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LINEAR TEMPORAL LOGIC Fall 2013 Dr. Eric Rozier

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**Propositional Temporal Logic**

Does the following hold? yes

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**Propositional Temporal Logic**

Does the following hold? no

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**Examples: What do they mean?**

Remember: Gp p holds in all states Fp p holds eventually Xp p holds in the next state G F p p holds infinitely often F G p 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

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**Examples: Write in Temporal Logic**

“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) “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) 1. G (p => F q) 2. G (p => X \neg q)

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**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.

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**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.

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**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.

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**What does this property mean?**

F(p => Xq) Is it satisfied by this trace? p -> p -> p -> __ -> q -> p -> …

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**What does this property mean?**

F(p => Xq) Is it satisfied by this trace? p -> p -> p -> __ -> q -> p -> q -> …

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**Does this automaton satisfy the property?**

pUq

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**Does this automaton satisfy the property?**

pUq

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**Does this automaton satisfy the property?**

qRp

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**Does this automaton satisfy the property?**

qRp

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**Does this automaton satisfy the property?**

qRp

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**Does this automaton satisfy the property?**

qRp

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**Does this automaton satisfy the property?**

F(p & XXX !q)

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**Does this automaton satisfy the property?**

F(p & XXX !q)

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**Does this automaton satisfy the property?**

F(p & XXX !q)

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