Fundamentals of Software Engineering
Formal Modeling with Linear Temporal Logic
Ina Schaefer
Institute for Software Systems Engineering
TU Braunschweig, Germany
Slides by Wolfgang Ahrendt, Richard Bubel, Reiner Hähnle
(Chalmers University of Technology, Gothenburg, Sweden)
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Syntax of Propositional Logic
Signature
A set of Propositional Variables P
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(with typical elements p, q, r , . . .)
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Syntax of Propositional Logic
Signature
A set of Propositional Variables P
(with typical elements p, q, r , . . .)
Propositional Connectives
true, false, ∧, ∨, ¬, →, ↔
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Syntax of Propositional Logic
Signature
A set of Propositional Variables P
(with typical elements p, q, r , . . .)
Propositional Connectives
true, false, ∧, ∨, ¬, →, ↔
Set of Propositional Formulas For0
• Truth constants true, false and variables P are formulas
• If φ and ψ are formulas then
¬φ,
(φ ∧ ψ),
(φ ∨ ψ),
(φ → ψ), (φ ↔ ψ)
are also formulas
• There are no other formulas (inductive definition)
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Remark on Concrete Syntax
Negation
Conjunction
Disjunction
Implication
Equivalence
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¬
∧
∨
→, ⊃
↔
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Spin
!
&&
||
−>
<−>
5
Remark on Concrete Syntax
Negation
Conjunction
Disjunction
Implication
Equivalence
Text book
¬
∧
∨
→, ⊃
↔
Spin
!
&&
||
−>
<−>
We use mostly the textbook notation.
Except for tool specific slides.
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Examples
Let P = {p, q, r } be the set of propositional variables.
Are the following character sequences also propositional formulas?
• (true → p)
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Examples
Let P = {p, q, r } be the set of propositional variables.
Are the following character sequences also propositional formulas?
• (true → p)
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Examples
Let P = {p, q, r } be the set of propositional variables.
Are the following character sequences also propositional formulas?
• (true → p)
4
• ((p(q ∧ r )) ∨ p)
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Examples
Let P = {p, q, r } be the set of propositional variables.
Are the following character sequences also propositional formulas?
• (true → p)
4
• ((p(q ∧ r )) ∨ p)
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Examples
Let P = {p, q, r } be the set of propositional variables.
Are the following character sequences also propositional formulas?
• (true → p)
4
• ((p(q ∧ r )) ∨ p)
8
• (p → (q∧))
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Examples
Let P = {p, q, r } be the set of propositional variables.
Are the following character sequences also propositional formulas?
• (true → p)
4
• ((p(q ∧ r )) ∨ p)
• (p → (q∧))
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8
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Examples
Let P = {p, q, r } be the set of propositional variables.
Are the following character sequences also propositional formulas?
• (true → p)
4
• ((p(q ∧ r )) ∨ p)
• (p → (q∧))
8
8
• (false ∧ (p → (q ∧ r )))
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Examples
Let P = {p, q, r } be the set of propositional variables.
Are the following character sequences also propositional formulas?
• (true → p)
4
• ((p(q ∧ r )) ∨ p)
• (p → (q∧))
8
8
• (false ∧ (p → (q ∧ r )))
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Semantics of Propositional Logic
Interpretation I
Assigns a truth value to each propositional variable
I : P → {T , F }
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Semantics of Propositional Logic
Interpretation I
Assigns a truth value to each propositional variable
I : P → {T , F }
Example
Let P = {p, q}
(p → (q → p))
I1
I2
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p q
F F
T F
...
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Semantics of Propositional Logic
Interpretation I
Assigns a truth value to each propositional variable
I : P → {T , F }
Example
Let P = {p, q}
(p → (q → p))
I1
I2
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p q
F F
T F
...
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Semantics of Propositional Logic
Interpretation I
Assigns a truth value to each propositional variable
I : P → {T , F }
Valuation function
valI : Continuation of I on For0
valI : For0 → {T , F }
valI (pi ) = I(pi )
valI (true) = T
valI (false) = F
(cont’d next page)
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Semantics of Propositional Logic (Cont’d)
Valuation function (Cont’d)
valI (¬φ) =
T
F
valI (φ ∧ ψ) =
valI (φ ∨ ψ) =
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T
F
if valI (φ) = T and valI (ψ) = T
otherwise
T
F
if valI (φ) = T or valI (ψ) = T
otherwise
T
F
if valI (φ) = F or valI (ψ) = T
otherwise
T
F
if valI (φ) = valI (ψ)
otherwise
valI (φ → ψ) =
valI (φ ↔ ψ) =
if valI (φ) = F
otherwise
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Examples - Cont’d
Example
Let P = {p, q}
(p → (q → p))
I1
I2
p q
F F
T F
...
How to evaluate (p → (q → p)) in I2 ?
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Examples - Cont’d
Example
Let P = {p, q}
(p → (q → p))
I1
I2
p q
F F
T F
...
How to evaluate (p → (q → p)) in I2 ?
Valuation in I2
valI2 ( q → p )
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=
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Examples - Cont’d
Example
Let P = {p, q}
(p → (q → p))
I1
I2
p q
F F
T F
...
How to evaluate (p → (q → p)) in I2 ?
Valuation in I2
valI2 ( q → p )
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=
T
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Examples - Cont’d
Example
Let P = {p, q}
(p → (q → p))
I1
I2
p q
F F
T F
...
How to evaluate (p → (q → p)) in I2 ?
Valuation in I2
valI2 ( q → p ) = T
valI2 ( p → (q → p) ) =
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Examples - Cont’d
Example
Let P = {p, q}
(p → (q → p))
I1
I2
p q
F F
T F
...
How to evaluate (p → (q → p)) in I2 ?
Valuation in I2
valI2 ( q → p ) = T
valI2 ( p → (q → p) ) =
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T
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Semantic Notions of Propositional Logic
Let φ ∈ For0 , Γ ⊂ For0
Definition (Satisfying Interpretation, Consequence Relation)
I satisfies φ (write: I |= φ) iff valI (φ) = T
φ follows from Γ (write: Γ |= φ) iff for all interpretations I:
If I |= ψ for all ψ ∈ Γ then also I |= φ
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Semantic Notions of Propositional Logic
Let φ ∈ For0 , Γ ⊂ For0
Definition (Satisfying Interpretation, Consequence Relation)
I satisfies φ (write: I |= φ) iff valI (φ) = T
φ follows from Γ (write: Γ |= φ) iff for all interpretations I:
If I |= ψ for all ψ ∈ Γ then also I |= φ
Definition (Satisfiability, Validity)
A formula is satisfiable if it is valid in some interpretation.
If every interpretation satisfies φ (write: |= φ) then φ is called valid.
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Examples
Formula (same as before)
p → (q → p)
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Examples
Formula (same as before)
p → (q → p)
Is this formula valid?
|= p → (q → p) ?
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Examples
p ∧ ((¬p) ∨ q)
Satisfiable?
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Examples
p ∧ ((¬p) ∨ q)
Satisfiable?
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Examples
p ∧ ((¬p) ∨ q)
Satisfiable?
Satisfying Interpretation?
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Examples
p ∧ ((¬p) ∨ q)
Satisfiable?
