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Rational and Recognizable Sets in the Qeue Monoid Highlights of Games, Logic and Automata 2018, Berlin Chris K¨ ocher 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 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 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 a 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 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 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 b 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 ⇒ p t − → q ⇐ → 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 ∗ aA ∗ ) and η ( A ∗ aA ∗ ) for a ∈ A start from η ( 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: a a a a a b b 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: 3 2 1 a a b a a a b 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: ≤ − , ≤ + , P n for any n ∈ N 3 2 1 a a b a a a b P 3 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