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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)