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A Logical Characterization of Timed Pushdown Languages Manfred Droste and Vitaly Perevoshchikov 1 Leipzig University CSR 2015, Listvyanka 1 Supported by the DFG Research Training Group "QuantLA" (Dense-)Timed Pushdown Automata 1 (TPDA)

SLIDE 1

A Logical Characterization

f Timed Pushdown Languages

Manfred Droste and Vitaly Perevoshchikov1

Leipzig University

CSR 2015, Listvyanka

1Supported by the DFG Research Training Group "QuantLA"

SLIDE 2

(Dense-)Timed Pushdown Automata1 (TPDA)

TPDA are nondeterministic finite automata (NFA) equipped with: real-valued clocks timed stack NFA TA PDA TPDA ➋➌

➋ ➋ ➋

➋➌

1Abdulla, Atig, Stenman ’12

SLIDE 3

Timed Pushdown Automata1 (TPDA)

Definition A TPDA over an alphabet Σ: A = (Q❀C❀Γ❀I❀T❀F) where Q is a finite set of states C is a finite set of clocks Γ is a stack alphabet I❀F ⊆ Q are initial and final state T is a finite set of edges of the form q

a❀ ✣❀ Λ

s

q′ where:

q❀q′ ∈ Q, a ∈ Σ ✣ is a clock constraint over C, Λ ⊆ C is a set of clocks to be reset s is: pushI(✌), # or popI(✌) where ✌ ∈ Γ and I is an interval

1Abdulla, Atig, Stenman ’12

SLIDE 4

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

SLIDE 5

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0

SLIDE 6

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0 0.3 delay 1 x = 0✿3

SLIDE 7

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0 0.3 delay 1 x = 0✿3 a switch 1 x = 0✿3 ✌ ∶ 0

SLIDE 8

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0 0.3 delay 1 x = 0✿3 a switch 1 x = 0✿3 ✌ ∶ 0 0.7 delay 1 x = 1 ✌ ∶ 0✿7

SLIDE 9

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0 0.3 delay 1 x = 0✿3 a switch 1 x = 0✿3 ✌ ∶ 0 0.7 delay 1 x = 1 ✌ ∶ 0✿7 a switch 1 x = 1 ✌ ∶ 0 ✌ ∶ 0✿7

SLIDE 10

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0 0.3 delay 1 x = 0✿3 a switch 1 x = 0✿3 ✌ ∶ 0 0.7 delay 1 x = 1 ✌ ∶ 0✿7 a switch 1 x = 1 ✌ ∶ 0 ✌ ∶ 0✿7 1 delay 1 x = 2 ✌ ∶ 1 ✌ ∶ 1✿7

SLIDE 11

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0 0.3 delay 1 x = 0✿3 a switch 1 x = 0✿3 ✌ ∶ 0 0.7 delay 1 x = 1 ✌ ∶ 0✿7 a switch 1 x = 1 ✌ ∶ 0 ✌ ∶ 0✿7 1 delay 1 x = 2 ✌ ∶ 1 ✌ ∶ 1✿7 b switch 2 x = 0 ✌ ∶ 1 ✌ ∶ 1✿7

SLIDE 12

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0 0.3 delay 1 x = 0✿3 a switch 1 x = 0✿3 ✌ ∶ 0 0.7 delay 1 x = 1 ✌ ∶ 0✿7 a switch 1 x = 1 ✌ ∶ 0 ✌ ∶ 0✿7 1 delay 1 x = 2 ✌ ∶ 1 ✌ ∶ 1✿7 b switch 2 x = 0 ✌ ∶ 1 ✌ ∶ 1✿7 0.1 delay 2 x = 0✿1 ✌ ∶ 1✿1 ✌ ∶ 1✿8

SLIDE 13

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0 0.3 delay 1 x = 0✿3 a switch 1 x = 0✿3 ✌ ∶ 0 0.7 delay 1 x = 1 ✌ ∶ 0✿7 a switch 1 x = 1 ✌ ∶ 0 ✌ ∶ 0✿7 1 delay 1 x = 2 ✌ ∶ 1 ✌ ∶ 1✿7 b switch 2 x = 0 ✌ ∶ 1 ✌ ∶ 1✿7 0.1 delay 2 x = 0✿1 ✌ ∶ 1✿1 ✌ ∶ 1✿8 a switch 2 x = 0✿1 ✌ ∶ 1✿8

