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Rational and Recognizable Sets in the Qeue Monoid

Highlights of Games, Logic and Automata 2018, Berlin

Chris K¨

• cher

Automata and Logics Group Technische Universit¨ at Ilmenau

September 20, 2018

1

Qeues

Let A be an alphabet (|A| ≥ 2). Two actions for each a ∈ A:

write leter a a read leter a a

A := {a | a ∈ A} Σ := A ⊎ A

Example

q = abaa t = babb a b a a

2

Qeues

Let A be an alphabet (|A| ≥ 2). Two actions for each a ∈ A:

write leter a a read leter a a

A := {a | a ∈ A} Σ := A ⊎ A

Example

q = abaa b t = babb a b a a

2

Qeues

Let A be an alphabet (|A| ≥ 2). Two actions for each a ∈ A:

write leter a a read leter a a

A := {a | a ∈ A} Σ := A ⊎ A

Example

q = abaa b t = babb a b a a

2

Qeues

Let A be an alphabet (|A| ≥ 2). Two actions for each a ∈ A:

write leter a a read leter a a

A := {a | a ∈ A} Σ := A ⊎ A

Example

q = abaa a t = babb a b a a b

2

Qeues

Let A be an alphabet (|A| ≥ 2). Two actions for each a ∈ A:

write leter a a read leter a a

A := {a | a ∈ A} Σ := A ⊎ A

Example

q = abaa t = babb b a a b

2

Qeues

Let A be an alphabet (|A| ≥ 2). Two actions for each a ∈ A:

write leter a a read leter a a

A := {a | a ∈ A} Σ := A ⊎ A

Example

q = abaa b t = babb b a a b

2

Qeues

Let A be an alphabet (|A| ≥ 2). Two actions for each a ∈ A:

write leter a a read leter a a

A := {a | a ∈ A} Σ := A ⊎ A

Example

q = abaa b t = babb b a a b

2

Qeues

Let A be an alphabet (|A| ≥ 2). Two actions for each a ∈ A:

write leter a a read leter a a

A := {a | a ∈ A} Σ := A ⊎ A

Example

q = abaa b t = babb b a a b b

2

Qeues

Let A be an alphabet (|A| ≥ 2). Two actions for each a ∈ A:

write leter a a read leter a a

A := {a | a ∈ A} Σ := A ⊎ A

Example

q = abaa t = babb a a b b

2

Qeue Monoids

Definition

s, t ∈ Σ∗ act equally (in symbols s ≡ t) if, and only if, ∀p, q ∈ A∗ : p s − → q ⇐ ⇒ p t − → q

Remark

≡ is the least congruence on Σ∗ satisfying certain commutations of write and read actions, e.g., aab ≡ aab for a, b ∈ A.

Definition

Q := Σ∗/

≡ … queue monoid

η: Σ∗ → Q: t → [t]≡ … natural homomorphism

3

Rational and Recognizable Sets

Definition

Let S ⊆ Q.

1 S is rational if there is a regular language L ⊆ Σ∗ with

η(L) = S.

closure properties: ∪, ·, ∗ generalizes regular expressions

2 S is recognizable if η−1(S) is regular.

closure properties: ∪, ∩, \ generalizes acceptance by finite automata

Theorem (Kleene 1951)

In the free monoid, a set is rational if, and only if, it is recognizable.

4

Q-Rational Subsets

Here: There are rational sets that are not recognizable! But: Each recognizable set is rational [McKnight 1964]. Restrict the rational sets in an appropriate way

q-rational sets start from η(A

∗aA ∗) and η(A∗aA∗) for a ∈ A

closure under union and complementation restricted closure under product and iteration

5

Main Theorem

Theorem

Let S ⊆ Q. Then the following are equivalent:

1 S is recognizable 2 S is q-rational

6

Structures for Qeue Transformations

Let a, b ∈ A be distinct. Consider t = [babaaaa]≡. We model t as a structure t with infinitely many relations: b a b a a a a babaaaa ≡ babaaaa

7

Structures for Qeue Transformations

Let a, b ∈ A be distinct. Consider t = [babaaaa]≡. We model t as a structure t with infinitely many relations: b a b a a a a

3 2 1

7

Structures for Qeue Transformations

Let a, b ∈ A be distinct. Consider t = [babaaaa]≡. We model t as a structure t with infinitely many relations:

≤−, ≤+, Pn for any n ∈ N

b a b a a a a

3 2 1

P3

7

Main Theorem

Theorem

Let S ⊆ Q. Then the following are equivalent:

1 S is recognizable 2 S is q-rational 3 S = {t ∈ Q |

t | = φ} for some φ ∈ MSO Similar results for aperiodic sets and for (partially) lossy queues

Thank you!

8