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Feasible Automata for Two-Variable Logic
with Successor on Data Words
Ahmet Kara, Thomas Schwentick, and Tony Tan
LATA 2012, A Coruña
Systems with Unboundedly Many Processes
Server
process 1
process 2 process 3
process 4
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 1
Systems with Unboundedly Many Processes
Server
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 2
Systems with Unboundedly Many Processes
Server
process 1
• A system run
req
1
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 2
Systems with Unboundedly Many Processes
Server
process 1
process 2
• A system run
req
1
req
2
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 2
Systems with Unboundedly Many Processes
Server
process 1
process 2
• A system run
req
1
req
2
ack
1
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 2
Systems with Unboundedly Many Processes
Server
process 1
process 2 process 3
• A system run
req
1
req
2
ack
1
req
3
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 2
Systems with Unboundedly Many Processes
Server
process 1
process 2 process 3
process 4
• A system run
req
1
req
2
ack
1
req
3
req
4
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 2
Systems with Unboundedly Many Processes
Server
process 1
process 2 process 3
process 4
• A system run
req
1
req
2
ack
1
req
3
req
4
ack
3
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 2
Systems with Unboundedly Many Processes
Server
process 1
process 2 process 3
process 4
• A system run
req
1
req
2
ack
1
req
3
req
4
ack
3
req
1
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 2
Systems with Unboundedly Many Processes
Server
process 1
process 2 process 3
process 4
• A system run
req
1
req
2
ack
1
req
3
req
4
ack
3
req
1. . .
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 2
Systems with Unboundedly Many Processes
Server
process 1
process 2 process 3
process 4
• A system run
req
1
req
2
ack
1
req
3
req
4
ack
3
req
1. . .
• A system property
„Every req of a process is followed by some ack for the same process.”∀x∃y(req(x) → (x < y ∧ ack(y) ∧ x ∼ y))
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 2
Words and Data Words
A Word over Σ = {a, b, c}
c c a c a b c b
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 3
Words and Data Words
A Word over Σ = {a, b, c}
c c a c a b c b
A Data Word over Σ = {a, b, c}
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
Definition: Data Words
• Let◮ Σ be a finite alphabet◮ D be an infinite set of data values
• w ∈ (Σ × D)∗ is a data word over Σ
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 3
Words and Data Words
A Word over Σ = {a, b, c}
c c a c a b c b
A Data Word over Σ = {a, b, c}
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
• D = {1, 2, 3, . . .}
Definition: Data Words
• Let◮ Σ be a finite alphabet◮ D be an infinite set of data values
• w ∈ (Σ × D)∗ is a data word over Σ
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 3
Logics on Data Words – Data Logics
• Even very weak logics on data words have an undecidablesatisfiability problem.
◮ LTL in general not decidable [Demri et al. 06]
◮ FO3 (with only three variables) not decidable[Bojanczyk et al. 06]
◮ FO2 decidable but exact complexity not known[Bojanczyk et al. 06]
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 4
Known Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 5
Known Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
is captured
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 5
Known Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
is captured ?
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 5
Known Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
2NEXPTIME
EMSO2(∼, Suc)
is captured ?
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 5
Known Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
2NEXPTIME
EMSO2(∼, Suc)
is captured ?
?
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 5
Known Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
2NEXPTIME
EMSO2(∼, Suc)
is captured ?
?
?
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 5
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x
∀x
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∀x∃y
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∀x∃y(
a(x)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∀x∃y(
a(x) →(x < y
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∀x∃y(
a(x) →(x < y ∧ b(y)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∀x∃y(
a(x) →
(x < y ∧ b(y) ∧ x ∼ y))
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∀x∃y(
a(x) →
(x < y ∧ b(y) ∧ x ∼ y))
Example
„There are two neighbouring positions withdifferent data values."
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∀x∃y(
a(x) →
(x < y ∧ b(y) ∧ x ∼ y))
Example
„There are two neighbouring positions withdifferent data values."
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x
∃x
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∀x∃y(
a(x) →
(x < y ∧ b(y) ∧ x ∼ y))
Example
„There are two neighbouring positions withdifferent data values."
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∃x∃y
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∀x∃y(
a(x) →
(x < y ∧ b(y) ∧ x ∼ y))
Example
„There are two neighbouring positions withdifferent data values."
