Random Fruits on the Zielonka Tree

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RANDOM FRUITS ON THE ZIELONKA TREE

FLORIANHORN

1

CWI,Amsterdam,TheNetherlands E-mailaddress: f.horn wi.nl

Abstra t. Sto hasti games are a natural model for the synthesis of ontrollers on-frontedto adversarialand/or random a tions. Inparti ular,

ω

-regulargamesof innite length anrepresentrea tivesystemswhi harenotexpe tedtorea ha orre tstate,but rathertohandlea ontinuousstreamofevents. One riti alresour einsu happli ations isthememoryusedbythe ontroller. Inthispaper,westudytheamountofmemorythat anbesavedthroughtheuseofrandomisationinstrategies, andpresentmat hingupper andlowerboundsforsto hasti Mullergames.

1. Introdu tion

A sto hasti gamearena is a dire ted graph with three kindsof states: Eve's, Adam's and random states. A token ir ulates on this arena: when it is in one of Eve's states, she hooses its next lo ation among the su essors of the urrent state; when it is in one of Adam's states, he hooses its next lo ation; and when it is ina random state, the next lo ation is hosen a ording to a xed probability distribution. The result of playing the game for

ω

moves is an innite path of the graph. A play is winning either for Eve or for Adam, and the winner problem onsists in determining whether one of the players has a winning strategy, from a given initial state. Closely related problems on ern the omputationofwinningstrategies,aswellasdeterminingthenatureofthesestrategies: pure orrandomised,with niteor innitememory. Therehasbeenalonghistoryofusingarenas without random states (2-player arenas) for modelling and synthesising rea tive pro esses [BL69 , PR89℄: Eve represents the ontroller, and Adam theenvironment. Sto hasti (2

1

2

-player) arenas [deA97℄, withthe addition of random states, an also model un ontrollable a tionsthathappen a ordingtoarandom law,rather thanby hoi e ofana tively hostile environment. The desired behaviour of the system is traditionally represented as an

ω

-regularwinning ondition,whi hnaturallyexpressesthetemporalspe i ationsandfairness assumptions of transition systems [MP92 ℄. From this point of view, the omplexity of the winningstrategiesisa entralquestion,sin etheyrepresentpossibleimplementationsofthe ontrollers inthe synthesisproblem. In this paper, we fo uson animportant normalform of

ω

-regular onditions,namely Muller winning onditions (see [Tho95℄for a survey).

1998ACMSubje tClassi ation: D.2.4. ModelChe king(Theory).

Keywordsandphrases: model he king, ontrollersynthesis,sto hasti games,randomisation.

Thisworkwas arriedoutduringthetenureofanERCIM"AlainBensoussan"FellowshipProgramme.

c

Florian Horn

CC

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In the ase of 2-player Muller games, a fundamental determina y result of Bü hi and Landweber states that, from any initial state, one of the players has a winning strategy [BL69 ℄. Gurevi handHarrington usedthelatest appearan e re ord (LAR)stru tureof M -Naughton to extend this result to strategies withmemory fa torial inthesize ofthe game [GH82℄. Zielonka renes the LAR onstru tion into a tree, and derives from it an elegant algorithmto omputethewinningregions[Zie98 ℄. AninsightfulanalysisoftheZielonkatree by Dziembowski, Jurdzinski, and Walukiewi z leads to optimal (and asymmetri al) mem-ory bounds for pure (non-randomised) winning strategies [DJW97 ℄. Chatterjee extended these bounds to the ase of pure strategies over 2

1

2

-player arenas [Cha07b℄. However, the lowerboundon memorydoesnotholdforrandomisedstrategies, eveninnon-sto hasti are-nas: Chatterjee, de Alfaro,and Henzingershowthatmemoryless randomised strategiesare enough for to deal with upward- losed winning onditions [CdAH04 ℄. Chatterjee extends thisresultin[Cha07a℄,showingthat onditionswithnon-trivialupward- losedsubsetsadmit randomised strategieswithlessmemory than pure ones.

Our ontributions. The memory bounds of [Cha07a ℄ are not tight in general, even for 2-player arenas. We give here mat hing upperand lower boundsfor any Muller ondition

F

,intheform ofa number

r

F

omputed fromthe Zielonka treeof

F

:

•

if Eve has a winning strategy in a

2

1

2

-player game

(A, F)

, she has a randomised winning strategy withmemory

r

F

(Theorem 4.2);

•

there is a 2-player game

(A

F

, F)

where any randomised winning strategy for Eve hasat least

r

F

memory states(Theorem 5.2).

Furthermore, the witness arenas we build in the proof of Theorem 5.2 are signi antly smallerthan in[DJW97 ℄, even thoughtheproblemof polynomial arenas remainsopen. Outline of the paper. Se tion 2 re alls the lassi al notions inthe area, whileSe tion 3 presentsformerresultsonmemoryboundsandrandomisedstrategies. Thenexttwose tions present our main results. In Se tion 4, we introdu e the number

r

F

and show that itis an upperbound on thememory needed to win inany2

1

2

-game

(A, F)

. InSe tion 5, we show thatthisboundistight. Finally,inSe tion 6,we hara terisethe lassofMuller onditions thatadmit memorylessrandomised strategies, and show thatfor ea h Muller ondition, at leastone ofthe players annot improve itsmemory throughrandomisation.

