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.