In general, a finite-state machine can be modeled as a discrete time system. Following the example in the previous slide in the previous video, we're going to use FSM for short for finite-state machine and it's going to be given by the following objects. The first one is an input alphabet, which we will denote as Sigma capital, and this is where the input v takes values. In addition, we're going to have a finite set of states. We are going to label that as capital Q, where the state q takes values. Since we might also issue an output that might be different than the state itself, we will have also an output set Delta where the output, we're going to use theta as we did earlier, takes values. And then we need to define this function that allows us to transition between different states according to the given input, and this is the transition function which we denoted as delta. It has two arguments. The first argument is the state, the second argument is the input and the value that they return should map back to the set of the states. And then we can also define an output function. The output function is going to be labeled as kappa. It's going to take values of the state and generate output values. This function can also depend on the input. With this model, we can now realize that we have a system with an input, a system with a state and a system with an output. So we have input, output, and inside the system which is the finite-state machine, we have a state which is Q. The dynamics of Q come according to the transition function, and the value of the output Is generated by the output function. If we relate this to the previous example in the previous video, this will correspond to zero,one and this would correspond to a, b, c. We didn't define an output. We could just use Q as the output set, the output function we'll choose the identity map and we defined this in the previous video for that example. Now we can interpret finite-state machines as a dynamic system, where if someone gives you an initial state and an input, in this case, one value or a sequence of values, one can generate the transitions of the finite-state machine and according to that, the output of the finite-state machine. The initial value of the state is typically denoted as q_sub_zero. And the property of a finite-state machine to reaching a state and staying in the state no matter what the input is is called invariance of that state.