2. Denitions

We onsider turn-based sto hasti two-player Muller games. We re all here several lassi al notions inthe eld,and refer thereaderto [Tho95,deA97℄ formore details. ProbabilityDistribution. Aprobabilitydistribution

γ

overaset

X

isafun tionfrom

X

to

[0, 1]

su h that

P

x∈X

γ(x) = 1

. The set of probability distributions over

X

is denoted by

D(X)

.

Arenas. A 2

1

2

-player arena

A

over a set of olours

C

onsists of a dire ted nite graph

(S, T )

,apartition

(S

E

, S

A

, S

R

)

of

S

,aprobabilisti transitionfun tion

δ : S

R

→ D(S)

su h that

δ(s)(t) > 0 ⇔ (s, t) ∈ T

, anda partial olouring fun tion

χ : S ⇀ C

. Thestatesin

S

E

(resp.

S

A

,

S

R

) are Eve's states (resp. Adam's states, random states), and are graphi ally representedas

#

's (resp.

2

,

△

). A 2-player arena isan arenawhere

S

R

= ∅

.

Aset

U ⊆ S

ofstatesis

δ

- losed ifforeveryrandomstate

u ∈ U ∩S

R

,

(u, t) ∈ T → t ∈ U

. It islive iffor every non-randomstate

u ∈ U ∩ (S

E

∪ S

A

)

,there isa state

t ∈ U

su h that

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Plays and Strategies. Aninnite path, or play, over thearena

A

is an innitesequen e

ρ = ρ

0 ρ

1 . . .

of states su h that

(ρ

i

, ρ

i+1

) ∈ T

for all

i ∈

N

. The set of states o urring innitelyofteninaplay

ρ

isdenoted by

Inf(ρ) = {s | ∃

∞

_{i ∈}

_{N, ρ}

i

= s}

. Wewrite

Ω

forthe setof all plays,and

Ω

s

for theset ofplays thatstart fromthestate

s

.

A strategy with memory

M

for Eve on the arena

A

is a (possibly innite) transdu er

σ = (M, σ

n

, σ

u

)

, where

σ

n

is the next-move fun tion from

(S

E

× M )

to

D(S)

and

σ

u

is the memory-update fun tion, from

(S × M )

to

D(M )

. Noti e that both the move and the update are randomised: strategies whose memory is deterministi are a dierent, less ompa t, model. The strategies for Adam are dened likewise. A strategy

σ

is pure if it doesnot userandomisation. It is nite-memory if

M

is a nite set, and memoryless if

M

is a singleton. Noti e that strategies dened in the usual way as fun tions from

S

∗

to

S

an be dened as strategies withinnite memory: theset of memory states is

S

∗

and the memory update is

σ

u

(s, w) 7→ ws

.

On eastartingstate

s ∈ S

andstrategies

σ ∈ Σ

for bothplayersarexed,theout ome of the game is a random walk

ρ

σ,τ

s

for whi h the probabilities of events are uniquely xed (an event is a measurable set of paths). For an event

P ∈ Ω

, we denote by

P

σ,τ

s

(P )

the probabilitythataplaybelongsto

P

ifitstartsfrom

s

andEveandAdamfollowthestrategies

σ

and

τ

.

A playis onsistent with

σ

iffor ea h position

i

su hthat

w

i

∈ S

E

,

P

σ,τ

w

0 (ρ

i+1

= w

i+1

|

ρ

0 = w

0 . . . ρ

i

= w

i

) > 0

. The set of plays onsistent with

σ

is denoted by

Ω

σ

. Similar notions an be dened for Adam'sstrategies.

Traps andAttra tors. Theattra tor ofEve tothe set

U

,denoted

Attr

E

(U )

,isthesetof stateswhereEve anguaranteethatthetokenrea hestheset

U

withapositiveprobability. It isdened indu tively by:

Attr

0 _E

(U )

=

U

Attr

i+1

_E

(U ) = Attr

i

_E

(U ) ∪{s ∈ S

E

∪ S

R

, ∃t ∈ Attr

i

E

(U ) | (s, t) ∈ T }

∪{s ∈ S

A

| ∀t, (s, t) ∈ E ⇒ t ∈ Attr

i

E

(U )}

Attr

E

(U )

=

S

_i>0

Attr

i

E

(U )

The orresponding attra tor strategy to

U

for Eve is a pure and memoryless strategy

a

U

su hthat for anystate

s ∈ S

E

∩ (Attr

E

(U ) \ U )

,

s ∈ Attr

i+1

E

(U ) ⇒ a

U

(s) ∈ Attr

i

E

(U )

. The dual notion of trap for Eve denotes a set from where Eve annot es ape, unless Adamallowshertodoso: aset

U

isatrapforEveifandonlyif

∀s ∈ U ∩ (S

E

∪ S

R

), (s, t) ∈

T ⇒ t ∈ U

and

∀s ∈ U ∩ S

A

, ∃t ∈ U, (s, t) ∈ T

. Noti e that a trap is a strong notion the token an neverleave itifAdam does not allowit to do so whilean attra tor is a weak one thetoken an avoidthetarget even ifEveuses theattra torstrategy. Noti e also thata trap(foreitherplayer) isalways a subarena.