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∃x∃y(
Suc(x, y)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Existential Monadic Second Order Logic on Data Words
Existential Monadic Second Order Logic with twoposition variables (EMSO2(∼, Suc, ≤)):
• formulas are of the form∃M1 . . . ∃Mn ϕwhere ϕ is a first order formula
◮ uses only two position variables
◮ can test whether a position is containedin some set Mi
◮ can test whether a position is labelled bysome symbol
◮ can use order and successor relations onpositions
◮ can test whether the data values at twopositions are equal
Example
„Every a is followed by some b with the samedata value.”
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∀x∃y(
a(x) →
(x < y ∧ b(y) ∧ x ∼ y))
Example
„There are two neighbouring positions withdifferent data values."
c
1
c
4
a
3
c
2
a
2
b
3
c
7
b
2
x y
∃x∃y(
Suc(x, y) ∧ ¬(x ∼ y))
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 6
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 7
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
σ1
1
σ2
1
σ3
3
σ4
2
σ5
2
σ6
3
σ7
1
σ8
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 8
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
σ1
1
σ2
1
σ3
3
σ4
2
σ5
2
σ6
3
σ7
1
σ8
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 8
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
σ2
1
σ3
3
σ4
2
σ5
2
σ6
3
σ7
1
σ8
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 8
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
σ3
3
σ4
2
σ5
2
σ6
3
σ7
1
σ8
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 8
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
σ4
2
σ5
2
σ6
3
σ7
1
σ8
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 8
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
σ5
2
σ6
3
σ7
1
σ8
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 8
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
σ6
3
σ7
1
σ8
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 8
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
σ7
1
σ8
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 8
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
σ8
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 8
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 8
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
γ1
1
γ2
1
γ3
3
γ4
2
γ5
2
γ6
3
γ7
1
γ8
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 9
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 10
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 10
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 10
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 10
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 10
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 10
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 10
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 10
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 10
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 10
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„There is an a which is followed by some b with a different data value.”
−
1
−
1
X
3
−
2
−
2
−
3
−
1
X
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 11
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 12
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 12
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 12
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 12
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 12
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 12
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 12
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 12
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 12
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 12
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C) C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Data Automata
Data Automata (DA) [Bojanczyk et al. 06]
A data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a class automaton C: a usual string automaton over Γ
Example
„Every a is followed by some b with the same data value.”
c
1
c
1
a
3
c
2
a
2
b
3
c
1
b
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 13
Known Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
2NEXPTIME
EMSO2(∼, Suc)
is captured ?
?
?
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 14
Weak Data Automata
Data Automata
Base
Automaton
B
Class
Automaton
C
Definition
A data automaton D = (B, C) over analphabet Σ consists of
• a base automaton B: a nondeterministicletter-to-letter string transducer with inputalphabet Σ and output alphabet Γ
• a class automaton C: a usual stringautomaton over Γ
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 15
Weak Data Automata
Data Automata
Base
Automaton
B
Class
Automaton
C
Definition
A data automaton D = (B, C) over analphabet Σ consists of
• a base automaton B: a nondeterministicletter-to-letter string transducer with inputalphabet Σ and output alphabet Γ
• a class automaton C: a usual stringautomaton over Γ
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 15
Weak Data Automata
Data Automata
Base
Automaton
B
Class
Automaton
C
Definition
A data automaton D = (B, C) over analphabet Σ consists of
• a base automaton B: a nondeterministicletter-to-letter string transducer with inputalphabet Σ and output alphabet Γ
• a class automaton C: a usual stringautomaton over Γ
Weak Data Automata
Base
Automaton
B
Set of
Constraints
C
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 15
Weak Data Automata
Data Automata
Base
Automaton
B
Class
Automaton
C
Definition
A data automaton D = (B, C) over analphabet Σ consists of
• a base automaton B: a nondeterministicletter-to-letter string transducer with inputalphabet Σ and output alphabet Γ
• a class automaton C: a usual stringautomaton over Γ
Weak Data Automata
Base
