old.automata.rwth-aachen.deold.automata.rwth-aachen.de/users/loeding/inf-games-and-automata.pdfOrigin Circuit synthesis and Church’s problem (1957) Setting: •Sequence of input

Infinite games and automata theory

Christof LodingRWTH Aachen, Germany

Spring School 2009June 1–5, Bertinoro, Italy

Infinite games and automata theory · Bertinoro, June 2009 1 / 84

In this tutorial

1. How can automata theory help to solve problems for games?

2. How can games help to solve problems in automata theory?


In this tutorial

1. How can automata theory help to solve problems for games?

2. How can games help to solve problems in automata theory?

automata oninfinite words

games of infiniteduration

automata oninfinite trees


1 Introduction

2 Basics on games

3 Transformation of winning conditions

ω-Automata

Game reductions

Logical winning conditions

4 Tree automata

Complementation

Emptiness

5 Beyond finite automata


Origin

Circuit synthesis and Church’s problem (1957)

Setting:

• Sequence of input signals arrives

• Circuit produces a sequence of output signals (depending onthe inputs it has seen)

• Result is a non-terminating sequence of input and outputsignals

• A logical specification describes the desired properties ofthese sequences

circuit finput output

Task: Automatically synthesize a circuit from the specification


More formally

circuit finput ∈ Σ1 output ∈ Σ2

• Input sequence α ∈ Σω1 and output sequence β ∈ Σω

2

• Specification ϕ(α, β)

Problem:

• Decide if the there is a sequential transformation f : Σ∗1 → Σ2

realizing ϕ, and construct one if possible.


More formally



2


Problem:

• Decide if the there is a sequential transformation f : Σ∗1 → Σ2

realizing ϕ, and construct one if possible.

Modeled as a game:

• One player plays input signals, the other player output signals

• The specification is the winning condition for the output player

• A transformation f realizing the specification is a winningstrategy for the output player


Reactive Systems

• Games can serve as general model for reactive systems, i.e.,systems with input/output behavior

• To obtain more realistic models it is often required to addfeatures like time, probabilities, concurrency, ...

• To study these more complex models a good understanding ofthe core theory originating from Church’s problem isfundamental{ this tutorial presents the automata theoretic aspects


1 Introduction

2 Basics on games


ω-Automata

Game reductions


4 Tree automata

Complementation

Emptiness



Notations

For a set X:

• |X| = size of X

• X∗ = the set of finite sequences over X

• Xω = the set of infinite sequences over X

• For α ∈ Xω:

Inf(α) = {x ∈ X | x occurs infinitely often in α}.


Game Graph / Arena

G = (VE,VA, E, c)

• VE: vertices of Eva (player 1, circle)• VA: vertices of Adam (player 2, box)• E ⊆ V ×V: edges with V = VE ∪VA

• c : V → C with a finite set of colors C


Game Graph / Arena

G = (VE,VA, E, c)

• VE: vertices of Eva (player 1, circle) {x1, . . . , x7}

• VA: vertices of Adam (player 2, box) {y1, . . . , y7}

• E ⊆ V ×V: edges with V = VE ∪VA . . .

• c : V → C with a finite set of colors C {•, •, •, •}

y1 x1 y2 x2

y3 x3 y4 x4 x5 y5

x6 y6 y7 x7


Plays

A play in G is an infinite sequence α = v0v1v2 · · · of vertices suchthat (vi, vi+1) ∈ E for all i ≥ 0.

By c(α) we denote the corresponding sequence of colorsc(v0)c(v1)c(v2) · · ·


Plays



y1 x1 y2 x2

y3 x3 y4 x4 x5 y5

x6 y6 y7 x7


Plays



y1 x1 y2 x2

y3 x3 y4 x4 x5 y5

x6 y6 y7 x7

α : y7c(α) : •


Plays



y1 x1 y2 x2

y3 x3 y4 x4 x5 y5

x6 y6 y7 x7

α : y7 x6c(α) : • •


Plays



y1 x1 y2 x2

y3 x3 y4 x4 x5 y5

x6 y6 y7 x7

α : y7 x6 y3c(α) : • • •


Plays



y1 x1 y2 x2

y3 x3 y4 x4 x5 y5

x6 y6 y7 x7

α : y7 x6 y3 y6c(α) : • • • •


Plays



y1 x1 y2 x2

y3 x3 y4 x4 x5 y5

x6 y6 y7 x7

α : y7 x6 y3 y6 x3c(α) : • • • • •


Plays



y1 x1 y2 x2

y3 x3 y4 x4 x5 y5

x6 y6 y7 x7

α : y7 x6 y3 y6 x3 y4 · · ·c(α) : • • • • • • · · ·


Game

G = (G,Win)

• G: game graph

• Win ⊆ Cω: winning condition

Eva wins a play α if c(α) ∈ Win. Otherwise Adam wins.

