Feasible Automata for Two-Variable Logic with Successor on Data ...

Feasible Automata for Two-Variable Logic

with Successor on Data Words

Ahmet Kara, Thomas Schwentick, and Tony Tan

LATA 2012, A Coruña

Systems with Unboundedly Many Processes

Server

process 1

process 2 process 3

process 4

Ahmet Kara Feasible Automata for Two-Variable Logic with Successor on Data Words � � Slide 1


Server



Server

process 1

• A system run

req

1



Server

process 1

process 2

• A system run

req

1

req

2



Server

process 1

process 2

• A system run

req

1

req

2

ack

1



Server

process 1

process 2 process 3

• A system run

req

1

req

2

ack

1

req

3



Server

process 1

process 2 process 3

process 4

• A system run

req

1

req

2

ack

1

req

3

req

4



Server

process 1

process 2 process 3

process 4

• A system run

req

1

req

2

ack

1

req

3

req

4

ack

3



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



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. . .



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))


Words and Data Words

A Word over Σ = {a, b, c}

c c a c a b c b




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




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


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]


Known Results

DECIDABLE

EMSO2(∼, Suc, ≤)


Known Results

DECIDABLE


Data Automata

is captured


Known Results

DECIDABLE


Data Automata

is captured ?


Known Results

DECIDABLE


Data Automata

2NEXPTIME

EMSO2(∼, Suc)

is captured ?


Known Results

DECIDABLE


Data Automata

2NEXPTIME

EMSO2(∼, Suc)

is captured ?

?


Known Results

DECIDABLE


Data Automata

2NEXPTIME

EMSO2(∼, Suc)

is captured ?

?

?


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










Example


c

1

c

4

a

3

c

2

a

2

b

3

c

7

b

2

x

∀x










Example


c

1

c

4

a

3

c

2

a

2

b

3

c

7

b

2

x y

∀x∃y










Example


c

1

c

4

a

3

c

2

a

2

b

3

c

7

b

2

x y

∀x∃y(

a(x)










Example


c

1

c

4

a

3

c

2

a

2

b

3

c

7

b

2

x y

∀x∃y(

a(x) →(x < y










Example


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)










Example


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


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










Example


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


c

1

c

4

a

3

c

2

a

2

b

3

c

7

b

2

x

∃x










Example


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


c

1

c

4

a

3

c

2

a

2

b

3

c

7

b

2

x y

∃x∃y










Example


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


c

1

c

4

a

3

c

2

a

2

b

3

c

7

b

2

x y

∃x∃y(

Suc(x, y)










Example


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


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))


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


Data Automata





σ1

1

σ2

1

σ3

3

σ4

2

σ5

2

σ6

3

σ7

1

σ8

2

(B, C)


Data Automata





σ1

1

σ2

1

σ3

3

σ4

2

σ5

2

σ6

3

σ7

1

σ8

2

(B, C) B


Data Automata





γ1

1

σ2

1

σ3

3

σ4

2

σ5

2

σ6

3

σ7

1

σ8

2

(B, C) B


Data Automata





γ1

1

γ2

1

σ3

3

σ4

2

σ5

2

σ6

3

σ7

1

σ8

2

(B, C) B


Data Automata





γ1

1

γ2

1

γ3

3

σ4

2

σ5

2

σ6

3

σ7

1

σ8

2

(B, C) B


Data Automata





γ1

1

γ2

1

γ3

3

γ4

2

σ5

2

σ6

3

σ7

1

σ8

2

(B, C) B


Data Automata





γ1

1

γ2

1

γ3

3

γ4

2

γ5

2

σ6

3

σ7

1

σ8

2

(B, C) B


Data Automata





γ1

1

γ2

1

γ3

3

γ4

2

γ5

2

γ6

3

σ7

1

σ8

2

(B, C) B


Data Automata





γ1

1

γ2

1

γ3

3

γ4

2

γ5

2

γ6

3

γ7

1

σ8

2

(B, C) B


Data Automata





γ1

1

γ2

1

γ3

3

γ4

2

γ5

2

γ6

3

γ7

1

γ8

2

(B, C)


