Learn the basic of linear temporal logic. This presentation slide includes the explanation of syntax and semantics of linear temporal logic in model checking. This is research presentation submitted to Prof Gihwon Kwon, Kyonggi University. Includes ; Syantax of LTL Semantics of LTL LTL Formula Examples

1. Understanding LTL
By :
Anit Thapaliya
Software Engineering
Department of Computer Science
Kyonggi University, South Korea

2. ∗ It is temporal logic with connectives that allow us to
refer to the future.
∗ It models the time as a sequence of states, extending
infinitely to the future.
Definition

3. ∗ ϕ ::= true | p | ¬ϕ | ϕ1^ϕ2 | Xϕ | ϕ1Uϕ2
Where, p belongs to AP
X= ‘next’: ϕ is true at next step
U= ‘until’: ϕ2 is true at some point, ϕ1 is true until that
time
Syntax

4. ∗ ϕ := true |
Explanation LTL
…
{p1,p2} {p1,p2} {p2} {p1,p2} {p2}

5. ∗ ϕ := true | p |
Explanation LTL
…
p = p1, p2, p3, p4, …
{p1,p2} {p2} {p1,p2} {p2}
Where p = AP (Every atomic proposition is LTL Formula)
{p1,p2}

6. ∗ ϕ := true | p | ¬ϕ |
Explanation LTL
…¬P1
{p1} {p2} {p2} {p2}
Where p = AP
{p2}
¬ϕ = if ϕ is an LTL formula then not of phi (¬ϕ) is also an LTL formula
Look at the first state it does not satisfy p1. hence, ¬P1 is true

7. ∗ ϕ := true | p | ¬ϕ | ϕ1^ϕ2 |
Explanation LTL
…
P1 ^ P2
{p1,p2} {p2} {p1,p2} {p2}
Where p = AP
{p1,p2}
Φ1 & Φ1 are LTL Formual, then p1 & p2 are LTL formula
Look at the first state it satisfy p1 and p2. hence, P1 & P2 is true
^ stands for And

8. ∗ ϕ := true | p | ¬ϕ | ϕ1^ϕ2 | Xϕ |
Explanation LTL
…
Xp1 is true
Xp2 is not true
X¬p2 is true
{p1} {p2} {p2} {p2}
Where p = AP
{p2}
If Φ is an LTL formula then, XΦ is also an LTL formula
Earlier, we are verifying the states by looking the first part now
with Xp1 operator we have to look to next part. If the following part
satisfy p1 then it is true. Note: Focused on second part
following the first in sequence.
X stands for Next

9. ∗ ϕ := true | p | ¬ϕ | ϕ1^ϕ2 | Xϕ | ϕ1Uϕ2
Explanation LTL
…
p1 U p2
{p1} {p1} {p2} {p1}
Where p = AP
{p1}
If Φ1, Φ2 are LTL formula then, p1, p2 also LTL formula
We going further states in this part. That is p2 is true at some
point in the future, until that point where p2 is p1 must be true.
Or p2 should definitely true at some point until when p1 must
be true.
U stands for Until

10. ∗ ϕ := true | p | ¬ϕ | ϕ1^ϕ2 | Xϕ | ϕ1Uϕ2
Some Example
…
¬(p1 U p2)
{p1} {} {p2} {p1}
LTL Formula
{p1}
¬ (p1 U p2)
In this formula, p2 is true at some point which is true
but until where p2 is true p1 is not completely true.
Meaning
Here p2 is true
Here p1 is not true

11. ∗ ϕ := true | p | ¬ϕ | ϕ1^ϕ2 | Xϕ | ϕ1Uϕ2
Some LTL Formula
…
¬(p1 U p2)
{p1} {} {p2} {p1}
LTL Formula
{p1}
¬ (p1 U p2)
In this formula, p2 is true at some point which is true
but until where p2 is true p1 is not completely true.
Meaning
Here p2 is true
Here p1 is not true

12. ∗ ϕ := true | p | ¬ϕ | ϕ1^ϕ2 | Xϕ | ϕ1Uϕ2
Some LTL Formula
…
{p1} {p1} {p2} {p1,p3}
LTL Formula
{p1, p3}
p1 U (p2 ^ X p3)
In this formula, (p2 ^ X p3) is true at some point in the future until where p1 is also
true. At the black state Xp3 is true because there is p3 in next state where as p2 is
also true there. Lastly in all the yellow state p1 is present so p1 is true until (p2 ^ X
p3).
Meaning
(p2 ^ X p3) is true
Here p1 is true in all yellow state