Examples for winning conditions:

• Buchi : given by F ⊆ CWin contains all plays α with Inf(α) ∩ F , ∅

• Muller : given by F ⊆ 2C

Win contains all plays α with Inf(α) ∈ F

• Parity : set of colors is a finite set of natural numbers (priorities)Win contains all plays α withmax(Inf(c(α))) even


Trying to Win – Strategies

y1

y2

x1

x2

Winning condition for Eva: y2 is visited infinitely often iff x1 and x2are both visited infinitely often.

y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)

Eva can win by playing as follows when the play is at x1:

• If the previous xi was x2: move to y2• Otherwise: move to y1



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



α : y1



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



α : y1 x1



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



α : y1 x1 y1



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



α : y1 x1 y1 x1



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



α : y1 x1 y1 x1 y1



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



α : y1 x1 y1 x1 y1 x2



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



α : y1 x1 y1 x1 y1 x2 y1



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



α : y1 x1 y1 x1 y1 x2 y1 x1



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



α : y1 x1 y1 x1 y1 x2 y1 x1 y2



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



α : y1 x1 y1 x1 y1 x2 y1 x1 y2 x1



y1

y2

x1

x2


y2 ∈ Inf(α) ⇔ {x1, x2} ⊆ Inf(α)



α : y1 x1 y1 x1 y1 x2 y1 x1 y2 x1 y1 · · ·


Strategies – Formal

• A strategy for Eva is a function

σ : V∗VE → V

with σ(xv) = v′ implies (v, v′) ∈ E




σ : V∗VE → V


• A play v0v1v2 · · · is played according to σ if

∀i : vi ∈ VE → σ(v0 · · · vi) = vi+1

• Out(σ, v0) = set of all plays starting in v0 that are playedaccording to σ (the possible outcomes of σ).




σ : V∗VE → V


• A play v0v1v2 · · · is played according to σ if

∀i : vi ∈ VE → σ(v0 · · · vi) = vi+1

• Out(σ, v0) = set of all plays starting in v0 that are playedaccording to σ (the possible outcomes of σ).

• A strategy for Eva is a winning strategy from vertex v0 ifOut(σ, v0) ⊆ Win

• A strategy for Adam (defined similarly) is a winning strategyfrom vertex v0 if Out(σ, v0) ∩Win = ∅


Determinacy

A game G = (G,Win) is determined if from each node either Evaor Adam has a winning strategy.


Determinacy


Determinacy is about swapping quantifiers

Eva does not have a winning strategy:

∀σE∃σA ( σA wins against σE)


Determinacy


Determinacy is about swapping quantifiers

Eva does not have a winning strategy:

∀σE∃σA ( σA wins against σE)

By determinacy:

∃σA∀σE ( σA wins against σE)


Special strategies

• Computable : The function σ : V∗VE → V is computable.

• Positional : The strategy only depends on the current vertex(not on the past), i.e., σ : VE → V.

• Finite memory : The strategy is implemented by adeterministic finite automaton that reads the colors of the play.It depends on the current vertex and the state of theautomaton.


Finite memory strategy – example

Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

σ(m0, x1) = y1σ(m1, x1) = y2

“Move to y2 if the previous xi was x2”Infinite games and automata theory · Bertinoro, June 2009 16 / 84


Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

y1

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

m0

σ(m0, x1) = y1σ(m1, x1) = y2



Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

y1 x1

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

m0 m0

σ(m0, x1) = y1σ(m1, x1) = y2



Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

y1 x1 y1

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

m0 m0 m0

σ(m0, x1) = y1σ(m1, x1) = y2



Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

y1 x1 y1 x1

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

m0 m0 m0 m0

σ(m0, x1) = y1σ(m1, x1) = y2



Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

y1 x1 y1 x1 y1

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

m0 m0 m0 m0 m0

σ(m0, x1) = y1σ(m1, x1) = y2



Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

y1 x1 y1 x1 y1 x2

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

m0 m0 m0 m0 m0 m0

σ(m0, x1) = y1σ(m1, x1) = y2



Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

y1 x1 y1 x1 y1 x2 y1

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

m0 m0 m0 m0 m0 m0 m1

σ(m0, x1) = y1σ(m1, x1) = y2



Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

y1 x1 y1 x1 y1 x2 y1 x1

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

m0 m0 m0 m0 m0 m0 m1 m1

σ(m0, x1) = y1σ(m1, x1) = y2



Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

y1 x1 y1 x1 y1 x2 y1 x1 y2

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

m0 m0 m0 m0 m0 m0 m1 m1 m0

σ(m0, x1) = y1σ(m1, x1) = y2



Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

y1 x1 y1 x1 y1 x2 y1 x1 y2 x1

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

m0 m0 m0 m0 m0 m0 m1 m1 m0 m0

σ(m0, x1) = y1σ(m1, x1) = y2



Muller game:

y1

y2

x1

x2F = {{x1, y1}, {x2, y1}, {x1, x2, y1, y2}}

y1 x1 y1 x1 y1 x2 y1 x1 y2 x1 y1

m0 m1

x1, y1, y2 x2, y1, y2x2

x1

m0 m0 m0 m0 m0 m0 m1 m1 m0 m0 m0

σ(m0, x1) = y1σ(m1, x1) = y2


Some results (see tutorial of Marcin Jurdzinski)

Theorem (Buchi/Landweber’69, Gurevich/Harrington’82,Zielonka’98)Muller games are determined with finite memory strategies.




Theorem (Emerson/Jutla’88,Mostowski’91)Parity games are positionally determined.





What about other types of winning conditions, e.g.

• Every blue state is followed by a green state, there are at most3 red states, and if there is no yellow state, then there areinfinitely many green ones...

Do we have to design a new algorithm for each winning condition?


