Chapter 1 in “ - PowerPoint Presentation

Slide1

Chapter 1 in “Automata, Logic and infinite games”, edited by Gradel, Thomas and WilkeGames, logic and Automata SeminarAssaf Ben Shimon

ω

-

AutomataSlide2
Intro

The main topic covered in this chapter is the question how to define acceptance of infinite words by finite automata.In contrast to the case of finite words, there are many possibilities, and it is a nontrivial problem to compare them with respect to expressive power.Connections were established with other specification formalisms, e.g. regular expressions, grammars, and logical systems. In this chapter, we confine ourselves to the automata theoretic view.Slide3
Intro

Automata on infinite words have gained a great deal of importance since their first definition some forty years ago. Apart from the interests from a theoretical point of view, they have practical importance for the specification and verification of reactive systems that are not supposed to terminate at some point of time.Operating systems are an example of such systems, as they should be ready to process any user input as it is entered, without terminating after or during some task.Slide4
Topics of this lecture

Formal definition of ω-Automata Nondeterministic modelsDeterministic modelsLower bounds for transformationsWeak acceptance conditionsSlide5
Some Notations

ω := {0, 1, 2, 3, . . . }Σ* – the set of finite words over Σ

–

the set of infinite words over

Σ

… – finite words

… – infinite words

– number of a’s in

– the

I’th

letter of REG – the class of regular languages

 Slide6
Some Notations

Given an ω-word , we will also define:

 Slide7
ω-Automata

In the present context, we are interested only in the acceptance of words by automata (and not in generation of ω-words by grammars).We only consider finite automata.The usual definitions of deterministic and nondeterministic automata are adapted to the case of ω-input-words by the introduction of new acceptance conditions.For this purpose one introduces an “acceptance component” in the specification of automata, which will arise in different formats.Slide8
ω-Automata

Definition: An ω-automaton is a quintuple

where:

Q

is a finite set of states

Σ

is a finite alphabet

is the state transition function

is the initial stateAcc is the acceptance componentIn a deterministic ω-automaton, a transition function δ : Q × Σ → Q is used. Slide9
ω-Automata

Let A =

be an

ω

-automaton.

A

run

of

A

on an

ω-word is an infinite state sequence , such that the following conditions hold:

for

if A is nondeterministic,

for

if A is deterministic

 Slide10
ω-Automata

The size of an automaton , denoted by , is measured by the number of its states, i.e. for

the size is

 Slide11
Nondeterministic ModelsSlide12
Büchi Acceptance

Acceptance component – a set of statesDefinition: An ω-automaton

with acceptance component

is called

Büchi

automaton

if it is used with the following acceptance condition (

Büchi

acceptance

):

A word is accepted by A There exists a run of on satisfying the condition: i.e. at least one of the states in F has to be visited infinitely often during the run. Slide13
Büchi Acceptance

For example, which language does the following automaton accepts?(The states from F are drawn with a double circle, i.e

)

This

Büchi

automaton accepts the words from

 Slide14
Büchi Acceptance

Consider a Büchi automaton

Using this automaton with initial state

and final state

we obtain a regular language

of finite words.

An

ω

-word is accepted by some run of on visits some final state infinitely often. This is equivalent to

 Slide15
Büchi Acceptance

Taking the union over these sets for , we obtain the following representation result for Büchi recognizable ω-languages:Theorem: The

B

ü

chi

recognizable ω-languages are the ω-languages of the form

This family of ω-languages is also called the

ω-Kleene closure

of the class of regular languages.

 Slide16
Muller Acceptance

Acceptance component – a set of state sets

Definition

: An

ω

-automaton

with acceptance component

is called

Muller automaton

if it is used with the following acceptance condition (

Muller acceptance): A word is accepted by A There exists a run of on satisfying the condition: i.e. the set of infinitely recurring states of is exactly one of the sets in F. Slide17
Muller Acceptance

For example, which language does the following automaton accepts,When

?

And when

?

 Slide18
Büchi and Muller Equivalence

We will show Büchi and Muller automaton are equivalent in terms of expressive power.Buchi  Muller: Let

be a

Büchi

automaton.

Define the Muller automaton

with

Then

 Slide19
Büchi and Muller Equivalence

Muller  Buchi : The idea:We will guess the set

which should turn out to be

nondeterministically

For each set, we will create a unique copy of the states

We will guess the time during the run from which we will no longer see any state that is seen only finite amount of times

We will simulate a “memory” that remembers all the states we have been to, and reset it every time we seen all the states in G ( =

)

If all of our guesses were correct, we will visit the reset state infinitely many times.

