[MUSIC] With the addition of the gar function, now the finite state machine can make transitions when certain conditions are true, and those conditions might involve the input, the state, and the output. One other feature that sometimes is needed is to be able to have multiple possible states at the transition time. Let's consider the following, Chessboard game. It's a simplified version of the real game. So we're going to do a grid of nine cells. Each cell will have a number. And according to the choice of the player, which for us will be the input, we will characterize where the state of the system, which corresponds to which cell one currently is at, would be. So let's label these ones as red. And this one as green. The idea is as follows. Suppose that one is the start. And that if v is equal to red, then new state. Is any of the cells that is adjacent to one and red. Well, if v is equal to green, Then the new state could be any. State that is adjacent to current state. With green. What does it mean? Logically, he says that if q = 1 and v = red, then from this plot q plus can be either 1 again or 5. If q = 1 and v = green, Then the new value could be 2 or 4. And we can continue this. Now we can go to the next state, which will be q = 2. Consider first v = red, The new value of this state could be any value among 1, 3, and 5. And then similarly if q = 2 and v = green, Then the new value from 2, could be 2 itself, 4, or 6. And we can continue these for all possible values of q and all corresponding values of 3. The point here is as you see here, this is telling us that the new state is not unique. So such simple problems lead to finite state machines, with set-valued, Transition function. The way we are going to write that down is as we did before. The new value of the state q is given by the transition function. But now, because this might be a set that is more than a point, we might have an E symbol right there. And this delta is now not necessarily a function but a set value map for which we will use the double-arrow symbol, the symbol denoting that you might be mapping points into sets that are more than a point. So these are type of machines, and one can add an output and one can add a gar function as we did before, but these are the type of machines that have nondetermination into the cyber-physical model. [MUSIC]
