Two-Way Alternating Automata and Finite Models Tedious proofs of - - PowerPoint PPT Presentation

Two-Way Alternating Automata and Finite Models

Tedious proofs of irrelevant results

Mikolaj Bojanczyk Warsaw University

Two-Way Alternating Automata and Finite Models – p.1/18

Intuition on the automaton

A two-way alternating automaton recognizes a property of graphs with a distinguished starting vertex. In other words, such an automaton looks at a graph and the starting vertex and says “yes” or “no”.

Two-Way Alternating Automata and Finite Models – p.2/18

Intuition on the automaton

A two-way alternating automaton recognizes a property of graphs with a distinguished starting vertex. In other words, such an automaton looks at a graph and the starting vertex and says “yes” or “no”. Some example properties recognized by alternating two-way automata:

Two-Way Alternating Automata and Finite Models – p.2/18

Intuition on the automaton

A two-way alternating automaton recognizes a property of graphs with a distinguished starting vertex. In other words, such an automaton looks at a graph and the starting vertex and says “yes” or “no”. Some example properties recognized by alternating two-way automata: There is a vertex labelled by “a” in the graph

Two-Way Alternating Automata and Finite Models – p.2/18

Intuition on the automaton

A two-way alternating automaton recognizes a property of graphs with a distinguished starting vertex. In other words, such an automaton looks at a graph and the starting vertex and says “yes” or “no”. Some example properties recognized by alternating two-way automata: There is a vertex labelled by “a” in the graph There is an infinite path in the graph

Two-Way Alternating Automata and Finite Models – p.2/18

Intuition on the automaton

A two-way alternating automaton recognizes a property of graphs with a distinguished starting vertex. In other words, such an automaton looks at a graph and the starting vertex and says “yes” or “no”. Some example properties recognized by alternating two-way automata: There is a vertex labelled by “a” in the graph There is an infinite path in the graph There is an infinite path in the graph and no vertex of this path is the starting point of some infinite backward path

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The automaton

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An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

An example:

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Two-Way Alternating Automata and Finite Models – p.4/18

Parity condition

An infinite sequence

• f elements from a finite set
• f natural numbers satisfies the parity condition if the lowest

number occurring infinitely often is even.

Two-Way Alternating Automata and Finite Models – p.5/18

accepts only infinite graphs

Fact 0 For any graph

, the automaton accpets in a vertex

and state

iff

• 1. No infinite backward path condition.

is not the beginning of a sequence

where for all

,

is an edge in

.

• 2. Infinite forward path condition.

is the beginning of a sequence

where for all

,

is an edge in

and accepts in

and

. Cor:

accepts only infinite graphs.

Two-Way Alternating Automata and Finite Models – p.6/18

Finite model problems

Automata Instance: A two-way alternating automaton . Question: Does accept some finite graph?

Two-Way Alternating Automata and Finite Models – p.7/18

Finite model problems

Automata Instance: A two-way alternating automaton . Question: Does accept some finite graph?

• calculus

Instance: A formula

• f the two-way modal

• calculus

Question: Is

satisfiable in some finite structure?

Two-Way Alternating Automata and Finite Models – p.7/18

Finite model problems

Automata Instance: A two-way alternating automaton . Question: Does accept some finite graph?

• calculus

Instance: A formula

• f the two-way modal

• calculus

Question: Is

satisfiable in some finite structure? Guarded fragment with fixed points Instance: A formula

• f the guarded fragment with

fixed points Question: Is

satisfiable in some finite structure?

Two-Way Alternating Automata and Finite Models – p.7/18

Finite model problems

Automata Instance: A two-way alternating automaton . Question: Does accept some finite graph?

• calculus

Instance: A formula

• f the two-way modal

• calculus

Question: Is

satisfiable in some finite structure? Guarded fragment with fixed points Instance: A formula

• f the guarded fragment with

fixed points Question: Is

satisfiable in some finite structure? All three are equivalent

Two-Way Alternating Automata and Finite Models – p.7/18

A strategy for the good player

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Two-Way Alternating Automata and Finite Models – p.8/18

Memoryless strategies

Thm:[Emmerson-Jutla/Mostowski] One of the players has a

winning strategy and, moreover, it is a memoryless strategy

Two-Way Alternating Automata and Finite Models – p.9/18

The graph

Two-Way Alternating Automata and Finite Models – p.10/18

Its unwinding

Two-Way Alternating Automata and Finite Models – p.10/18

A strategy

for the green player

Two-Way Alternating Automata and Finite Models – p.10/18

Locally possible moves under

Two-Way Alternating Automata and Finite Models – p.10/18

Locally possible moves under

with accessible positions

Two-Way Alternating Automata and Finite Models – p.10/18

The graph

Two-Way Alternating Automata and Finite Models – p.10/18

Parity length

The

• length of a sequence of numbers

is the length of the longest sequence of

• s in the

sequence

resulting from

by taking out all numbers greater than

. For example, the

• length of

is

.