1 Introduction

2 Basics on games


ω-Automata

Game reductions


4 Tree automata

Complementation

Emptiness



Translation of winning conditions

Goal: general mechanism for transforming winning conditions

Example:

Game graph with colors C = {a, b, c, d}

Winning condition: Eva wins if the play never matches the regularexpression r (over the alphabet C)

a

c

b

d

C∗c(a+ b)∗cC∗ + C∗d(a+ b)∗dC∗




Example:



a

c

b

d


strategy with two states of memory




Example:



a

c

b

d


strategy with two states of memory

How to solve such games in general?


Safety game

Product of game graph and DFA for r: the automaton reads the play

Eva wins if she can avoid the final states of the DFA.

a

c

b

d

0

1 2

3

c d

a, b

a, b a, b

a, b, c, d

d

c

dc


Safety game



a

c

b

d

0

1 2

3

c d

a, b

a, b a, b

a, b, c, d

d

c

dc

a, 0

c, 0

b, 0

d, 0

a, 1

c, 1

b, 1

d, 1

a, 2

c, 2

b, 2

d, 2


Safety game



a

c

b

d

0

1 2

3

c d

a, b

a, b a, b

a, b, c, d

d

c

dc

a, 0

c, 0

b, 0

d, 0

a, 1

c, 1

b, 1

d, 1

a, 2

c, 2

b, 2

d, 2


Safety game



a

c

b

d

0

1 2

3

c d

a, b

a, b a, b

a, b, c, d

d

c

dc

a, 0

c, 0

b, 0

d, 0

a, 1

c, 1

b, 1

d, 1

a, 2

c, 2

b, 2

d, 2


Safety game



a

c

b

d

0

1 2

3

c d

a, b

a, b a, b

a, b, c, d

d

c

dc

a, 0

c, 0

b, 0

d, 0

a, 1

c, 1

b, 1

d, 1

a, 2

c, 2

b, 2

d, 2


Safety game



a

c

b

d

0

1 2

3

c d

a, b

a, b a, b

a, b, c, d

d

c

dc

a, 0

c, 0

b, 0

d, 0

a, 1

c, 1

b, 1

d, 1

a, 2

c, 2

b, 2

d, 2c, 1 d, 2


Safety game



a

c

b

d

0

1 2

3

c d

a, b

a, b a, b

a, b, c, d

d

c

dc

a, 0

c, 0

b, 0

d, 0

a, 1

c, 1

b, 1

d, 1

a, 2

c, 2

b, 2

d, 2c, 1 d, 2

a, 2


Safety game



a

c

b

d

0

1 2

3

c d

a, b

a, b a, b

a, b, c, d

d

c

dc

a, 0

c, 0

b, 0

d, 0

a, 1

c, 1

b, 1

d, 1

a, 2

c, 2

b, 2

d, 2c, 1 d, 2

a, 2


Safety game



a

c

b

d

0

1 2

3

c d

a, b

a, b a, b

a, b, c, d

d

c

dc

a, 0

c, 0

b, 0

d, 0

a, 1

c, 1

b, 1

d, 1

a, 2

c, 2

b, 2

d, 2c, 1 d, 2

a, 2

Strategy for Eva with three memory states.Infinite games and automata theory · Bertinoro, June 2009 20 / 84

Summary of the method

• Transform the regular expression into a DFA.

• Take the product with the game graph.

• The resulting game is a safety game: Eva wins if she can avoidthe set R of red states.

• Solve the safety game by an attractor construction:• Compute stepwise the set AttrA(R) of nodes from where Adam

can force a visit to a red state.• For the nodes outside of AttrA(R) Eva has a strategy to stay

outside (by the definition of AttrA(R)).

• Translate the strategy for Eva (or for Adam) back to the originalgame, using the DFA as memory.


Infinitary conditions

• For the above example we used a translation from regularexpressions to deterministic finite automata (DFAs).

• For infinitary conditions (something happens infinitely/finitelyoften) standard DFAs are not enough.

• To treat such conditions we can use ω-automata.


ω-Automata


Buchi automata

An ω-automaton is of the form A = (Q,Σ, qin,∆,Acc), whereQ,Σ, qin,∆ are as for standard finite automata, and Acc defines theacceptance condition.

Acceptance conditions:

• Buchi automata : Acc given as set F ⊆ Q of final states.A run is accepting if it contains infinitely often a state from F

• Parity automata : Acc given as priority mapping pri : Q→ N

A run is accepting if the maximal priority appearing infinitelyoften is even.

Deterministic automata : as usual (at most one transition per stateand letter)


Basic examples

Σ = {a, b}

• A nondeterministic Buchi automaton for “finitely many b”:

q0 q1

a, b

a

a


Basic examples

Σ = {a, b}


q0 q1

a, b

a

a

There is no deterministic Buchi automaton for this language


Basic examples

Σ = {a, b}


q0 q1

a, b

a

a

There is no deterministic Buchi automaton for this language

• A deterministic parity automaton for the same language:

q00

q11

a

b

b

a


The determinization problem

• Classical subset construction fails

q0 q1

a, b

a, b

a

aaaaaa · · · and abababab · · · induce the same sequence ofsets:

{q0}a−→a

{q0, q1}a−→b

{q0, q1}a−→a

{q0, q1}a−→b

{q0, q1} · · ·




q0 q1

a, b

a, b

a


{q0}a−→a

{q0, q1}a−→b

{q0, q1}a−→a

{q0, q1}a−→b

{q0, q1} · · ·

• A deterministic automaton has to store more information on thepossible runs of the Buchi automaton: Besides the statesreached some information on the visits to final states are isnecessary.




q0 q1

a, b

a, b

a


{q0}a−→a

{q0, q1}a−→b

{q0, q1}a−→a

{q0, q1}a−→b

{q0, q1} · · ·

• A deterministic automaton has to store more information on thepossible runs of the Buchi automaton: Besides the statesreached some information on the visits to final states are isnecessary.