Winning Conditions. A winning ondition is a subset

Φ

of

Ω

. A play

ρ

is winning for Eve if

ρ ∈ Φ

, and winning for Adam otherwise. We onsider

ω

-regular winning onditions formalised as Muller onditions. A Muller ondition is determined by a subset

F

of the power set

P(C)

of olours, and Eve wins a play if and only if the set of olours visited innitelyoften belongs to

F

:

Φ

F

= {ρ ∈ Ω|χ(Inf(ρ)) ∈ F}

. An exampleof Muller game is giveninFigure1(a). Weuseitthroughoutthepapertodes ribevariousnotions andresults. WinningStrategies. Astrategy

σ

forEveissurelywinning(orsure)fromastate

s

forthe winning ondition

Φ

ifanyplay onsistent with

σ

belongs to

Φ

,and almost-surely winning (or almost-sure) if for any strategy

τ

for Adam,

P

σ,τ

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c

d

a

b

a

b

c

F

= {{a, b}, {a, b, c}, {a, b, c, d}}

(a)ThegameG

= (

A

,

F

)

abcd

bcd acd abd

ab

a

b

(b)ZielonkaTreeofF

Figure1: Re urringExample

3. Former results in memory bounds and randomisation

3.1. Pure strategies

Therehasbeenintenseresear hsin ethesixtiesonthenon-sto hasti setting,i.e. pure strategies and 2-player arenas. Bü hi and Landweber showed the determina y of Muller games in[BL69 ℄. Gurevi h and Harrington used theLAR (Latest Appearan e Re ord) of M Naughton to prove their Forgetful Determina y theorem [GH82 ℄, whi h shows that a memory of size

|C|!

issu ient for any gamethat uses only oloursfrom

C

,even whenthe arenais innite. Thisresult waslater rened byZielonka in[Zie98 ℄, using arepresentation of theMuller onditions astrees:

Denition 3.1(ZielonkaTreeofaMuller ondition). TheZielonkaTree

Z

F ,C

ofawinning ondition

F ⊆ P(C)

isdened indu tively asfollows:

(1) If

C /

∈ F

,then

Z

F ,C

= Z

F ,C

,where

F = P(C) \ F

.

(2) If

C ∈ F

, then the root of

Z

F ,C

is labelled with

C

. Let

C

1 , C

2 , . . . , C

k

be all the maximal sets in

{U /

∈ F | U ⊆ C}

. Then we atta h to theroot, asits subtrees, the Zielonka trees of

F ↾ C

i

,i.e. the

Z

F ↾C

i

,C

i

,for

i = 1 . . . k

.

Hen e, the Zielonka tree is atree with nodeslabelled by sets of olours. A node of

Z

F ,C

is an Eve node ifitislabelled witha set from

F

,otherwise itisan Adamnode.

A later analysis of this onstru tion by Dziembowski, Jurdzinski and Walukiewi z in [DJW97 ℄led to an optimaland asymmetri al boundon thememory needed bythe players to dene surestrategies:

Denition 3.2 (Number

m

F

ofa Muller ondition). Let

F ⊆ P(C)

bea Muller ondition, and

Z

F

1 ,C

1 , Z

F

2 ,C

2 , . . . , Z

F

k

,C

k

be the subtrees atta hed to the root of the tree

Z

F ,C

. We dene thenumber

m

F

indu tively asfollows:

m

F

=











1

if

Z

F ,C

doesnot have anysubtrees,

max{m

F

1 , m

F

2 , . . . , m

F

k

}

if

C /

∈ F

(Adamnode),

k

X

i=1

m

F

i

if

C ∈ F

(Eve node).

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Theorem 3.4 ([Cha07b℄). If Eve has an almost-sure strategy in a 2

1

2

-player Muller game withthewinning ondition

F

,she hasa purealmost-surewithatmost

m

F

memory states.

3.2. Memory redu tion through randomisation

Randomisedstrategiesaremoregeneralthanpurestrategies,andinsome ases,theyare alsomore ompa t. In[CdAH04 ℄,arstresultshowedthatupward- losed onditionsadmit memorylessrandomised strategies, while theydon't admit memorylesspurestrategies: Theorem 3.5 ([CdAH04℄). If Eve has an almost-sure strategy in a 2

1

2

-player Muller game withan upward- losed winning ondition,she has a randomised almost-sure strategy.

Thisresult was later extendedin [Cha07a ℄,byremoving theleavesatta hed to a node of theZielonkaTreerepresenting anupward- losed sub ondition:

Denition 3.6 ([Cha07a ℄). Let

F ⊆ P(C)

be a Muller ondition, and

Z

F

1 ,C

1 , Z

F

2 ,C

2 , . . . ,

Z

F

k

,C

k

be the subtrees atta hed to the root of the tree

Z

F ,C

. We dene the number

m

U

F

indu tively asfollows:

m

U

F

=











1

if

Z

F ,C

1

if

F

isupward- losed,

max{m

U

_F

1 , m

U

F

2 , . . . , m

U

F

k

}

if

C /

∈ F

(Adamnode),

k

X

i=1

m

U

_F

i

if

C ∈ F

(Eve node).