Satisfying Interpretation?
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I(p) = T , I(q) = T
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Examples
p ∧ ((¬p) ∨ q)
Satisfiable?
4
Satisfying Interpretation?
I(p) = T , I(q) = T
Other Satisfying Interpretations?
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Examples
p ∧ ((¬p) ∨ q)
Satisfiable?
4
Satisfying Interpretation?
I(p) = T , I(q) = T
Other Satisfying Interpretations? 8
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Examples
p ∧ ((¬p) ∨ q)
Satisfiable?
4
Satisfying Interpretation?
I(p) = T , I(q) = T
Other Satisfying Interpretations? 8
Therefore, also not valid!
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Examples
p ∧ ((¬p) ∨ q)
Satisfiable?
4
Satisfying Interpretation?
I(p) = T , I(q) = T
Other Satisfying Interpretations? 8
Therefore, also not valid!
p ∧ ((¬p) ∨ q) |= q ∨ r
Does it hold?
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Examples
p ∧ ((¬p) ∨ q)
Satisfiable?
4
Satisfying Interpretation?
I(p) = T , I(q) = T
Other Satisfying Interpretations? 8
Therefore, also not valid!
p ∧ ((¬p) ∨ q) |= q ∨ r
Does it hold?
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Yes.
Why?
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Formalisation
byte n ;
a c t i v e proctype [2] p () {
n = 0;
n = n + 1
5 }
1
2
3
4
How can we model the above PROMELA program P?
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Formalisation
byte n ;
a c t i v e proctype [2] p () {
n = 0;
n = n + 1
5 }
1
2
3
4
How can we model the above PROMELA program P?
A model of P is a propositional formula φP , which is true if and only if
(iff) it describes a possible state of P.
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Formalisation
byte n ;
a c t i v e proctype [2] p () {
n = 0;
n = n + 1
5 }
1
2
3
4
P : N0, N1, N2, . . . , N255
PC 0 3, PC 0 4, PC 0 5, PC 1 3, PC 1 4, PC 1 5
Which interpretations do we need to ’exclude’ ?
φP :=


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Formalisation
byte n ;
a c t i v e proctype [2] p () {
n = 0;
n = n + 1
5 }
1
2
3
4
P : N0, N1, N2, . . . , N255
PC 0 3, PC 0 4, PC 0 5, PC 1 3, PC 1 4, PC 1 5
Which interpretations do we need to ’exclude’ ?
• The variable n has exactly one value at one time.
φP :=

((N0 ∧ ¬N1 ∧ . . . ∧ ¬N255) ∨ . . . ∨ (¬N0 ∧ . . . ∧ ¬N254 ∧ N255))

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Formalisation
byte n ;
a c t i v e proctype [2] p () {
n = 0;
n = n + 1
5 }
1
2
3
4
P : N0, N1, N2, . . . , N255
PC 0 3, PC 0 4, PC 0 5, PC 1 3, PC 1 4, PC 1 5
Which interpretations do we need to ’exclude’ ?
• The variable n has exactly one value at one time.
• A process cannot be at two positions at the same time.
φP :=

((N0 ∧ ¬N1 ∧ . . . ∧ ¬N255) ∨ . . . ∨ (¬N0 ∧ . . . ∧ ¬N254 ∧ N255))
 ∧ ((PC 0 3 ∧ ¬PC 0 4 ∧ ¬PC 0 5) ∨ . . .)
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Formalisation
byte n ;
a c t i v e proctype [2] p () {
n = 0;
n = n + 1
5 }
1
2
3
4
P : N0, N1, N2, . . . , N255
PC 0 3, PC 0 4, PC 0 5, PC 1 3, PC 1 4, PC 1 5
Which interpretations do we need to ’exclude’ ?
•
•
•
•
The variable n has exactly one value at one time.
A process cannot be at two positions at the same time.
If neither process 0 nor process 1 are at position 5, then n is zero.
...
φP :=


((N0 ∧ ¬N1 ∧ . . . ∧ ¬N255) ∨ . . . ∨ (¬N0 ∧ . . . ∧ ¬N254 ∧ N255))
 ∧ ((PC 0 3 ∧ ¬PC 0 4 ∧ ¬PC 0 5) ∨ . . .)

...
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Transition systems (aka. Kripke Structures)
p=
T
sa
x
p=T ;
sb
FF
TF
q=p;
F;
q=
sc
TT
p=F ;
;
sd
FT
Notation
name
interp.
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update
x
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Transition systems (aka. Kripke Structures)
p=
T
sa
x
FF
p=T ;
sb
TF
q=p;
F;
q=
sc
p=F ;
TT
;
sd
FT
• Each state has its own propositional interpretation!
• Computations, or runs, are infinite paths through states
• Infinitely many different runs
• How to express (for example) that either p or q changes its value
infinitely often in each run?
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Temporal Logic
An extension of propositional logic that allows to
specify properties of sets of runs
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Temporal Logic— Syntax
An extension of propositional logic that allows to
specify properties of sets of runs
Syntax
Based on propositional signature and syntax.
Extension with three connectives:
Always If φ is a formula then so is φ
Sometimes If φ is a formula then so is ♦φ
Until If φ and ψ are formulas then so is φ Uψ
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Temporal Logic — Syntax
Concrete Syntax
Always
Sometimes
Until
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♦
U
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[]
<>
U
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Semantics of Temporal Logic
A run σ is an infinite chain of states
s0
s1
s2
s3
s4
I0
I1
I2
I3
I4
···
Ij propositional interpretation of variables in j-th state
Write more compactly s0 s1 s2 s3 . . .
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Semantics of Temporal Logic
A run σ is an infinite chain of states
s0
s1
s2
s3
s4
I0
I1
I2
I3
I4
···
Ij propositional interpretation of variables in j-th state
Write more compactly s0 s1 s2 s3 . . .
If σ = s0 s1 . . ., then σ|i denotes the suffix si si+1 . . . of σ.
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Semantics of Temporal Logic (Cont’d)
Definition (Validity Relation)
Validity of temporal formula depends on runs σ = s0 s1 . . . for which the
formula may, or may not, hold:
σ |= p
iff I0 (p) = T , for p ∈ P.
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Semantics of Temporal Logic (Cont’d)
Definition (Validity Relation)
Validity of temporal formula depends on runs σ = s0 s1 . . . for which the
formula may, or may not, hold:
σ |= p
iff I0 (p) = T , for p ∈ P.
σ |= ¬φ
iff not σ |= φ (write σ 6|= φ)
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Semantics of Temporal Logic (Cont’d)
Definition (Validity Relation)
Validity of temporal formula depends on runs σ = s0 s1 . . . for which the
formula may, or may not, hold:
σ |= p
iff I0 (p) = T , for p ∈ P.
σ |= ¬φ
iff not σ |= φ (write σ 6|= φ)
σ |= φ ∧ ψ iff σ |= φ and σ |= ψ
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Semantics of Temporal Logic (Cont’d)
Definition (Validity Relation)
Validity of temporal formula depends on runs σ = s0 s1 . . . for which the
formula may, or may not, hold:
σ |= p
iff I0 (p) = T , for p ∈ P.
