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 cs-486:homework-7 [2017/02/21 17:59]egm cs-486:homework-7 [2018/11/02 22:21] (current)egm [Problems] Both sides previous revision Previous revision 2018/11/02 22:21 egm [Problems] 2017/02/21 18:05 egm [Problems] 2017/02/21 18:00 egm [Problems] 2017/02/21 17:59 egm 2017/02/21 17:58 egm created2017/01/11 19:12 egm removed2016/04/06 19:44 egm [Paper Pencil] 2014/09/02 16:43 egm created Next revision Previous revision 2018/11/02 22:21 egm [Problems] 2017/02/21 18:05 egm [Problems] 2017/02/21 18:00 egm [Problems] 2017/02/21 17:59 egm 2017/02/21 17:58 egm created2017/01/11 19:12 egm removed2016/04/06 19:44 egm [Paper Pencil] 2014/09/02 16:43 egm created Line 5: Line 5: == Problems == == Problems == - # '''​(12 points)'''​ For the transition system, which states satisfy the formulas. L(s1) = {a}. L(s2) = {a}. L(s3) = {a,b}. L(s4) = {b}. s1 -> s2. s2 -> s3. s3 -> s1. s4 -> s2. s3 and s4 are initial states. ​ + '''​1. (12 points)'''​ For the transition system, which states satisfy the formulas. L(s1) = {a}. L(s2) = {a}. L(s3) = {a,b}. L(s4) = {b}. s1 -> s2. s2 -> s3. s3 -> s1. s4 -> s2. s3 and s4 are initial states. ​ - ## X a + * A(X a) - ## XXX a + * A(XXX a) - ## [] b + * A([] b) - ## []<>​a + * A([]<>a) - ## [](b U a) + * A([](b U a)) - ## <>(b U a) + * A(<>(b U a)) - # '''​(12 points)'''​ This problem is Exercise 5.5 in [http://​is.ifmo.ru/​books/​_principles_of_model_checking.pdf Principles of Model Checking ] (page 321 of the PDF document), consider an elevator system that services N > 0 floors numbered 0 through N-1. There is an elevator door at each floor with a call-button and an indicator light that signals whether or not the elevator has been called. For simplicity, consider N=4. Present a set of atomic propositions-try to minimize that set-that are needed to describe the following properties of the elevator as LTL formulae and give the LTL formulae: + Hint: consider each state as the starting state for the formula. - ## Any door on any floor is never open when the elevator is not present + - ## A requested floor will be served eventually + '''​2. (16 points)'''​ This problem is Exercise 5.5 in [http://​is.ifmo.ru/​books/​_principles_of_model_checking.pdf Principles of Model Checking ] (page 321 of the PDF document), consider an elevator system that services N > 0 floors numbered 0 through N-1. There is an elevator door at each floor with a call-button and an indicator light that signals whether or not the elevator has been called. For simplicity, consider N=4. Present a set of atomic propositions-try to minimize that set-that are needed to describe the following properties of the elevator as LTL formulae and give the LTL formulae: - ## The elevator repeatedly returns to floor 0 + * Any door on any floor is never open when the elevator is not present - ## When the top floor is requested, the elevator serves it immediately and does not stop on any floor on the way up + * A requested floor will be served eventually + * The elevator repeatedly returns to floor 0 + * When the top floor is requested, the elevator serves it immediately and does not stop on any floor on the way up + + '''​3.'''​ ('''​16 points'''​) For each of the following properties: label as either safety or liveness, express in a CTL* formula, and create a minimal label trace to satisfy written formula. + + * The number of items in a buffer may not exceed ''​n''​ + * If a process reaches a location ''​li''​ just before a critical section, it should not be blocked forever + * Clients may not retain resources forever + * It is always cloudy before it rains + + '''​4.'''​ ('''​4 points'''​) Express the following property in LTL (the '​!'​ operator is a logical not): + + ''​the transition from !p to p will happen at most once''​ + + '''​5.'''​ ('''​4 points'''​) Express the following property in LTL: + + ''​there are infinitely many transitions from !p to p (and vice versa)''​ + + '''​6.'''​ ('''​8 points'''​) Express the following property in LTL using ''​r'',​ ''​y'',​ ''​g''​ as atomic propositions denoting the states of the traffic light (r = red, y = yellow, g = green): + + ''​The lights of a traffic signal light in the following periodic sequence: red, yellow, green, red, yellow, green,... and only one light is on at any given time, starting with red. You cannot make any assumptions on how long a light is on other than it will be on for a finite length of time.''​ + + '''​7.'''​ ('''​5 points'''​) Express the following property in LTL: + + ''​if q holds in a state si and r holds in some subsequent state (sj , j > i) then p must not hold in any state sk in between si and sj (i < k < j , where j is the first such subsequent state)''​