Theorem 3.7 ([Cha07a℄). If Eve has an almost-sure strategy in a 2

1

2

-player Muller game with the winning ondition

F

, she has a randomised almost-sure strategy with at most

m

U

F

memory states.

4. Randomised Upper Bound

TheupperboundofTheorem3.7isnottightforall onditions. Forexample,thenumber

m

U

F

ofthe onditionFinFigure1(b)isthree,whilethereisalwaysanalmost-sure strategy withtwo memory states. We present hereyet anothernumber for any Muller ondition

F

, denoted

r

F

,thatwe ompute fromthe Zielonka Tree:

Denition 4.1 (Number

r

F

of a Muller ondition). Let

F ⊆ P(C)

be a Muller ondition, where theroot has

k + l

hildren,

l

of thembeingleaves. We denote by

Z

F

1 ,C

1 , Z

F

2 ,C

2 , . . . ,

Z

F

k

,C

k

the non-leaves subtrees atta hed to the root of

Z

F ,C

. We dene

r

F

indu tively as follows:

r

F

=











1

if

Z

F ,C

max{1, r

F

1 , r

F

2 , . . . , r

F

k

}

if

C /

∈ F

(Adamnode),

k

X

i=1

r

F

i

if

C ∈ F

(Eve node)and

l = 0,

k

X

i=1

(7)

The rst remark is that if

∅ ∈ F

,

r

F

is equal to

m

F

: asthe leaves belong to Eve, the fourth ase annoto ur. Intheother ase, theintuitionisthatwemergeleavesiftheyare siblings. For example,the number

r

F

for our re urring example istwo: one for the leaves labelled

bcd

and

acd

,and one for theleaveslabelled

a

and

b

. Thenumber

m

F

isfour (one for ea h leaf),and

m

U

F

isthree (onefor the leaves labelled

a

and

b

,and one for ea h other leaf). Thisse tion willbedevotedto theproofof Theorem 4.2:

Theorem 4.2 (Randomised upper bound). If Eve has an almost-sure strategy in a

2-1

2

player Muller game with the winning ondition

F)

, she has an almost-sure strategy with memory

r

F

.

Let

G = (F, A)

be agame dened on thesetof olours

C

su h thatEve winsfrom any initial node. We des ribeinthenextthreesubse tionsare ursive pro edureto ompute an almost-surestrategyforEvewith

r

F

memorystatesinea hnon-trivial aseinthedenition of

r

F

. WeusetwolemmasLemmas4.3and4.5thatderivedire tlyfromsimilarresults in [DJW97 ℄ and [Cha07b ℄. The appli ation of these prin iples to the game G in Figure 1 builds a randomised strategy with two memory statesleft and right. Inleft, Eve sendsthe token to(

տ

or

ւ

)andinright,to(

ր

or

ց

). Thememory swit hesfromright toleft with probabilityone when the token visits a

c

,and from left to right withprobability

1

2

at ea h step.

4.1.

C

is winning for Adam

Inthe ase where Adam winstheset

C

,the onstru tion of

σ

relieson Lemma 4.3: Lemma 4.3. Let

F ⊆ P(C)

be a Muller winning ondition su h that

C /

∈ F

, and

A

be a 2

1

2

-player arena su h that Eve winseverywhere. There are subarenas

A

1 . . . A

n

su h that:

• i 6= j ⇒ A

i

∩ A

j

= ∅

;

• ∀i, A

i

is a trap for Adam inthe subarena

A \ Attr

E

∪

i−1

_j=1

A

j

;

• ∀i, χ(A

i

)

is in luded in the label

E

i

of a hild of the root of

Z

F ,C

, and Eve wins everywhere in

(A

i

, F ↾ E

i

)

;

• A = Attr

E

(∪

n

j=1

A

j

)

.

Letthe subarenas

A

i

bethe oneswhose existen eis proved inthis lemma. We denote by

σ

i

thealmost-sure strategyfor Eve in

A

i

,and by

a

i

theattra torstrategy for Eveto

A

i

inthearena

A \ Attr(∪

i−1

j=1

A

j

)

. We identify the memorystates ofthe

σ

i

,sotheirunionhas thesame ardinalas thelargest of them. For a state

s

,if

i = min{j | s ∈ Attr

E

(∪

j

k=1

A

k

)}

, we dene

σ(s, m)

by:

•

if

s ∈ A

i

σ

u

(s, m) = σ

u

i

(s, m)

σ

n

(s, m) = σ

n

i

(s, m)

•

if

s ∈ Attr

E

(∪

i

k=1

A

k

) \ A

i

σ

u

(s, m) = m

σ

n

(s, m) = a

i

(s)

Byindu tion hypothesis overthe number of olours, we an assumethatthestrategies

σ

i

have

r

F

i

memorystates. Thestrategy

σ

uses

max{r

F

i

}

memory states. Proposition 4.4.