Data Automata





γ1

1

γ2

1

γ3

3

γ4

2

γ5

2

γ6

3

γ7

1

γ8

2

(B, C) C


Data Automata





γ1

1

γ2

1

γ3

3

γ4

2

γ5

2

γ6

3

γ7

1

γ8

2

(B, C)


Data Automata





γ1

1

γ2

1

γ3

3

γ4

2

γ5

2

γ6

3

γ7

1

γ8

2

(B, C) C


Data Automata





γ1

1

γ2

1

γ3

3

γ4

2

γ5

2

γ6

3

γ7

1

γ8

2

(B, C)


Data Automata





γ1

1

γ2

1

γ3

3

γ4

2

γ5

2

γ6

3

γ7

1

γ8

2

(B, C) C


Data Automata





γ1

1

γ2

1

γ3

3

γ4

2

γ5

2

γ6

3

γ7

1

γ8

2

(B, C)


Data Automata





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)


Data Automata





Example


c

1

c

1

a

3

c

2

a

2

b

3

c

1

b

2

(B, C) B


Data Automata





Example


−

1

c

1

a

3

c

2

a

2

b

3

c

1

b

2

(B, C) B


Data Automata





Example


−

1

−

1

a

3

c

2

a

2

b

3

c

1

b

2

(B, C) B


Data Automata





Example


−

1

−

1

X

3

c

2

a

2

b

3

c

1

b

2

(B, C) B


Data Automata





Example


−

1

−

1

X

3

−

2

a

2

b

3

c

1

b

2

(B, C) B


Data Automata





Example


−

1

−

1

X

3

−

2

−

2

b

3

c

1

b

2

(B, C) B


Data Automata





Example


−

1

−

1

X

3

−

2

−

2

−

3

c

1

b

2

(B, C) B


Data Automata





Example


−

1

−

1

X

3

−

2

−

2

−

3

−

1

b

2

(B, C) B


Data Automata





Example


−

1

−

1

X

3

−

2

−

2

−

3

−

1

X

2

(B, C)


Data Automata





Example


−

1

−

1

X

3

−

2

−

2

−

3

−

1

X

2

(B, C) C


Data Automata





Example


−

1

−

1

X

3

−

2

−

2

−

3

−

1

X

2

(B, C)


Data Automata





Example


−

1

−

1

X

3

−

2

−

2

−

3

−

1

X

2

(B, C) C


Data Automata





Example


−

1

−

1

X

3

−

2

−

2

−

3

−

1

X

2

(B, C)


Data Automata





Example


−

1

−

1

X

3

−

2

−

2

−

3

−

1

X

2

(B, C) C


Data Automata





Example


−

1

−

1

X

3

−

2

−

2

−

3

−

1

X

2

(B, C)


Data Automata





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)


Data Automata





Example


c

1

c

1

a

3

c

2

a

2

b

3

c

1

b

2

(B, C) B


Data Automata





Example


c

1

c

1

a

3

c

2

a

2

b

3

c

1

b

2

(B, C)


Data Automata





Example


c

1

c

1

a

3

c

2

a

2

b

3

c

1

b

2

(B, C) C


Data Automata





Example


c

1

c

1

a

3

c

2

a

2

b

3

c

1

b

2

(B, C)


Data Automata





Example


c

1

c

1

a

3

c

2

a

2

b

3

c

1

b

2

(B, C) C


Data Automata





Example


c

1

c

1

a

3

c

2

a

2

b

3

c

1

b

2

(B, C)


Data Automata





Example


c

1

c

1

a

3

c

2

a

2

b

3

c

1

b

2

(B, C) C


Data Automata





Example


c

1

c

1

a

3

c

2

a

2

b

3

c

1

b

2

(B, C)


Known Results

DECIDABLE


Data Automata

2NEXPTIME

EMSO2(∼, Suc)

is captured ?

