel,in order toavoidonfusion,agame withawinningondition

dened asinsetion2isalledgame withinternal winningondition.

As we are going to show, games with external winning ondition are not

essentiallymoreexpressivethangameswithinternalone.

Theorem5 (Internalization). Forany

n

^{-proess game}

G

^{withexternal win-}

ning ondition, thereexistsa

n + 1

^{-proess game}

G ^′

^{withinternal winningon-}

dition suh that

G ^′ [1, . . . , n] = G

^{, and suh that the proesses have a winning}

strategy

σ

ⁱⁿ

G

^{ifand onlyif the proesses havea winningstrategyof the form}

σ ⊗ σ ^′

ⁱⁿ

G ^′

^.

Proof. (sketh)Let

G = hP, E, T P , T E , e 0 , Ai

^(where

A = hQ ^∀ ⊎ Q ^∃ , P, E, q 0 , δ = δ ^∀ ∪ δ ^∃ , Acci

^)beadistributedgamewithexternalwinningondition.Thegame

G ^′ = hP ^′ , E ^′ , T _P ^′ , T _E ^′ , e ^′ ₀ , Wi

^{isdenedasfollows.Thepositionsandthewinning}

onditionaregivenby:

P ^′ = (E × (Q ^∃ × E)) ∪ (P × (Q ^∃ × {#}))

^,

E ^′ = (E × Q ^∃ ) ∪ (E × Q ^∀ )

^,

e ^′ ₀ = (e 0 , q 0 )

^,

W = {w ∈ (E ^′ .P ^′ ) ^ω | π _Q ^∀ _∪Q ^∃ (w) ∈ Acc}

and movesare (repeatedly) dened by: from an environmentposition

(e, q) ∈ E × Q ^∃

^{(ortheinitialposition):}

1. rst, Environment (deterministially) moves to the proess position

(e, (q, e)) ∈ E × (Q ^∃ × E)

^,

2. then, the new(automaton) proess loally hooses

q ^′ ∈ δ ^∃ (q, e)

^{, the other}

proessesstayidle,thustheplayproeedsin

G ^′

^,totheenvironmentposition

(e, q ^′ ) ∈ E × Q ^∀

^,

3. then,Environmenthooses

p ∈ T _E (e)

^and

q 1 ∈ δ ^∀ (q ^′ , p)

^{, andtheplaypro-}

eedstotheProessposition

(p, (q 1 , #)) ∈ P × Q ^∃

^,

4. nally, proesses

1

^to

n

^{(on game}

G

^{) hoose some}

e 1 ∈ T _P (p)

^{, the new}

(automaton) proessstays almost idle (he simplydeletes the

#

^{sign), and}

theplayproeedstotheEnvironmentposition

(e 1 , q 1 ) ∈ E × Q ^∃

^.

If

ρ

^{is an aepting runof}

A

^over

t _σ

^{(for some strategy}

σ

ⁱⁿ

G

^{), one dedue}

from

ρ

^astrategy

σ ^′

^{suh that}

σ ⊗ σ ^′

^{iswinning in}

G ^′

^.Conversely,if

σ ⊗ σ ^′

^isa

winning strategyin

G ^′

^{,oneaninferanaeptingrun of}

A

^over

t _σ

^from

σ ^′

^.

Moreover, when

G

^{is a simple game with external winning ondition, the}

internalization proedure anbe furthersimplied (and amounts essentiallyto

buildtheprodutof

G

^withtheautomaton),andtheresultinggamewithinternal onditionisasimplegameaswell.

Example 2 (Pipeline ExampleContinued).Followingthepresentationfrom

[3℄, the synthesisproblem for distributed arhitetures is presentedas follows:

given adistributed arhiteture

A

and avetorof programs

(f i ) 1≤i≤n

^{(one for}

eah site of

A

), the omputation tree of the system is a

( Q

1≤i≤n X _i )

^-labeled

X n

^{-tree,whereeahnode}

w

^{islabeledbythevaluesheldbythe}ommuniation

hannelsafter input

w

^tothesystem.

Aspeiation forthesystemisalanguageof suh treesspeiedbyatree

automaton

A

^(orequivalentlybyaMSO-formula).

Thesynthesisproblem is then: doesthere exists avetorof programssuh

that theresultingomputationtreebelongsto thespeiation?

Building upon thepipeline arena

G _A

^{asdened in example 1,wean now}

deneadistributed gameinwhihtheproesseshaveawinningstrategyifand

only if there is a solution to the synthesis problem in the pipeline arhite-

ture. Suppose thespeiation for

A

is given by thenite

(X n , Q

0≤i≤n−1 X i )

^-

automaton

A

^{, wean easily denea}

( Q

1≤i≤n X i , Q

0≤i≤n−1 X i )

^-automaton

A ^′

thataeptsatree

t ^′

^{ifandonlyifitisthewideningofsometree}

t ∈ L(A)

^(i.e.

if

t ^′ (w) = t(w[n])

^forall

w ∈ Dom(t ^′ )

^).