P

σ,τ

(8)

Proof. The subarenas

A

i

are embedded traps, dened in su h a way that the token an es ape an

A

i

only by going to the attra tor of a smaller one. Eve has thus a positive probabilityofrea hingan

A

j

with

j < i

. Thus, ifthetokenes apesoneof the

A

i

innitely often, thetoken hasprobabilityone to go toan

A

j

with

j < i

. Byargument ofminimality, aftera nite prex,the token will stayinone of the trapsforever.

The strategy

σ

i

is almost-sure from any state in

A

i

. As Muller onditions are prex-independent, it follows from Proposition 4.4 that

σ

is also almost-sure from any state in

A

.

4.2.

C

iswinningfor Eve,and the rootof

Z

F ,C

has no leaves among its hildren. Inthis ase, the onstru tion relies onthe following lemma:

Lemma 4.5. Let

F ⊆ P(C)

be a Muller winning ondition su h that

C ∈ F

,

A

a 2

1

2

-player arena oloured by

C

su hthat Eve winseverywhere, and

A

i

the labelof a hild of therootin

Z

F ,C

. Then,Evewinseverywhere onthesubarena

A \ Attr

E

(χ

−1

_{(C \ A}

i

))

withthe ondition

F ↾ A

i

.

Eve has a strategy

σ

i

that is almost-sure from ea h state in

A \ Attr

E

(χ

−1

_{(C \ A}

i

))

. In this ase, the set of memory states of

σ

is

M = ∪

k

i=1

(i × M

i

)

. The next-move and memory-update fun tions

σ

n

and

σ

u

for a memory state

m = (i, m

i

₎

aredened below:

•

if

s ∈ χ

−1

_{(C \ A}

i

)

σ

u

(s, (i, m

i

_{)) = (i + 1, m}

i+1

₎

where

m

i+1

isanystate in

M

i+1

if

s ∈ S

E

,

σ

n

(s, (i, m

i

₎₎

isany su essorof

s

in

A

•

if

s ∈ Attr

E

(χ

−1

_{(C \ A}

i

))

σ

u

(s, (i, m

i

_{)) = (i, m}

i

₎

σ

n

(s, (i, m

i

_{)) = a}

i

(s)

•

if

s ∈ A \ Attr

E

(χ

−1

_{(C \ A}

i

))

σ

u

(s, (i, m

i

)) = (i, σ

u

i

(s, m

i

))

σ

n

(s, (i, m

i

_{)) = σ}

n

i

(s, m

i

)

On eagain, we anassumethatthememory

M

i

ofthestrategy

σ

i

isofsize

r

F ↾A

i

. Here, however, the memory set of

σ

is thedisjoint unionof the

M

i

',so

σ

's needsthe sumof the

{r

F ↾A

i

}

's.

Proposition 4.6. Let

uc

be the event the top-level memory of

σ

is ultimately onstant. Then,

P

σ,τ

s

0 (ρ ∈ Φ

F

| uc) = 1

.

Proof. We all

i

thevalueofthetop-levelmemoryatthelimit. Afteraniteprex,thetoken stopsvisiting

χ

−1

_{(C \ A}

i

)

. Thus,withprobabilityone,italsostops visiting

Attr

E

(χ

−1

_{(C \}

A

i

))

. Fromthispointon,thetokenstaysinthearena

A

i

,whereEveplays withthe almost-sure strategy

σ

i

. Thus,

P

σ,τ

_{(ρ ∈ Φ}

F ↾A

i

| uc) = 1

, and, as

Φ

F ↾A

i

⊆ Φ

F

, Proposition 4.6 follows.

Proposition 4.7. If the top-level memory takes ea h value in

1 . . . k

innitely often, then surely,

∀i ∈ 1 . . . k, χ(Inf(ρ))

* A

i

.

Proof. The update on the top-level memory follows a y le on

1 . . . k

, leaving

i

only when thetokenvisits

χ

−1

_{(C \A}

i

)

. Thus,inorderforthetop-levelmemoryto hange ontinuously, thetokenhasto visit ea h ofthe

χ

−1

_{(C \ A}

(9)

4.3.

C

iswinningforEve,andtherootof

Z

F ,C

hasatleast oneleafinits hildren. As in the previous se tion, the onstru tion relies on Lemma 4.5. In fa t, the on-stru tion for hildren whi h are not leaves, labelled

A

1 , . . . , A

k

, is exa tly the same. The dieren eisthatweaddhereasinglememorystate

0

thatrepresentsalltheleaves (la-belled

A

−1

, . . . , A

−l

). Thememorystatesarethusupdatedmodulo

k +1

,andnotmodulo

k

. The next-move fun tion of

σ

whenthe top-level memory is

0

is aneven distribution over all thesu essorsin

A

of the urrent state. The memory-update fun tion hasprobability

1

2

to stay into

0

, and

1

2

to go to

(1, m

1 )

, for some memory state

m

1 ∈ M

1

. Thus,

σ

uses memory

P

k

i=1

r

F

i

+ 1

. We prove now that

σ

is almost-sure. The stru ture of the proof is thesame asintheformer se tion, withsome extra onsiderations for thememory state

0

. Proposition4.8. Let

uc

be the eventthetop-level memory of

σ

isultimately onstant and dierentfrom 0. Then,

P

σ,τ

s

(ρ ∈ Φ

F

| uc) = 1

.

