Probabilistic Automaton - p-adic Languages

p-adic Languages

The p-adic languages provide an example of a stochastic language that is not regular, and also show that the number of stochastic languages is uncountable. A p-adic language is defined as the set of strings in the letters such that

That is, a p-adic language is merely the set of real numbers, written in base-p, such that they are greater than . It is straightforward to show that all p-adic languages are stochastic. However, a p-adic language is regular if and only if is rational. In particular, this implies that the number of stochastic languages is uncountable.