 Slide20
Büchi and Muller Equivalence

Let

be a Muller automaton.

For each set

, we introduce a separate copy of

. We’ll indicate it as

We will define

, where

The automaton will make the 2 nondeterministic guesses at once – and switch from a state

to a state

corresponding to some group G, with memory =

(reset state)

 Slide21
Büchi and Muller Equivalence

We will define to contain all the reset states- states of the form

The

Büchi

automaton will be defines as

,

With

defined as above.

Then

 Slide22
Büchi and Muller Equivalence

If has n states, and contains m sets, then

Summarizing-

Theorem

: nondeterministic

B

ü

chi

automaton with

states can be converted into an equivalent Muller automaton of equal size, and a nondeterministic Muller automaton with

states and accepting sets can be transformed into an equivalent Büchi automaton with states. Slide23
So far:

Nondeterministic models Büchi AutomatonMuller Automaton

=Slide24
Rabin Acceptance

Acceptance component – a finite family of pairs

of state sets

Definition

: An

ω

-automaton

with acceptance component

with

is called

Rabin automaton if it is used with the following acceptance condition (Rabin acceptance): A word

is accepted by

A

there exists a run

of

on

such that

 Slide25
Rabin Acceptance

For example, which language does the following Rabin automaton accepts,When

?

This Rabin automaton accepts all words that consist of infinitely many

a

’s but only finitely many

b

’s.

 Slide26
Rabin Acceptance

If we want to specify the language consisting of all words that contain infinitely many b’s only if they also contain infinitely many a’s, using this state graph, which should we choose?

 Slide27
Streett Acceptance

Dual to the Rabin condition. Acceptance component – same as in Rabin. Definition: An ω-automaton

with acceptance component

with

is called

Streett automaton

if it is used with the following acceptance condition (

Streett acceptance

):

A word

is accepted by A there exists a run of on such that

(equivalently: if

then

)

 Slide28
Streett Acceptance

For example, which language does the following Streett automaton accepts,When

?

This Streett automaton accepts all words that contain infinitely many

b

’s if they contain infinitely many

a

’s.

 Slide29
Rabin+Streett and Muller Equivalance

Rabin/Streett  Muller:Let

be a Rabin automaton (respectively, Streett automaton)

Define a Muller automaton

with

,

(respectively, with

)

Then

 Slide30
Rabin+Streett and Muller Equivalance

Muller  Rabin/Streett:Perform the transition from Muller automaton to Büchi Automaton.Observe that Büchi Acceptance can be viewed as a special case of Rabin Acceptance (with

)

As well as Streett Acceptance (with

 Slide31
So far:

Nondeterministic models Büchi AutomatonMuller Automaton

Rabin Automaton

Streett Automaton

=

=

=Slide32
So far:

Nondeterministic models Büchi AutomatonMuller Automaton

Rabin Automaton

Streett Automaton

=Slide33
Parity Condition

A different formalization for the Rabin Acceptance, if the special case where the accepting pair

form a chain with respect to set inclusion- meaning

.

We will associate indices (colors) with states as follows: states of

receive color 1, states of

receive color 2, and so on with the rule that states of

have color

and states of

have color

.

An

ω

-word

is then accepted by

the parity

automaton

the least color occurring infinitely often in a run on

is even (hence the term “parity condition”).

 Slide34
Parity Condition

Definition: An ω-automaton

with acceptance component

is called

parity automaton

if it is used with the following acceptance condition (

parity condition

):

A word

is accepted by

A there exists a run of where

is even

 Slide35
Parity Condition

For example, in the following parity automaton, what language is accepted when:

?

?

 Slide36
Parity and Rabin Equivalence

Parity Rabin:Let

be a parity automaton with

.

An equivalent Rabin automaton

has the acceptance component

with

,

and

.

Rabin 

Parity

:

Can just

Rabin 

B

ü

chi

, and then define

if

,

otherwise.