σ |= ¬φ
iff not σ |= φ (write σ 6|= φ)
σ |= φ ∧ ψ iff σ |= φ and σ |= ψ
σ |= φ ∨ ψ iff σ |= φ or σ |= ψ
σ |= φ → ψ iff σ 6|= φ or σ |= ψ
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Semantics of Temporal Logic Cont’d
s0
s1
···
sk−1
sk
···
Definition (Validity Relation for Temporal Connectives)
Given a run σ = s0 s1 . . .
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Semantics of Temporal Logic Cont’d
s0
φ
s1
φ
···
sk−1
···
φ
sk
φ
···
···
Definition (Validity Relation for Temporal Connectives)
Given a run σ = s0 s1 . . .
σ |= φ
iff σ|k |= φ for all k ≥ 0
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Semantics of Temporal Logic Cont’d
s0
s1
···
sk−1
sk
···
φ
Definition (Validity Relation for Temporal Connectives)
Given a run σ = s0 s1 . . .
σ |= φ
iff σ|k |= φ for all k ≥ 0
σ |= ♦φ
iff σ|k |= φ for some k ≥ 0
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Semantics of Temporal Logic Cont’d
s0
φ
s1
φ
···
sk−1
···
φ
sk
···
ψ
Definition (Validity Relation for Temporal Connectives)
Given a run
σ |= φ
σ |= ♦φ
σ |= φ Uψ
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σ = s0 s1 . . .
iff σ|k |= φ for all k ≥ 0
iff σ|k |= φ for some k ≥ 0
iff σ|k |= ψ for some k ≥ 0, and σ|j |= φ for all 0≤j<k
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Safety and Liveness Properties
Safety Properties
Always-formulas called safety property: something bad never happens
Let mutex be variable that is true when two process do not access a
critical resource at the same time
mutex expresses that simultaneous access never happens
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Safety and Liveness Properties
Safety Properties
Always-formulas called safety property: something bad never happens
Let mutex be variable that is true when two process do not access a
critical resource at the same time
mutex expresses that simultaneous access never happens
Liveness Properties
Sometimes-formulas called liveness property: something good happens
eventually
Let s be variable that is true when a process delivers a service
♦ s expresses that service is eventually provided
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Complex Properties
What does this mean?
♦φ
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Complex Properties
Infinitely Often
♦φ
During a run the formulas φ will become true infinitely often.
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Validity Temporal Logic
Definition (Validity)
φ is valid, write |= φ, iff φ is valid in all runs σ = s0 s1 . . ..
Recall that each run s0 s1 . . . essentially is an infinite sequence of
interpretations I0 I1 . . ..
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Examples
♦φ
Valid?
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Examples
♦φ
Valid?
No, there is a run in where it is not valid:
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Examples
♦φ
Valid?
No, there is a run in where it is not valid:
(¬φ, ¬φ, ¬φ, . . .)
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Examples
♦φ
Valid?
No, there is a run in where it is not valid:
(¬φ, ¬φ, ¬φ, . . .)
Valid in some run?
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Examples
♦φ
Valid?
No, there is a run in where it is not valid:
(¬φ, ¬φ, ¬φ, . . .)
Valid in some run?
Yes: (φ, φ, φ, . . .)
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Examples
♦φ
Valid?
No, there is a run in where it is not valid:
(¬φ, ¬φ, ¬φ, . . .)
Valid in some run?
Yes: (φ, φ, φ, . . .)
φ → φ
(¬φ) ↔ (♦¬φ)
Both are valid!
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Examples
♦φ
Valid?
No, there is a run in where it is not valid:
(¬φ, ¬φ, ¬φ, . . .)
Valid in some run?
Yes: (φ, φ, φ, . . .)
φ → φ
(¬φ) ↔ (♦¬φ)
Both are valid!
• is reflexive
• and ♦ are dual connectives
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Transition Systems Revisited
Definition (Transition System)
A Transition System T = (S, Ini, δ, I) is given by a set of states S, a
non-empty subset Ini ⊆ S of initial states, and a transition relation
δ ⊆ S × S, and I labeling each state s ∈ S with a propositional
interpretation Is .
Definition (Runs of Transition System)
A run of T is a is a run σ = s0 s1 . . ., with si ∈ S, such that s0 ∈ Ini and
(si , si+1 ) ∈ δ for all i.
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Semantics of Temporal Logic (Cont’d)
Validity of temporal formula is extended to transition systems in the
following way:
Definition (Validity Relation)
Given a transition systems T = (S, Ini, δ, I), a temporal formula φ is
valid in T (write T |= φ) iff σ |= φ for all runs σ of T .
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ω-Languages
Given a finite alphabet (vocabulary) Σ.
A word w ∈ Σ∗ is a finite sequence
w = ao · · · an
with ai ∈ Σ, i ∈ {0, . . . , n}.
L ⊆ Σ∗ is called a language.
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ω-Languages
Given a finite alphabet (vocabulary) Σ.
An ω word w ∈ Σω is an infinite sequence
w = ao · · · ak · · ·
with ai ∈ Σ, i ∈ N.
Lω ⊆ Σω is called an ω-language.
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Büchi Automaton
Definition (Büchi Automaton)
A (non-deterministic) Büchi automaton over an alphabet Σ consists of a
• finite, non-empty set of locations Q
• a non-empty set of initial/start locations I ⊆ Q
• a set of accepting locations F = {F1 , . . . , Fn } ⊆ Q
• a transition relation δ ⊆ Q × Σ × Q
Example
Σ = {a, b}, Q = {q1 , q2 , q3 }, I = {q1 }, F = {q2 }
a, b
b
start
q1
a
q2
q3
a
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Büchi Automaton – Acceptance
Definition (Run and Accepted Run)
An infinite word w = ao . . . ak . . . ∈ Σω is a run of a Büchi automaton if
qi+1 ∈ δ(qi , ai )
for some initial location q0 ∈ I .
A Büchi automaton accepts a run w ∈ Σω , if some accepting location
f ∈ F is infinitely often visited (i.e., ∃ infinite set R ⊆ N:
f ∈ δ(qi , ai ), ∀i ∈ R)
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Büchi Automaton – Acceptance
Definition (Run and Accepted Run)
An infinite word w = ao . . . ak . . . ∈ Σω is a run of a Büchi automaton if
qi+1 ∈ δ(qi , ai )
for some initial location q0 ∈ I .
A Büchi automaton accepts a run w ∈ Σω , if some accepting location
f ∈ F is infinitely often visited (i.e., ∃ infinite set R ⊆ N:
f ∈ δ(qi , ai ), ∀i ∈ R)
Let B be a Büchi automaton, then
Lω (B) = {w ∈ Σω |w ∈ Σω is an accepted run of B}
denotes the ω-language recognised by B.
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Büchi Automaton – Acceptance
Definition (Run and Accepted Run)
An infinite word w = ao . . . ak . . . ∈ Σω is a run of a Büchi automaton if
qi+1 ∈ δ(qi , ai )
for some initial location q0 ∈ I .