SLIDE 14

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0 0.3 delay 1 x = 0✿3 a switch 1 x = 0✿3 ✌ ∶ 0 0.7 delay 1 x = 1 ✌ ∶ 0✿7 a switch 1 x = 1 ✌ ∶ 0 ✌ ∶ 0✿7 1 delay 1 x = 2 ✌ ∶ 1 ✌ ∶ 1✿7 b switch 2 x = 0 ✌ ∶ 1 ✌ ∶ 1✿7 0.1 delay 2 x = 0✿1 ✌ ∶ 1✿1 ✌ ∶ 1✿8 a switch 2 x = 0✿1 ✌ ∶ 1✿8 0.1 delay 2 x = 0✿2 ✌ ∶ 1✿9

SLIDE 15

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0 0.3 delay 1 x = 0✿3 a switch 1 x = 0✿3 ✌ ∶ 0 0.7 delay 1 x = 1 ✌ ∶ 0✿7 a switch 1 x = 1 ✌ ∶ 0 ✌ ∶ 0✿7 1 delay 1 x = 2 ✌ ∶ 1 ✌ ∶ 1✿7 b switch 2 x = 0 ✌ ∶ 1 ✌ ∶ 1✿7 0.1 delay 2 x = 0✿1 ✌ ∶ 1✿1 ✌ ∶ 1✿8 a switch 2 x = 0✿1 ✌ ∶ 1✿8 0.1 delay 2 x = 0✿2 ✌ ∶ 1✿9 a switch 2 x = 0✿1

SLIDE 16

TPDA: Example

Σ = {a❀b} C = {x} Γ = {✌}. A 1 2

a, push[0❀0](✌) a, pop[1❀2)(✌) b, x ≤ 2, x ∶= 0, #

A run of A:

1 x = 0 0.3 delay 1 x = 0✿3 a switch 1 x = 0✿3 ✌ ∶ 0 0.7 delay 1 x = 1 ✌ ∶ 0✿7 a switch 1 x = 1 ✌ ∶ 0 ✌ ∶ 0✿7 1 delay 1 x = 2 ✌ ∶ 1 ✌ ∶ 1✿7 b switch 2 x = 0 ✌ ∶ 1 ✌ ∶ 1✿7 0.1 delay 2 x = 0✿1 ✌ ∶ 1✿1 ✌ ∶ 1✿8 a switch 2 x = 0✿1 ✌ ∶ 1✿8 0.1 delay 2 x = 0✿2 ✌ ∶ 1✿9 a switch 2 x = 0✿1

Accepted timed word: (a❀0✿3)(a❀1)(b❀2)(a❀2✿1)(a❀2✿2)

SLIDE 17

Relative Distance Logic (RDL)1

Let Σ be an alphabet. Definition Relative distance logic RDL(Σ): consists of formulas of the form ∃X1✿✿✿∃Xn✿✬ where ✬ ∶∶= Pa(x) ∣ x ≤ y ∣ x ∈ X ∣ d(X❀x) ∼ c ∣ ✬ ∨ ✬ ∣ ¬✬ ∣ ∃x✿✬ with a ∈ Σ, ∼ ∈ {<❀=❀>}, c ∈ N.

1Wilke ’94

SLIDE 18

Relative Distance Logic (RDL)1

Let Σ be an alphabet. Definition Relative distance logic RDL(Σ): consists of formulas of the form ∃X1✿✿✿∃Xn✿✬ where ✬ ∶∶= Pa(x) ∣ x ≤ y ∣ x ∈ X ∣ d(X❀x) ∼ c ∣ ✬ ∨ ✬ ∣ ¬✬ ∣ ∃x✿✬ with a ∈ Σ, ∼ ∈ {<❀=❀>}, c ∈ N. Model: a timed word w = (a1❀t1)✿✿✿(an❀tn) ∈ TΣ+. (w❀✛) ⊧ d(X❀x) ∼ c 1 2 3 i ti tj ✛(x) j

position time

t1 t2 t3

t✛(x) − ti ∼ c

✛(X) ✛(X)

1Wilke ’94

SLIDE 19

Relative Distance Logic (RDL)1

Let Σ be an alphabet. Definition Relative distance logic RDL(Σ): consists of formulas of the form ∃X1✿✿✿∃Xn✿✬ where ✬ ∶∶= Pa(x) ∣ x ≤ y ∣ x ∈ X ∣ d(X❀x) ∼ c ∣ ✬ ∨ ✬ ∣ ¬✬ ∣ ∃x✿✬ with a ∈ Σ, ∼ ∈ {<❀=❀>}, c ∈ N. Theorem (Wilke ’94) Let L ⊆ TΣ+ be a timed language. TFAE:

1 L is recognizable by a timed automaton 2 L is definable by a RDL(Σ)-sentence. 1Wilke ’94

SLIDE 20

Logic for Pushdown Automata1

Matching logic ML(Σ): ∃match✖✿FO(Σ❀<❀✖) Definition (Matching). A relation M ⊆ {1❀✿✿✿❀n}2 is a matching if: (x❀y) ∈ M ⇒ x < y; every x ∈ {1❀✿✿✿❀n} belongs to at most one pair in M;

1Lautemann, Schwentick, Thérien ’94

SLIDE 21

Logic for Pushdown Automata1

Matching logic ML(Σ): ∃match✖✿FO(Σ❀<❀✖) Definition (Matching). A relation M ⊆ {1❀✿✿✿❀n}2 is a matching if: (x❀y) ∈ M ⇒ x < y; every x ∈ {1❀✿✿✿❀n} belongs to at most one pair in M; M is non-crossing: ✔ ✖

1Lautemann, Schwentick, Thérien ’94

SLIDE 22

Logic for Pushdown Automata1

Matching logic ML(Σ): ∃match✖✿FO(Σ❀<❀✖) Definition (Matching). A relation M ⊆ {1❀✿✿✿❀n}2 is a matching if: (x❀y) ∈ M ⇒ x < y; every x ∈ {1❀✿✿✿❀n} belongs to at most one pair in M; M is non-crossing: ✔ ✖

1Lautemann, Schwentick, Thérien ’94

SLIDE 23

Timed Matching Logic (TML)

Definition TML(Σ) is the set of formulas of the form ∃match✖✿∃X1✿ ✿✿✿ ∃Xn✿✬ where ✬ is defined by the grammar: ✬ ∶∶= Pa(x) ∣ x ≤ y ∣ x ∈ X ∣ ✖(x❀y) ∼ c ∣ d(X❀x) ∼ c ∣ ✬ ∨ ✬ ∣ ¬✬ ∣ ∃x✿✬ where a ∈ Σ, ∼ ∈ {<❀=❀>} and c ∈ N.

SLIDE 24

Timed Matching Logic (TML)

Definition TML(Σ) is the set of formulas of the form ∃match✖✿∃X1✿ ✿✿✿ ∃Xn✿✬ where ✬ is defined by the grammar: ✬ ∶∶= Pa(x) ∣ x ≤ y ∣ x ∈ X ∣ ✖(x❀y) ∼ c ∣ d(X❀x) ∼ c ∣ ✬ ∨ ✬ ∣ ¬✬ ∣ ∃x✿✬ where a ∈ Σ, ∼ ∈ {<❀=❀>} and c ∈ N. Let w = (a1❀t1)✿✿✿(an❀tn) ∈ TΣ+. Then, (w❀✛) ⊧ ✖(x❀y) ∼ c iff: (✛(x)❀✛(y)) ∈ ✛(✖) 1 ✛(x) ✛(y) n

t✛(y) − t✛(x) ∼ c

SLIDE 25

Example: Timed Dyck Languages

Definition Let Σ = {a1❀✿✿✿❀am} be a set of opening brackets Let Σ = {a1❀✿✿✿❀am} be a set of corresponding closing brackets ❀✿✿✿❀ ❀ ❀ ❀ ❀✿✿✿❀ ❀ ✿✿✿ ❀

✿✿✿

SLIDE 26

Example: Timed Dyck Languages

Definition Let Σ = {a1❀✿✿✿❀am} be a set of opening brackets Let Σ = {a1❀✿✿✿❀am} be a set of corresponding closing brackets Let I1❀✿✿✿❀Im be intervals (e.g., (0❀3]❀[2❀∞), etc.) ❀✿✿✿❀ ❀ ✿✿✿ ❀

✿✿✿

SLIDE 27

Example: Timed Dyck Languages

Definition Let Σ = {a1❀✿✿✿❀am} be a set of opening brackets Let Σ = {a1❀✿✿✿❀am} be a set of corresponding closing brackets Let I1❀✿✿✿❀Im be intervals (e.g., (0❀3]❀[2❀∞), etc.) A timed Dyck language DΣ(I1❀✿✿✿❀Im) consists of all timed words (b1❀t1)✿✿✿(bn❀tn) ∈ T(Σ ∪ Σ)+ such that:

✿✿✿

SLIDE 28

Example: Timed Dyck Languages

Definition Let Σ = {a1❀✿✿✿❀am} be a set of opening brackets Let Σ = {a1❀✿✿✿❀am} be a set of corresponding closing brackets Let I1❀✿✿✿❀Im be intervals (e.g., (0❀3]❀[2❀∞), etc.) A timed Dyck language DΣ(I1❀✿✿✿❀Im) consists of all timed words (b1❀t1)✿✿✿(bn❀tn) ∈ T(Σ ∪ Σ)+ such that:

b1✿✿✿bn ∈ (Σ ∪ Σ)+ is a correctly nested sequence of brackets

SLIDE 29

Example: Timed Dyck Languages

Definition Let Σ = {a1❀✿✿✿❀am} be a set of opening brackets Let Σ = {a1❀✿✿✿❀am} be a set of corresponding closing brackets Let I1❀✿✿✿❀Im be intervals (e.g., (0❀3]❀[2❀∞), etc.) A timed Dyck language DΣ(I1❀✿✿✿❀Im) consists of all timed words (b1❀t1)✿✿✿(bn❀tn) ∈ T(Σ ∪ Σ)+ such that:

b1✿✿✿bn ∈ (Σ ∪ Σ)+ is a correctly nested sequence of brackets the time distance between any two matching brackets aj and aj is in Ij.

SLIDE 30

Example: Timed Dyck Languages

Definition Let Σ = {a1❀✿✿✿❀am} be a set of opening brackets Let Σ = {a1❀✿✿✿❀am} be a set of corresponding closing brackets Let I1❀✿✿✿❀Im be intervals (e.g., (0❀3]❀[2❀∞), etc.) A timed Dyck language DΣ(I1❀✿✿✿❀Im) consists of all timed words (b1❀t1)✿✿✿(bn❀tn) ∈ T(Σ ∪ Σ)+ such that:

b1✿✿✿bn ∈ (Σ ∪ Σ)+ is a correctly nested sequence of brackets the time distance between any two matching brackets aj and aj is in Ij.

SLIDE 31

Example: Timed Dyck Languages

Definition Let Σ = {a1❀✿✿✿❀am} be a set of opening brackets Let Σ = {a1❀✿✿✿❀am} be a set of corresponding closing brackets Let I1❀✿✿✿❀Im be intervals (e.g., (0❀3]❀[2❀∞), etc.) A timed Dyck language DΣ(I1❀✿✿✿❀Im) consists of all timed words (b1❀t1)✿✿✿(bn❀tn) ∈ T(Σ ∪ Σ)+ such that:

b1✿✿✿bn ∈ (Σ ∪ Σ)+ is a correctly nested sequence of brackets the time distance between any two matching brackets aj and aj is in Ij.

DΣ(I1❀✿✿✿❀Im) is defined by the sentence: ✬ = ∃match✖✿(∀x✿∃y✿(✖(x❀y) ∨ ✖(y❀x)) ∧ ∀x✿∀y✿(✖(x❀y) →

m

⋁

j=1

(Paj(x) ∧ Paj(y) ∧ ✖Ij(x❀y))))

SLIDE 32

Main Result

Theorem Let Σ be an alphabet and L ⊆ TΣ+ a timed language. TFAE:

1 L is recognizable by a TPDA. 2 L is definable by a TML(Σ)-sentence.

SLIDE 33

Main Result

Theorem Let Σ be an alphabet and L ⊆ TΣ+ a timed language. TFAE:

1 L is recognizable by a TPDA. 2 L is definable by a TML(Σ)-sentence.

Corollary It is decidable, given an alphabet Σ and a sentence ✥ ∈ TML(Σ), whether there exists a timed word w ∈ TΣ+ with w ⊧ ✥.

SLIDE 34

Proof

NFA TIMED PDA TPDA NFA_TREE TIMED_TREE PDA_TREE TPDA_TREE

MSO RDL ML TML MSO ? ? ?

SLIDE 35

Proof

NFA TIMED PDA TPDA NFA_TREE TIMED_TREE PDA_TREE TPDA_TREE

MSO RDL ML TML MSO ? ? ?

? ? ? ? ?

SLIDE 36

Proof

NFA TIMED PDA TPDA vPDA1 NFA_TREE TIMED_TREE PDA_TREE TPDA_TREE

MSO RDL ML TML MSO ? ? ? MSO

1Alur, Madhusudan ’04

SLIDE 37

Visibly Pushdown Automata1 (vPDA)

Let Σpush, Σ# and Σpop be pairwise disjoint alphabets Let Σ = Σpush ∪ Σ# ∪ Σpop and ˜ Σ = ⟨Σpush❀Σ#❀Σpop⟩ Definition A vPDA over ˜ Σ is a tuple A = (Q❀Γ❀I❀T❀F) where: Q is a finite set of states, Γ is a stack alphabet I❀F ⊆ Q are sets of initial resp. final states T = T push ∪ ⋅ T # ∪ ⋅ T pop where:

T push ⊆ Q × Σpush × Γ × Q T # ⊆ Q × Σ# × Q T pop ⊆ Q × Σpop × (Γ ∪ ⋅ {}) × Q

Accepted language: L(A) ⊆ Σ+.