Automaton
B
Set of
Constraints
C
Definition
A weak data automaton D = (B, C) over analphabet Σ consists of
• a base automaton B: a nondeterministicletter-to-letter string transducer with inputalphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 15
Constraints on Data Words
Constraints on Data Words
• key constraintskey(σ): all σ-positions have different datavalues
Example
w =
c
1
c
1
c
1
c
1
b
4
c
2
c
2
a
2
a
3
c
3
c
3
c
3
b
1
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 16
Constraints on Data Words
Constraints on Data Words
• key constraintskey(σ): all σ-positions have different datavalues
Example
w =
c
1
c
1
c
1
c
1
b
4
c
2
c
2
a
2
a
3
c
3
c
3
c
3
b
1
• w |= key(a)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 16
Constraints on Data Words
Constraints on Data Words
• key constraintskey(σ): all σ-positions have different datavalues
• inclusion constraintsV (σ1) ⊆ V (σ2): all data values onσ1-positions occur on σ2-positions
Example
w =
c
1
c
1
c
1
c
1
b
4
c
2
c
2
a
2
a
3
c
3
c
3
b
1
• w |= key(a)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 16
Constraints on Data Words
Constraints on Data Words
• key constraintskey(σ): all σ-positions have different datavalues
• inclusion constraintsV (σ1) ⊆ V (σ2): all data values onσ1-positions occur on σ2-positions
Example
w =
c
1
c
1
c
1
c
1
b
4
c
2
c
2
a
2
a
3
c
3
c
3
b
1
• w |= key(a)
• w |= V (a) ⊆ V (c)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 16
Constraints on Data Words
Constraints on Data Words
• key constraintskey(σ): all σ-positions have different datavalues
• inclusion constraintsV (σ1) ⊆ V (σ2): all data values onσ1-positions occur on σ2-positions
• denial constraintsV (σ1) ∩ V (σ2) = ∅: σ1-positionsand σ2-positions do not share any datavalue
Example
w =
c
1
c
1
c
1
c
1
b
4
c
2
c
2
a
2
a
3
c
3
c
3
b
1
• w |= key(a)
• w |= V (a) ⊆ V (c)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 16
Constraints on Data Words
Constraints on Data Words
• key constraintskey(σ): all σ-positions have different datavalues
• inclusion constraintsV (σ1) ⊆ V (σ2): all data values onσ1-positions occur on σ2-positions
• denial constraintsV (σ1) ∩ V (σ2) = ∅: σ1-positionsand σ2-positions do not share any datavalue
Example
w =
c
1
c
1
c
1
c
1
b
4
c
2
c
2
a
2
a
3
c
3
c
3
b
1
• w |= key(a)
• w |= V (a) ⊆ V (c)
• w |= V (a) ∩ V (b) = ∅
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 16
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 17
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„There is an a and a b with different data values.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 18
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„There is an a and a b with different data values.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 18
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„There is an a and a b with different data values.”
−
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 18
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„There is an a and a b with different data values.”
−
1
−
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 18
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„There is an a and a b with different data values.”
−
1
−
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 18
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„There is an a and a b with different data values.”
−
1
−
1
b
3
−
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 18
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„There is an a and a b with different data values.”
−
1
−
1
b
3
−
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 18
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„There is an a and a b with different data values.”
−
1
−
1
b
3
−
2
a
2
−
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 18
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„There is an a and a b with different data values.”
−
1
−
1
b
3
−
2
a
2
−
3
−
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 18
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„There is an a and a b with different data values.”
−
1
−
1
b
3
−
2
a
2
−
3
−
1
−
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 18
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„There is an a and a b with different data values.”
−
1
−
1
b
3
−
2
a
2
−
3
−
1
−
2
(B, C)
• C = {V (a) ∩ V (b) = ∅}
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 19
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 20
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 20
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 20
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 20
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 20
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 20
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 20
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 20
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C) B
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 20
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 20
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C)
• C = {V (a) ⊆ V (b)}
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 21
Weak Data Automata
Weak Data Automata (WDA)
A weak data automaton D = (B, C) over an alphabet Σ consists of
• a base automaton B: a nondeterministic letter-to-letter string transducer withinput alphabet Σ and output alphabet Γ
• a set C of constraints over Γ
Example
„For every a there exists some b with the same data value.”
c
1
c
1
b
3
c
2
a
2
a
3
c
1
b
2
(B, C)
• C = {V (a) ⊆ V (b)}
• Main difference between class automata and constraints:constraints cannot access the order of positions
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 21
Our Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
2NEXPTIME
EMSO2(∼, Suc)
is captured ?