A Büchi automaton accepts a run w ∈ Σω , if some accepting location
f ∈ F is infinitely often visited (i.e., ∃ infinite set R ⊆ N:
f ∈ δ(qi , ai ), ∀i ∈ R)
Let B be a Büchi automaton, then
Lω (B) = {w ∈ Σω |w ∈ Σω is an accepted run of B}
denotes the ω-language recognised by B.
An ω-language for which an accepting Büchi automaton exists is called
ω-regular language.
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Examples
Which language is accepted by the following Büchi automaton?
a, b
b
start
q1
a
q2
q3
a
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Examples
Which language is accepted by the following Büchi automaton?
a, b
b
start
q1
a
q2
q3
a
Solution: (a + b)∗ (ab)ω
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80
Examples
Which language is accepted by the following Büchi automaton?
a, b
b
start
q1
a
q2
q3
a
Solution: (a + b)∗ (ab)ω
ω-regular expressions like standard regular expression
ab a then b
a + b a or b
a∗ arbitrarily, but finitely often a
new: aω infinitely often a
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Decidability, Closure Properties
Many properties for regular finite automata hold also for Büchi automata.
Theorem (Decidability)
It is decidable whether the accepted language Lω (B) of a Büchi
automaton B is empty or not.
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Decidability, Closure Properties
Many properties for regular finite automata hold also for Büchi automata.
Theorem (Decidability)
It is decidable whether the accepted language Lω (B) of a Büchi
automaton B is empty or not.
Theorem (Closure properties)
The set of ω-regular languages is closed with respect to intersection,
union and complement, i.e.,
• L1 , L2 are ω-reg. then L1 ∩ L2 and L1 ∪ L2 are ω-reg.
• L is ω-reg. then Σω \V ω is ω-reg.
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Decidability, Closure Properties
Many properties for regular finite automata hold also for Büchi automata.
Theorem (Decidability)
It is decidable whether the accepted language Lω (B) of a Büchi
automaton B is empty or not.
Theorem (Closure properties)
The set of ω-regular languages is closed with respect to intersection,
union and complement, i.e.,
• L1 , L2 are ω-reg. then L1 ∩ L2 and L1 ∪ L2 are ω-reg.
• L is ω-reg. then Σω \V ω is ω-reg.
But in contrast to regular finite automata:
Non-deterministic Büchi automata are strictly more expressive than
deterministic ones.
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84
Examples
Language:
a
0
1
a
b
Ina Schaefer
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85
Examples
Language: a(a + ba)ω
a
0
1
a
b
Ina Schaefer
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86
Examples
Language: a(a + ba)ω
a
0
a
1
b
Language:
b
a
0
1
a
Ina Schaefer
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87
Examples
Language: a(a + ba)ω
a
0
a
1
b
Language: (a∗ ba)ω
b
a
0
1
a
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88
Linear Temporal Logic and Büchi Automata
LTL and Büchi Automata are connected.
Short reminder
Definition (Validity Relation)
Given a transition systems T = (S, Ini, δ, I), a temporal formula φ is
valid in T (write T |= φ) iff σ |= φ for all runs σ of T .
A run of the transition system is an infinite sequence of interpretations I .
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Linear Temporal Logic and Büchi Automata
LTL and Büchi Automata are connected.
Short reminder
Definition (Validity Relation)
Given a transition systems T = (S, Ini, δ, I), a temporal formula φ is
valid in T (write T |= φ) iff σ |= φ for all runs σ of T .
A run of the transition system is an infinite sequence of interpretations I .
Goal
Given an LTL formula φ.
Ina Schaefer
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90
Linear Temporal Logic and Büchi Automata
LTL and Büchi Automata are connected.
Short reminder
Definition (Validity Relation)
Given a transition systems T = (S, Ini, δ, I), a temporal formula φ is
valid in T (write T |= φ) iff σ |= φ for all runs σ of T .
A run of the transition system is an infinite sequence of interpretations I .
Goal
Given an LTL formula φ.
Construct a Büchi automaton accepting exactly those runs (infinite
sequences of interpretations) that satisfy φ .
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91
Encoding an LTL Formula as Büchi Automaton
P set of propositional variables, e.g. P = {r , s}
Alphabet Σ
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92
Encoding an LTL Formula as Büchi Automaton
P set of propositional variables, e.g. P = {r , s}
Alphabet Σ
Define Σ as set of all interpretations over P (i.e. Σ := 2P )
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Encoding an LTL Formula as Büchi Automaton
P set of propositional variables, e.g. P = {r , s}
Alphabet Σ
Define Σ as set of all interpretations over P (i.e. Σ := 2P )
E.g. Σ = ∅, {r }, {s}, {r , s}
Each element of Σ corresponds exactly to one propositional
interpretation over P, e.g.,
I∅ (r ) = F , I∅ (s) = F , I{r } (r ) = T , I{r } (s) = F , . . .
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Encoding an LTL Formula as Büchi Automaton
P set of propositional variables, e.g. P = {r , s}
Alphabet Σ
Define Σ as set of all interpretations over P (i.e. Σ := 2P )
E.g. Σ = ∅, {r }, {s}, {r , s}
Each element of Σ corresponds exactly to one propositional
interpretation over P, e.g.,
I∅ (r ) = F , I∅ (s) = F , I{r } (r ) = T , I{r } (s) = F , . . .
Using this alphabet we can now construct a Büchi automaton for a
formula φ over P
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95
Examples – Büchi Automaton for LTL Formulas
Example (Büchi automaton for formula r )
Give a Büchi automaton B accepting exactly those runs σ satisfying r
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Examples – Büchi Automaton for LTL Formulas
Example (Büchi automaton for formula r )
Give a Büchi automaton B accepting exactly those runs σ satisfying r
start
Ina Schaefer
{r },{r , s}
Σ
FSE
97
Examples – Büchi Automaton for LTL Formulas
Example (Büchi automaton for formula r )
Give a Büchi automaton B accepting exactly those runs σ satisfying r
start
{r },{r , s}
Σ
Example (Büchi automaton for formula r )
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98
Examples – Büchi Automaton for LTL Formulas
Example (Büchi automaton for formula r )
Give a Büchi automaton B accepting exactly those runs σ satisfying r
{r },{r , s}
start
Σ
Example (Büchi automaton for formula r )
{r },{r , s}
start
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99
Examples – Büchi Automaton for LTL Formulas
Example (Büchi automaton for formula r )
Give a Büchi automaton B accepting exactly those runs σ satisfying r
{r },{r , s}
start
Σ
Example (Büchi automaton for formula r )
start
R
(R := {I |I ∈ Σ, r ∈ I })
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100
Examples – Büchi Automaton for LTL Formulas
Example (Büchi automaton for formula ♦r )
start
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101
Examples – Büchi Automaton for LTL Formulas
Example (Büchi automaton for formula ♦r )
{r },{r , s}
start
{r },{r , s}
Σ
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102
Model Checking
How can we now check if a formula is valid in all runs of a transition
system?
Given a transition system T (e.g., derived from a Promela program).
Verification task: Is the LTL formula φ satisfied in all runs of T , i.e.,
T |= φ
Ina Schaefer
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?
103
Model Checking
How can we now check if a formula is valid in all runs of a transition
system?
Given a transition system T (e.g., derived from a Promela program).
Verification task: Is the LTL formula φ satisfied in all runs of T , i.e.,
T |= φ
?