• The details are rather technical...


Determinization

Theorem (McNaughton’66,Safra’88)For each nondeterministic Buchi automaton with n states there isan equivalent deterministic parity automaton with 2O(n log n) states.


Determinization


Deterministic parity automata are easy to complement: increase allpriorities by 1.

CorollaryThe class of regular ω-languages is closed under complementation.


Game reductions


Game reductions

We use the following scheme:

(G,Win)parity gameG×AWin

productgame

• AWin is a determinisitic parity automaton forWin

• It can be seen as a transducer that transforms sequences fromCω into sequences of priorities


Game reductions



productgame

positionalwinning strategy

compute




Game reductions



productgame


compute

winning strategywith memory AWin

translate




Example: Muller to Parity

• There is a direct algorithm for solving Muller games.

• For illustrating the concept of game reduction we provide atranslation of Muller games to parity games.

• For this purpose we construct a deterministic parity automatonthat• reads sequences α from Cω and• accepts if α satisfies the given Muller condition

• The construction is based on “latest appearance records”.


Latest Appearance Record – Idea

• Recall the order in which the colors (or vertices) of the playappeared (starting with an arbitrary order)

• The colors appearing finitely often will gather at one endd b b d c b a b a c b a b a c




• The colors appearing finitely often will gather at one end

abcd

d b b d c b a b a c b a b a c





abcd

ddabc

b b d c b a b a c b a b a c





abcd

ddabc

bbdac

b d c b a b a c b a b a c

• When a color is moved to the front (top), mark its previousposition

• This allows to infer from the LAR which colors are visitedinfinitely often





abcd

ddabc

bbdac

bbdac

d c b a b a c b a b a c







abcd

ddabc

bbdac

bbdac

ddbac

c b a b a c b a b a c







abcd

ddabc

bbdac

bbdac

ddbac

c

cdba

b a b a c b a b a c







abcd

ddabc

bbdac

bbdac

ddbac

c

cdba

bbcda

a b a c b a b a c







abcd

ddabc

bbdac

bbdac

ddbac

c

cdba

bbcda

a

abcd

b a c b a b a c







abcd

ddabc

bbdac

bbdac

ddbac

c

cdba

bbcda

a

abcd

bbacd

a c b a b a c







abcd

ddabc

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bbdac

ddbac

c

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bbcda

a

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bbacd

a

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c b a b a c







abcd

ddabc

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bbdac

ddbac

c

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bbcda

a

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bbacd

a

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c

cabd

b a b a c







abcd

ddabc

bbdac

bbdac

ddbac

c

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a

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a

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c

cabd

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a b a c







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b a c







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cdba

bbcda

a

abcd

bbacd

a

abcd

c

cabd

bbcad

a

abcd

bbacd

a

abcd

c

cabd




Latest Appearance Record

• A latest appearance record (LAR) over C is an ordering ofthe elements of C.

LAR(C) =

{[d1 · · · dn, h] di ∈ C, di , dj for all i , j,

and 1 ≤ h ≤ n

}

• LAR update:

δLAR([d1 · · · dn, h], d) = [dd1 · · · di−1di+1 · · · dn, i]

for the unique i with d = di• This defines the states (the LARs) and the transition structure

(LAR update) of the deterministic parity automaton.


Assigning Priorities

• Priority depends on the size of the part of the LAR that haschanged in the last transition.

• The biggest part that changes infinitely often decides

• Example for F = {{b, d}, {a, b, c}}

abcd

7

ddabc

7

bbdac

5

bbdac

1

ddbac

4

c

cdba

7

bbcda

5

a

abcd

7

bbacd

3

a

abcd

3

c

cabd

6

bbcad

6

a

abcd

6

bbacd

3

a

abcd

3

c

cabd

6


The LAR automaton

• Let F be a Muller condition over |C| = n colors.

• Define the deterministic parity automatonALAR = (LAR(C),C, qin, δLAR, cLAR) with

cLAR([d1 · · · dn, h]) =

{2h− 1 {d1, . . . , dh} < F ,2h {d1, . . . , dh} ∈ F .

TheoremFor a Muller condition F over C the corresponding deterministicparity automaton ALAR accepts precisely those α ∈ Cω that satisfythe Muller condition F .


From Muller to Parity Games

Muller game (G,F )parity gameG×ALAR

productgame



productgame


compute



productgame


compute

winning strategywith memory ALAR

translate




productgame


compute

winning strategywith memory ALAR

translate

Theorem (Gurevich/Harrington’82)Muller games are determined with the LAR automaton as memory.