13. ∗ Word σ : A0 A1 A2 … ε AP
∗ Each Ai is a set of atomic proposition
∗ Every words satisfies true
∗ Every sigma satisfy LTL formula
∗ Words (true) = AP
∗ σ satisfies Pi if Pi ε A0
∗ If the first letter A0 contain pi.
∗ Word s(Pi) = {A0 A1 A2 A3…. | Pi ε A0} ie Pi must be in A0
Semantics of LTL Formula
∗ ϕ := true | p | ¬ϕ | ϕ1^ϕ2 | Xϕ | ϕ1Uϕ2

14. ∗ Word σ : A0 A1 A2 … ε AP
∗ Each Ai is a set of atomic proposition
∗ σ satisfy ¬ϕ if σ does not satisfy ϕ
∗ Words(¬ϕ) = (Words (ϕ))’
∗ σ satisfies ϕ1^ϕ2 if σ satisfy ϕ1 and σ satisfy ϕ2
∗ Words (ϕ1^ϕ2) = Words (ϕ1) Intersection Words (ϕ2)
∗ It means words must be common in ϕ1, ϕ2
Semantics of LTL Formula
∗ ϕ := true | p | ¬ϕ | ϕ1^ϕ2 | Xϕ | ϕ1Uϕ2

15. Word σ : A0 A1 A2 A3 … ε AP
 Each Ai is a set of atomic proposition
σ satisfies Xϕ if A1 A2 A3 ….. ϕ
What is words expect A0 must satisfy ϕ
σ satisfy ϕ1 U ϕ2 if there exists j Aj Aj+1….. Satisfy ϕ2 and for
all Aj-1 (0<i<j Ai Ai+1 ) ….satisfy ϕ1
Semantics of LTL Formula
∗ ϕ := true | p | ¬ϕ | ϕ1^ϕ2 | Xϕ | ϕ1Uϕ2
Except A0

16. Word σ : A0 A1 A2 A3 … ε AP
 Each Ai is a set of atomic proposition
σ satisfies Xϕ if A1 A2 A3 ….. ϕ
What is words expect A0 must satisfy ϕ
Words (X ϕ)={A0 A1 A2…| A1 A2 .. ε Words (ϕ) }
σ satisfy ϕ1 U ϕ2 if there exists j Aj Aj+1….. Satisfy ϕ2 and for
all Ai and Aj-1 (0<i<j Ai Ai+1 ) ….satisfy ϕ1
Words (ϕ1 U ϕ2) means all the suffix starting from Aj belongs to ϕ2
And all suffixes starting from Ai and Aj-1 belongs to ϕ1.
Semantics of LTL Formula
∗ ϕ := true | p | ¬ϕ | ϕ1^ϕ2 | Xϕ | ϕ1Uϕ2

17.  σ satisfy true Uϕ if there exists j Aj Aj+1….. Satisfy ϕ
This is because ture is always true for all Ai and Aj-1 (0<i<j Ai Ai+1 )
….satisfy true
Semantics for Fϕ: true U ϕ
Semantics for Gϕ: ¬F ¬ϕ
 σ satisfy F ¬ϕ if there exists j Aj Aj+1….. Satisfy ¬ϕ
σ satisfy ¬F ¬ϕ if σ does not Satisfy F ¬ϕ

18. ∗ X & U are called temporal operators.
∗ Temporal operators means they are related to time.
∗ G global true now and forever (Rectangle in temporal
logic )
∗ F Eventually true now and some time in future (like
diamond in temporal logic)

19. Primary Temporal Logic Operators
Eventually ◊ ϕ := true U ϕ (ϕ will become true at some point in
the future)
Always □ ϕ := ¬◊¬ϕ ϕ is always true; (never (eventually (¬ϕ)))
∗ p ◊q p implies eventually q (response)→
∗ P p U r p implies q until r (precedence)→
∗ □ ◊p always eventually p (process)
∗ ◊□p eventually always p (stability)
∗ ◊p ◊q eventually p implies eventually q (correlation)→
More Operators & Formulas

20. Thank You