Temporal model checking with SPIN: Topic of next lecture.
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104
Model Checking
How can we now check if a formula is valid in all runs of a transition
system?
Given a transition system T (e.g., derived from a Promela program).
Verification task: Is the LTL formula φ satisfied in all runs of T , i.e.,
T |= φ
?
Temporal model checking with SPIN: Topic of next lecture.
Today: Basic principle behind model checking only.
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105
Model Checking – Overview
T |= φ
?
1. Represent transition system T as Büchi automaton BT such that
BT accepts exactly those words corresponding to runs through T
Ina Schaefer
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106
Model Checking – Overview
T |= φ
?
1. Represent transition system T as Büchi automaton BT such that
BT accepts exactly those words corresponding to runs through T
2. Construct Büchi automaton B¬φ for negation of formula φ
Ina Schaefer
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107
Model Checking – Overview
T |= φ
?
1. Represent transition system T as Büchi automaton BT such that
BT accepts exactly those words corresponding to runs through T
2. Construct Büchi automaton B¬φ for negation of formula φ
3. If
Lω (BT ) ∩ Lω (B¬φ ) = ∅
then φ holds, otherwise we have a counterexample.
Ina Schaefer
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108
Model Checking – Overview
T |= φ
?
1. Represent transition system T as Büchi automaton BT such that
BT accepts exactly those words corresponding to runs through T
2. Construct Büchi automaton B¬φ for negation of formula φ
3. If
Lω (BT ) ∩ Lω (B¬φ ) = ∅
then φ holds, otherwise we have a counterexample.
To check Lω (BT ) ∩ Lω (B¬φ ) construct intersection automaton and
search for cycle through accepting state.
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109
Representing a model as Büchi Automaton
First Step: Represent transition system T as Büchi automaton BT
accepting exactly those words representing a run of T .
Example
active proctype P () {
do
:: atomic {
!wantQ;
wantP = true
}
pcs = true;
atomic {
pcs = false;
wantP = false
}
od }
(the very first location skipped and second atomic only to keep the automaton small; similar for Q)
Ina Schaefer
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110
Representing a model as Büchi Automaton
First Step: Represent transition system T as Büchi automaton BT
accepting exactly those words representing a run of T .
Example
active proctype P () {
do
:: atomic {
!wantQ;
wantP = true
}
pcs = true;
atomic {
pcs = false;
wantP = false
}
od }
start
0
{wp}
1
{wq}
∅
∅
2
{wp, pcs}
{wq, qcs}
3
4
(the very first location skipped and second atomic only to keep the automaton small; similar for Q)
Ina Schaefer
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111
Representing a model as Büchi Automaton
First Step: Represent transition system T as Büchi automaton BT
accepting exactly those words representing a run of T .
Example
active proctype P () {
do
:: atomic {
!wantQ;
wantP = true
}
pcs = true;
atomic {
pcs = false;
wantP = false
}
od }
start
0
{wp}
1
{wq}
∅
∅
2
{wp, pcs}
{wq, qcs}
3
4
What are the accepting locations?
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112
Representing a model as Büchi Automaton
First Step: Represent transition system T as Büchi automaton BT
accepting exactly those words representing a run of T .
Example
active proctype P () {
do
:: atomic {
!wantQ;
wantP = true
}
pcs = true;
atomic {
pcs = false;
wantP = false
}
od }
start
0
{wp}
1
{wq}
∅
∅
2
{wp, pcs}
{wq, qcs}
3
4
What are the accepting locations? All
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113
Representing a model as Büchi Automaton
First Step: Represent transition system T as Büchi automaton BT
accepting exactly those words representing a run of T .
Example
active proctype P () {
do
:: atomic {
!wantQ;
wantP = true
}
pcs = true;
atomic {
pcs = false;
wantP = false
}
od }
start
0
{wp}
1
{wq}
∅
∅
2
{wp, pcs}
{wq, qcs}
3
4
The property we want to check is φ = ¬pcs
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114
Büchi Automaton B¬φ for ¬φ
T |= φ holds iff. there is no accepting run for ¬φ
¬φ = ¬¬pcs = ♦pcs
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115
Büchi Automaton B¬φ for ¬φ
T |= φ holds iff. there is no accepting run for ¬φ
¬φ = ¬¬pcs = ♦pcs
Büchi Automaton B¬φ
P = {wp, wq, pcs, qcs}, Σ = 2P
start
N
a
b
Σ
Nc
N = {I |I ∈ Σ, pcs ∈ I },
Ina Schaefer
Nc = Σ − N
FSE
116
Checking for the empty language
Lω (B¬φ ) ∩ Lω (BT ) = ∅
Ina Schaefer
FSE
?
117
Checking for the empty language
Lω (B¬φ ) ∩ Lω (BT ) = ∅
?
Intersection Automaton
start
{wp}
0a
{wp, pcs}
1a0
∅
3b
0b 0
{wp, pcs}
{wq}
2a0
Ina Schaefer
{wp}
0a0
{wq}
{wq, qcs}
1b 0
∅
{wp}
{wp}
4a0
0b
FSE
∅
3b 0
{wp, pcs}
1b
118
Checking for the empty language
Lω (B¬φ ) ∩ Lω (BT ) 6= ∅
Counterexample
Intersection Automaton
start
{wp}
0a
{wp, pcs}
1a0
∅
3b
0b 0
{wp, pcs}
{wq}
2a0
Ina Schaefer
{wp}
0a0
{wq}
{wq, qcs}
1b 0
∅
{wp}
{wp}
4a0
0b
FSE
∅
3b 0
{wp, pcs}
1b
119
Literature for this Lecture
Ben-Ari Section 5.2.1
(Promela examples on the surface only)
Baier and Katoen Principles of Model Checking, ISBN: 026202649X,
May, 2008, The MIT Press
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Intersection Automaton
–
Construction
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Insersection Automaton (I)
Given: Two Büchi automata Bi = (Qi , δi , Ii , Fi )
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122
Insersection Automaton (I)
Given: Two Büchi automata Bi = (Qi , δi , Ii , Fi )
Wanted: A Büchi automaton
B1∩2 = (Q1∩2 , δ1∩2 , I1∩2 , F1∩2 )
accepting a word w iff. w accepted by B1 and B2
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Insersection Automaton (I)
Given: Two Büchi automata Bi = (Qi , δi , Ii , Fi )
Wanted: A Büchi automaton
B1∩2 = (Q1∩2 , δ1∩2 , I1∩2 , F1∩2 )
accepting a word w iff. w accepted by B1 and B2
Maybe just the productautomaton as for regular automata?
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124
Productautomata as Intersection Automata
Σ = {a, b}
(a∗ ba)ω :
a(a + ba)ω :
a
0
b
1
a
a
b
Ina Schaefer
0
1
a
FSE
125
Productautomata as Intersection Automata
Σ = {a, b}, a(a + ba)ω ∩ (a∗ ba)ω = ∅?