?
?
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 22
Our Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
2NEXPTIME
EMSO2(∼, Suc)
is captured ?
?
Weak Data Automata
is captured:
translatable
in 2EXPTIME
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 22
Our Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
2NEXPTIME
EMSO2(∼, Suc)
is captured ?
?
Weak Data Automata
is captured:
translatable
in 2EXPTIME
<
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 22
Our Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
2NEXPTIME
EMSO2(∼, Suc)
is captured ?
?
Weak Data Automata
is captured:
translatable
in 2EXPTIME
<Theorem
WDA < DA
• inclusion: the class automaton can check all constraints
◮ e.g. key(a): class automaton checks that all class strings contain at mostone a
• separating property: „Every a is followed by some b with the same data value.”
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 23
Our Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
2NEXPTIME
EMSO2(∼, Suc)
is captured ?
<
Weak Data Automata
is captured:
translatable
in 2EXPTIME
<
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 24
Weak ω-Data Automata
Data ω-words
• Let
◮ Σ be a finite alphabet
◮ D be an infinite set of data values
• w ∈ (Σ × D)ω is a data ω-word overΣ
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 25
Weak ω-Data Automata
Data ω-words
• Let
◮ Σ be a finite alphabet
◮ D be an infinite set of data values
• w ∈ (Σ × D)ω is a data ω-word overΣ
Data ω-Automata [Bojanczyk et al. 06]
A data ω-automaton D = (B, Cfin, Cinf) overan alphabet Σ consists of
• a base Büchi automaton B
• a class automaton Cfin
• a class Büchi automaton Cinf
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 25
Weak ω-Data Automata
Data ω-words
• Let
◮ Σ be a finite alphabet
◮ D be an infinite set of data values
• w ∈ (Σ × D)ω is a data ω-word overΣ
Data ω-Automata [Bojanczyk et al. 06]
A data ω-automaton D = (B, Cfin, Cinf) overan alphabet Σ consists of
• a base Büchi automaton B
• a class automaton Cfin
• a class Büchi automaton Cinf
Weak Data ω-Automata
A weak data ω-automaton D = (B, C) overan alphabet Σ consists of
• a base Büchi automaton B
• a set C of constraints
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 25
Weak ω-Data Automata
Data ω-words
• Let
◮ Σ be a finite alphabet
◮ D be an infinite set of data values
• w ∈ (Σ × D)ω is a data ω-word overΣ
Data ω-Automata [Bojanczyk et al. 06]
A data ω-automaton D = (B, Cfin, Cinf) overan alphabet Σ consists of
• a base Büchi automaton B
• a class automaton Cfin
• a class Büchi automaton Cinf
Weak Data ω-Automata
A weak data ω-automaton D = (B, C) overan alphabet Σ consists of
• a base Büchi automaton B
• a set C of constraints
E∞MSO2(∼, Suc)
Extension of EMSO2(∼, Suc):
• quantified sets can be restricted to bebound to infinite sets only
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 25
Our Results on Data ω-Words
DECIDABLE
EMSO2(∼, Suc, ≤)
Data ω-Automata
2NEXPTIME
E∞MSO2(∼, Suc)
is captured ?
<
Weak Data ω-Automata
is captured:
translatable
in 2EXPTIME
<
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 26
Conclusion
• WDA: automata model for EMSO2(∼, Suc) (and
E∞MSO2(∼, Suc)) with reasonable complexity
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 27
Conclusion
• WDA: automata model for EMSO2(∼, Suc) (and
E∞MSO2(∼, Suc)) with reasonable complexity
• Open Questions:
◮ Are WDA the best automata model for these logics?
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 27
Conclusion
• WDA: automata model for EMSO2(∼, Suc) (and
E∞MSO2(∼, Suc)) with reasonable complexity
• Open Questions:
◮ Are WDA the best automata model for these logics?
◮ What is the exact complexity of the emptiness problem for weak dataautomata?� The 2NEXPTIME upper bound yields only a 3NEXPTIME upper bound for
the satisfiability of EMSO2(∼, Suc)� which is in 2NEXPTIME [Niewerth et al. 09]
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 27
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