Logic for infinite words

• Used for specifying properties of system executions

• In practice usually temporal logics are used

• From a theoretical point of view and as reference forexpressive power predicate logic is interesting

• Here we consider• the linear temporal logic LTL and• monadic second-order logic over infinite words, called S1S.


Linear temporal logic – LTL

LTL formulas: Build over a set P = {p1, . . . , pn} of atomicpropositions.

Models: Infinite sequences of vectors of size n. Entry i of a vectorcodes truth value of pi (1 = true, 0 = false).

• Atomic formulas: pi (pi is true in the first position)• Boolean combinations• Temporal operators:

Fϕ · · ·ϕ

· · · “eventually ϕ”

Gϕϕ ϕ ϕ

· · ·ϕ ϕ ϕ

· · · “always ϕ”

Xϕϕ

· · · · · · “next ϕ”

ϕUψ ϕ ϕ ϕ· · ·

ϕ ψ· · · “ϕ until ψ”


Examplesp1 ∧ X¬p2

(11

)(10

)(01

)(00

)

· · ·



(11

)(10

)(01

)(00

)

· · ·

Gp2 ∧ Fp1

(01

)(01

)(01

)

· · ·

(01

)(11

)(01

)(01

)(11

)

· · ·



(11

)(10

)(01

)(00

)

· · ·

Gp2 ∧ Fp1

(01

)(01

)(01

)

· · ·

(01

)(11

)(01

)(01

)(11

)

· · ·

F(p3 ∧ X(¬p2Up1))

010

001

· · ·

101

001

001

000

110

· · ·


From LTL to automata

Buchi automaton “guesses” valuations of the subformulas andverifies its guesses:

• Atomic formulas, Boolean combinations: verified directly

• Operators X, G: verified using the transitions

• Operators F, U: verified by acceptance condition







α = (10) (01) (11) (00) (10) (01) · · ·

¬p1

¬p2

¬p2Up1

X(¬p2Up1)

¬p1 ∧ X(¬p2Up1)

F(¬p1 ∧ X(¬p2Up1))







α = (10) (01) (11) (00) (10) (01) · · ·

¬p1

¬p2

¬p2Up1

X(¬p2Up1)

¬p1 ∧ X(¬p2Up1)

F(¬p1 ∧ X(¬p2Up1))

0

1

1

0

0

1







α = (10) (01) (11) (00) (10) (01) · · ·

¬p1

¬p2

¬p2Up1

X(¬p2Up1)

¬p1 ∧ X(¬p2Up1)

F(¬p1 ∧ X(¬p2Up1))

0

1

1

0

0

1

1

0

0

1

1

1







α = (10) (01) (11) (00) (10) (01) · · ·

¬p1

¬p2

¬p2Up1

X(¬p2Up1)

¬p1 ∧ X(¬p2Up1)

F(¬p1 ∧ X(¬p2Up1))

0

1

1

0

0

1

1

0

0

1

1

1

0

0

1

1

0

1







α = (10) (01) (11) (00) (10) (01) · · ·

¬p1

¬p2

¬p2Up1

X(¬p2Up1)

¬p1 ∧ X(¬p2Up1)

F(¬p1 ∧ X(¬p2Up1))

0

1

1

0

0

1

1

0

0

1

1

1

0

0

1

1

0

1

1

1

1

1

1

1







α = (10) (01) (11) (00) (10) (01) · · ·

¬p1

¬p2

¬p2Up1

X(¬p2Up1)

¬p1 ∧ X(¬p2Up1)

F(¬p1 ∧ X(¬p2Up1))

0

1

1

0

0

1

1

0

0

1

1

1

0

0

1

1

0

1

1

1

1

1

1

1

0

1

1

0

0

·







α = (10) (01) (11) (00) (10) (01) · · ·

¬p1

¬p2

¬p2Up1

X(¬p2Up1)

¬p1 ∧ X(¬p2Up1)

F(¬p1 ∧ X(¬p2Up1))

0

1

1

0

0

1

1

0

0

1

1

1

0

0

1

1

0

1

1

1

1

1

1

1

0

1

1

0

0

·

1

0

0

·

·

·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·


Games with LTL conditions

Theorem• For each LTL formula ϕ one can construct an equivalent Buchi

automaton Aϕ of size exponential in ϕ. (Vardi/Wolper’86)

• For each LTL formula ϕ one can construct an equivalentdeterministic parity automaton of size doubly exponential in ϕ.(determinization theorem)

• Finite games (G, ϕ) with a winning condition given by an LTLformula can be solved in doubly exponential time.(game reduction)


S1S

“Second-order theory of one successor”:

• Monadic second-order logic over the structure (N,+1)

0 1 2 · · · i i+ 1 i+ 2 · · ·

• Extension of first-order logic over (N,+1) by quantifiers forsets of positions (numbers).


S1S



0 1 2 · · · i i+ 1 i+ 2 · · ·


• Example: ϕ(X) = ∃Y(

0 ∈ Y ∧

∀x(x ∈ Y ↔ x+ 1 < Y) ∧

∀x(x ∈ X → x ∈ Y))

ϕ defines the set of all ω-words over {0, 1} such that 1 canonly occur on even positions.


S1S



0 1 2 · · · i i+ 1 i+ 2 · · ·


• Example: ϕ(X) = ∃Y(

0 ∈ Y ∧

∀x(x ∈ Y ↔ x+ 1 < Y) ∧

∀x(x ∈ X → x ∈ Y))

ϕ defines the set of all ω-words over {0, 1} such that 1 canonly occur on even positions.

