LINEAR TEMPORAL LOGIC

Fall 2013Dr. Eric Rozier

Propositional Temporal Logic

Does the following hold?

yes

Propositional Temporal Logic

Does the following hold?

no

Examples: What do they mean?

G F pp holds infinitely often

F G pEventually, p holds henceforth

G( p => F q ) Every p is eventually followed by a q

F( p => (X X q) )Every p is followed by a q two reactions

later

Remember:Gp p holds in all statesFp p holds eventuallyXp p holds in the next state

Examples: Write in Temporal Logic

1. “Whenever the iRobot is at the ramp-edge (cliff), eventually it moves 5 cm away from the cliff.”• p – iRobot is at the cliff• q – iRobot is 5 cm away from the cliff

• G (p => F q)

2. “Whenever the distance between cars is less than 2m, cruise control is deactivated”• p – distance between cars is less than 2 m• q – cruise control is active

• G (p => X ! q)

Remember, LTL Formulas are Formulas

• Suppose the robot must visit a set of n locations l1, l2, …, ln. Let pi be an atomic formula that is true if and only if the robot visits location li.

• Express the following: – The robot must eventually visit at least one of the

n locations.

Remember, LTL Formulas are Formulas

• Suppose the robot must visit a set of n locations l1, l2, …, ln. Let pi be an atomic formula that is true if and only if the robot visits location li.

• Express the following: – The robot must eventually visit all n locations, but

in any order.

Remember, LTL Formulas are Formulas

• Suppose the robot must visit a set of n locations l1, l2, …, ln. Let pi be an atomic formula that is true if and only if the robot visits location li.

• Express the following: – The robot must eventually visit all n locations, in

numeric order.

What does this property mean?

• F(p => Xq)

• Is it satisfied by this trace?

p -> p -> p -> __ -> q -> p -> …

What does this property mean?

• F(p => Xq)

• Is it satisfied by this trace?

p -> p -> p -> __ -> q -> p -> q -> …

Does this automaton satisfy the property?

• pUq

Does this automaton satisfy the property?

• pUq

Does this automaton satisfy the property?

• qRp

Does this automaton satisfy the property?

• qRp

Does this automaton satisfy the property?

• qRp

Does this automaton satisfy the property?

• qRp

Does this automaton satisfy the property?

• F(p & XXX !q)

Does this automaton satisfy the property?

• F(p & XXX !q)

Does this automaton satisfy the property?

• F(p & XXX !q)