Usingthisautomatonasanexternalwinningondition,wegettheenoding

ofthesynthesisproblemforthepipelinearhitetureinadistributedgame.

Observethat in this game, foreah

i ∈ {1, . . . , n}

^{, provided that Proess}

i

knowsthestrategyforalltheproessesfrom

1

^to

i − 1

^{,thenheanpreditthe}

positionineahoftheloalarenasfrom

1

^to

i − 1

^.

Usingtheaboveobservation,onemayasknowwhetheraninverseonstru-

tion to internalization is possible ornot. Intuitively, assuming that there is a

proessinan

n + 1

-distributedgamethatanpredit,ateverystep,whatisthe globalpositioninthegame,anweexternalizeitintoanexternalwinningondi-

tionsuhthat,theresulting

n

^-proessdistributedgamewithexternalondition isequivalent,in someeetivesense,to theinitialgame?

Thenotionofleaderdenedbelowfollowsthis intuition.Infat,itprovides

aloalonditionthatissuientforsuhaglobalknowledgetobeavailableto

aProessplayer.

Denition10 (Leader). Given a

2

^{-proess game}

G = hP, E, T P , T E , e 0 , Ai

^,

wesaythatProess

2

^{isaleaderwhen,forany}Environmentposition

e ∈ E

^,any

Proessespositions

x

^and

y ∈ P

^suhthatboth

(e, x) ∈ T E

^and

(e, y) ∈ T E

^,

if

x[2] = y[2]

^then

x[1] = y[1]

^,

if

x[2] ∈ E[2]

^or

y[2] ∈ E[2]

^then

x = y

^.

Intuitively,Proess

2

^{isaleaderwhen,assoonasheknowsaglobalEnviron-}

mentpositionthen,afteranEnvironmentmove(orseveralonseutivemovesif

Proess

2

^{staysidleforsometime),Proess}

2

^{anpredit,fromhisownposition,}

theglobalProessespositionofthegame.

Thisloalpropertyhasthefollowingformulationwhenitomestoonsider-

ingplays:

Lemma1. Let

G = hP, E, T P , T E , e 0 i

^bea

2

^{-proessarenawithinitialposition}

e 0

^{.Foranystrategy}

σ

^{fortheproesses,therestritionof}

view 2

^{tothe playsthat}

are onsistentwith

σ

^{andativefor Proess}

2

^isone-to-one.

Proof. Immediatefrom thedenition.

Rephrasedinamoreusefulway,thisobservationleadstothefollowingresult:

Lemma2. Forany

2

^-proessgame

G = hP, E, T P , T E , e 0 , Wi

^{suhthatProess}

2

^{isaleader,thereexistsa}

(P [1], E[1])

^-automaton

A 2

^{suhthatforanystrategies}

σ

^on

G

^,

σ 1

^on

G 1

^{,the following} propositionsare equivalent:

(1) thereexistsastrategy

σ 2

^on

G[2]

^suhthat

σ = σ 1 ⊗ σ 2

(2) thereisanaepting run

ρ

^of

A 2

^over

t _{σ 1}

^suhthat

ρ = t _{σ 1}

^.

Proof. (sketh)Werstgivehereaonstrutionfor

A 2

^{intheasebothProess}

1

^andProess

2

^{arealwaysativeinthepositionsforProesses.}

Automaton

A 2 = hQ 2 , P [1], E[1], q 0,2 , δ 2 , Acc 2 i

^{anbedenedasfollows:}

Q ^∀ ₂ = E

^;

Q ^∃ ₂ = P [2] ∪ {q 0,2 }

^,

δ ₂ ^∀ (q, p 1 ) = {p 2 ∈ Q ^∃ ₂ : (q, (p 1 , p 2 )) ∈ T _E } (q ∈ Q ^∀ ₂ , p 1 ∈ P [1])

^,

δ ₂ ^∃ (p 2 , e 1 ) = {q ∈ Q ^∀ ₂ : q[1] = e 1 ∧ (p 2 , q[2]) ∈ T P [2]} (p 2 ∈ Q ^∃ ₂ , e 1 ∈ E[1])

with

δ ₂ ^∃ (q 0,2 , e 1 ) = {e 0 [2]}

^,

Acc 2 = Q ^ω ₂

^.

Theorrespondenebetweenruns of

A 2

^{onstrategytrees in}

G[1]

^{and strategy}

treesin

G

^{easilyfollowsfromthis} onstrution,andfrom thefat that Proess

2

^isaleaderin

G

^.

IntheaseProess

2

^{maybeinativatedby}Environmentoneanhekthat, sineProess

2

^{isaleader,game}

G

^{anberstnormalizedsothatthisnolonger}

happens(detailsarenotgivendue tolakofspae).

IntheaseProess

1

^{maybeinativatedby}Environment,thentheonstru- tionbelowanbeextended,dening(quiteeasilythoughtediously)anautoma-

ton

A 2

^with

ǫ

-transition.However,themainargumentsremainthesame.