• In general we consider formulas ϕ(X1, . . . ,Xn) with n free setvariables defining ω-languages over {0, 1}n.


Equivalence to automata

Theorem (Buchi’62)A language L ⊆ ({0, 1}n)ω is definable by an S1S formula iff it canbe accepted by a nondeterministic Buchi automaton.

Proof:

• From formulas to automata use an inductive translation, basedon the closure properties of automata.

• From automata to formulas: Write a formula that describes theexistence of an accepting run.Sets Xq for each state q code the positions of the run wherethe automaton is in state q.


Games with S1S conditions

Using determinization of Buchi automata and game reduction:

Theorem (Buchi/Landweber’69)For finite games (G, ϕ) with a winning condition given by an S1Sformula one can decide the winner and can compute acorresponding winning strategy.

Complexity: The size of the memory required for such a winningstrategy cannot be bounded by function

222···2n

}

k

for a fixed k.


Summary of this part

Solving infinite games with the help of automata: Extend the gameby running a deterministic automaton along the plays.


product

game


compute

winning strategywith memory AWin

translate

Examples:

• Muller conditions: LAR construction• LTL conditions: nondeterministic automaton guessing

valuations of subformulas + determinization• S1S conditions: inductive translation to automata


Solving Church’s synthesis problem



2


Assume Σ1 = Σ2 = {0, 1}.

• A computation of f is an infinite sequence of the form

(01

)(11

)(00

)(00

)(11

)

· · ·

with the input signals in the first row, the output signals in thesecond row.

• The specification can be written as LTL formula over {p1, p2}or as S1S formula ϕ(X1,X2).


Church’s synthesis problem as game(00

)

(01

)

(10

)

(11

)

Winning condition: Specification ϕ rewritten such that it onlyconsiders the red vertices.

Finite state winning strategy for Eva gives the desired program for f .


1 Introduction

2 Basics on games


ω-Automata

Game reductions


4 Tree automata

Complementation

Emptiness



Reminder



automata on infinitewords

games of infiniteduration

automata on infinitetrees


Why infinite trees?

• Discrete systems (circuits, protocols etc.) can be described bytransitions graphs.

• The possible behavior of such a system is captured by aninfinite tree: the unraveling of the transition graph.

• Properties of the system behavior can be specified asproperties of infinite trees.


Infinite trees

• For simplicity we restrict ourselves to complete binary trees.

• The nodes are labeled from a finite alphabet Σ.

• Formally, a tree is a mapping t : {0, 1}∗ → Σ.

b

a

b

a a

b

a a

b

b

b a

b

a a...


Infinite trees

• For simplicity we restrict ourselves to complete binary trees.

• The nodes are labeled from a finite alphabet Σ.

• Formally, a tree is a mapping t : {0, 1}∗ → Σ.

b

a

b

a a

b

a a

b

b

b a

b

a a...

t(ε) = b

t(0) = a

t(01) = b

t(110) = a


Automata on infinite trees

automata onfinite words

automata onfinite trees

automata oninfinite words

automata oninfinite trees

• Robust model (good closure and algorithmic properties)

• Captures many known specification logics


Parity tree automata

• A = (Q,Σ, qin,∆, pri)

• Transitions of the form (q, a, q′, q′′)

Run of A : b

a

b

a a

b

a a

b

b

b a

b

a a...



• A = (Q,Σ, qin,∆, pri)


Run of A : bqin

a

b

a a

b

a a

b

b

b a

b

a a...



• A = (Q,Σ, qin,∆, pri)


Run of A : bqin

(qin, b, q1, q2) ∈ ∆

aq1

b

a a

b

a a

bq2

b

b a

b

a a...



• A = (Q,Σ, qin,∆, pri)


Run of A : bqin

(qin, b, q1, q2) ∈ ∆

aq1

(q1, a, q3, q4) ∈ ∆

bq3

a a

bq4

a a

bq2

b

b a

b

a a...



• A = (Q,Σ, qin,∆, pri)


Run of A : bqin

(qin, b, q1, q2) ∈ ∆

aq1

(q1, a, q3, q4) ∈ ∆

bq3

a a

bq4

a a

bq2

(q2, b, q5, q6) ∈ ∆

bq5

b a

bq6

a a...



• A = (Q,Σ, qin,∆, pri)


Run of A : bqin

(qin, b, q1, q2) ∈ ∆

aq1

(q1, a, q3, q4) ∈ ∆

bq3

a a

bq4

a a

bq2

(q2, b, q5, q6) ∈ ∆

bq5

b a

bq6

a a...

• Priority function pri : Q→ N

• Run accepting if on each path the maximal priority appearinginfinitely often is even.

• Tree accepted if there is an accepting run on this tree. T(A)denotes the language of accepted trees.


Example

• Trees over {a, b} such that on each infinite path there are onlyfinitely many b

• Use parity word automaton and run it on all branches

q00

q11

a

b

b

a


Example



q00

q11

a

b

b

a

• Required transitions: (q0, a, q0, q0), (q0, b, q1, q1), (q1, a, q0, q0),and (q1, b, q1, q1)


Example



q00

q11

a

b

b

a

• Required transitions: (q0, a, q0, q0), (q0, b, q1, q1), (q1, a, q0, q0),and (q1, b, q1, q1)

• There is no (nondeterministic) Buchi tree automaton for thislanguage.


