Algorithms for CTL B. Srivathsan Chennai Mathematical Institute - PowerPoint PPT Presentation

Algorithm for E ( φ 1 U φ 2 ) s … If any state is labelled with φ 2 , label it with E ( φ 1 U φ 2 ) … Repeat : Label any state with E ( φ 1 U φ 2 ) if it is labelled with φ 1 and at least one successor is labelled with E ( φ 1 U φ 2 ) until no change 11 / 16

Algorithm for E ( φ 1 U φ 2 ) E ( φ 1 U φ 2 ) φ 1 s … If any state is labelled with φ 2 , label it with E ( φ 1 U φ 2 ) … Repeat : Label any state with E ( φ 1 U φ 2 ) if it is labelled with φ 1 and at least one successor is labelled with E ( φ 1 U φ 2 ) until no change 11 / 16

Algorithm for E ( φ 1 U φ 2 ) E ( φ 1 U φ 2 ) E ( φ 1 U φ 2 ) φ 1 s … If any state is labelled with φ 2 , label it with E ( φ 1 U φ 2 ) … Repeat : Label any state with E ( φ 1 U φ 2 ) if it is labelled with φ 1 and at least one successor is labelled with E ( φ 1 U φ 2 ) until no change 11 / 16

Part 3 : Algorithm for E G 12 / 16

E G p 1 { p 1 } { p 1 } { p 1 } { } s 1 s 2 s 3 s 4 s 8 s 7 s 6 s 5 { p 1 } { p 1 } { p 1 } { } 13 / 16

E G p 1 E G p 1 E G p 1 E G p 1 E G p 1 { p 1 } { p 1 } { p 1 } { } s 1 s 2 s 3 s 4 s 8 s 7 s 6 s 5 { p 1 } { p 1 } { p 1 } { } E G p 1 E G p 1 E G p 1 E G p 1 13 / 16

E G p 1 E G p 1 E G p 1 E G p 1 { p 1 } { p 1 } { p 1 } { } s 1 s 2 s 3 s 4 s 8 s 7 s 6 s 5 { p 1 } { p 1 } { p 1 } { } E G p 1 E G p 1 E G p 1 13 / 16

E G p 1 E G p 1 E G p 1 E G p 1 { p 1 } { p 1 } { p 1 } { } s 1 s 2 s 3 s 4 s 8 s 7 s 6 s 5 { p 1 } { p 1 } { p 1 } { } E G p 1 E G p 1 13 / 16

E G p 1 E G p 1 E G p 1 { p 1 } { p 1 } { p 1 } { } s 1 s 2 s 3 s 4 s 8 s 7 s 6 s 5 { p 1 } { p 1 } { p 1 } { } E G p 1 13 / 16

E G p 1 E G p 1 { p 1 } { p 1 } { p 1 } { } s 1 s 2 s 3 s 4 s 8 s 7 s 6 s 5 { p 1 } { p 1 } { p 1 } { } 13 / 16

E G p 1 { p 1 } { p 1 } { p 1 } { } s 1 s 2 s 3 s 4 s 8 s 7 s 6 s 5 { p 1 } { p 1 } { p 1 } { } 13 / 16

E G p 1 { p 1 } { p 1 } { p 1 } { } s 1 s 2 s 3 s 4 s 8 s 7 s 6 s 5 { p 1 } { p 1 } { p 1 } { } No state of the above transition system satisfies E G p 1 13 / 16

E G p 1 { p 1 } { p 1 } { p 2 } s 6 s 1 s 2 s 5 s 4 s 3 { p 1 } { } { p 1 } 14 / 16

E G p 1 E G p 1 E G p 1 E G p 1 { p 1 } { p 1 } { p 2 } s 6 s 1 s 2 s 5 s 4 s 3 { p 1 } { } { p 1 } E G p 1 E G p 1 E G p 1 14 / 16