Sinethepreviousresultholdsforarbitraryexternalonditionandarbitrary

strategiesin

G[1]

^(evenif

G[1]

^isitselfadistributed game),itfollows:

Theorem6 (Externalization).

For any

n

^-proess distributed game

G = hP, E, T P , T _E , e 0 , Ai

^{with non deter-}

ministiexternalwinningondition

A

^{suhthatProess}

n

^{isaleader,thereisa}

(P [1 . . . n − 1], E[1 . . . n − 1])

^-automaton

A n

^{suhthat thefollowing}propositions are equivalent:

(1) theproesses haveadistributedwinning strategy on

G

^.

(2) theproesseshaveadistributedwinningstrategyin

hG[1 . . . n−1], e 0 [1 . . . n−

1], A ◦ A n i

^.

Example 3 (The Pipeline: End). Wehavealreadymentionedthat,in the

n

^-

proesspipelinearena,fromanyinitialposition,Proess

n

^{isaleader.Itfollows}

that Theorem6applies.

Moreover,observethattheresulting

(n−1)

^{-proessgamearena}

G[1 . . . n− 1]

is nothingbut a

(n − 1)

^{-proesspipeline arena.This says that Theorem6an}

beapplied repeatedlytill thenumberof proessesis reduedto one.Now,one

aninternalizetheautomaton,andomputeawinningstrategyintheresulting

simplegameusingTheorem3.

Transposed onour moreabstrat setting, this anbe expressed asthe fol-

lowingorollaryofthetheorem.

Corollary 1. Forany

n

^-proess(

n ≥ 2

⁾ distributedgame

G

^{suhthat for eah}

i ∈ {2, . . . , n}

^proess

i

^{is a leader in}

G[1 . . . i]

^{, the problem of} determining whether the proesseshave awinning strategy isdeidable.

is an alternating automaton that needs to be simulated by a non alternating

one sothat the omposition anbe iterated. This means that the omplexity

ofsolvingthepipelinearhiteturesynthesisproblem bymeansofitsenoding

into adistributed game is atowerof exponents of depth at least the number

of omponentsin the pipeline. This (bad) omplexitywasexpeted, sine this

problemisnon-elementary[10℄.

4 Conluding Remarks

We have dened a set of automata theoreti tools that an be used to solve

variousdistributed synthesisproblems,e.g.thepipelinearhiteture[3℄.

Comparedto[6℄wedoobtainanautomatatheoreti interpretationofmost

oftheoperationsdenedthere:intheirapproah,applyingsuessivelyDivide

andGluetoagamewhereboth

0

^and

n

^{areleadersamounts,inoursetting,to}

externalize

0

^{,toapplythesimulationtheorem,to}externalize

n

^{,andeventually}

tointernalizetheresultingautomaton.

Still,oneappliationasepresentedbytheauthorstosolvetheloalspei-

ationase[5℄isnotsolvedin thispaper.Thisisleftforfurtherstudies. There

isahanethattreeautomata theorywillstillprovidearguments.

Referenes

1. A.Arnold,A.Vinent,andI.Walukiewiz. Gamesforsynthesisofontrollerswith

partialobservation. toappearinTheoretialComputerSienes,2002.

2. E.A. Emersonand C.S. Jutla. Treeautomata, mu-alulusand determinay. In

Pro.32thSymp.onFoudationsofComputerSienes,pages368377.IEEE,1991.

3. O. Kupferman and M. Y. Vardi. Synthesizingdistributed systems. In Logi in

Computer Sienes,pages389398,2001.

4. F. Lin and M. Wonham. Deentralized ontrol and oordination of disrete

eventsystemswithpartialobservation. IEEE Transations onautomationtrol,

33(12):13301337, 1990.

5. P.Madhusudanand P.S. Thiagarajan. Distributed ontroller synthesis for loal

speiations.In28thInternationalColloquiumonAutomata,LanguagesandPro-

gramming(ICALP),volume2076 ofLNCS,pages396407,2001.

6. S.Mohalik and I. Walukiewiz. Distributedgames. InFoundations of Software

TehnologyandTheoretialComputerSiene,pages338351,2003.

7. D.E. Muller and P.E. Shupp. Simulating alternating tree automata by non-

deterministiautomata. TheoretialComputerSienes,141:67107,1995.

8. G.L. Petersonand J.H.Reif. Multiple-personalternation. In20th AnnualIEEE

SymposiumonFoundationsofComputer Sienes,pages348363,otober1979.

9. G.L.Peterson,J.H.Reif, andS.Azhar. Deisionalgorithmsfor multiplayernon-

ooperative gamesof inomplete information. Computers andMathematis with

Appliations,43:179206,january2002.

10. AmirPnueliandRoniRosner.Distributedreativesystemsarehardtosynthesize.

InIEEESymposiumonFoundations ofComputerSiene,pages746757,1990.

11. M.O.Rabin. Deidabilityofseondordertheoriesandautomataoninnitetrees.

TransationsoftheAmerianMathematialSoiety,141:135, 1969.