Example

The set of all trees t over {a, b} such that t contains at least onenode labeled b:

(qb, a, qb, q) (qb, a, q, qb) (qb, b, q, q)

pri(qb) = 1 and pri(q) = 0

Nondeterminsm is required


Acceptance conditions

• As for games we can define Muller, Rabin, Streett, Buchi, etc.automata.

• The LAR construction can be used to transform Muller treeautomata into parity tree automata.

• Call a language of infinite trees regular if it can be accepted bya parity tree automaton.


Basic closure properties

PropositionThe class of regular languages of infinite trees is closed underunion, intersection, and projection (relabeling).




Proof:

• Union, intersection: A standard product construction yields aMuller automaton over pairs of priorities.

• Projection h : Σ → Γ: simply apply the projection to the labelsin the transitions.




Proof:

• Union, intersection: A standard product construction yields aMuller automaton over pairs of priorities.

• Projection h : Σ → Γ: simply apply the projection to the labelsin the transitions.

What about complementation?


Complementation


Complementation

Tree accepted:

∃run∀path.(path satisfies acceptance condition)

Tree not accepted:

∀run∃path.(path does not satisfy acceptance condition)

This exchange of quantifiers makes the problem difficult.


Complementation

Tree accepted:


Tree not accepted:



Idea for solution

Reformulate acceptance as:

∃strategy for Eva ∀strategies for Adam (...)


Complementation

Tree accepted:


Tree not accepted:



Idea for solution

Reformulate acceptance as:

∃strategy for Eva ∀strategies for Adam (...)

Determinacy yields for non-acceptance:

∃strategy for Adam ∀strategies for Eva (...)


Membership game

Use a game to characterize when a tree t is accepted by A:

• Player C tries to show that t is accepted byconstructing a run.

• Player S tries to show the contrary by selecting a path onwhich the acceptance condition is not satisfied.


Membership game

Use a game to characterize when a tree t is accepted by A:

• Player C tries to show that t is accepted byconstructing a run.

• Player S tries to show the contrary by selecting a path onwhich the acceptance condition is not satisfied.

• The game starts a the root of the tree in the initial state of A.• The moves of the game from a position (u, q) where u is a

node of t, and q is a state of A:1. C picks a transition (q, a, q0, q1) that matches q and

the label a of t at u2. S chooses a direction and the game moves on to position

(u0, q0) or (u1, q1).

• Winning condition for C: acceptance condition of A

Parity game on an infinite graph


Shape of the membership game

...



...



...



...



...



...

winning strategy of C{ accepting run



...

winning strategy of C{ accepting run

accepting run{ winning strategy of C


Applying positional determinacy

LemmaA tree t is not in T(A) iffthere is a positional winning strategy for Sin the membership game.


Strategies for S

...


Strategies for S

...


Strategies for S

...


Strategies for S

...

Postional strategy for S coded as

σ : {0, 1}∗ → (∆ → {0, 1})︸ÃÃÃÃÃÃÃÃÃÃÃÃÃ︷︷ÃÃÃÃÃÃÃÃÃÃÃÃÃ︸

Γ


Strategies for S

...

Postional strategy for S coded as

σ : {0, 1}∗ → (∆ → {0, 1})︸ÃÃÃÃÃÃÃÃÃÃÃÃÃ︷︷ÃÃÃÃÃÃÃÃÃÃÃÃÃ︸

Γ

This is a tree over the alphabet Γ


Next steps

• Construct an automaton Astrat that reads trees of the formt× σ, i.e., annotated with a positional strategy for S

• Astrat accepts t× σ if σ is winning for S

• Obtain C from Astrat by omitting the strategy annotation in thelabels (C guesses the strategy for S).


Construction of Astrat

...

Astrat has to check: the paths following the blue edges do notsatisfy the acceptance condition


Construction of Astrat

0

1

...

Astrat has to check: the paths following the blue edges do notsatisfy the acceptance condition

Focus on single branches: infinite words over Σ × Γ × {0, 1}


Verifying single branches

0

1

...

• Construct a nondeterministic Buchi automaton that guesses ablue path and accepts if it does satisfy the acceptancecondition of A.

• Determinize and complement, and obtain a deterministic parity

word automaton Apathstrat that accepts those branches of a tree

on which S’s strategy is winning.


Back to trees

• Run Apathstrat along each branch:

Apathstrat : δ(q, (a,γ, 0)) = q′ δ(q, (a,γ, 1)) = q′′

Astrat : (q, (a,γ), q′, q′′)

• Then obtain C by omitting the strategy encoding:

(q, (a,γ), q′, q′′) becomes (q, a, q′, q′′)


Summary of the complementation method

• Characterize acceptance in terms of winning strategy inmembership game

• Positional determinacy for parity games yields: t not acceptediff S has positional winning strategy

• Construct an automaton that checks if a given strategy ofS is winning.• This construction is based on the determinization of

ω-automata.

• Obtain the desired automaton by projection (removing thestrategy annotations).


Closure under complementation

Theorem (Rabin’69)For a given tree automaton one can construct a tree automaton forthe complement language.