 Slide37
So far:

Nondeterministic models Büchi AutomatonMuller Automaton

Rabin Automaton

Streett Automaton

Parity Automaton

=Slide38
Result

Theorem: Nondeterministic Büchi automata, Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive power, i.e. they recognize the same ω-languages.The ω-languages recognized by these ω-automata form the class ω-KC(REG), i.e. the ω-Kleene closure of the class of regular languages.The ω-languages in this class are commonly referred to as the regular ω-languages, denoted by

ω

-

REG.Slide39
Deterministic ModelsSlide40
Büchi automaton

We will see that a deterministic Büchi automata are too weak to recognize even very simple ω-languages from ω-REG. Let’s look at the following nondeterministic automaton:According to Büchi Acceptance with

, it recognizes the language

 Slide41
Büchi automaton

It is easy to provide an equivalent deterministic Muller automaton, using the following automaton with as acceptance component.

If one would work with the

Büchi

acceptance condition, using a set

of accepting states, then one has a specification of states which should be visited infinitely often, but it is not directly possible to specify which states should be seen only finitely often.

 Slide42
Büchi automaton

We will show that deterministic Büchi automata are too weak for recognizing the language

, by contradiction:

Assuming that the deterministic

Büchi

automaton

with final state set

recognizes

, it will on input visit an -state after a finite prefix, say after the -th letter. It will also accept , visiting -states infinitely often and hence after the , say when finishing the prefix .Continuing this construction the ω-word

is generated which causes

A

to pass through an

F

-state before each letter

a,

but which should of course be rejected.

 Slide43
So far:

Nondeterministic models Deterministic modelsBüchi Automaton

Muller Automaton

Rabin Automaton

Streett Automaton

Parity Automaton

Büchi

Automaton

Muller Automaton

Rabin Automaton

Streett Automaton=Parity Automaton

 Slide44
Muller  Rabin

We use a technique called latest appearance record (LAR).In a list of (distinct) states, we use the last entry for the current state in the run on the given Muller automaton. The hit position (the position of the marker ♮) indicates where the last change occurred in the record.For every transition from one state p to q in the original automaton, the state q is moved to the last position of the record while the symbols which were to the right of q are shifted one position to the left (so the previous place of q is filled again). The marker is inserted in front of the position where

q

was taken from.Slide45

 Slide46
Muller  Rabin

Formally: Let

be a deterministic Muller automaton.

Assume

w.l.o.g

. that

and

. Let ♮

be a new symbol, i.e. .An equivalent Rabin automaton A is given by the following definition: is the set of all order vector words with hit position over , i.e.

The initial state is

 Slide47
Muller  Rabin

The transition function is constructed as follows: Assume

and

. Then

is defined for any word

with

. Supposing that

, define

The acceptance component is given by

, defined as follows:

-

-

 Slide48
Muller  Parity

In this transformation, we have

We can remove pairs where

And present the automaton as a Parity automaton

Theorem

: By this transformation,

a deterministic Muller automaton with

states is transformed into a deterministic Rabin automaton with

states and

accepting pairs, and also into a deterministic parity automaton with states and colors. Slide49
Rabin  Streett

Note that the negation of the Rabin acceptance condition:

Is equivalent to the Streett condition:

 Slide50
Rabin  Streett

Let

be a deterministic Muller automaton. Then the Muller automaton

recognizes the complement of

.

From Rabin to Streett:

(Rabin automaton

with language

)

From A construct an equivalent Muller automatonComplement the Muller automatonTransform it back to a Rabin automaton , recognizing the complement of LUsed as Streett automaton, recognizes L Slide51
So far:

Nondeterministic models Deterministic modelsBüchi Automaton

Muller Automaton

Rabin Automaton

Streett Automaton

Parity Automaton

Büchi

Automaton

Muller Automaton

Rabin Automaton

Streett Automaton=Parity Automaton

 Slide52
So far:

Nondeterministic models Deterministic modelsBüchi Automaton

Muller Automaton

Rabin Automaton

Streett Automaton

Parity Automaton

Büchi

Automaton

Muller Automaton

Rabin Automaton

Streett Automaton=Parity Automaton

=Slide53
Complement

We showed Muller automaton is closed under complement, and therefor Rabin and Streett are too.It is not hard to show this for Parity as well (directly):Let

be a deterministic

ω

-

automaton with parity condition.

Then the complement of

is recognized by the parity automaton

where

.

 Slide54
Deterministic Models

Theorem: Deterministic Muller automata, Rabin automata, Streett automata and parity automata recognize the same ω-languages, and the class of ω-languages recognized by any of these types of ω-automata is closed under complementation.Slide55
Lower BoundsSlide56
Lower Bounds

Lemma: Let

be an ω-automaton with Rabin condition, and assume

are two nonaccepting runs. Then any run with

is also non-accepting.