?

?


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

Data Automata

Base

Automaton

B

Class

Automaton

C

Definition




Weak Data Automata

Base

Automaton

B

Set of

Constraints

C


Weak Data Automata

Data Automata

Base

Automaton

B

Class

Automaton

C

Definition




Weak Data Automata

Base

Automaton

B

Set of

Constraints

C

Definition

A weak data automaton D = (B, C) over analphabet Σ consists of


• a set C of constraints over Γ


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





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)





• 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)






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)






• 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)






• 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) = ∅


Weak Data Automata

Weak Data Automata (WDA)

A weak data automaton D = (B, C) over an alphabet Σ consists of




Weak Data Automata





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)


Weak Data Automata





Example


c

1

c

1

b

3

c

2

a

2

a

3

c

1

b

2

(B, C) B


Weak Data Automata





Example


−

1

c

1

b

3

c

2

a

2

a

3

c

1

b

2

(B, C) B


Weak Data Automata





Example


−

1

−

1

b

3

c

2

a

2

a

3

c

1

b

2

(B, C) B


Weak Data Automata





Example


−

1

−

1

b

3

−

2

a

2

a

3

c

1

b

2

(B, C) B


Weak Data Automata





Example


−

1

−

1

b

3

−

2

a

2

−

3

c

1

b

2

(B, C) B


Weak Data Automata





Example


−

1

−

1

b

3

−

2

a

2

−

3

−

1

b

2

(B, C) B


Weak Data Automata





Example


−

1

−

1

b

3

−

2

a

2

−

3

−

1

−

2

(B, C)


Weak Data Automata





Example


−

1

−

1

b

3

−

2

a

2

−

3

−

1

−

2

(B, C)

• C = {V (a) ∩ V (b) = ∅}


Weak Data Automata





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)


Weak Data Automata





Example


c

1

c

1

b

3

c

2

a

2

a

3

c

1

b

2

(B, C) B


Weak Data Automata





Example


c

1

c

1

b

3

c

2

a

2

a

3

c

1

b

2

(B, C)


Weak Data Automata





Example


c

1

c

1

b

3

c

2

a

2

a

3

c

1

b

2

(B, C)

• C = {V (a) ⊆ V (b)}


Weak Data Automata





Example


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


Our Results

DECIDABLE


Data Automata

2NEXPTIME

EMSO2(∼, Suc)

is captured ?

?

?


Our Results

DECIDABLE


Data Automata

2NEXPTIME

EMSO2(∼, Suc)

is captured ?

?

Weak Data Automata

is captured:

translatable

in 2EXPTIME


Our Results

DECIDABLE


Data Automata

2NEXPTIME

EMSO2(∼, Suc)

is captured ?

?

Weak Data Automata

is captured:

translatable

in 2EXPTIME

<


Our Results

DECIDABLE


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.”


Our Results

DECIDABLE


Data Automata

2NEXPTIME

EMSO2(∼, Suc)

is captured ?

<

Weak Data Automata

is captured:

translatable

in 2EXPTIME

<


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-words

• Let




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



Data ω-words

• Let









Weak Data ω-Automata

A weak data ω-automaton D = (B, C) overan alphabet Σ consists of


• a set C of constraints



Data ω-words

• Let










A weak data ω-automaton D = (B, C) overan alphabet Σ consists of


• a set C of constraints

E∞MSO2(∼, Suc)

Extension of EMSO2(∼, Suc):

• quantified sets can be restricted to bebound to infinite sets only


Our Results on Data ω-Words

DECIDABLE


Data ω-Automata

2NEXPTIME

E∞MSO2(∼, Suc)

is captured ?

<


is captured:

translatable

in 2EXPTIME

<


Conclusion

• WDA: automata model for EMSO2(∼, Suc) (and

E∞MSO2(∼, Suc)) with reasonable complexity


Conclusion



• Open Questions:

◮ Are WDA the best automata model for these logics?


Conclusion



• 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]


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