(a∗ ba)ω :
a(a + ba)ω :
a
0
b
1
a
a
b
Ina Schaefer
0
1
a
FSE
126
Productautomata as Intersection Automata
Σ = {a, b}, a(a + ba)ω ∩ (a∗ ba)ω = ∅? No, e.g. a(ba)ω
(a∗ ba)ω :
a(a + ba)ω :
a
0
b
1
a
a
b
Ina Schaefer
0
1
a
FSE
127
Productautomata as Intersection Automata
Σ = {a, b}, a(a + ba)ω ∩ (a∗ ba)ω = ∅? No, e.g. a(ba)ω
(a∗ ba)ω :
a(a + ba)ω :
a
0
b
a
1
a
0
1
b
a
Productautomaton:
a
00
10
a
b
a
01
Ina Schaefer
11
FSE
128
First Try: Productautomata as Intersection Automata
Σ = {a, b}, a(a + ba)ω ∩ (a∗ ba)ω = ∅? No, e.g. a(ba)ω
(a∗ ba)ω :
a(a + ba)ω :
a
0
b
a
1
a
0
1
b
a
Productautomaton: Oops, accepting location 11 never reached.
a
00
10
a
b
a
01
Ina Schaefer
11
FSE
129
Next Try: Constructing the Intersection Automata
(a∗ ba)ω :
a(a + ba)ω :
a
0
b
a
1
a
0
1
b
a
Construction
a
00
10
a
b
a
01
11
Q∩ = Q1 × Q2
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130
Next Try: Constructing the Intersection Automata
(a∗ ba)ω :
a(a + ba)ω :
a
0
b
a
1
a
0
1
b
a
Construction
a
00
10
a
b
a
01
Q∩ = Q1 × Q2
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131
Next Try: Constructing the Intersection Automata
(a∗ ba)ω :
a(a + ba)ω :
a
b
0
a
1
a
0
1
b
a
Construction
a
a
00 1
10 1
a
00 2
b
10 2
a
b
a
a
01 1
01 2
Q∩ = Q1 × Q2 × {1, 2}
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132
Next Try: Constructing the Intersection Automata
(a∗ ba)ω :
a(a + ba)ω :
a
b
0
a
1
a
0
1
b
a
Construction
a
a
00 1
10 1
a
00 2
b
10 2
a
b
a
a
01 1
01 2
Q∩ = Q1 × Q2 × {1, 2}, I∩ = I1 × I2 × {1}
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133
Next Try: Constructing the Intersection Automata
(a∗ ba)ω :
a(a + ba)ω :
a
b
0
a
1
a
0
1
b
a
Construction
a
a
00 1
10 1
a
00 2
10 2
a
b
b
a
a
01 1
01 2
Q∩ = Q1 × Q2 × {1, 2}, I∩ = I1 × I2 × {1}, F = F1 × Q2 × {1}
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134
Next Try: Constructing the Intersection Automata
Construction
a
a
00 1
10 1
a
00 2
10 2
a
b
b
a
a
01 1
01 2
Q∩ = Q1 × Q2 × {1, 2}, I∩ = I1 × I2 × {1}, F = F1 × Q2 × {1}
Ina Schaefer
FSE
135
Next Try: Constructing the Intersection Automata
Construction
a
a
00 1
10 1
a
00 2
10 2
a
b
b
a
a
01 1
01 2
Q∩ = Q1 × Q2 × {1, 2}, I∩ = I1 × I2 × {1}, F = F1 × Q2 × {1}
s1 ∈ Q1 , s2 ∈ Q2 , α ∈ Σ :
if s1 ∈ F1 : δ∩ ((s1 , s2 , 1), α) = {(s10 , s20 , 2)|s10 ∈ δ1 (s1 , α), s20 ∈ δ2 (s2 , α)}
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136
Next Try: Constructing the Intersection Automata
Construction
a
a
a
00 1
10 1
00 2
10 2
a
b
b
a
a
01 1
01 2
Q∩ = Q1 × Q2 × {1, 2}, I∩ = I1 × I2 × {1}, F = F1 × Q2 × {1}
s1 ∈ Q1 , s2 ∈ Q2 , α ∈ Σ :
if s1 ∈ F1 : δ∩ ((s1 , s2 , 1), α) = {(s10 , s20 , 2)|s10 ∈ δ1 (s1 , α), s20 ∈ δ2 (s2 , α)}
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137
Next Try: Constructing the Intersection Automata
Construction
a
a
a
00 1
10 1
00 2
10 2
a
b
b
a
a
01 1
01 2
Q∩ = Q1 × Q2 × {1, 2}, I∩ = I1 × I2 × {1}, F = F1 × Q2 × {1}
s1 ∈ Q1 , s2 ∈ Q2 , α ∈ Σ :
if s1 ∈ F1 : δ∩ ((s1 , s2 , 1), α) = {(s10 , s20 , 2)|s10 ∈ δ1 (s1 , α), s20 ∈ δ2 (s2 , α)}
Ina Schaefer
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138
Next Try: Constructing the Intersection Automata
Construction
a
a
a
00 1
10 1
00 2
10 2
a
b
a
a
01 1
01 2
b
Q∩ = Q1 × Q2 × {1, 2}, I∩ = I1 × I2 × {1}, F = F1 × Q2 × {1}
s1 ∈ Q1 , s2 ∈ Q2 , α ∈ Σ :
if s1 ∈ F1 : δ∩ ((s1 , s2 , 1), α) = {(s10 , s20 , 2)|s10 ∈ δ1 (s1 , α), s20 ∈ δ2 (s2 , α)}
Ina Schaefer
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139
Next Try: Constructing the Intersection Automata
Construction
a
a
a
00 1
10 1
00 2
10 2
a
b
a
a
01 1
01 2
b
Q∩ = Q1 × Q2 × {1, 2}, I∩ = I1 × I2 × {1}, F = F1 × Q2 × {1}
s1 ∈ Q1 , s2 ∈ Q2 , α ∈ Σ :
if s1 ∈ F1 : δ∩ ((s1 , s2 , 1), α) = {(s10 , s20 , 2)|s10 ∈ δ1 (s1 , α), s20 ∈ δ2 (s2 , α)}
if s2 ∈ F2 : δ∩ ((s1 , s2 , 2), α) = {(s10 , s20 , 1)|s10 ∈ δ1 (s1 , α), s20 ∈ δ2 (s2 , α)}
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140
Next Try: Constructing the Intersection Automata
Construction
a
a
00 1
a
10 1
a
00 2
10 2
a
b
a
01 1
01 2
b
Q∩ = Q1 × Q2 × {1, 2}, I∩ = I1 × I2 × {1}, F = F1 × Q2 × {1}
s1 ∈ Q1 , s2 ∈ Q2 , α ∈ Σ :
if s1 ∈ F1 : δ∩ ((s1 , s2 , 1), α) = {(s10 , s20 , 2)|s10 ∈ δ1 (s1 , α), s20 ∈ δ2 (s2 , α)}
if s2 ∈ F2 : δ∩ ((s1 , s2 , 2), α) = {(s10 , s20 , 1)|s10 ∈ δ1 (s1 , α), s20 ∈ δ2 (s2 , α)}
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141
Next Try: Constructing the Intersection Automata
Construction
a
a
00 1
a
10 1
a
00 2
10 2
a
b
a
01 1
01 2
b
Q∩ = Q1 × Q2 × {1, 2}, I∩ = I1 × I2 × {1}, F = F1 × Q2 × {1}
s1 ∈ Q1 , s2 ∈ Q2 , α ∈ Σ :
if s1 ∈ F1 : δ∩ ((s1 , s2 , 1), α) = {(s10 , s20 , 2)|s10 ∈ δ1 (s1 , α), s20 ∈ δ2 (s2 , α)}
if s2 ∈ F2 : δ∩ ((s1 , s2 , 2), α) = {(s10 , s20 , 1)|s10 ∈ δ1 (s1 , α), s20 ∈ δ2 (s2 , α)}
else:
δ∩ ((s1 , s2 , i), α) = {(s10 , s20 , i)|s10 ∈ δ1 (s1 , α), s20 ∈ δ2 (s2 , α)}
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142
Construction of a Büchi
Automaton Bφ
for an
LTL-Formula φ
Ina Schaefer
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143
Generalised Büchi Automaton
A generalised Büchi automaton is defined as:
B g = (Q, δ, I , F)
Q, δ, I as for normal Büchi automata.
