### Lecture 6 - Linear Temporal Logic

```LINEAR TEMPORAL LOGIC
Fall 2013
Dr. Eric Rozier
Propositional Temporal Logic
Does the following hold?
yes
Propositional Temporal Logic
Does the following hold?
no
Examples: What do they mean?
GFp
p holds infinitely often
Remember:
Gp p holds in all states
Fp p holds eventually
Xp p holds in the next state
FGp
Eventually, 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
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)
```