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Objectives:

• Write temporal logic properties
• Master the semantics of temporal logic

## Problems

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.
1. X a
2. XXX a
3. [] b
4. []<>a
5. [](b U a)
6. <>(b U a)
2. (12 points) This problem is Exercise 5.5 in 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:
1. Any door on any floor is never open when the elevator is not present
2. A requested floor will be served eventually
3. The elevator repeatedly returns to floor 0
4. When the top floor is requested, the elevator serves it immediately and does not stop on any floor on the way up