F = {F1 , . . . , Fn }, where Fi = {qi1 , . . . , qimi }, qik ∈ Q
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144
Generalised Büchi Automaton
A generalised Büchi automaton is defined as:
B g = (Q, δ, I , F)
Q, δ, I as for normal Büchi automata.
F = {F1 , . . . , Fn }, where Fi = {qi1 , . . . , qimi }, qik ∈ Q
Definition (Acceptance for generalised Büchi automata)
A generalised Büchi automata accepts an ω-word w ∈ Σω iff. for every
i ∈ {1, . . . , n} some qik ∈ Fi are visited infinitely often.
Ina Schaefer
FSE
145
Example – Generalised Büchi Automaton
a
start
1
b
b
Ina Schaefer
a
0
FSE
2
146
Example – Generalised Büchi Automaton
a
start
1
b
a
0
b
2
F1
B normal with F = {1, 2}, B general
Ina Schaefer
FSE
F2
z}|{ z}|{
with F = { {1} , {2} }
147
Example – Generalised Büchi Automaton
a
start
1
b
a
0
b
2
F1
B normal with F = {1, 2}, B general
F2
z}|{ z}|{
with F = { {1} , {2} }
Which ω-word is accepted by which automata?
ω-word B normal
Ina Schaefer
FSE
B general
148
Example – Generalised Büchi Automaton
a
start
1
b
a
0
b
2
F1
B normal with F = {1, 2}, B general
F2
z}|{ z}|{
with F = { {1} , {2} }
Which ω-word is accepted by which automata?
ω-word B normal
(ab)ω
Ina Schaefer
FSE
B general
149
Example – Generalised Büchi Automaton
a
start
1
b
a
0
b
2
F1
B normal with F = {1, 2}, B general
F2
z}|{ z}|{
with F = { {1} , {2} }
Which ω-word is accepted by which automata?
ω-word B normal
(ab)ω
4
Ina Schaefer
FSE
B general
150
Example – Generalised Büchi Automaton
a
start
1
b
a
0
b
2
F1
B normal with F = {1, 2}, B general
F2
z}|{ z}|{
with F = { {1} , {2} }
Which ω-word is accepted by which automata?
ω-word B normal
(ab)ω
4
Ina Schaefer
FSE
B general
8
151
Example – Generalised Büchi Automaton
a
start
1
b
a
0
b
2
F1
B normal with F = {1, 2}, B general
F2
z}|{ z}|{
with F = { {1} , {2} }
Which ω-word is accepted by which automata?
ω-word B normal
(ab)ω
4
ω
(aab)
Ina Schaefer
FSE
B general
8
152
Example – Generalised Büchi Automaton
a
start
1
b
a
0
b
2
F1
B normal with F = {1, 2}, B general
F2
z}|{ z}|{
with F = { {1} , {2} }
Which ω-word is accepted by which automata?
ω-word B normal
(ab)ω
4
ω
(aab)
4
Ina Schaefer
FSE
B general
8
153
Example – Generalised Büchi Automaton
a
start
1
b
a
0
b
2
F1
B normal with F = {1, 2}, B general
F2
z}|{ z}|{
with F = { {1} , {2} }
Which ω-word is accepted by which automata?
ω-word B normal
(ab)ω
4
ω
(aab)
4
Ina Schaefer
FSE
B general
8
4
154
Fischer-Ladner Closure
Fischer-Ladner closure of an LTL-formula φ
FL(φ) = {ϕ|ϕ is subformula or negated subformula of φ}
Example
FL(r Us) = {r , ¬r , s, ¬s, r Us, ¬(r Us)}
Ina Schaefer
FSE
155
Bφ -Construction: Locations
Assumption: U the only temporal logic operator in LTL-formula φ (, ♦
expressable with U)
Ina Schaefer
FSE
156
Bφ -Construction: Locations
Assumption: U the only temporal logic operator in LTL-formula φ (, ♦
expressable with U)
Locations Q ⊆ 2FL(φ) where each q ∈ Q satisfies:
1. ψ ∈ FL(φ): either ψ ∈ q or ¬ψ ∈ q, but not both
Ina Schaefer
FSE
157
Bφ -Construction: Locations
Assumption: U the only temporal logic operator in LTL-formula φ (, ♦
expressable with U)
Locations Q ⊆ 2FL(φ) where each q ∈ Q satisfies:
1. ψ ∈ FL(φ): either ψ ∈ q or ¬ψ ∈ q, but not both
2. ψ1 ∧ ψ2 ∈ q: ψ1 ∈ q and ψ2 ∈ q
Ina Schaefer
FSE
158
Bφ -Construction: Locations
Assumption: U the only temporal logic operator in LTL-formula φ (, ♦
expressable with U)
Locations Q ⊆ 2FL(φ) where each q ∈ Q satisfies:
1. ψ ∈ FL(φ): either ψ ∈ q or ¬ψ ∈ q, but not both
2. ψ1 ∧ ψ2 ∈ q: ψ1 ∈ q and ψ2 ∈ q
3. other propositional connectives similar
Ina Schaefer
FSE
159
Bφ -Construction: Locations
Assumption: U the only temporal logic operator in LTL-formula φ (, ♦
expressable with U)
Locations Q ⊆ 2FL(φ) where each q ∈ Q satisfies:
1. ψ ∈ FL(φ): either ψ ∈ q or ¬ψ ∈ q, but not both
2. ψ1 ∧ ψ2 ∈ q: ψ1 ∈ q and ψ2 ∈ q
3. other propositional connectives similar
4. ψ1 Uψ2 ∈ q then ψ1 ∈ q or ψ2 ∈ q
Ina Schaefer
FSE
160
Bφ -Construction: Locations
Assumption: U the only temporal logic operator in LTL-formula φ (, ♦
