Symbolic representations have been introduced on slide 53 of the lecture notes.
Here, your variables are V = {p, q}. Your initial states Are described by ΨI = ¬q. You transition relation is defined by ΨR = (p → (¬(p' ↔ q') ∨ (q' ↔ q))). You'd do the following steps:
- Create all possible states. The variables p and q can either be present or absent. These yields four states: {}, {q}, {p}, {p,q}
- Marks the initial states. Those are the states satisfying ¬q. That means that q may not be present in the label. That's the case for {}, {p}.
- Find all satisfying assignments of the formula representing the transition relation. One satisfying assignment is p = q = q' = true, p' = false. The primed variables mark the destination, the unprimed ones the source of the source of the transition. This satisfying assignment corresponds to the transition {p, q} → {q}
I hope that helps. I am not quite sure at which point you got stuck when asking this question. The question is an exam question (Feb 2020, Page 8, Problem 3). You might have a look at the standard solutions to this and other exam questions. That might help understanding what is going on here.
In this solution, you can find a partial truth table. This means that for the unprimed variables, all combinations were plugged in and the formula was partially evaluated. For example p = q = 1 evaluated to p' ∨ q'. Every satisfying assignment to p', q' extends the partial assignment to a variable assignment that corresponds to a transition. A special case is the first line of the (partial) truth table. Here, the partial assignment p = q = 0 turned the whole formula true. This means that the transitions from {} (both p, q are false) lead to all states in the structure.