From logic to automata

• Closure properties allow translation of S2S into tree automata• S2S is the monadic second-order logic over the infinite binary

tree, i.e., the structure ({0, 1}∗, S0, S1) consisting of• the tree nodes as domain• the two successor functions

• The translation works in the same way as for S1S toω-automata

• The satisfiability problem for S2S becomes the emptinessproblem for tree automata


Emptiness


Emptiness problem

Decide for a given PTA A if T(A) is empty.

Idea for solution:

• In the membership game for t, C’s task is toconstruct an accepting run on t

• In the emptiness game, C’s task is to construct atree along with an accepting run on t


From membership to emptiness

How to adapt the membership game to obtain the emptiness game:

• The game starts at the root of the tree t in the initial state ofA.

• The moves of the game from a position ( u , q) where

u is a node of t, and q is a state of A:

1. C picks a transition (q, a, q0, q1) that matches q

and the label a of t at u

2. S chooses a direction and the game moves on to position( u0 , q0) or ( u1 , q1).























Emptiness game

The emptiness game for A:

• The game positions are Q ∪ ∆ (states and transitions of A)• The moves of the game from a position q (a state of A):

1. C picks a transition (q, a, q0, q1)2. S chooses a direction and the game moves on to positionq0 or q1


This is a finite game graph.

LemmaC has a winning strategy in the emptiness game for A iffT(A) , ∅


Example automaton

(q1, a, q1, q+)(q1, a, q2, q+)(q2, a, q1, q

′1)

(q′1, a, q+, q′1)

(q′1, b, q+, q+)(q+, a, q+, q+)(q+, b, q+, q+)

c(q1) = c(q′1) = 1c(q2) = c(q+) = 2

a

a

aa

ab

a

ab

q1

q1q1q1q2

q1q2

q1

q′1

q′1q′1q′1

q′1q′1


Emptiness game

1q1

2q2

1

q′1

2q+

(q1, a, q1, q+) (q1, a, q2, q+)

(q2, a, q1, q′1)

(q′1, a, q+, q′1)

(q′1, b, q+, q+)(q+, a, q+, q+) (q+, b, q+, q+)


Emptiness game

1q1

2q2

1

q′1

2q+

(q1, a, q1, q+) (q1, a, q2, q+)

(q2, a, q1, q′1)

(q′1, a, q+, q′1)

(q′1, b, q+, q+)(q+, a, q+, q+) (q+, b, q+, q+)

A winning strategy for C


Regular trees

From the winning strategy: a finitely generated (regular) tree in thelanguage

(q1, (q1, a, q2, q+))

(q2, (q2, a, q1, q′1))

(q′1, (q′1, b, q+, q+))

(q+, (q+, a, q+, q+))

0

10

1

0,10,1


Emptiness decidable

Theorem (Rabin’69)The emptiness problem for parity tree automata is decidable (in NP∩ co-NP). If the language is not empty, then one can construct afinite representation of a tree in the language.

Corollary (Rabin’69)S2S is decidable.


Summary of this section

Games as a tool for tree automata constructions:

• For complementation of parity tree automata the positionaldeterminacy of parity games is used for the correctnessproof of the construction.The actual construction is based on determinization of wordautomata.

• The emptiness test directly reduces to a game . A positionalwinning strategy for C yields a regular tree in thelanguage.


1 Introduction

2 Basics on games


ω-Automata

Game reductions


4 Tree automata

Complementation

Emptiness



Context-free specifications

What happens if specifications are not ω-regular?

PropositionThe problem of finding the winner in a game (G,Win), whereWinis defined by a nondeterministic pushdown ω-automaton isundecidable.


Pushdown games

AssumeWin is specified by a deterministic pushdownω-automaton (with parity condition):

Taking the product of G with the pushdown automaton results in aparity game on a pushdown graph. These games can be solvedand winning strategies can be implemented by pushdownautomata.

Theorem (Walukiewicz’96)Parity games on pushdown graphs can be solved in exponentialtime and winning strategies can be implemented by pushdownautomata.

CorollaryThe problem of deciding the winner in a game (G,Win) for a finitegame graph, whereWin is defined by a deterministic pushdownω-automaton (with parity condition) is decidable.


Recursive games

In general, one can study games on recursive game graphs (infiniteset of vertices, computable edge relation and coloring).

For recursive parity games winning strategies need not becomputable:

· · ·i− 1 i i+ 1

· · ·

......

......

......

......

......

...

...

...

...

Switch at level k if Turing ma-

chine number i halts after k

steps for the empty input

Eva wins if she visits

finitely many green ver-

tices


Summary

1. Solving infinite games with the help of automata: Extend thegame by running a deterministic automaton along the plays• Determinization: nondeterministic Buchi → deterministic parity• Muller conditions: LAR construction• LTL conditions: nondeterministic automaton guessing

valuations of subformulas• S1S conditions: inductive translation to automata• Solution to Church’s synthesis problem

2. Games as a tool for tree automata constructions• Complementation• Emptiness test• Application: decidability of S2S

3. Nonregular conditions and infinite graphs• Pushdown games• Recursive games


Documents

old.automata.rwth-aachen.deold.automata.rwth-aachen.de/users/loeding/inf-games-and-automata.pdfOrigin Circuit synthesis and Church’s problem (1957) Setting: •Sequence of input