E G p 1 E G p 1 E G p 1 { p 1 } { p 1 } { p 2 } s 6 s 1 s 2 s 5 s 4 s 3 { p 1 } { } { p 1 } E G p 1 E G p 1 14 / 16

E G p 1 E G p 1 E G p 1 { p 1 } { p 1 } { p 2 } s 6 s 1 s 2 s 5 s 4 s 3 { p 1 } { } { p 1 } E G p 1 14 / 16

Algorithm for E G φ 15 / 16

Algorithm for E G φ … Label all states with E G φ 15 / 16

Algorithm for E G φ … Label all states with E G φ … If any state is not labelled with φ , delete the label E G φ 15 / 16

Algorithm for E G φ … Label all states with E G φ … If any state is not labelled with φ , delete the label E G φ … Repeat : Delete the label E G φ from a state if none of its successors is labelled with E G φ until no change 15 / 16

Algorithm for E G φ … Label all states with E G φ … If any state is not labelled with φ , delete the label E G φ E G φ s E G φ E G φ E G φ … Repeat : Delete the label E G φ from a state if none of its successors is labelled with E G φ until no change 15 / 16

Algorithm for E G φ … Label all states with E G φ … If any state is not labelled with φ , delete the label E G φ s E G φ … Repeat : Delete the label E G φ from a state if none of its successors is labelled with E G φ until no change 15 / 16

Algorithm for E G φ … Label all states with E G φ … If any state is not labelled with φ , delete the label E G φ s … Repeat : Delete the label E G φ from a state if none of its successors is labelled with E G φ until no change 15 / 16

Summary Algorithms EX, EU, EG 16 / 16

Module 3: Final algorithm 2 / 8

CTL model-checking problem Given transition system M and a CTL formula φ , find all states of M that satisfy φ 3 / 8

CTL model-checking problem Given transition system M and a CTL formula φ , find all states of M that satisfy φ … Module 1: Every CTL formula can be written using EX, EU, EG … Module 2: Labelling algorithms for EX, EU, EG 3 / 8

Coming next: Generic algorithm for a CTL formula State formulae φ := true | p i | φ 1 ∧ φ 2 | ¬ φ | E X φ | E ( φ 1 U φ 2 ) | E G φ p i ∈ AP φ , φ 1 , φ 2 : State formulae 4 / 8

E X E G ( p 1 ∧ p 2 ) { p 1 , p 2 } { p 1 , p 2 } s 0 { p 1 } s 3 s 6 s 2 s 1 s 5 { p 1 , p 2 } { p 1 , p 2 } { p 2 } s 4 s 7 { p 1 , p 2 } { p 1 , p 2 } 5 / 8

E X E G ( p 1 ∧ p 2 ) p 1 ∧ p 2 p 1 ∧ p 2 { p 1 , p 2 } { p 1 , p 2 } s 0 { p 1 } s 3 s 6 s 2 s 1 s 5 { p 1 , p 2 } { p 1 , p 2 } { p 2 } s 4 s 7 p 1 ∧ p 2 p 1 ∧ p 2 { p 1 , p 2 } { p 1 , p 2 } p 1 ∧ p 2 p 1 ∧ p 2 5 / 8

E X E G ( p 1 ∧ p 2 ) E G ( p 1 ∧ p 2 ) E G ( p 1 ∧ p 2 ) p 1 ∧ p 2 p 1 ∧ p 2 E G ( p 1 ∧ p 2 ) { p 1 , p 2 } { p 1 , p 2 } s 0 { p 1 } s 3 s 6 s 2 s 1 s 5 { p 1 , p 2 } { p 1 , p 2 } { p 2 } s 4 s 7 p 1 ∧ p 2 p 1 ∧ p 2 E G ( p 1 ∧ p 2 ) { p 1 , p 2 } { p 1 , p 2 } E G ( p 1 ∧ p 2 ) E G ( p 1 ∧ p 2 ) p 1 ∧ p 2 p 1 ∧ p 2 E G ( p 1 ∧ p 2 ) E G ( p 1 ∧ p 2 ) 5 / 8