Computation Tree Logic B. Srivathsan Chennai Mathematical Institute - PowerPoint PPT Presentation

State formulae φ := true | p i | φ 1 ∧ φ 2 | ¬ φ 1 | E α | A α α : Path formula p i ∈ AP φ 1 , φ 2 : State formulae … Every state satisfies true … State satisfies p i if its label contains p i … State satisfies φ 1 ∧ φ 2 if it satisfies both φ 1 and φ 2 9 / 25

State formulae φ := true | p i | φ 1 ∧ φ 2 | ¬ φ 1 | E α | A α α : Path formula p i ∈ AP φ 1 , φ 2 : State formulae … Every state satisfies true … State satisfies p i if its label contains p i … State satisfies φ 1 ∧ φ 2 if it satisfies both φ 1 and φ 2 … State satisfies ¬ φ if it does not satisfy φ 9 / 25

State formulae φ := true | p i | φ 1 ∧ φ 2 | ¬ φ 1 | E α | A α α : Path formula p i ∈ AP φ 1 , φ 2 : State formulae … Every state satisfies true … State satisfies p i if its label contains p i … State satisfies φ 1 ∧ φ 2 if it satisfies both φ 1 and φ 2 … State satisfies ¬ φ if it does not satisfy φ … State satisfies E α if there exists a path starting from the state satisfying α 9 / 25

State formulae φ := true | p i | φ 1 ∧ φ 2 | ¬ φ 1 | E α | A α α : Path formula p i ∈ AP φ 1 , φ 2 : State formulae … Every state satisfies true … State satisfies p i if its label contains p i … State satisfies φ 1 ∧ φ 2 if it satisfies both φ 1 and φ 2 … State satisfies ¬ φ if it does not satisfy φ … State satisfies E α if there exists a path starting from the state satisfying α … State satisfies A α if all paths starting from the state satisfy α 9 / 25

When does a path in a tree satisfy a path formula ? . . . . . . . . . . . . . . . 10 / 25

Path formulae α := φ | α 1 ∧ α 2 | ¬ α 1 | X α 1 | α 1 U α 2 | F α 1 | G α 1 φ : State formula α 1 , α 2 : Path formulae 11 / 25

Path formulae α := φ | α 1 ∧ α 2 | ¬ α 1 | X α 1 | α 1 U α 2 | F α 1 | G α 1 φ : State formula α 1 , α 2 : Path formulae … Path satisfies φ if the initial state of the path satisfies φ 11 / 25

Path formulae α := φ | α 1 ∧ α 2 | ¬ α 1 | X α 1 | α 1 U α 2 | F α 1 | G α 1 φ : State formula α 1 , α 2 : Path formulae … Path satisfies φ if the initial state of the path satisfies φ … Rest standard semantics like LTL 11 / 25

A tree satisfies state formula φ if its root satisfies φ ∈ . . . . . . . . . . . . . . . 12 / 25

… E F p 1 : Exists a path where p 1 is true sometime 13 / 25

… E F p 1 : Exists a path where p 1 is true sometime … A F A G p 1 : 13 / 25

… E F p 1 : Exists a path where p 1 is true sometime … A F A G p 1 : … In all paths, there exists a state where A G p 1 is true 13 / 25

… E F p 1 : Exists a path where p 1 is true sometime … A F A G p 1 : … In all paths, there exists a state where A G p 1 is true … In all paths, there exists a state from which all paths satisfy G p 1 13 / 25

… E F p 1 : Exists a path where p 1 is true sometime … A F A G p 1 : … In all paths, there exists a state where A G p 1 is true … In all paths, there exists a state from which all paths satisfy G p 1 … In all paths, there exists a state such that every state in the subtree below it contains p 1 13 / 25

… E F p 1 : Exists a path where p 1 is true sometime … A F A G p 1 : … In all paths, there exists a state where A G p 1 is true … In all paths, there exists a state from which all paths satisfy G p 1 … In all paths, there exists a state such that every state in the subtree below it contains p 1 … A F G p 2 : In all paths, there exists a state from which p 2 is true forever 13 / 25

… E F p 1 : Exists a path where p 1 is true sometime … A F A G p 1 : … In all paths, there exists a state where A G p 1 is true … In all paths, there exists a state from which all paths satisfy G p 1 … In all paths, there exists a state such that every state in the subtree below it contains p 1 … A F G p 2 : In all paths, there exists a state from which p 2 is true forever … A p 1 : 13 / 25

… E F p 1 : Exists a path where p 1 is true sometime … A F A G p 1 : … In all paths, there exists a state where A G p 1 is true … In all paths, there exists a state from which all paths satisfy G p 1 … In all paths, there exists a state such that every state in the subtree below it contains p 1 … A F G p 2 : In all paths, there exists a state from which p 2 is true forever … A p 1 : … All paths satisfy p 1 13 / 25

… E F p 1 : Exists a path where p 1 is true sometime … A F A G p 1 : … In all paths, there exists a state where A G p 1 is true … In all paths, there exists a state from which all paths satisfy G p 1 … In all paths, there exists a state such that every state in the subtree below it contains p 1 … A F G p 2 : In all paths, there exists a state from which p 2 is true forever … A p 1 : … All paths satisfy p 1 … All paths start with p 1 13 / 25

… E F p 1 : Exists a path where p 1 is true sometime … A F A G p 1 : … In all paths, there exists a state where A G p 1 is true … In all paths, there exists a state from which all paths satisfy G p 1 … In all paths, there exists a state such that every state in the subtree below it contains p 1 … A F G p 2 : In all paths, there exists a state from which p 2 is true forever … A p 1 : … All paths satisfy p 1 … All paths start with p 1 … Same as p 1 ! 13 / 25

E F A G ( red ) . . . . . . . . . . . . . . . 14 / 25

A F A G ( red ) . . . . . . . . . . . . . . . 15 / 25

E G E X ( red ) . . . . . . . . . . . . . . . 16 / 25

E G E X ( red ) . . . . . . . . . . . . . . . 17 / 25