Proof. The proofis exa tlythe sameastheone of Proposition4.6.

Proposition 4.9. The event the top-level memory is ultimately onstant and equal to 0 has probability 0.

Proof. When the top-level memory is 0, the memory-update fun tion has probability

1

2

at ea h stepto swit hto 1. Proposition 4.9follows.

Proposition 4.10 onsiders the ase where the top-level memory evolves ontinuously. By denition of the memory update, this an happen only if all the memory states are visitedinnitelyoften.

ec

be the event the top-level memory takes ea h value in

0 . . . k

innitely often. Then,

∀i ∈ −l . . . k,

P

σ,τ

s

(χ(Inf(ρ)) ⊆ C

i

| ec) = 0

.

Proof. Asintheproofof Proposition 4.7,fromthefa tthatthememory isequalto ea hof the

i ∈ 1 . . . k

innitelyoften, we an dedu ethatthe token surely visitsea h ofthe

C \ A

i

innitely often. We only need to show that, with probability one and for any

j ∈ 1 . . . l

, thesetoflimit statesisnot in luded in

A

−j

. The ZielonkaTrees ofthe onditions

F ↾ A

−j

areleaves. Thismeans that they aretrivial onditions,where allthe playsare winning for Adam. Consequently, in this ase, Lemma 4.5 guarantees that

Attr

E

(χ

−1

_{(C \ A}

−j

))

isthe wholearena. Thedenitionof

σ

inthe memory state

(0)

isto playlegal movesat random. ThereisthusapositiveprobabilitythatEvewillplaya ordingtotheattra torstrategy

a

j

long enough to guarantee a positive probability that thetoken visits

χ

−1

_{(C \ A}

−j

)

. To be pre ise, for any

s ∈ S

,this probability is greater than

(2 · |S|)

−|S|

. Thus, withprobability one,thetoken visitsea h

χ

−1

_{(C \ A}

−j

)

innitelyoften. Proposition 4.10follows.

Theinitial ase, where theZielonkatreeis redu edto aleaf, istrivial: thewinnerdoes not depend ontheplay. Thus, Theorem 4.2followsfrom Se tions4.1, 4.2, and4.3.

5. Lower Bound

(10)

Theorem 5.1 ([Maj03℄). For any set of olours

C

, there is a 2-player Muller game

G

C

=

(A

C

, F

C

)

su h that Eve has an almostsure, but none of heralmost-sure strategies have less than

|C|

2 !

memory states.

However, this is a general lower bound on all Muller onditions, while we aim to nd spe i lowerboundsforea h ondition. We prove herethatthereisalowerboundforea h Muller onditionthat mat hes the upperboundof Theorem 4.2:

Theorem 5.2. Let

F

be a Muller ondition on

C

. There is a 2-player arena

A

F

over

C

su h that Eve has a sure strategy, but none of her almost-sure strategies have less than

r

F

memory states.

Asthe onstru tionoftheupperboundwasbasedontheZielonkatree,thelowerbound isbased onthe Zielonka DAG:

Denition 5.3. The ZielonkaDAG

D

F ,C

ofawinning ondition

F ⊆ P(C)

isderived from

Z

F ,C

bymergingthe nodeswhi hshare thesame label.

5.1. Cropped DAGs

Therelation between

r

F

and theshapeof

D

F ,C

isasymmetri al: itdependsdire tlyon thenumber of hildrenof Eve's nodes, andnot at all onthe numberof hildren of Adam's nodes. Thenotion of ropped DAG isthe nextlogi al step: a sub-DAGwhere Eve's nodes keep all their hildren, while ea h nodeof Adamkeepsonly one hild:

Denition 5.4. ADAG

E

isa ropped DAG ofa ZielonkaDAG

D

F ,C

ifand onlyif

•

The nodes of

E

arenodesof

D

F ,C

,with thesame owner andlabel.

•

There isonly one node withoutprede essor in

E

, whi h we all the root of

E

. It is the root of

D

F ,C

,ifitbelongs to Eve;otherwise, itis oneof its hildren.

•

The hildren ofa node of Evein

E

areexa tlyits hildren in

D

F ,C

.

•

A node of Adam has exa tly one hild in

E

, hosen among his hildren in

D

F ,C

, provided there isone. If ithasno hildrenin

D

F ,C

,ithasno hildren in

E

.

CroppedDAGresembleZielonka DAGs: thenodesbelong to eitherEve or Adam, and they arelabelled by setsof states. We an thus ompute thenumber

r

E

ofa roppedDAG

E

inanaturalway. Infa t,this numberhasamoreintuitivemeaninginthe aseof ropped DAGs: iftheleavesbelongtoEve,itisthenumberof bran hes;ifAdamownstheleaves,it isthenumberofbran heswiththeleafremoved. Furthermore,thereisadire tlinkbetween the ropped DAGsof aZielonka DAG

D

F ,C

andthe number

r

F

:

F

be a Muller ondition on

C

, and

D

F ,C

be its Zielonka DAG.Then there isa ropped DAG

E

∗

su h that

r

E

∗

= r

F

.

