What is the concept of an Automaton?

An automaton is a system where the output $y$ is based on the input $x$, but it also depends on the system's internal state $h$. Mathematically, the concept of constructing an algorithm $f(x,h)$, where a program returns an output $y$ for a given input $x$, but the output can change based on the internal state $h$, is called an automaton. In simpler terms, the same input can produce different results depending on the system's current state.

For example:

Vending machine: Imagine a vending machine that sells canned drinks, and each can costs €2. If you insert 50 cents or €1, nothing happens because it’s not enough to buy a drink. But if you insert €2, the button to select a can lights up. In this case, the internal state of the vending machine changes depending on how much money has been inserted. If less than €2 is inserted, the machine waits for more coins, and when the total reaches €2, the selection button lights up, allowing you to choose a drink.