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# hw1 - CS 267 Spring 2011 Homework Assignment 1 Due Thursday...

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CS 267 – Spring 2011 – Homework Assignment 1 Due Thursday, April 14th Do not discuss the problems with anyone other than the instructor. 1. Consider the following two transition systems: M 1 = ( AP 1 ,S 1 ,R 1 ,I 1 ,L 1 ) with the set of states S 1 = { 0 , 1 , 2 , 3 } , the initial set of states I 1 = { 0 } , the transition relation R 1 = { (0 , 1) , (1 , 2) , (2 , 3) , (2 , 1) , (3 , 3) } , the set of atomic propositions AP 1 = { p,q } and the labeling function L 1 : S 1 2 AP where L 1 (0) = , L 1 (1) = { p } , L 1 (2) = { p,q } , and L 1 (3) = { q } . M 2 = ( AP 2 ,S 2 ,R 2 ,I 2 ,L 2 ) with the set of states S 2 = { 0 , 1 , 2 , 3 } , the initial set of states I 2 = { 0 } , the transition relation R 2 = { (0 , 1) , (1 , 2) , (2 , 3) , (1 , 1) , (3 , 2) } , the set of atomic propositions AP 2 = { p,q } and the labeling function L 2 : S 2 AP where L 2 (0) = { p } , L 2 (1) = { p,q } , L 2 (2) = { p,q } , and L 2 (3) = { q } . (a) For each of the CTL formulas show the states which satisfy them: EG p, AG p, EG q, AG q, EF p, AF p, p EU q, p AU q, EG(EF q ) , AG(EF q ) , AG(AF q ) , EG(AF q ) . For the ACTL properties above that fail for the given transition systems, give a counter- example. For the ECTL properties above that hold for the given transition systems, give a

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