expressable with U)
Locations Q ⊆ 2FL(φ) where each q ∈ Q satisfies:
1. ψ ∈ FL(φ): either ψ ∈ q or ¬ψ ∈ q, but not both
2. ψ1 ∧ ψ2 ∈ q: ψ1 ∈ q and ψ2 ∈ q
3. other propositional connectives similar
4. ψ1 Uψ2 ∈ q then ψ1 ∈ q or ψ2 ∈ q
5. ψ1 Uψ2 ∈ (FL(φ)\q) then ψ2 6∈ q
FL(r Us) = {r , ¬r , s, ¬s, r Us, ¬(r Us)}
∈Q
Ina Schaefer
FSE
161
Bφ -Construction: Locations
Assumption: U the only temporal logic operator in LTL-formula φ (, ♦
expressable with U)
Locations Q ⊆ 2FL(φ) where each q ∈ Q satisfies:
1. ψ ∈ FL(φ): either ψ ∈ q or ¬ψ ∈ q, but not both
2. ψ1 ∧ ψ2 ∈ q: ψ1 ∈ q and ψ2 ∈ q
3. other propositional connectives similar
4. ψ1 Uψ2 ∈ q then ψ1 ∈ q or ψ2 ∈ q
5. ψ1 Uψ2 ∈ (FL(φ)\q) then ψ2 6∈ q
FL(r Us) = {r , ¬r , s, ¬s, r Us, ¬(r Us)}
{r Us, ¬r , s}
Ina Schaefer
FSE
∈Q
4
162
Bφ -Construction: Locations
Assumption: U the only temporal logic operator in LTL-formula φ (, ♦
expressable with U)
Locations Q ⊆ 2FL(φ) where each q ∈ Q satisfies:
1. ψ ∈ FL(φ): either ψ ∈ q or ¬ψ ∈ q, but not both
2. ψ1 ∧ ψ2 ∈ q: ψ1 ∈ q and ψ2 ∈ q
3. other propositional connectives similar
4. ψ1 Uψ2 ∈ q then ψ1 ∈ q or ψ2 ∈ q
5. ψ1 Uψ2 ∈ (FL(φ)\q) then ψ2 6∈ q
FL(r Us) = {r , ¬r , s, ¬s, r Us, ¬(r Us)}
{r Us, ¬r , s}
{r Us, ¬r , ¬s}
Ina Schaefer
FSE
∈Q
4
8
163
Bφ -Construction: Locations
Assumption: U the only temporal logic operator in LTL-formula φ (, ♦
expressable with U)
Locations Q ⊆ 2FL(φ) where each q ∈ Q satisfies:
1. ψ ∈ FL(φ): either ψ ∈ q or ¬ψ ∈ q, but not both
2. ψ1 ∧ ψ2 ∈ q: ψ1 ∈ q and ψ2 ∈ q
3. other propositional connectives similar
4. ψ1 Uψ2 ∈ q then ψ1 ∈ q or ψ2 ∈ q
5. ψ1 Uψ2 ∈ (FL(φ)\q) then ψ2 6∈ q
FL(r Us) = {r , ¬r , s, ¬s, r Us, ¬(r Us)}
{r Us, ¬r , s}
{r Us, ¬r , ¬s}
{¬(r Us), r , s}
Ina Schaefer
FSE
∈Q
4
8
8
164
Bφ -Construction: Locations
Assumption: U the only temporal logic operator in LTL-formula φ (, ♦
expressable with U)
Locations Q ⊆ 2FL(φ) where each q ∈ Q satisfies:
1. ψ ∈ FL(φ): either ψ ∈ q or ¬ψ ∈ q, but not both
2. ψ1 ∧ ψ2 ∈ q: ψ1 ∈ q and ψ2 ∈ q
3. other propositional connectives similar
4. ψ1 Uψ2 ∈ q then ψ1 ∈ q or ψ2 ∈ q
5. ψ1 Uψ2 ∈ (FL(φ)\q) then ψ2 6∈ q
FL(r Us) = {r , ¬r , s, ¬s, r Us, ¬(r Us)}
∈Q
{r Us, ¬r , s}
4
{r Us, ¬r , ¬s}
8
{¬(r Us), r , s}
8
{¬(r Us), r , ¬s} 4
Ina Schaefer
FSE
165
Bφ -Construction: Transitions
{r Us, ¬r , s}, {r Us, r , ¬s}, {r Us, r , s}, {¬(r Us), r , ¬s}, {¬(r Us), ¬r , ¬s}
{z
} |
{z
} | {z } |
{z
} |
{z
}
|
q1
q2
q3
q4
q5
q4
q1
q2
q3
q5
Ina Schaefer
FSE
166
Bφ -Construction: Transitions
{r Us, ¬r , s}, {r Us, r , ¬s}, {r Us, r , s}, {¬(r Us), r , ¬s}, {¬(r Us), ¬r , ¬s}
{z
} |
{z
} | {z } |
{z
} |
{z
}
|
q1
q2
q3
q4
q5
Transitions (q, α, q 0 ) ∈ δφ :
q4
• α=q∩P
q1
q2
(P set of propositional
variables); e.g. outgoing
edges of q1 labeled {s}; of
q2 labeled {r } etc.
q3
• If ψ1 Uψ2 ∈ q and ψ2 6∈ q
then ψ1 Uψ2 ∈ q 0
• If ψ1 Uψ2 ∈ (FL(φ)\q) and
q5
Ina Schaefer
ψ2 6∈ q then ψ1 Uψ2 ∈ q 0
FSE
167
Bφ -Construction: Transitions
{r Us, ¬r , s}, {r Us, r , ¬s}, {r Us, r , s}, {¬(r Us), r , ¬s}, {¬(r Us), ¬r , ¬s}
{z
} |
{z
} | {z } |
{z
} |
{z
}
|
q1
q2
q3
q4
q5
Transitions (q, α, q 0 ) ∈ δφ :
q4
• α=q∩P
q1
q2
(P set of propositional
variables); e.g. outgoing
edges of q1 labeled {s}; of
q2 labeled {r } etc.
q3
• If ψ1 Uψ2 ∈ q and ψ2 6∈ q
then ψ1 Uψ2 ∈ q 0
• If ψ1 Uψ2 ∈ (FL(φ)\q) and
q5
Ina Schaefer
ψ2 6∈ q then ψ1 Uψ2 ∈ q 0
FSE
168
Bφ -Construction: Transitions
{r Us, ¬r , s}, {r Us, r , ¬s}, {r Us, r , s}, {¬(r Us), r , ¬s}, {¬(r Us), ¬r , ¬s}
{z
} |
{z
} | {z } |
{z
} |
{z
}
|
q1
q2
q3
q4
q5
Initial locations
q ∈ Iφ iff. φ ∈ q
q4
q1
q2
q3
q5
Ina Schaefer
FSE
169
Bφ -Construction: Transitions
{r Us, ¬r , s}, {r Us, r , ¬s}, {r Us, r , s}, {¬(r Us), r , ¬s}, {¬(r Us), ¬r , ¬s}
{z
} |
{z
} | {z } |
{z
} |
{z
}
|
q1
q2
q3
q4
q5
Initial locations
q ∈ Iφ iff. φ ∈ q
q4
Accepting locations
F = {F1 , . . . , Fn }
q1
q2
• for each subformula
q3
ψi1 Uψi2 ∈ FL(φ) exists an
Fi ; in example: F = {F1 }
• Fi set of locations not
containing ψi1 Uψi2 or that
contain ψi2
in ex. F1 = {q1 , q3 , q4 , q5 }
q5
Ina Schaefer
FSE
170