Two-Way Alternating Automata and Finite Models – p.11/18

Parity length

The

• length of a sequence of numbers

is the length of the longest sequence of

• s in the

sequence

resulting from

by taking out all numbers greater than

. For example, the

• length of

is

. The parity length of a sequence of numbers maximal

• length of the sequence for odd

.

Two-Way Alternating Automata and Finite Models – p.11/18

Parity length

The

• length of a sequence of numbers

is the length of the longest sequence of

• s in the

sequence

resulting from

by taking out all numbers greater than

. For example, the

• length of

is

. The parity length of a sequence of numbers maximal

• length of the sequence for odd

. The parity length of a path labelled by priorities is the parity length of the corresponding sequence of priorities.

Two-Way Alternating Automata and Finite Models – p.11/18

Properties of

is a winning strategy for the green player iff no infnite path in

violates the parity condition (the parity length of paths in

is finite).

Two-Way Alternating Automata and Finite Models – p.12/18

Properties of

is a winning strategy for the green player iff no infnite path in

violates the parity condition (the parity length of paths in

is finite).

can be wound back into a finite graph iff for some

, the parity length of paths in

is bounded, i. e. there is some

such that all paths in

have parity length not greater than .

Two-Way Alternating Automata and Finite Models – p.12/18

Properties of

is a winning strategy for the green player iff no infnite path in

violates the parity condition (the parity length of paths in

is finite).

can be wound back into a finite graph iff for some

, the parity length of paths in

is bounded, i. e. there is some

such that all paths in

have parity length not greater than . The finite graph question thus becomes: is there some tree

and strategy

such that the parity length of paths in

is bounded.

Two-Way Alternating Automata and Finite Models – p.12/18

Regular trees and languages

A tree language is regular iff it is recognized by some finite automaton. A tree is regular iff it contains a only finitely many non-isomorphic subtrees.

Thm:[Rabin]Every regular tree language contains some reg-

ular tree.

Two-Way Alternating Automata and Finite Models – p.13/18

Let

be the set of graphs

where the parity length of paths is bounded.

Two-Way Alternating Automata and Finite Models – p.14/18

Let

be the set of graphs

where the parity length of paths is bounded. Let

be the set of graphs

where the parity length (both ways) of paths is finite.

Two-Way Alternating Automata and Finite Models – p.14/18

Let

be the set of graphs

where the parity length of paths is bounded. Let

be the set of graphs

where the parity length (both ways) of paths is finite.

is not regular.

Two-Way Alternating Automata and Finite Models – p.14/18

Let

be the set of graphs

where the parity length of paths is bounded. Let

be the set of graphs

where the parity length (both ways) of paths is finite.

is not regular.

and

are not equal, but ...

Two-Way Alternating Automata and Finite Models – p.14/18

Let

be the set of graphs

where the parity length of paths is bounded. Let

be the set of graphs

where the parity length (both ways) of paths is finite.

is not regular.

and

are not equal, but ...

and

coincide on regular trees

Two-Way Alternating Automata and Finite Models – p.14/18

Let

be the set of graphs

where the parity length of paths is bounded. Let

be the set of graphs

where the parity length (both ways) of paths is finite.

is not regular.

and

are not equal, but ...

and

coincide on regular trees Since

is regular and

is a sum of regular languages, we obtain:

Two-Way Alternating Automata and Finite Models – p.14/18

Let

be the set of graphs

where the parity length of paths is bounded. Let

be the set of graphs

where the parity length (both ways) of paths is finite.

is not regular.

and

are not equal, but ...

and

coincide on regular trees Since

is regular and

is a sum of regular languages, we obtain:

Thm:

is nonempty iff

is nonempty.

Two-Way Alternating Automata and Finite Models – p.14/18

Thm: The finite graph problem is decidable

Two-Way Alternating Automata and Finite Models – p.15/18

Signature

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Two-Way Alternating Automata and Finite Models – p.16/18

Signature

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1

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1 1

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1 2

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1 3

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Two-Way Alternating Automata and Finite Models – p.16/18

Another graph

a a c b a a:+

a:+ a:+ a:+ a:+ c:o b:+ a:o a:- 1 2

Two-Way Alternating Automata and Finite Models – p.17/18

Another graph

a a c b a

a:+

a:+ a:+ a:+ a:+ c:o b:+ a:o a:- 1 2

Two-Way Alternating Automata and Finite Models – p.17/18

Another graph

a a c b a

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2 1

a:+

a:+ a:+ a:+ a:+ c:o b:+ a:o a:- 1 2

Two-Way Alternating Automata and Finite Models – p.17/18

Tree unwinding

a a c b a

Two-Way Alternating Automata and Finite Models – p.18/18

Tree unwinding

a a c b a a a c b a a a b a a a b c c c c

Two-Way Alternating Automata and Finite Models – p.18/18

Tree unwinding

a a c b a

a a c b a

a

a

b

a

a

a

b

c c c c

Two-Way Alternating Automata and Finite Models – p.18/18

Tree unwinding

a a c b a

4

3 2 1

a a c b a

4

3 2 1

a

4

a

3

b

1

a

2

a

2

a

3

b

1

c c c c

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