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Syantax of LTL

Semantics of LTL

LTL Formula Examples

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- 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

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