Proof

:

Assume that

are non accepting, but with

There is a pair

s.t.

and



and

,

And also

or

One of

would be accepting, in contradiction.

 Slide57
Lower Bounds

Duality of Rabin and Streett also gives us:Lemma: Let

be an ω-automaton with Streett condition, and assume

are two accepting runs.

Then any run with

is also accepting.

 Slide58
Büchi 

RabinWe will show a lower bound on the transformation from nondeterministic Büchi automata to deterministic Rabin automataWe will use the languages of the following automata family: (

 Slide59
Büchi 

RabinWe can encode the symbols by words over

such that

is encoded by

Now we can specify the same family of languages w.r.t. the encoding by the family of automata

over the fixed alphabet

The size of

(in either of the two versions) is

.

 Slide60
Büchi 

RabinLemma: There exists a family of languages

over the alphabet

recognizable by nondeterministic

B

ü

chi

automata of size

such that any nondeterministic Streett automaton accepting the complement language of has at least states.Proof:Let be different permutations of Words

and

are not accepted by

.

For any Streett automaton

accepting

, there have to exist accepting runs

and

with

and

.

 Slide61
Büchi 

RabinWe want to show that .Assume in the contrary that there is a state

Then there has to exist an accepting run

of

on a word

 Slide62
Büchi 

RabinThe run is indeed accepting, thanks to the Lemma we proved earlier.But

, in contradiction. (On Board)

Thus,

for every two words.

There are

different permutations of

, thus

words, with no intersection of their

sets.

has at least states. Slide63
Büchi 

RabinFrom the duality of Rabin and Streett condition, If there is a deterministic Rabin automaton of size that accepts

then there also exists a deterministic Streett automaton that accepts the complement language with same number of states.

Therefor-

Theorem

:

There exists a family of languages

over the alphabet

recognizable by nondeterministic

B

üchi automata of size such that any equivalent deterministic Rabin automaton must be of size or larger. Slide64
Streett  Rabin

We can show a lower bound on the transformation from deterministic Streett automata to deterministic Rabin automataThe proof uses the languages of the following automata family (

when the acceptance component is

and

 Slide65
Streett  Rabin

Like before, we can encode it with

Theorem

:

There exists a family of languages

over the alphabet

recognizable by deterministic Streett automata with

states and

pairs of designated state sets such that any deterministic Rabin automaton accepting requires at least states.We will only show the general guidelines. Slide66
Streett  Rabin

Proof: By induction trivialStep:

Any given deterministic Rabin automaton

accepting

can be modified to a deterministic automaton

over

by removing all arcs labelled by

. It has at least

states. Moreover, has a reachable SCC of size

For each we can construct with run s.t.

and

We can show that for each

,

Therefor we have at least

states

 Slide67
Weak Acceptance ConditionsSlide68
Weak Acceptance

Instead of , we will use the set

for acceptance.

Staiger

and Wagner acceptance:

-acceptance:

-acceptance:

Last two are special cases of

Staiger

and Wagner acceptance

 Slide69
Weak Acceptance

For example, in this automaton, what language is accepted When

ith

-acceptance?

When

ith

-acceptance?

 Slide70
Staiger-Wagner

 BüchiLet

. The language

recognized by

with the

Staiger

-Wagner acceptance condition is recognized by a

Büchi

automaton

wherecontains all states

with

and

contains all states

with

.

The exponential blow-up can be avoided if only

-acceptance or

-acceptance are involved.

 Slide71
Büchi 

Staiger-WagnerNot Possible!For example, the set L of words with infinitely many b’s cannot be recognized by an ω-automaton with Staiger-Wagner acceptance.Proof: assume otherwise, and be

the number of states of this automaton.

It should have an accepting run on the word

There is a

s.t.

after

it went through all the states to be visited.

In the next there must be a loop; which can also be taken in Total, in the run on

same states will be visited like in

Hence

would be accepted- a contradiction.

 Slide72
Conclusion

We saw today:Expressive equivalence of nondeterministic Büchi, Muller, Rabin, Streett, and parity automataExpressive equivalence of deterministic Muller, Rabin, Streett, and parity automataSome lower bounds on the transformations between themWeak acceptance condition and their inferiority in expressive power. Slide73
Spoiler to chapter 3

Nondeterministic models Deterministic modelsBüchi Automaton

Muller Automaton

Rabin Automaton

Streett Automaton

Parity Automaton

Büchi

Automaton

Muller Automaton

Rabin Automaton

Streett Automaton=Parity Automaton

=

=Slide74
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