5.2. From ropped DAGs to arenas

From any ropped DAG

E

of

D

F ,C

, we dene an arena

A

E

whi h follows roughly the stru ture of

E

: the token starts from the root, goes towards the leaves, and then restarts fromtheroot. Inhernodes,Eve an hoosetowhi h hildshewantstogo. Adam's hoi es, ontheotherhand, onsistsineitherstoppingthe urrenttraversalorallowingittopro eed.

(11)

•

in

Pick

∗

_(C)

,Adam an visit anysubset of oloursin

C

;

•

in

Pick(D)

,he mustvisit exa tlyone olour in

D

.

Bothare representedinFigure2,and theyaretheonly o asionswhere oloursarevisited in

A

E

: all the otherstates are olourless.

c

1 c

i

c

k

· · ·

C = {c

1 . . . c

k

}

(a)

Pick

∗

_(C)

d

1 · · ·

d

i

· · ·

d

k

D = {d

1 . . . d

k

}

(b)

Pick(D)

Figure2:

Pick

∗

_(C)

and

Pick(D)

Eve'sstatesinthearena

A

E

areinbije tionwithhernodesin

E

. Adam'snodes,onthe other hand, are in bije tion with the pairs parent- hild of

E

, where the parent belongs to Eve and the hild to Adam.

In the state orresponding to the node

n

, Eve an send the token to any state of the form

n − c

. In states orresponding to leaves, Eve has no de ision to take, and Adam an visit any olours inthe labelof the leaf(

Pick

∗

pro edure). The token isthen sent ba k to theroot.

Adam'smoves donot involve the hoi e of a hild: by Denition5.4, Adam'snodesin

E

have but one hild. Instead, he an either stop the urrent traversal, or, if the urrent node isnot a leaf, allow itto pro eed to its only hild. If he hooses to stop, Adamhas to visit some oloured states before the token is sent ba kto the root. The available hoi es depend on the labels of both the urrent and the former nodes whi h is why there are asmany opies of Adam's nodes in

A

E

as they have parents in

E

. If theparent islabelled by

E

,andthe urrent node by

A

,thetokengoesthrough

Pick

∗

_(E)

and

Pick(E \ A)

. Adam anthus hooseanynumberof oloursin

E

,aslongashe hoosesat leastoneoutsideof

A

.

E

A

E

′

E

− A

E

′

Pick

∗

(E)

Pick(E \ A)

root

(a)Edge

E

-

A

when

A

isanode

E

A

Pick

∗

(E)

Pick(E \ A)

root

(b)Edge

E

-

A

when

A

isaleaf

Figure 3: Adam'sstates in

A

E

.

5.3. Winning strategy and bran h strategies

(12)

b = E

1 A

1 . . . E

ℓ

(A

ℓ

)

, Eve's moves follow the bran h

b

: in

E

i

, she goes to

E

i

− A

i

. When Adamstops the traversalat the

i

th step, Eveupdates hermemory asfollows:

•

If

E

i

haszero orone hild in

E

,thememoryis un hanged;

•

otherwise, thenewmemorybran hhas

E

1 A

1 . . . E

i

A

asaprex,where

A

isthenext hild of

E

i

,or therst one if

A

i

wasthe last.

Proposition 5.6. The strategy

ς

is surely winning for Eve inthe game

(A

E

, F)

.

Proof. Let

ρ

be a play onsistent with

ς

. We denote by

i

the smallest integer su h that traversals stopsinnitelyoftenatthe

i

th step. After aniteprex,therst

2i − 1

nodesin the memory bran h are onstant, and we denote them by

E

1 A

1 E

2 . . . E

i

. From this point on, the oloursvisited belong to

E

i

. Furthermore, ea h time a traversal stops at step

i

, a stateisvisitedoutsideofthe urrent

A

i

,whi h hangesafterwards tothenext,ina ir ular way. It follows that

Inf(ρ) ⊆ E

i

, and, for any hild

A

of

E

i

in

E

,

Inf(ρ)

* A

. Thus

ρ

is winningfor Eve. Proposition 5.6follows.

Obviously, Adam has no winning strategy in

A

E

. However, we des ribe the lass of bran h strategies, whose point is to punish any attempt of Eve to win with less than

r

F

memory states. There is one su h strategy

τ

b

for ea h bran h

b

in

E

(when e the name), and theprin iple isthat

τ

b

stops thetraversalassoon asEve deviates from

b

:

Denition 5.7. The bran h strategy

τ

b

for Adam in

A

E

, orresponding to the bran h

b = E

1 A

1 E

2 . . . E

ℓ

(A

ℓ

)

in

E

,is a positionalstrategy whose moves aredes ribed below.

