feasible automata for two-variable logic with successor on data
TRANSCRIPT
![Page 1: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/1.jpg)
Feasible Automata for Two-Variable Logic
with Successor on Data Words
Ahmet Kara, Thomas Schwentick, and Tony Tan
LATA 2012, A Coruña
![Page 2: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/2.jpg)
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
![Page 3: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/3.jpg)
Systems with Unboundedly Many Processes
Server
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 2
![Page 4: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/4.jpg)
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
![Page 5: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/5.jpg)
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
![Page 6: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/6.jpg)
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
![Page 7: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/7.jpg)
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
![Page 8: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/8.jpg)
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
![Page 9: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/9.jpg)
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
![Page 10: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/10.jpg)
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
![Page 11: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/11.jpg)
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
![Page 12: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/12.jpg)
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
![Page 13: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/13.jpg)
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
![Page 14: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/14.jpg)
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
![Page 15: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/15.jpg)
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
![Page 16: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/16.jpg)
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
![Page 17: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/17.jpg)
Known Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 5
![Page 18: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/18.jpg)
Known Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
is captured
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 5
![Page 19: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/19.jpg)
Known Results
DECIDABLE
EMSO2(∼, Suc, ≤)
Data Automata
is captured ?
Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 5
![Page 20: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/20.jpg)
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
![Page 21: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/21.jpg)
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
![Page 22: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/22.jpg)
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
![Page 23: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/23.jpg)
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
![Page 24: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/24.jpg)
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
![Page 25: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/25.jpg)
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
![Page 26: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/26.jpg)
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
![Page 27: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/27.jpg)
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
![Page 28: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/28.jpg)
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
![Page 29: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/29.jpg)
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
![Page 30: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/30.jpg)
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
![Page 31: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/31.jpg)
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
![Page 32: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/32.jpg)
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
![Page 33: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/33.jpg)
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
![Page 34: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/34.jpg)
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
![Page 35: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/35.jpg)
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
![Page 36: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/36.jpg)
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
![Page 37: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/37.jpg)
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
![Page 38: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/38.jpg)
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
![Page 39: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/39.jpg)
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
![Page 40: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/40.jpg)
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
![Page 41: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/41.jpg)
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
![Page 42: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/42.jpg)
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
![Page 43: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/43.jpg)
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
![Page 44: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/44.jpg)
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
![Page 45: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/45.jpg)
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
![Page 46: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/46.jpg)
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
![Page 47: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/47.jpg)
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
![Page 48: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/48.jpg)
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
![Page 49: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/49.jpg)
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
![Page 50: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/50.jpg)
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
![Page 51: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/51.jpg)
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
![Page 52: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/52.jpg)
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
![Page 53: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/53.jpg)
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
![Page 54: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/54.jpg)
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
![Page 55: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/55.jpg)
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
![Page 56: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/56.jpg)
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
![Page 57: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/57.jpg)
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
![Page 58: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/58.jpg)
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
![Page 59: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/59.jpg)
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
![Page 60: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/60.jpg)
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
![Page 61: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/61.jpg)
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
![Page 62: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/62.jpg)
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
![Page 63: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/63.jpg)
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
![Page 64: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/64.jpg)
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
![Page 65: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/65.jpg)
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
![Page 66: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/66.jpg)
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
![Page 67: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/67.jpg)
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
![Page 68: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/68.jpg)
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
![Page 69: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/69.jpg)
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
![Page 70: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/70.jpg)
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
![Page 71: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/71.jpg)
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
![Page 72: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/72.jpg)
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
![Page 73: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/73.jpg)
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
![Page 74: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/74.jpg)
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
![Page 75: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/75.jpg)
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
![Page 76: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/76.jpg)
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
![Page 77: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/77.jpg)
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
![Page 78: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/78.jpg)
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
![Page 79: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/79.jpg)
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
![Page 80: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/80.jpg)
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
![Page 81: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/81.jpg)
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
![Page 82: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/82.jpg)
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
![Page 83: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/83.jpg)
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
![Page 84: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/84.jpg)
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
![Page 85: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/85.jpg)
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
![Page 86: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/86.jpg)
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
![Page 87: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/87.jpg)
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
![Page 88: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/88.jpg)
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
![Page 89: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/89.jpg)
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
![Page 90: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/90.jpg)
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
![Page 91: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/91.jpg)
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
![Page 92: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/92.jpg)
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
![Page 93: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/93.jpg)
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
![Page 94: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/94.jpg)
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
![Page 95: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/95.jpg)
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
![Page 96: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/96.jpg)
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
![Page 97: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/97.jpg)
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
![Page 98: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/98.jpg)
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
![Page 99: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/99.jpg)
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
![Page 100: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/100.jpg)
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
![Page 101: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/101.jpg)
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
![Page 102: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/102.jpg)
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
![Page 103: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/103.jpg)
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
![Page 104: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/104.jpg)
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
![Page 105: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/105.jpg)
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
![Page 106: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/106.jpg)
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
![Page 107: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/107.jpg)
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
![Page 108: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/108.jpg)
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
![Page 109: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/109.jpg)
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
![Page 110: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/110.jpg)
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
![Page 111: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/111.jpg)
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
![Page 112: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/112.jpg)
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
![Page 113: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/113.jpg)
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
![Page 114: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/114.jpg)
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
![Page 115: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/115.jpg)
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
![Page 116: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/116.jpg)
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
![Page 117: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/117.jpg)
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
![Page 118: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/118.jpg)
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
![Page 119: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/119.jpg)
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
![Page 120: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/120.jpg)
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
![Page 121: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/121.jpg)
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
![Page 122: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/122.jpg)
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
![Page 123: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/123.jpg)
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
![Page 124: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/124.jpg)
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
![Page 125: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/125.jpg)
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
![Page 126: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/126.jpg)
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
![Page 127: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/127.jpg)
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
![Page 128: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/128.jpg)
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
![Page 129: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/129.jpg)
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
![Page 130: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/130.jpg)
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
![Page 131: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/131.jpg)
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
![Page 132: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/132.jpg)
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
![Page 133: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/133.jpg)
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
![Page 134: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/134.jpg)
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
![Page 135: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/135.jpg)
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
![Page 136: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/136.jpg)
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
![Page 137: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/137.jpg)
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
![Page 138: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/138.jpg)
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
![Page 139: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/139.jpg)
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
![Page 140: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/140.jpg)
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
![Page 141: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/141.jpg)
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
![Page 142: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/142.jpg)
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
![Page 143: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/143.jpg)
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
![Page 144: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/144.jpg)
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
![Page 145: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/145.jpg)
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
![Page 146: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/146.jpg)
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
![Page 147: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/147.jpg)
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
![Page 148: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/148.jpg)
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
![Page 149: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/149.jpg)
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
![Page 150: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/150.jpg)
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
![Page 151: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/151.jpg)
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
![Page 152: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/152.jpg)
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
![Page 153: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/153.jpg)
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
![Page 154: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/154.jpg)
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
![Page 155: Feasible Automata for Two-Variable Logic with Successor on Data](https://reader031.vdocuments.us/reader031/viewer/2022030322/5875ed911a28ab135e8b5297/html5/thumbnails/155.jpg)
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