1Alur, Madhusudan ’04

SLIDE 38

Logic for Visibly Pushdown Languages1

Let Σpush, Σ# and Σpop be pairwise disjoint alphabets Let Σ = Σpush ∪ Σ# ∪ Σpop and ˜ Σ = ⟨Σpush❀Σ#❀Σpop⟩ Definition Logic MSO(˜ Σ) is defined as: ✬ ∶∶= Pa(x) ∣ x ≤ y ∣ x ∈ X ∣ match(x❀y) ∣ ✬ ∨ ✬ ∣ ¬✬ ∣ ∃x✿✬ ∣ ∃X✿✬ where a ∈ Σ. Defined language of a sentence ✬ ∈ MSO(˜ Σ): L(✬) ⊆ Σ+. Theorem1 Let L ⊆ Σ+ be a language. TFAE:

1 L is recognizable by a vPDA over ˜

Σ.

2 L is MSO(˜

Σ)-definable.

1Alur, Madhusudan ’04

SLIDE 39

Decomposition of TPDA

➋ ➋ ➋ ➋

➌ TPDA

ver Σ

✙ ∶ Γk❀n → Σ

∩

vPDA

ver ˜

Γk❀n

➋ ➋ ➋ ➋

➌ "Primitive" TPDL Tk❀n ⊆ TΓk❀n Extended alphabet Γk❀n = Σ× (P(k))n ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

clock constraints

× {0❀1}n ÜÜÜÜÜÜÜÜÜÜÜÜÜ

clock resets

× P(k)

stack constraints

×{push❀#❀pop} ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

stack commands

n ∶= number of global clocks k ∶= maximal number appearing in constraints P(k) ∶= {[0❀0]❀(0❀1)❀[1❀1]❀✿✿✿❀(k − 1❀k)❀[k❀k]❀(k❀∞)}

SLIDE 40

Decomposition of TPDA

➋ ➋ ➋ ➋

➌ TPDA

ver Σ

✙ ∶ Γk❀n → Σ

∩

vPDA

ver ˜

Γk❀n

➋ ➋ ➋ ➋

➌ "Primitive" TPDL Tk❀n ⊆ TΓk❀n Theorem Let L ⊆ TΣ+. TFAE:

1 L is a timed pushdown language. 2 There exist k❀n ∈ N and a vPDL L′ ⊆ Γk❀n+ with

L = ✙(L′ ∩ Tk❀n)

SLIDE 41

Decomposition of TML

✖(x❀ y) ∼ c d(X❀ x) ∼ c

TML

ver Σ

✙ ∶ Γk❀n → Σ

∩

match(x❀ y)

MSO

ver ˜

Γk❀n

➋ ➋ ➋ ➋

➌ "Primitive" TPDL Tk❀n ⊆ TΓk❀n Theorem Let L ⊆ TΣ+. TFAE:

1 L is TML-definable. 2 There exist k❀n ∈ N and a vPDL L′ ⊆ (Γk❀n)+ with

L = ✙(L′ ∩ Tk❀n)

SLIDE 42

Future work

1 Weighted TPDA 2 TPDA with ✧-transitions

✖✿ ❀✖

SLIDE 43

Future work

1 Weighted TPDA 2 TPDA with ✧-transitions 3 Are TDPA without global clocks expressively equivalent to

∃match✖✿FO(<❀✖∼c)?

SLIDE 44

Future work

1 Weighted TPDA 2 TPDA with ✧-transitions 3 Are TDPA without global clocks expressively equivalent to

∃match✖✿FO(<❀✖∼c)?

4 Connection to timed tree automata

SLIDE 45

Future work

1 Weighted TPDA 2 TPDA with ✧-transitions 3 Are TDPA without global clocks expressively equivalent to

∃match✖✿FO(<❀✖∼c)?

4 Connection to timed tree automata 5 A Chomsky-Schützenberger characterization

SLIDE 46

Future work

1 Weighted TPDA 2 TPDA with ✧-transitions 3 Are TDPA without global clocks expressively equivalent to

∃match✖✿FO(<❀✖∼c)?

4 Connection to timed tree automata 5 A Chomsky-Schützenberger characterization