•

In astate

E − A

su hthat

∃i, E = E

i

∧ A 6= A

i

: stop thetraversaland visit

A

i

;

•

ina state

E − A

su hthat

∃i, E = E

i

∧ A = A

i

: sendthetoken to

E

i+1

;

•

inthe state

E

ℓ

− A

ℓ

,or the leaf

E

ℓ

: visit the oloursof

E

ℓ

.

Nomoveisgivenforastate

E −A

su hthat

∀i, E 6= E

i

,asthesestatesarenotrea hable fromtheroot whenAdamplays

τ

b

. Noti ealsothatwhenAdam hooses tostopatraversal ina state

E

i

− A

,he an visit exa tlythe olours of

A

i

: as

A

and

A

i

aremaximal subsets of

E

i

,thereisat leastone state in

A

i

\ A

thathe an pi kinthe

Pick(E

i

\ A)

area.

5.4. Winning against bran h strategies

The key ideaof the proof of Theorem 5.2 isthat iftwo bran hes

b

and

b

′

of

E

are too dierent, Eve needsdierent memory statesto win against

τ

b

and

τ

b

′

.

σ = (M, σ

n

, σ

u

)

be an almost-sure strategy forEve in

(A

E

, F)

. Then

σ

has memory atleast

r

E

.

Proof. Let

b = E

1 A

1 . . . E

ℓ

(A

ℓ

)

beabran hof

E

and

τ

b

bethe orrespondingbran hstrategy for Adam. By denition of

τ

b

,the set of oloursvisited ina traversal onsistent with

τ

b

is one ofthe

A

i

's, or

E

ℓ

ifand onlyifEve playsalong

b

. As

σ

isalmost-sure,there mustbe a memorystate

m

su hthatEvehasapositiveprobabilitytoplayalong

b

. Itisalsone essary to ensure thatnone ofthe

A

i

'sis visitedinnitelyoften, withthepossible ex eption of

A

ℓ

. So, if Eve has a positive to play along a bran h

b

′

when she is in the memory state

m

,

E

1 A

1 . . . E

ℓ

must be a prex of

b

′

. It follows that a single memory state an be suitable against two strategies

τ

b

and

τ

b

′

with

b = E

1 A

1 . . . E

ℓ

(A

ℓ

)

and

b

′

_{= E}

′

1 A

′

1 . . . E

ℓ

′

(A

′

_ℓ

′

)

only if

ℓ = ℓ

′

and

∀i ≤ ℓ, E

i

= E

′

(13)

By Proposition 5.5, there is a ropped DAG

E

of

D

F ,C

su h that

r

E

= r

F

. So, in general, Eve needs randomised strategies with memory

r

F

in order to win games whose winning ondition is

F

. This ompletes theproofof Theorem5.2.

6. Con lusion

Wehaveprovidedbetterandtightboundsforthememoryneededto denealmostsure winningrandomised strategies. Thisallowsus to hara terisethe lassofMuller onditions whi h admit randomisedmemoryless strategies:

Corollary6.1. Eveadmitsrandomised memorylessalmost-surestrategies foraMuller on-dition

F

if andonly ifall her nodes in

Z

F ,C

have either one hild, or onlyleave hildren.

This yields a NPalgorithm for the winnerproblem of su h games, assolving 1

1

2

-player MullergamesisPtime[CdAH04℄. Another onsequen eofourresultisthatforea hMuller ondition, at leastone ofthe players annotimprove its memory throughrandomisation: Corollary 6.2. Let

F

be a Muller ondition. If

∅ ∈ F

, Eve needs as mu h memory for randomised strategy as for pure strategies. Otherwise, Adam does.

Our proof of lower bound also improves on the size of thewitness arena: itis roughly equivalentto thesizeoftheZielonkaDAG,insteadofthesizeoftheZielonkatree. Whether these bounds still hold for arenas of size polynomial in the number of olours remains an openquestion, ex eptforspe ial ases like Streett games[Hor07 ℄.

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[Cha07a℄ KrishnenduChatterjee.OptimalStrategySynthesisinSto hasti MullerGames.InFOSSACS, pages138152,2007.

[Cha07b℄ KrishnenduChatterjee.Sto hasti MüllerGamesarePSPACE-Complete.InFSTTCS,pages436 448,2007.

[deA97℄ Lu adeAlfaro.FormalVeri ationofProbabilisti Systems.PhDthesis,Stanford,1997. [DJW97℄ StefanDziembowski,Mar inJurdzinski,andIgorWalukiewi z.Howmu hmemoryisneededto

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[GH82℄ YuriGurevi handLeoHarrington.Trees,automata,andgames.InSTOC,pages6065,1982. [Hor07℄ FlorianHorn.Di ingontheStreett.IPL,104(1):19,2007.

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Spe i- ation.Springer,1992.

[PR89℄ AmirPnueli andRoniRosner.Onthesynthesisofarea tivemodule.InPOPL,pages179190, 1989.

[Tho95℄ WolfgangThomas.Onthesynthesisofstrategiesininnitegames.InSTACS,pages113,1995. [Zie98℄ WieslawZielonka.Innitegamesonnitely olouredgraphswith appli ations toautomataon

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